Joe Moeller Christina Vasilakopouloumath.ucr.edu/home/baez/SYCO4/moeller_syco4.pdf ·...

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

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Motivation

k ∈ Ring

Modk ∈ Cat

Modall???

Joe Moeller UCR


Motivation

k ∈ Ring Modk ∈ Cat

Modall???

Joe Moeller UCR


Motivation


Modall???

Joe Moeller UCR


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


Motivation


Mod: Ring→ Cat???

f : k → k ′ f ∗ : Modk ′ → Modk

Joe Moeller UCR


Motivation


Mod: Ring→ Cat???

f : k → k ′ f ∗ : Modk ′ → Modk

Joe Moeller UCR


Motivation


Mod: Ring→ Cat???

f : k → k ′

f ∗ : Modk ′ → Modk

Joe Moeller UCR


Motivation


Mod: Ring→ Cat???

f : k → k ′ f ∗ : Modk ′ → Modk

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Motivation


Mod: Ringop → Cat

f : k → k ′ f ∗ : Modk ′ → Modk

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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


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


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

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Fibrations

A b

X x y

P

f

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Fibrations

cartesian lift

A a b

X x y

P

φ

f

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Fibrations

pullback

A f ∗(b) b

X x y

P

φ

f

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Fibrations

reindexing functor

Ay Axf ∗

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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

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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

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2-Equivalence

Theorem

The Grothendieck construction gives an equivalence:

ICat(X ) ∼= Fib(X )

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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

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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.

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Theorem (Vasilakopoulou, M)

The Grothendieck construction lifts to an equivalence:

f MonFib(X ) ' f MonICat(X )

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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.

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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.

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The Grothendieck construction lifts to an equivalence:

gMonFib(X ) ' gMonICat(X )

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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))

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(F , φ) : (X op,⊗)→ (Cat,×)

φ : Fx ×Fy → F(x ⊗ y)

(x , a)⊗ (y , b) = (x ⊗ y ,

φx ,y (a, b))

Joe Moeller UCR




(F , φ) : (X op,⊗)→ (Cat,×)

φ : Fx ×Fy → F(x ⊗ y)

(x , a)⊗ (y , b) = (x ⊗ y , φx ,y (a, b))

Joe Moeller UCR


Shulman’s Monoidal Grothendieck Construction

Theorem (Shulman)

If X is cartesian monoidal, then

gMonFib(X ) ' f MonICat(X )

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Cartesian Case


If X is a cartesian monoidal category, then

gMonFib(X ) gMonICat(X )

f MonFib(X ) f MonICat(X )

''

'

'

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Example: Modules

Mod: Ringop → Cat

(f : k → k ′) 7→ (f ∗ : Modk ′ → Modk)

∫Mod:

(k,M) (k ′,N)(f ,g)

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Example: Modules

Mod: Ringop → Cat

(f : k → k ′) 7→ (f ∗ : Modk ′ → Modk)∫Mod:

(k,M) (k ′,N)(f ,g)

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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)

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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


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.

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