set-theoretic remarks on a possible definition of elementary...
TRANSCRIPT
![Page 1: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/1.jpg)
Set-theoretic remarks on a possible definition ofelementary ∞-topos
Giulio Lo Monaco
Masaryk University
Category TheoryEdinburgh, Scotland
11 July 2019
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 1 / 14
![Page 2: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/2.jpg)
Geometric ∞-toposes
Definition
An ∞-category X is called a geometric ∞-topos if there is a small∞-category C and an adjunction
P(C) Xa L
i
where i is full and faithful, L◦ i is accessible and L preserves all finite limits.
In particular, every geometric ∞-topos is presentable.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 2 / 14
![Page 3: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/3.jpg)
Geometric ∞-toposes
Definition
An ∞-category X is called a geometric ∞-topos if there is a small∞-category C and an adjunction
P(C) Xa L
i
where i is full and faithful, L◦ i is accessible and L preserves all finite limits.
In particular, every geometric ∞-topos is presentable.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 2 / 14
![Page 4: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/4.jpg)
Dependent sums and products
Let f : X → Y a morphism in an ∞-category E with pullbacks.
A dependent sum along f is a left adjoint of the base changef ∗ : E/Y → E/X .A dependent product along f , if it exists, is a right adjoint to the basechange f ∗ : E/Y → E/X .
E/X E/Y : f ∗
a
∑f
a∏f
Remark
Dependent sums always exist by universal property of pullbacks.
Proposition
In a geometric ∞-topos all dependent products exist.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 3 / 14
![Page 5: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/5.jpg)
Dependent sums and products
Let f : X → Y a morphism in an ∞-category E with pullbacks.
A dependent sum along f is a left adjoint of the base changef ∗ : E/Y → E/X .
A dependent product along f , if it exists, is a right adjoint to the basechange f ∗ : E/Y → E/X .
E/X E/Y : f ∗
a
∑f
a∏f
Remark
Dependent sums always exist by universal property of pullbacks.
Proposition
In a geometric ∞-topos all dependent products exist.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 3 / 14
![Page 6: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/6.jpg)
Dependent sums and products
Let f : X → Y a morphism in an ∞-category E with pullbacks.
A dependent sum along f is a left adjoint of the base changef ∗ : E/Y → E/X .A dependent product along f , if it exists, is a right adjoint to the basechange f ∗ : E/Y → E/X .
E/X E/Y : f ∗
a
∑f
a∏f
Remark
Dependent sums always exist by universal property of pullbacks.
Proposition
In a geometric ∞-topos all dependent products exist.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 3 / 14
![Page 7: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/7.jpg)
Dependent sums and products
Let f : X → Y a morphism in an ∞-category E with pullbacks.
A dependent sum along f is a left adjoint of the base changef ∗ : E/Y → E/X .A dependent product along f , if it exists, is a right adjoint to the basechange f ∗ : E/Y → E/X .
E/X E/Y : f ∗
a
∑f
a∏f
Remark
Dependent sums always exist by universal property of pullbacks.
Proposition
In a geometric ∞-topos all dependent products exist.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 3 / 14
![Page 8: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/8.jpg)
Dependent sums and products
Let f : X → Y a morphism in an ∞-category E with pullbacks.
A dependent sum along f is a left adjoint of the base changef ∗ : E/Y → E/X .A dependent product along f , if it exists, is a right adjoint to the basechange f ∗ : E/Y → E/X .
E/X E/Y : f ∗
a
∑f
a∏f
Remark
Dependent sums always exist by universal property of pullbacks.
Proposition
In a geometric ∞-topos all dependent products exist.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 3 / 14
![Page 9: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/9.jpg)
Dependent sums and products
Let f : X → Y a morphism in an ∞-category E with pullbacks.
A dependent sum along f is a left adjoint of the base changef ∗ : E/Y → E/X .A dependent product along f , if it exists, is a right adjoint to the basechange f ∗ : E/Y → E/X .
E/X E/Y : f ∗
a
∑f
a∏f
Remark
Dependent sums always exist by universal property of pullbacks.
Proposition
In a geometric ∞-topos all dependent products exist.Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 3 / 14
![Page 10: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/10.jpg)
Classifiers
Let S be a class of morphisms in an ∞-category E , which is closed underpullbacks.A classifier for the class S is a morphism t : U → U such that for everyobject X the operation of pulling back defines an equivalence of∞-groupoids
Map(X ,U) ' (ES/X )∼
Theorem (Rezk)
In a geometric ∞-topos, there are arbitrarily large cardinals κ such thatthe class Sκ of relatively κ-compact morphisms has a classifier.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 4 / 14
![Page 11: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/11.jpg)
Classifiers
Let S be a class of morphisms in an ∞-category E , which is closed underpullbacks.A classifier for the class S is a morphism t : U → U such that for everyobject X the operation of pulling back defines an equivalence of∞-groupoids
Map(X ,U) ' (ES/X )∼
Theorem (Rezk)
In a geometric ∞-topos, there are arbitrarily large cardinals κ such thatthe class Sκ of relatively κ-compact morphisms has a classifier.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 4 / 14
![Page 12: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/12.jpg)
Elementary ∞-toposes
Definition (Shulman)
An elementary ∞-topos is an ∞-category E such that
1 E has all finite limits and colimits.
2 E is locally Cartesian closed.
3 The class of all monomorphisms in E admits a classifier.
4 For each morphism f in E there is a class of morphisms S 3 f suchthat S has a classifier and is closed under limits and colimits taken inovercategories and under dependent sums and products.
We will only focus on a subaxiom of (4):
Definition
We say that a class of morphisms S satisfies (DepProd) if it has aclassifier and it is closed under dependent products
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 5 / 14
![Page 13: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/13.jpg)
Elementary ∞-toposes
Definition (Shulman)
An elementary ∞-topos is an ∞-category E such that
1 E has all finite limits and colimits.
2 E is locally Cartesian closed.
3 The class of all monomorphisms in E admits a classifier.
4 For each morphism f in E there is a class of morphisms S 3 f suchthat S has a classifier and is closed under limits and colimits taken inovercategories and under dependent sums and products.
We will only focus on a subaxiom of (4):
Definition
We say that a class of morphisms S satisfies (DepProd) if it has aclassifier and it is closed under dependent products
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 5 / 14
![Page 14: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/14.jpg)
Elementary ∞-toposes
Definition (Shulman)
An elementary ∞-topos is an ∞-category E such that
1 E has all finite limits and colimits.
2 E is locally Cartesian closed.
3 The class of all monomorphisms in E admits a classifier.
4 For each morphism f in E there is a class of morphisms S 3 f suchthat S has a classifier and is closed under limits and colimits taken inovercategories and under dependent sums and products.
We will only focus on a subaxiom of (4):
Definition
We say that a class of morphisms S satisfies (DepProd) if it has aclassifier and it is closed under dependent products
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 5 / 14
![Page 15: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/15.jpg)
Uniformization
We will need a fundamental result:
Theorem (Adamek, Rosicky for the 1-dimensional case)
Given a small family (fi : Ki → Li )i∈I of accessible functors betweenpresentable ∞-categories, there are arbitrarily large cardinals κ such thatall functors fi ’s preserve κ-compact objects.
Example
We may assume that κ-compact objects in a presheaf ∞-category areprecisely the objectwise κ-compact presheaves.
Given a diagram shape R, we may assume that κ-compact objects arestable under R-limits.
We may assume that many such properties hold for the same cardinal.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 6 / 14
![Page 16: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/16.jpg)
Uniformization
We will need a fundamental result:
Theorem (Adamek, Rosicky for the 1-dimensional case)
Given a small family (fi : Ki → Li )i∈I of accessible functors betweenpresentable ∞-categories, there are arbitrarily large cardinals κ such thatall functors fi ’s preserve κ-compact objects.
Example
We may assume that κ-compact objects in a presheaf ∞-category areprecisely the objectwise κ-compact presheaves.
Given a diagram shape R, we may assume that κ-compact objects arestable under R-limits.
We may assume that many such properties hold for the same cardinal.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 6 / 14
![Page 17: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/17.jpg)
Uniformization
We will need a fundamental result:
Theorem (Adamek, Rosicky for the 1-dimensional case)
Given a small family (fi : Ki → Li )i∈I of accessible functors betweenpresentable ∞-categories, there are arbitrarily large cardinals κ such thatall functors fi ’s preserve κ-compact objects.
Example
We may assume that κ-compact objects in a presheaf ∞-category areprecisely the objectwise κ-compact presheaves.
Given a diagram shape R, we may assume that κ-compact objects arestable under R-limits.
We may assume that many such properties hold for the same cardinal.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 6 / 14
![Page 18: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/18.jpg)
Uniformization
We will need a fundamental result:
Theorem (Adamek, Rosicky for the 1-dimensional case)
Given a small family (fi : Ki → Li )i∈I of accessible functors betweenpresentable ∞-categories, there are arbitrarily large cardinals κ such thatall functors fi ’s preserve κ-compact objects.
Example
We may assume that κ-compact objects in a presheaf ∞-category areprecisely the objectwise κ-compact presheaves.
Given a diagram shape R, we may assume that κ-compact objects arestable under R-limits.
We may assume that many such properties hold for the same cardinal.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 6 / 14
![Page 19: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/19.jpg)
Main result
Theorem
Fixing a Grothendieck universe U , every geometric ∞-topos satisfies(DepProd) if and only if there are unboundedly many inaccessiblecardinals below the cardinality of U .
First, prove ⇐.
Step 1. In the ∞-category S of spaces, if κ is inaccessible then κ-compactobjects are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 7 / 14
![Page 20: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/20.jpg)
Main result
Theorem
Fixing a Grothendieck universe U , every geometric ∞-topos satisfies(DepProd) if and only if there are unboundedly many inaccessiblecardinals below the cardinality of U .
First, prove ⇐.
Step 1. In the ∞-category S of spaces, if κ is inaccessible then κ-compactobjects are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 7 / 14
![Page 21: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/21.jpg)
Main result
Theorem
Fixing a Grothendieck universe U , every geometric ∞-topos satisfies(DepProd) if and only if there are unboundedly many inaccessiblecardinals below the cardinality of U .
First, prove ⇐.
Step 1. In the ∞-category S of spaces, if κ is inaccessible then κ-compactobjects are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 7 / 14
![Page 22: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/22.jpg)
Main result
Step 2. In P(C), given presheaves F and G , their exponential FG is givenby the formula
FG (C ) =
∫D∈C
Map(Map(D,C )× G (D),F (D))
By uniformization, we may choose a cardinal κ such that:
κ-compactness is detected objectwise
all representables are κ-compact
κ-compact spaces are stable under binary products
κ-compact spaces are stable under exponentiation (Step 1)
κ-compact spaces are stable under C-ends
⇒ κ-compact presheaves are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 8 / 14
![Page 23: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/23.jpg)
Main result
Step 2. In P(C), given presheaves F and G , their exponential FG is givenby the formula
FG (C ) =
∫D∈C
Map(Map(D,C )× G (D),F (D))
By uniformization, we may choose a cardinal κ such that:
κ-compactness is detected objectwise
all representables are κ-compact
κ-compact spaces are stable under binary products
κ-compact spaces are stable under exponentiation (Step 1)
κ-compact spaces are stable under C-ends
⇒ κ-compact presheaves are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 8 / 14
![Page 24: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/24.jpg)
Main result
Step 2. In P(C), given presheaves F and G , their exponential FG is givenby the formula
FG (C ) =
∫D∈C
Map(Map(D,C )× G (D),F (D))
By uniformization, we may choose a cardinal κ such that:
κ-compactness is detected objectwise
all representables are κ-compact
κ-compact spaces are stable under binary products
κ-compact spaces are stable under exponentiation (Step 1)
κ-compact spaces are stable under C-ends
⇒ κ-compact presheaves are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 8 / 14
![Page 25: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/25.jpg)
Main result
Step 2. In P(C), given presheaves F and G , their exponential FG is givenby the formula
FG (C ) =
∫D∈C
Map(Map(D,C )× G (D),F (D))
By uniformization, we may choose a cardinal κ such that:
κ-compactness is detected objectwise
all representables are κ-compact
κ-compact spaces are stable under binary products
κ-compact spaces are stable under exponentiation (Step 1)
κ-compact spaces are stable under C-ends
⇒ κ-compact presheaves are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 8 / 14
![Page 26: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/26.jpg)
Main result
Step 2. In P(C), given presheaves F and G , their exponential FG is givenby the formula
FG (C ) =
∫D∈C
Map(Map(D,C )× G (D),F (D))
By uniformization, we may choose a cardinal κ such that:
κ-compactness is detected objectwise
all representables are κ-compact
κ-compact spaces are stable under binary products
κ-compact spaces are stable under exponentiation (Step 1)
κ-compact spaces are stable under C-ends
⇒ κ-compact presheaves are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 8 / 14
![Page 27: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/27.jpg)
Main result
Step 2. In P(C), given presheaves F and G , their exponential FG is givenby the formula
FG (C ) =
∫D∈C
Map(Map(D,C )× G (D),F (D))
By uniformization, we may choose a cardinal κ such that:
κ-compactness is detected objectwise
all representables are κ-compact
κ-compact spaces are stable under binary products
κ-compact spaces are stable under exponentiation (Step 1)
κ-compact spaces are stable under C-ends
⇒ κ-compact presheaves are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 8 / 14
![Page 28: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/28.jpg)
Main result
Step 2. In P(C), given presheaves F and G , their exponential FG is givenby the formula
FG (C ) =
∫D∈C
Map(Map(D,C )× G (D),F (D))
By uniformization, we may choose a cardinal κ such that:
κ-compactness is detected objectwise
all representables are κ-compact
κ-compact spaces are stable under binary products
κ-compact spaces are stable under exponentiation (Step 1)
κ-compact spaces are stable under C-ends
⇒ κ-compact presheaves are stable under exponentiation.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 8 / 14
![Page 29: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/29.jpg)
Main result
Step 3. Given an adjunction
P(C) XLa
i
making X a geometric ∞-topos, choose κ such that (Step 2) holds inP(C).
The properties of L a i will transfer stability of κ-compact objects underexponentiation to X .
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 9 / 14
![Page 30: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/30.jpg)
Main result
Step 3. Given an adjunction
P(C) XLa
i
making X a geometric ∞-topos, choose κ such that (Step 2) holds inP(C).
The properties of L a i will transfer stability of κ-compact objects underexponentiation to X .
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 9 / 14
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Main result
Step 4. Given an object p : Z → X in X/X , its dependent product along aterminal morphism X → ∗ is given by∏
X
p = ZX ×XX {p}
Choose κ such that (Step 3) holds and κ-compact objects are stable underpullbacks ⇒ relatively κ-compact morphisms are stable under dependentproducts along terminal morphisms.
Step 5. For generic dependent products, decompose the codomain as acolimit of compact objects Yi ’s and then choose κ such that (Step 4)holds in all ∞-toposes X/Yi
.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 10 / 14
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Main result
Step 4. Given an object p : Z → X in X/X , its dependent product along aterminal morphism X → ∗ is given by∏
X
p = ZX ×XX {p}
Choose κ such that (Step 3) holds and κ-compact objects are stable underpullbacks
⇒ relatively κ-compact morphisms are stable under dependentproducts along terminal morphisms.
Step 5. For generic dependent products, decompose the codomain as acolimit of compact objects Yi ’s and then choose κ such that (Step 4)holds in all ∞-toposes X/Yi
.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 10 / 14
![Page 33: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/33.jpg)
Main result
Step 4. Given an object p : Z → X in X/X , its dependent product along aterminal morphism X → ∗ is given by∏
X
p = ZX ×XX {p}
Choose κ such that (Step 3) holds and κ-compact objects are stable underpullbacks ⇒ relatively κ-compact morphisms are stable under dependentproducts along terminal morphisms.
Step 5. For generic dependent products, decompose the codomain as acolimit of compact objects Yi ’s and then choose κ such that (Step 4)holds in all ∞-toposes X/Yi
.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 10 / 14
![Page 34: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/34.jpg)
Main result
Step 4. Given an object p : Z → X in X/X , its dependent product along aterminal morphism X → ∗ is given by∏
X
p = ZX ×XX {p}
Choose κ such that (Step 3) holds and κ-compact objects are stable underpullbacks ⇒ relatively κ-compact morphisms are stable under dependentproducts along terminal morphisms.
Step 5. For generic dependent products, decompose the codomain as acolimit of compact objects Yi ’s and then choose κ such that (Step 4)holds in all ∞-toposes X/Yi
.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 10 / 14
![Page 35: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/35.jpg)
Main result
Now prove ⇒.
It suffices to prove it assuming that S satisfies (DepProd).
For a discrete space X , the terminal morphism X → ∗ is contained in aclass S having a classifier t : U → U such that
∃Y U
Z U
pp
t ,Z U
W U
fp
t =⇒ ∃
∏f p U
W U.
p t
Assume that all fibers of t are discrete.
For each point in U, its fiber along t can be regarded as a set.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 11 / 14
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Main result
Now prove ⇒. It suffices to prove it assuming that S satisfies (DepProd).
For a discrete space X , the terminal morphism X → ∗ is contained in aclass S having a classifier t : U → U such that
∃Y U
Z U
pp
t ,Z U
W U
fp
t =⇒ ∃
∏f p U
W U.
p t
Assume that all fibers of t are discrete.
For each point in U, its fiber along t can be regarded as a set.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 11 / 14
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Main result
Now prove ⇒. It suffices to prove it assuming that S satisfies (DepProd).
For a discrete space X , the terminal morphism X → ∗ is contained in aclass S having a classifier t : U → U such that
∃Y U
Z U
pp
t ,Z U
W U
fp
t
=⇒ ∃
∏f p U
W U.
p t
Assume that all fibers of t are discrete.
For each point in U, its fiber along t can be regarded as a set.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 11 / 14
![Page 38: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/38.jpg)
Main result
Now prove ⇒. It suffices to prove it assuming that S satisfies (DepProd).
For a discrete space X , the terminal morphism X → ∗ is contained in aclass S having a classifier t : U → U such that
∃Y U
Z U
pp
t ,Z U
W U
fp
t =⇒ ∃
∏f p U
W U.
p t
Assume that all fibers of t are discrete.
For each point in U, its fiber along t can be regarded as a set.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 11 / 14
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Main result
Now prove ⇒. It suffices to prove it assuming that S satisfies (DepProd).
For a discrete space X , the terminal morphism X → ∗ is contained in aclass S having a classifier t : U → U such that
∃Y U
Z U
pp
t ,Z U
W U
fp
t =⇒ ∃
∏f p U
W U.
p t
Assume that all fibers of t are discrete.
For each point in U, its fiber along t can be regarded as a set.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 11 / 14
![Page 40: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/40.jpg)
Main result
Now prove ⇒. It suffices to prove it assuming that S satisfies (DepProd).
For a discrete space X , the terminal morphism X → ∗ is contained in aclass S having a classifier t : U → U such that
∃Y U
Z U
pp
t ,Z U
W U
fp
t =⇒ ∃
∏f p U
W U.
p t
Assume that all fibers of t are discrete.
For each point in U, its fiber along t can be regarded as a set.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 11 / 14
![Page 41: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/41.jpg)
Main result
Fx U
{x} U
pt
X U
{x0} U
pt
Define κ := supx∈U |Fx |.
κ > |X |.For λ, µ < κ, closure under dependent products ⇒ µλ < κ.
In non-trivial cases,∑
i∈I αi ≤∏
i∈I αi ⇒ κ is regular.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 12 / 14
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Main result
Fx U
{x} U
pt
X U
{x0} U
pt
Define κ := supx∈U |Fx |.
κ > |X |.For λ, µ < κ, closure under dependent products ⇒ µλ < κ.
In non-trivial cases,∑
i∈I αi ≤∏
i∈I αi ⇒ κ is regular.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 12 / 14
![Page 43: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/43.jpg)
Main result
Fx U
{x} U
pt
X U
{x0} U
pt
Define κ := supx∈U |Fx |.
κ > |X |.
For λ, µ < κ, closure under dependent products ⇒ µλ < κ.
In non-trivial cases,∑
i∈I αi ≤∏
i∈I αi ⇒ κ is regular.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 12 / 14
![Page 44: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/44.jpg)
Main result
Fx U
{x} U
pt
X U
{x0} U
pt
Define κ := supx∈U |Fx |.
κ > |X |.For λ, µ < κ, closure under dependent products ⇒ µλ < κ.
In non-trivial cases,∑
i∈I αi ≤∏
i∈I αi ⇒ κ is regular.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 12 / 14
![Page 45: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/45.jpg)
Main result
Fx U
{x} U
pt
X U
{x0} U
pt
Define κ := supx∈U |Fx |.
κ > |X |.For λ, µ < κ, closure under dependent products ⇒ µλ < κ.
In non-trivial cases,∑
i∈I αi ≤∏
i∈I αi
⇒ κ is regular.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 12 / 14
![Page 46: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/46.jpg)
Main result
Fx U
{x} U
pt
X U
{x0} U
pt
Define κ := supx∈U |Fx |.
κ > |X |.For λ, µ < κ, closure under dependent products ⇒ µλ < κ.
In non-trivial cases,∑
i∈I αi ≤∏
i∈I αi ⇒ κ is regular.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 12 / 14
![Page 47: Set-theoretic remarks on a possible definition of elementary -toposconferences.inf.ed.ac.uk/ct2019/slides/38.pdf · 4 For each morphism f in Ethere is a class of morphisms S 3f such](https://reader033.vdocuments.us/reader033/viewer/2022060600/60547d21626b067d8752169d/html5/thumbnails/47.jpg)
Construction
Definition
We call a cardinal µ 1-inaccessible if it is inaccessible and there areunboundedly many inaccessibles below it.
Assume the existence of a 1-inaccessible cardinal µ inside theGrothendieck universe.
Given a geometric ∞-topos X , take
X µ ⊂ X .
⇒ X µ is not a geometric ∞-topos (it doesn’t have all small colimits), butit is an elementary ∞-topos.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 13 / 14
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Construction
Definition
We call a cardinal µ 1-inaccessible if it is inaccessible and there areunboundedly many inaccessibles below it.
Assume the existence of a 1-inaccessible cardinal µ inside theGrothendieck universe.
Given a geometric ∞-topos X , take
X µ ⊂ X .
⇒ X µ is not a geometric ∞-topos (it doesn’t have all small colimits), butit is an elementary ∞-topos.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 13 / 14
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Construction
Definition
We call a cardinal µ 1-inaccessible if it is inaccessible and there areunboundedly many inaccessibles below it.
Assume the existence of a 1-inaccessible cardinal µ inside theGrothendieck universe.
Given a geometric ∞-topos X , take
X µ ⊂ X .
⇒ X µ is not a geometric ∞-topos (it doesn’t have all small colimits), butit is an elementary ∞-topos.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 13 / 14
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Construction
Definition
We call a cardinal µ 1-inaccessible if it is inaccessible and there areunboundedly many inaccessibles below it.
Assume the existence of a 1-inaccessible cardinal µ inside theGrothendieck universe.
Given a geometric ∞-topos X , take
X µ ⊂ X .
⇒ X µ is not a geometric ∞-topos (it doesn’t have all small colimits), butit is an elementary ∞-topos.
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 13 / 14
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Thank you!
Giulio Lo Monaco (Masaryk University) Set-theoretic remarks on a possible definition of elementary ∞-toposCT2019 14 / 14