sketches in higher category theories and the homotopy ...tuyeras/th/ct2016_slides.pdf · sketches...
TRANSCRIPT
![Page 1: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/1.jpg)
Sketches in Higher Category Theories and the HomotopyHypothesis
Remy Tuyeras
Macquarie University
August 8th, 2016
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 1
![Page 2: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/2.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 2
![Page 3: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/3.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 3
![Page 4: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/4.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 4
![Page 5: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/5.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 5
![Page 6: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/6.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 6
![Page 7: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/7.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 7
![Page 8: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/8.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 8
![Page 9: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/9.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 9
![Page 10: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/10.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 10
![Page 11: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/11.jpg)
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 11
![Page 12: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/12.jpg)
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 12
![Page 13: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/13.jpg)
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 13
![Page 14: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/14.jpg)
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 14
![Page 15: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/15.jpg)
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 15
![Page 16: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/16.jpg)
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 16
![Page 17: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/17.jpg)
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 17
![Page 18: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/18.jpg)
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 18
![Page 19: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/19.jpg)
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 19
![Page 20: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/20.jpg)
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 20
![Page 21: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/21.jpg)
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 21
![Page 22: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/22.jpg)
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 22
![Page 23: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/23.jpg)
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right
^(•$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 23
![Page 24: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/24.jpg)
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right
^(•$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 24
![Page 25: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/25.jpg)
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right
^(•$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 25
![Page 26: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/26.jpg)
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Properness properties;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 26
![Page 27: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/27.jpg)
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Properness properties;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 27
![Page 28: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/28.jpg)
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Properness properties;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 28
![Page 29: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/29.jpg)
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Properness properties;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 29
![Page 30: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/30.jpg)
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Left or right Properness;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 30
![Page 31: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/31.jpg)
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Left or right Properness;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 31
![Page 32: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/32.jpg)
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Left or right Properness;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 32
![Page 33: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/33.jpg)
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Left or right Properness;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 33
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Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 34
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Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
((((((hhhhhhInduction (Left)
'/em
Deduction (Right)
^(•
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 35
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Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
((((((hhhhhhInduction (Left)
'/em
Deduction (Right)
^(•
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 36
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Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
((((((hhhhhhInduction (Left)
'/em
Deduction (Right)
^(•
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 37
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Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
((((((hhhhhhInduction (Left)
'/em
Deduction (Right)
^(•
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 38
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A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 39
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A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 40
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A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 41
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A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 42
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A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 43
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A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 44
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A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
∀// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 45
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A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
∀// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 46
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A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
∀// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 47
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A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 48
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A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 49
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Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
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Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 51
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Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 52
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Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 53
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Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
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Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
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Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
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Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 57
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 58
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 59
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 60
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 61
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 62
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 63
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 64
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 65
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 66
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Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 67
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Loading .
... After looking into how the internal logics work ...
... we notice that a very elementary structure appears constantly ...
... in any formulation of the internal axioms ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 68
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Loading ..
... After looking into how the internal logics work ...
... we notice that a very elementary structure appears constantly ...
... in any formulation of the internal axioms ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 69
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Loading ...
... After looking into how the internal logics work ...
... we notice that a very elementary structure appears constantly ...
... in any formulation of the internal axioms ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 70
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Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 71
![Page 72: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/72.jpg)
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 72
![Page 73: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/73.jpg)
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 73
![Page 74: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/74.jpg)
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 74
![Page 75: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/75.jpg)
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 75
![Page 76: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/76.jpg)
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 76
![Page 77: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/77.jpg)
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 77
![Page 78: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/78.jpg)
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 78
![Page 79: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/79.jpg)
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 79
![Page 80: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/80.jpg)
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 80
![Page 81: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/81.jpg)
Elementary Structure for Homotopy Theory
We can also retrieve the representatives of the so-called generatingstructure seen previously as follows:
. generating cofibrations;
. generating trivial cofibrations;
. generating squares;
• //
��x•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 81
![Page 82: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/82.jpg)
Elementary Structure for Homotopy Theory
We can also retrieve the representatives of the so-called generatingstructure seen previously as follows:
. generating cofibrations;
. generating trivial cofibrations;
. generating squares;
• //
��x•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 82
![Page 83: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/83.jpg)
Elementary Structure for Homotopy Theory
We can also retrieve the representatives of the so-called generatingstructure seen previously as follows:
. generating cofibrations;
. generating trivial cofibrations;
. generating squares;
• //
��x•
������• // • //// •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 83
![Page 84: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/84.jpg)
Elementary Structure for Homotopy Theory
We can also retrieve the representatives of the so-called generatingstructure seen previously as follows:
. generating cofibrations;
. generating trivial cofibrations;
. generating squares;
• //
��x•
����• //
88• // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 84
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Examples
In the category of topological spaces Top:
Sn−1 //
��
x
Dn
��
Dn // Sn // Dn+1
In the category of simplicial sets sSet:
∂∆n−1//
��
xΛkn
��
∆n−1// ∂∆n
// ∆n
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 85
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Examples
In the category of topological spaces Top:
Sn−1 //
��
x
Dn
��
Dn // Sn // Dn+1
In the category of simplicial sets sSet:
∂∆n−1//
��
xΛkn
��
∆n−1// ∂∆n
// ∆n
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 86
![Page 87: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/87.jpg)
Examples
In the category of topological spaces Top:
Sn−1 //
��
x
Dn
��
Dn // Sn // Dn+1
In the category of simplicial sets sSet:
∂∆n−1//
��
xΛkn
��
∆n−1// ∂∆n
// ∆n
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 87
![Page 88: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/88.jpg)
Examples
In the category of topological spaces Top:
Sn−1 //
��
x
Dn
��
Dn // Sn // Dn+1
In the category of simplicial sets sSet:
∂∆n−1//
��
xΛkn
��
∆n−1// ∂∆n
// ∆n
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 88
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Examples
In the topos Sh(C∞Ringop):
{0} //
��
D
��
D // D(2) // D
... many other examples exist.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 89
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Examples
In the topos Sh(C∞Ringop):
{0} //
��
D
��
D // D(2) // D
... many other examples exist.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 90
![Page 91: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/91.jpg)
Examples
In the topos Sh(C∞Ringop):
{0} //
��
D
��
D // D(2) // D
... many other examples exist.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 91
![Page 92: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/92.jpg)
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 92
![Page 93: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/93.jpg)
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 93
![Page 94: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/94.jpg)
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 94
![Page 95: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/95.jpg)
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 95
![Page 96: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/96.jpg)
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 96
![Page 97: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/97.jpg)
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
S //
��x
D1
��
D2// S′ // D′
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 97
![Page 98: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/98.jpg)
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a trivial fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 98
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A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a trivial fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 99
![Page 100: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/100.jpg)
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a trivial fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 100
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A Mini-Homotopy Theory
So, for every structure S //
��
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 101
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A Mini-Homotopy Theory
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 102
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A Mini-Homotopy Theory
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 103
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A Mini-Homotopy Theory
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 104
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A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 105
![Page 106: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/106.jpg)
A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 106
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A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 107
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A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 108
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A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 109
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A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 110
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A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 111
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A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 112
![Page 113: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/113.jpg)
A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 113
![Page 114: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/114.jpg)
A Mini-Homotopy Theory
So, for every structure S //
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 114
![Page 115: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/115.jpg)
A Mini-Homotopy Theory
So, for every structure S //
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•
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>>
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 115
![Page 116: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/116.jpg)
A Mini-Homotopy Theory
So, for every structure S //
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•
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>>
an intraction if S′
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 116
![Page 117: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/117.jpg)
A Mini-Homotopy Theory
weak equivalence = surtraction + intraction
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 117
![Page 118: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/118.jpg)
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 118
![Page 119: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/119.jpg)
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 119
![Page 120: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/120.jpg)
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 120
![Page 121: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/121.jpg)
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 121
![Page 122: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/122.jpg)
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 122
![Page 123: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/123.jpg)
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 123
![Page 124: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/124.jpg)
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 124
![Page 125: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/125.jpg)
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 125
![Page 126: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/126.jpg)
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 126
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A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 127
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A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 128
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A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 129
![Page 130: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/130.jpg)
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 130
![Page 131: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/131.jpg)
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 131
![Page 132: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/132.jpg)
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 132
![Page 133: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/133.jpg)
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 133
![Page 134: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/134.jpg)
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��⊆
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 134
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Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��⊆
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 135
![Page 136: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/136.jpg)
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��⊆
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 136
![Page 137: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/137.jpg)
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��⊆
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 137
![Page 138: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/138.jpg)
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 138
![Page 139: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/139.jpg)
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 139
![Page 140: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/140.jpg)
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 140
![Page 141: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/141.jpg)
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 141
![Page 142: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/142.jpg)
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 142
![Page 143: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/143.jpg)
Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 143
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Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 144
![Page 145: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/145.jpg)
Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 145
![Page 146: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/146.jpg)
Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 146
![Page 147: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/147.jpg)
Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 147
![Page 148: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/148.jpg)
How do we encode dimensions?
... and we need to extend the elementary structures S //
��
D1��
D2// S′ // D′
to
longer ones made out of the elementary ones:
S0
��
//
x
D01
��
D02
// S1 . . .Sn //
��
x
Dn1
��
Dn2
// Sn+1// D′
... this encodes the dimension of a cell!Sketches in Higher Category Theories and the H.H. Remy Tuyeras 148
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How do we encode dimensions?
... and we need to extend the elementary structures S //
��
D1��
D2// S′ // D′
to
longer ones made out of the elementary ones:
S0
��
//
x
D01
��
D02
// S1 . . .Sn //
��
x
Dn1
��
Dn2
// Sn+1// D′
... this encodes the dimension of a cell!Sketches in Higher Category Theories and the H.H. Remy Tuyeras 149
![Page 150: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/150.jpg)
How do we encode dimensions?
... and we need to extend the elementary structures S //
��
D1��
D2// S′ // D′
to
longer ones made out of the elementary ones:
S0
��
//
x
D01
��
D02
// S1 . . .Sn //
��
x
Dn1
��
Dn2
// Sn+1// D′
... this encodes the dimension of a cell!Sketches in Higher Category Theories and the H.H. Remy Tuyeras 150
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Vertebrae & Spines
I call every elementary structure · //
��
x
·��
· // · // ·
a vertebra and I call every
structure·��
//
x
·��
· // · //
��
x
·��
· // ·��
//
x
·��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 151
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Vertebrae & Spines
I call every elementary structure · //
��
x
·��
· // · // ·
a vertebra and I call every
structure·��
//
x
·��
· // · //
��
x
·��
· // ·��
//
x
·��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 152
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Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure•��
//
x
•��
• // • //
��
x
•��
· // ·��
//
x
·��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 153
![Page 154: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/154.jpg)
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure•��
//
x
•��
• // • //
��
x
·��
• // ·��
//
x
·��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 154
![Page 155: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/155.jpg)
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // • //
��
x
•��
• // •��
//
x
•��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 155
![Page 156: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/156.jpg)
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // • //
��
x
•��
• // •��
//
x
·��
• // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 156
![Page 157: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/157.jpg)
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // · //
��
x
·��
· // •��
//
x
•��
• // • . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 157
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Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // · //
��
x
·��
· // ·��
//
x
·��
· // ·. . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 158
![Page 159: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/159.jpg)
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // · //
��
x
·��
· // ·��
//
x
·��
· // · . . .• //
��
x
•��
• // • // •
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 159
![Page 160: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/160.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 160
![Page 161: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/161.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 161
![Page 162: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/162.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 162
![Page 163: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/163.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 163
![Page 164: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/164.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 164
![Page 165: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/165.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 165
![Page 166: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/166.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 166
![Page 167: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/167.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 167
![Page 168: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/168.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 168
![Page 169: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/169.jpg)
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 169
![Page 170: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/170.jpg)
Spinal Categories Model Categories
“acyclic fibrations” = cofibrations + weak equivalences
Theorem
If pushouts of acyclic cofibrations are weak equivalences (and somesmallness condition holds), then a spinal category is a model category.
The proof does not even need the Jeff Smith’s criteria as all the desiredproperties come from my definitions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 170
![Page 171: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/171.jpg)
Spinal Categories Model Categories
“acyclic fibrations” = cofibrations + weak equivalences
Theorem
If pushouts of acyclic cofibrations are weak equivalences (and somesmallness condition holds), then a spinal category is a model category.
The proof does not even need the Jeff Smith’s criteria as all the desiredproperties come from my definitions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 170
![Page 172: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/172.jpg)
Spinal Categories Model Categories
“acyclic fibrations” = cofibrations + weak equivalences
Theorem
If pushouts of acyclic cofibrations are weak equivalences (and somesmallness condition holds), then a spinal category is a model category.
The proof does not even need the Jeff Smith’s criteria as all the desiredproperties come from my definitions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 170
![Page 173: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/173.jpg)
More General Structures
... more general structures are sometimes needed ...
Local (e.g. stacks) non-algebraic algebraic
• //
��
•
��
• //
??
��
x
•
��
??
• // • // •
• //
??
• //
??
•
??
• //
��
x
•��
•...• // • //
??
•
• //
��
x
•��
• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 171
![Page 174: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/174.jpg)
More General Structures
... more general structures are sometimes needed ...
Local (e.g. stacks) non-algebraic algebraic
• //
��
•
��
• //
??
��
x
•
��
??
• // • // •
• //
??
• //
??
•
??
• //
��
x
•��
•...• // • //
??
•
• //
��
x
•��
• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 172
![Page 175: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/175.jpg)
More General Structures
... more general structures are sometimes needed ...
Local (e.g. stacks) non-algebraic algebraic
• //
��
•
��
• //
??
��
x
•
��
??
• // • // •
• //
??
• //
??
•
??
• //
��
x
•��
•...• // • //
??
•
• //
��
x
•��
• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 173
![Page 176: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/176.jpg)
Homotopy Hypothesis
Can we apply our theory to the category of Grothendieck’s ∞-groupoids∞-Grp?
It follows from the definition of a Grothendieck’s ∞-groupoid that any pairof parallel arrows in ∞-Grp ...
Dn+1∃
!!
Dn
f //
g//
sn
OO
tn
OO
B
...can be filled by a homotopy.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 174
![Page 177: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/177.jpg)
Homotopy Hypothesis
Can we apply our theory to the category of Grothendieck’s ∞-groupoids∞-Grp?
It follows from the definition of a Grothendieck’s ∞-groupoid that any pairof parallel arrows in ∞-Grp ...
Dn+1∃
!!
Dn
f //
g//
sn
OO
tn
OO
B
...can be filled by a homotopy.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 175
![Page 178: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/178.jpg)
Homotopy Hypothesis
Can we apply our theory to the category of Grothendieck’s ∞-groupoids∞-Grp?
It follows from the definition of a Grothendieck’s ∞-groupoid that any pairof parallel arrows in ∞-Grp ...
Dn+1∃
!!
Dn
f //
g//
sn
OO
tn
OO
B
...can be filled by a homotopy.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 176
![Page 179: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/179.jpg)
Homotopy Hypothesis
Equivalently, this amount to saying that ...
Sn−1
x
γn//
γn
��
Dn
δn1��
g
$$Dn δn2
//
f
44Sn //γn+1
// Dn+1// B
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 177
![Page 180: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/180.jpg)
Homotopy Hypothesis
Equivalently, this amount to saying that ...
Sn−1
x
γn//
γn
��
Dn
δn1��
g
$$Dn δn2
//
f
44Sn //Dn+1 B
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 178
![Page 181: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/181.jpg)
Homotopy Hypothesis
Equivalently, this amount to saying that ...
Sn−1
x
γn//
γn
��
Dn
δn1��
g
$$Dn δn2
//
f
44Snγn+1
// Dn+1// B
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 179
![Page 182: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/182.jpg)
Homotopy Hypothesis
On the one hand, the operations on ∞-groupoids come from theexistence of a certain homotopy filling of a certain pair of parallelarrows going to a certain globular sum.
On the other hand, all the operations defined on vertebrae and spinescome from existence of a certain vertebra filling a certain givencommutative square of parallel arrows going to a certain globular sum.
Theorem
The category of Grothendieck’s ∞-groupoids admits a spinal structure.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 180
![Page 183: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/183.jpg)
Homotopy Hypothesis
On the one hand, the operations on ∞-groupoids come from theexistence of a certain homotopy filling of a certain pair of parallelarrows going to a certain globular sum.
On the other hand, all the operations defined on vertebrae and spinescome from existence of a certain vertebra filling a certain givencommutative square of parallel arrows going to a certain globular sum.
Theorem
The category of Grothendieck’s ∞-groupoids admits a spinal structure.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 180
![Page 184: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/184.jpg)
Homotopy Hypothesis
On the one hand, the operations on ∞-groupoids come from theexistence of a certain homotopy filling of a certain pair of parallelarrows going to a certain globular sum.
On the other hand, all the operations defined on vertebrae and spinescome from existence of a certain vertebra filling a certain givencommutative square of parallel arrows going to a certain globular sum.
Theorem
The category of Grothendieck’s ∞-groupoids admits a spinal structure.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 180
![Page 185: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/185.jpg)
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
![Page 186: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/186.jpg)
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
![Page 187: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/187.jpg)
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
![Page 188: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/188.jpg)
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
![Page 189: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/189.jpg)
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
![Page 190: Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches in Higher Category Theories and the Homotopy Hypothesis R emy Tuy eras Macquarie](https://reader036.vdocuments.us/reader036/viewer/2022070916/5fb6dcdc55c8972ce44a1790/html5/thumbnails/190.jpg)
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
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The End
Thank you!
References:
[1] R. Tuyeras, Sketches in Higher Category Theory, version 1.01,http://www.normalesup.org/~tuyeras/th/thesis.html.
[2] R. Tuyeras, Elimination of quotients in various localisation of modelsinto premodels, arXiv:1511.09332
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 182