joe moeller christina vasilakopouloumath.ucr.edu/home/baez/syco4/moeller_syco4.pdf ·...
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![Page 1: Joe Moeller Christina Vasilakopouloumath.ucr.edu/home/baez/SYCO4/moeller_syco4.pdf · 2019-05-21 · MotivationGrothendieck C.Monoidal Grothendieck C.Applications Monoidal Grothendieck](https://reader033.vdocuments.us/reader033/viewer/2022041808/5e55d8f3e46ee1134f0bec73/html5/thumbnails/1.jpg)
Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Monoidal Grothendieck Construction
Joe Moeller Christina Vasilakopoulou
University of California, Riverside
Fourth Symposium on Compositional Structures23 May 2019
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
k ∈ Ring
Modk ∈ Cat
Modall???
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
k ∈ Ring Modk ∈ Cat
Modall???
Joe Moeller UCR
![Page 4: Joe Moeller Christina Vasilakopouloumath.ucr.edu/home/baez/SYCO4/moeller_syco4.pdf · 2019-05-21 · MotivationGrothendieck C.Monoidal Grothendieck C.Applications Monoidal Grothendieck](https://reader033.vdocuments.us/reader033/viewer/2022041808/5e55d8f3e46ee1134f0bec73/html5/thumbnails/4.jpg)
Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
k ∈ Ring Modk ∈ Cat
Modall???
Joe Moeller UCR
![Page 5: Joe Moeller Christina Vasilakopouloumath.ucr.edu/home/baez/SYCO4/moeller_syco4.pdf · 2019-05-21 · MotivationGrothendieck C.Monoidal Grothendieck C.Applications Monoidal Grothendieck](https://reader033.vdocuments.us/reader033/viewer/2022041808/5e55d8f3e46ee1134f0bec73/html5/thumbnails/5.jpg)
Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
Grothendieck: Yes!
I objects (k ,M), where M ∈ Modk
I maps (f , g) : (k ,M)→ (k ′,M ′)where f : k → k ′ andg : M → f ∗(M ′)
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
k ∈ Ring Modk ∈ Cat
Mod: Ring→ Cat???
f : k → k ′ f ∗ : Modk ′ → Modk
Joe Moeller UCR
![Page 7: Joe Moeller Christina Vasilakopouloumath.ucr.edu/home/baez/SYCO4/moeller_syco4.pdf · 2019-05-21 · MotivationGrothendieck C.Monoidal Grothendieck C.Applications Monoidal Grothendieck](https://reader033.vdocuments.us/reader033/viewer/2022041808/5e55d8f3e46ee1134f0bec73/html5/thumbnails/7.jpg)
Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
k ∈ Ring Modk ∈ Cat
Mod: Ring→ Cat???
f : k → k ′ f ∗ : Modk ′ → Modk
Joe Moeller UCR
![Page 8: Joe Moeller Christina Vasilakopouloumath.ucr.edu/home/baez/SYCO4/moeller_syco4.pdf · 2019-05-21 · MotivationGrothendieck C.Monoidal Grothendieck C.Applications Monoidal Grothendieck](https://reader033.vdocuments.us/reader033/viewer/2022041808/5e55d8f3e46ee1134f0bec73/html5/thumbnails/8.jpg)
Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
k ∈ Ring Modk ∈ Cat
Mod: Ring→ Cat???
f : k → k ′
f ∗ : Modk ′ → Modk
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
k ∈ Ring Modk ∈ Cat
Mod: Ring→ Cat???
f : k → k ′ f ∗ : Modk ′ → Modk
Joe Moeller UCR
![Page 10: Joe Moeller Christina Vasilakopouloumath.ucr.edu/home/baez/SYCO4/moeller_syco4.pdf · 2019-05-21 · MotivationGrothendieck C.Monoidal Grothendieck C.Applications Monoidal Grothendieck](https://reader033.vdocuments.us/reader033/viewer/2022041808/5e55d8f3e46ee1134f0bec73/html5/thumbnails/10.jpg)
Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
k ∈ Ring Modk ∈ Cat
Mod: Ringop → Cat
f : k → k ′ f ∗ : Modk ′ → Modk
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
Given
Mod: Ringop → Cat
we defined Modall to have
I objects (k ,M), whereM ∈ Modk
I maps(f , g) : (k ,M)→ (k ′,M ′)where f : k → k ′ andg : M → f ∗(M ′)
Given
F : X op → Cat
we define∫F to have
I objects (x , a), where x ∈ X ,a ∈ F(x)
I maps(f , g) : (x , a)→ (x ′, a′)where f : x → x ′ andg : a→ F f (a′)
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Motivation
Given
Mod: Ringop → Cat
we defined Modall to have
I objects (k ,M), whereM ∈ Modk
I maps(f , g) : (k ,M)→ (k ′,M ′)where f : k → k ′ andg : M → f ∗(M ′)
Given
F : X op → Cat
we define∫F to have
I objects (x , a), where x ∈ X ,a ∈ F(x)
I maps(f , g) : (x , a)→ (x ′, a′)where f : x → x ′ andg : a→ F f (a′)
Joe Moeller UCR
![Page 13: Joe Moeller Christina Vasilakopouloumath.ucr.edu/home/baez/SYCO4/moeller_syco4.pdf · 2019-05-21 · MotivationGrothendieck C.Monoidal Grothendieck C.Applications Monoidal Grothendieck](https://reader033.vdocuments.us/reader033/viewer/2022041808/5e55d8f3e46ee1134f0bec73/html5/thumbnails/13.jpg)
Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Indexed Categories
2-category ICat(X ):
I an indexed category is a pseudofunctor F : X op → Cat.
I an indexed functor is a pseudonatural transformationα : F ⇒ G
I an indexed natural transformation is a modificationm : αV β
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Fibrations
A b
X x y
P
f
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Fibrations
cartesian lift
A a b
X x y
P
φ
f
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Fibrations
pullback
A f ∗(b) b
X x y
P
φ
f
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Fibrations
reindexing functor
Ay Axf ∗
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Fibrations
2-category Fib(X ):
I an object is a fibrationP : A → X
I a 1-morphism is a functor F
A
X
B
P
F
Q
which preserves cartesianliftings
I a 2-morphism is a naturaltransformation
A
X
B
P
F ′F α⇒Q
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
The Grothendieck Construction
For indexed category F : X op → Cat,∫F is naturally fibred over
X :
PF :
∫F → X
(x , a) 7→ x
(f , k) 7→ f
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
2-Equivalence
Theorem
The Grothendieck construction gives an equivalence:
ICat(X ) ∼= Fib(X )
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Fibre-wise Monoidal Grothendieck Construction
A (fibre-wise) monoidal indexed category is
I a pseudofunctor F : X op → MonCat
Let f MonICat(X ) denote the 2-category of fibre-wise monoidalindexed categories
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Fibre-wise Monoidal Grothendieck Construction
A (fibre-wise) monoidal fibration is
I fibration P : A → XI the fibres Ax are monoidal
I the reindexing functors are monoidal
Let f MonFib(X ) denote the 2-category of fibre-wise monoidalfibrations.
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Fibre-wise Monoidal Grothendieck Construction
Theorem (Vasilakopoulou, M)
The Grothendieck construction lifts to an equivalence:
f MonFib(X ) ' f MonICat(X )
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Global Monoidal Grothendieck Construction
A (global) monoidal indexed category is
I an indexed category F : X op → Cat
I X is monoidal
I F is lax monoidal (F , φ) : (X op,⊗)→ (Cat,×)
Let gMonICat denote the 2-category of global monoidal indexedcategories.
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Global Monoidal Grothendieck Construction
A (global) monoidal fibration is a fibration P : A → XI A and X are monoidal
I P is a strict monoidal functor
I ⊗A preserves cartesian liftings.
Let gMonFib(X ) denote the 2-category of global monoidalfibrations.
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Global Monoidal Grothendieck Construction
Theorem (Vasilakopoulou, M)
The Grothendieck construction lifts to an equivalence:
gMonFib(X ) ' gMonICat(X )
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Monoidal structure on the total category
Given a lax monoidal functor
(F , φ) : (X op,⊗)→ (Cat,×)
φ : Fx ×Fy → F(x ⊗ y)
(x , a)⊗ (y , b) =
(x ⊗ y , φx ,y (a, b))
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Monoidal structure on the total category
Given a lax monoidal functor
(F , φ) : (X op,⊗)→ (Cat,×)
φ : Fx ×Fy → F(x ⊗ y)
(x , a)⊗ (y , b) = (x ⊗ y ,
φx ,y (a, b))
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Monoidal structure on the total category
Given a lax monoidal functor
(F , φ) : (X op,⊗)→ (Cat,×)
φ : Fx ×Fy → F(x ⊗ y)
(x , a)⊗ (y , b) = (x ⊗ y , φx ,y (a, b))
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Shulman’s Monoidal Grothendieck Construction
Theorem (Shulman)
If X is cartesian monoidal, then
gMonFib(X ) ' f MonICat(X )
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Cartesian Case
Theorem (Vasilakopoulou, M)
If X is a cartesian monoidal category, then
gMonFib(X ) gMonICat(X )
f MonFib(X ) f MonICat(X )
''
'
'
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Example: Modules
Mod: Ringop → Cat
(f : k → k ′) 7→ (f ∗ : Modk ′ → Modk)
∫Mod:
(k,M) (k ′,N)(f ,g)
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Example: Modules
Mod: Ringop → Cat
(f : k → k ′) 7→ (f ∗ : Modk ′ → Modk)∫Mod:
(k,M) (k ′,N)(f ,g)
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Example: Modules
(Mod, µ) : (Ringop,⊗)→ (Cat,×)
µ : Modk ×Modk ′ → Modk⊗k ′
(M,N) 7→ M ⊗Z N
(∫
Mod,⊗µ):
(k ,M)⊗µ (k ′,N) = (k ⊗ k ′,M ⊗Z N)
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
Example: Modules
(Mod, µ) : (Ringop,⊗)→ (Cat,×)
µ : Modk ×Modk ′ → Modk⊗k ′
(M,N) 7→ M ⊗Z N
(∫
Mod,⊗µ):
(k ,M)⊗µ (k ′,N) = (k ⊗ k ′,M ⊗Z N)
Joe Moeller UCR
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Motivation Grothendieck C. Monoidal Grothendieck C. Applications
John Baez, John Foley, Joseph Moeller, and Blake Pollard.Network models.arXiv:1711.00037 [math.CT], 2017.
Joe Moeller and Christina Vasilakopoulou.Monoidal grothendieck construction.arXiv:1809.00727 [math.CT], 2019.
Michael Shulman.Framed bicategories and monoidal fibrations.Theory Appl. Categ., 20:No. 18, 650–738, 2008.
Joe Moeller UCR