The Grothendieck Construction for Enriched, Internal and -Categories - - PowerPoint PPT Presentation

The Grothendieck Construction for Enriched, Internal and ∞-Categories

Liang Ze Wong

Final Exam

26 Feb 2019

Publications

BW1 Jonathan Beardsley and Liang Ze Wong. The enriched Grothendieck construction. Advances in Math, 2019. BW2 . The operadic nerve, relative nerve, and the Grothendieck construction. arXiv:1808.08020, 2018. W Liang Ze Wong. Smash products for Non-cartesian Internal Prestacks, 2019. Alex Chirvasitu, S Paul Smith and Liang Ze Wong. Noncommutative geometry of homogenized quantum sl(2, C), Pacific Journal of Math, 2017. Krzysztof Kapulkin, Zachery Lindsey and Liang Ze Wong. A co-reflection of cubical sets into simplicial sets with applications to model structures, 2019. Simon Cho, Cory Knapp, Clive Newstead and Liang Ze Wong. Weak equivalences between categories of models of type

• theory. (in preparation)

Semi-direct Products

Let G be a group, and N another group with a G-action G × N → N.

Semi-direct Products

Let G be a group, and N another group with a G-action G × N → N. Can form N ⋊ G

Semi-direct Products

Let G be a group, and N another group with a G-action G × N → N. Can form N ⋊ G = the set N × G with multiplication (n, g)(m, f ) = (n (g · m), gf ).

Semi-direct Products

Let G be a group, and N another group with a G-action G × N → N. Can form N ⋊ G = the set N × G with multiplication (n, g)(m, f ) = (n (g · m), gf ). Also have a split surjection: N ⋊ G G

π

Semi-direct Products

Let G be a group, and N another group with a G-action G × N → N. Can form N ⋊ G = the set N × G with multiplication (n, g)(m, f ) = (n (g · m), gf ). Also have a split surjection: N = ker π N ⋊ G G

π

And we can recover N by taking the kernel of π.

Semi-direct Products

Splitting Lemma (Classical) There is a bijective correspondence:    G-actions G × N → N    ∼ =

⋊ ker

     Split surjections N ⋊ G ։ G     

Semi-direct Products

Splitting Lemma (Classical) There is a bijective correspondence:    G-actions G × N → N    ∼ =

⋊ ker

     Split surjections N ⋊ G ։ G      Today, we’ll see that G and N don’t have to be groups:

Semi-direct Products

Splitting Lemma (Classical) There is a bijective correspondence:    G-actions G × N → N    ∼ =

⋊ ker

     Split surjections N ⋊ G ։ G      Today, we’ll see that G and N don’t have to be groups: They can be algebras, categories, ∞-categories, and more!

The Grothendieck Construction

A group G can be treated as a category C = ∗

G .

The Grothendieck Construction

A group G can be treated as a category C = ∗

G .

A group action G × N → N can be treated as a group hom G → Aut(N)

The Grothendieck Construction

A group G can be treated as a category C = ∗

G .

A group action G × N → N can be treated as a group hom G → Aut(N), or a functor C → Grp, ∗ → N.

The Grothendieck Construction

A group G can be treated as a category C = ∗

G .

A group action G × N → N can be treated as a group hom G → Aut(N), or a functor C → Grp, ∗ → N. Generalizing, we may start with a category C (with many objects)

The Grothendieck Construction

A group G can be treated as a category C = ∗

G .

A group action G × N → N can be treated as a group hom G → Aut(N), or a functor C → Grp, ∗ → N. Generalizing, we may start with a category C (with many objects) acting on a collection of categories {Nc}c∈C.

The Grothendieck Construction

A group G can be treated as a category C = ∗

G .

A group action G × N → N can be treated as a group hom G → Aut(N), or a functor C → Grp, ∗ → N. Generalizing, we may start with a category C (with many objects) acting on a collection of categories {Nc}c∈C. i.e. a functor N• : C → Cat c → Nc, (c

g

− → d) → (Nc

g∗

− → Nd).

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d)

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

c Nc x

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

c Nc x d Nd y

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

c Nc x d Nd y g

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

c Nc x d Nd y g g∗

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

c Nc x d Nd y g g∗ g∗x n

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

c Nc x d Nd y g g∗ g∗x n b Nb w

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

c Nc x d Nd y g g∗ g∗x n b Nb w f f∗

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

c Nc x d Nd y g g∗ g∗x n b Nb w f f∗ f∗w m

The Grothendieck Construction

Given N• : C → Cat, we can define a new category N• ⋊ C:

• bjects are (x, c) where x ∈ Nc

arrows are (g∗x

n

− → y, c

g

− → d) with composition: (n, g) ◦ (m, f ) = (n (g∗m), gf ).

c Nc x d Nd y g g∗ g∗x n b Nb w f f∗ f∗w m (gf )∗w g∗m

The Grothendieck Construction

Splitting Lemma (Classical) There is a bijective correspondence:      G-actions G → Aut(N)      ∼ =

⋊ ker

     Split surjections N ⋊ G ։ G      Theorem (Grothendieck 1959) There is an isomorphism of categories:      Functors N• : C → Cat      ∼ =

⋊ fibers

     Split opfibrations N• ⋊ C → C     

Examples

For c ∈ C, let C/c be the slice category over c:

Examples

For c ∈ C, let C/c be the slice category over c: x c

Examples

For c ∈ C, let C/c be the slice category over c: x y c

Examples

For c ∈ C, let C/c be the slice category over c: x y c

Examples

For c ∈ C, let C/c be the slice category over c: x y c Have C/•: C → Cat sending g : c → d to C/c

g◦−

− − − − − − → C/d.

Examples

For c ∈ C, let C/c be the slice category over c: x y c Have C/•: C → Cat sending g : c → d to C/c

g◦−

− − − − − − → C/d. (C/•) ⋊ C has objects (x → c, c) and morphisms: x y c d

g

Examples

For c ∈ C, let C/c be the slice category over c: x y c Have C/•: C → Cat sending g : c → d to C/c

g◦−

− − − − − − → C/d. (C/•) ⋊ C has objects (x → c, c) and morphisms: x y c d

g

Examples

For c ∈ C, let C/c be the slice category over c: x y c Have C/•: C → Cat sending g : c → d to C/c

g◦−

− − − − − − → C/d. (C/•) ⋊ C has objects (x → c, c) and morphisms: x y c d

g

Examples

For c ∈ C, let C/c be the slice category over c: x y c Have C/•: C → Cat sending g : c → d to C/c

g◦−

− − − − − − → C/d. (C/•) ⋊ C has objects (x → c, c) and morphisms: x y c d

g

(C/•) ⋊ C = ArrC and ArrC → C is the codomain functor.

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to f ∗ : ModS → ModR.

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to f ∗ : ModS → ModR. Mod• ⋊ Ringop has objects (M, R) and morphisms:

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to f ∗ : ModS → ModR. Mod• ⋊ Ringop has objects (M, R) and morphisms: (M, R)

( , )

− − − − − − − − − − − → (N, S)

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to f ∗ : ModS → ModR. Mod• ⋊ Ringop has objects (M, R) and morphisms: (M, R)

( , R

f

− →S ) − − − − − − − − − − − → (N, S)

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to f ∗ : ModS → ModR. Mod• ⋊ Ringop has objects (M, R) and morphisms: (M, R)

( M→f ∗N, R

f

− →S ) − − − − − − − − − − − → (N, S)

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to f ∗ : ModS → ModR. Mod• ⋊ Ringop has objects (M, R) and morphisms: (M, R)

( M→f ∗N, R

f

− →S ) − − − − − − − − − − − → (N, S) This is the global module category Mod.

The Skew Group Ring and Smash Products

Let G be a group.

The Skew Group Ring and Smash Products

Let G be a group. Instead of G acting on another group, suppose it acts on a k-algebra A G × A → A.

The Skew Group Ring and Smash Products

Let G be a group. Instead of G acting on another group, suppose it acts on a k-algebra A G × A → A. Can form the skew group ring A ⋊ G

The Skew Group Ring and Smash Products

Let G be a group. Instead of G acting on another group, suppose it acts on a k-algebra A G × A → A. Can form the skew group ring A ⋊ G =

g∈G A where

(a, g) · (b, h) = (a (g · b), gh).

The Skew Group Ring and Smash Products

Let G be a group. Instead of G acting on another group, suppose it acts on a k-algebra A G × A → A. Can form the skew group ring A ⋊ G =

g∈G A where

(a, g) · (b, h) = (a (g · b), gh). But we don’t have an algebra map A ⋊ G → kG . . .

Interlude: Comonoids and Comodules

kG is both an algebra and a coalgebra, with comultiplication: ∆: kG → kG ⊗ kG, g → g ⊗ g.

Interlude: Comonoids and Comodules

kG is both an algebra and a coalgebra, with comultiplication: ∆: kG → kG ⊗ kG, g → g ⊗ g. Can define comodules for any coalgebra C, with coactions M → M ⊗ C.

Interlude: Comonoids and Comodules

kG is both an algebra and a coalgebra, with comultiplication: ∆: kG → kG ⊗ kG, g → g ⊗ g. Can define comodules for any coalgebra C, with coactions M → M ⊗ C. We can similarly define comonoids and their comodules in any monoidal category (V, ⊗, 1).

Interlude: Comonoids and Comodules

kG is both an algebra and a coalgebra, with comultiplication: ∆: kG → kG ⊗ kG, g → g ⊗ g. Can define comodules for any coalgebra C, with coactions M → M ⊗ C. We can similarly define comonoids and their comodules in any monoidal category (V, ⊗, 1). Any X ∈ Set has a unique comonoid structure

Interlude: Comonoids and Comodules

kG is both an algebra and a coalgebra, with comultiplication: ∆: kG → kG ⊗ kG, g → g ⊗ g. Can define comodules for any coalgebra C, with coactions M → M ⊗ C. We can similarly define comonoids and their comodules in any monoidal category (V, ⊗, 1). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X

Interlude: Comonoids and Comodules

kG is both an algebra and a coalgebra, with comultiplication: ∆: kG → kG ⊗ kG, g → g ⊗ g. Can define comodules for any coalgebra C, with coactions M → M ⊗ C. We can similarly define comonoids and their comodules in any monoidal category (V, ⊗, 1). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X an X-grading W =

x∈X Wx

Interlude: Comonoids and Comodules

kG is both an algebra and a coalgebra, with comultiplication: ∆: kG → kG ⊗ kG, g → g ⊗ g. Can define comodules for any coalgebra C, with coactions M → M ⊗ C. We can similarly define comonoids and their comodules in any monoidal category (V, ⊗, 1). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X an X-grading W =

x∈X Wx

an X-coaction W → W × X

Interlude: Comonoids and Comodules

kG is both an algebra and a coalgebra, with comultiplication: ∆: kG → kG ⊗ kG, g → g ⊗ g. Can define comodules for any coalgebra C, with coactions M → M ⊗ C. We can similarly define comonoids and their comodules in any monoidal category (V, ⊗, 1). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X an X-grading W =

x∈X Wx

an X-coaction W → W × X

Interlude: Comonoids and Comodules

kG is both an algebra and a coalgebra, with comultiplication: ∆: kG → kG ⊗ kG, g → g ⊗ g. Can define comodules for any coalgebra C, with coactions M → M ⊗ C. We can similarly define comonoids and their comodules in any monoidal category (V, ⊗, 1). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X an X-grading W =

x∈X Wx

an X-coaction W → W × X In Vectk, these are not equivalent.

The Skew Group Ring and Smash Products

We don’t have an algebra map from A ⋊ G =

g∈G A to kG.

The Skew Group Ring and Smash Products

We don’t have an algebra map from A ⋊ G =

g∈G A to kG.

But we do have a G-grading on A ⋊ G,

The Skew Group Ring and Smash Products

We don’t have an algebra map from A ⋊ G =

g∈G A to kG.

But we do have a G-grading on A ⋊ G, or equivalently, a kG-coaction on A ⋊ G (a, g) → (a, g) ⊗ g.

The Skew Group Ring and Smash Products

We don’t have an algebra map from A ⋊ G =

g∈G A to kG.

But we do have a G-grading on A ⋊ G, or equivalently, a kG-coaction on A ⋊ G (a, g) → (a, g) ⊗ g. The coaction perspective allows us to replace kG with any bialgebra or Hopf algebra H.

The Skew Group Ring and Smash Products

Theorem (Cohen-Montgomery 1984) For G a group, there is a bijective correspondence:    G-actions G × A → A    ∼ =

⋊ fibers

     G-graded algebras A ⋊ G     

The Skew Group Ring and Smash Products

Theorem (Cohen-Montgomery 1984) For G a group, there is a bijective correspondence:    G-actions G × A → A    ∼ =

⋊ fibers

     G-graded algebras A ⋊ G      Theorem (v.d.Bergh 1984, Blattner-Montgomery 1985) For H a Hopf algebra, there is a bijective correspondence:      H-module algebras H ⊗ A → A      ∼ =

⋊ coinv

     H-comodule algebras A ⋊ H     

Smash Product A⋊H Grothendieck Construction N•⋊C Semi-direct Product N⋊G k-linear many objects

?

Smash Product A⋊H Grothendieck Construction N•⋊C Semi-direct Product N⋊G k-linear many objects

Enriched and Internal Categories

A small category C has: a set of objects C0 for all x, y ∈ C0, a set of arrows HomC(x, y)

Enriched and Internal Categories

A small category C has: a set of objects C0 for all x, y ∈ C0, a set of arrows HomC(x, y) Can replace (Set, ×, {∗}) with any monoidal category (V, ⊗, 1):

Enriched and Internal Categories

A small category C has: a set of objects C0 for all x, y ∈ C0, a set of arrows HomC(x, y) Can replace (Set, ×, {∗}) with any monoidal category (V, ⊗, 1): A V-enriched category C has: a set of objects C0 for all x, y ∈ C0, arrows HomC(x, y) ∈ V

Enriched and Internal Categories

A small category C has: a set of objects C0 for all x, y ∈ C0, a set of arrows HomC(x, y) Can replace (Set, ×, {∗}) with any monoidal category (V, ⊗, 1): A V-enriched category C has: a set of objects C0 for all x, y ∈ C0, arrows HomC(x, y) ∈ V A V-internal category C has:

• bjects C0 ∈ V

arrows C1 ∈ V

Enriched and Internal Categories

A small category C has: a set of objects C0 for all x, y ∈ C0, a set of arrows HomC(x, y) Can replace (Set, ×, {∗}) with any monoidal category (V, ⊗, 1): A V-enriched category C has: a set of objects C0 for all x, y ∈ C0, arrows HomC(x, y) ∈ V A V-internal category C has:

• bjects C0 ∈ V objects C0 ∈ Comon(V)

arrows C1 ∈ V arrows C1 ∈ C0ComodC0

Enriched and Internal Categories for (Vectk, ⊗k, k)

Enriched and Internal Categories for (Vectk, ⊗k, k)

A Vectk-enriched category is a k-linear category C with: a set of objects C0 for all x, y ∈ C0, a k-vector space HomC(x, y)

Enriched and Internal Categories for (Vectk, ⊗k, k)

A Vectk-enriched category is a k-linear category C with: a set of objects C0 for all x, y ∈ C0, a k-vector space HomC(x, y) e.g. a k-algebra A gives a k-linear category ∗

A

Enriched and Internal Categories for (Vectk, ⊗k, k)

A Vectk-enriched category is a k-linear category C with: a set of objects C0 for all x, y ∈ C0, a k-vector space HomC(x, y) e.g. a k-algebra A gives a k-linear category ∗

A

A many-object enriched category replaces ∗ with any set.

Enriched and Internal Categories for (Vectk, ⊗k, k)

A Vectk-enriched category is a k-linear category C with: a set of objects C0 for all x, y ∈ C0, a k-vector space HomC(x, y) e.g. a k-algebra A gives a k-linear category ∗

A

A many-object enriched category replaces ∗ with any set. Any k-linear category gives rise to a Vectk-internal category with:

• bjects kC0

arrows ⊕x,yHomC(x, y)

Enriched and Internal Categories for (Vectk, ⊗k, k)

A Vectk-enriched category is a k-linear category C with: a set of objects C0 for all x, y ∈ C0, a k-vector space HomC(x, y) e.g. a k-algebra A gives a k-linear category ∗

A

A many-object enriched category replaces ∗ with any set. Any k-linear category gives rise to a Vectk-internal category with:

• bjects kC0

arrows ⊕x,yHomC(x, y) e.g. a k-algebra A gives an internal category k

A

Enriched and Internal Categories for (Vectk, ⊗k, k)

A Vectk-enriched category is a k-linear category C with: a set of objects C0 for all x, y ∈ C0, a k-vector space HomC(x, y) e.g. a k-algebra A gives a k-linear category ∗

A

A many-object enriched category replaces ∗ with any set. Any k-linear category gives rise to a Vectk-internal category with:

• bjects kC0

arrows ⊕x,yHomC(x, y) e.g. a k-algebra A gives an internal category k

A

A ‘many-object’ internal category replaces k with a k-coalgebra.

Enriched and Internal Categories for (Vectk, ⊗k, k)

A Vectk-enriched category is a k-linear category C with: a set of objects C0 for all x, y ∈ C0, a k-vector space HomC(x, y) e.g. a k-algebra A gives a k-linear category ∗

A

A many-object enriched category replaces ∗ with any set. Any k-linear category gives rise to a Vectk-internal category with:

• bjects kC0

arrows ⊕x,yHomC(x, y) e.g. a k-algebra A gives an internal category k

A

A ‘many-object’ internal category replaces k with a k-coalgebra. (possibly with other properties, e.g. cocommutativty)

Smash Product A⋊H Grothendieck Construction N•⋊C Semi-direct Product N⋊G

• cocomm. comon.
• f objects

k-linear objects k-linear Homs k-linear many objects

Enriched version Smash Product A⋊H Grothendieck Construction N•⋊C Semi-direct Product N⋊G

• cocomm. comon.
• f objects

k-linear objects k-linear Homs k-linear many objects

Internal version Smash Product A⋊H Enriched version Smash Product A⋊H Grothendieck Construction N•⋊C Semi-direct Product N⋊G

• cocomm. comon.
• f objects

k-linear objects k-linear Homs k-linear many objects

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them.

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them. Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C   

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them. Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C    Want to replace the ordinary category C with a V-category C.

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them. Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C    Want to replace the ordinary category C with a V-category C. Theorem (W) Let C be a comonoidal V-category.

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them. Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C    Want to replace the ordinary category C with a V-category C. Theorem (W) Let C be a comonoidal V-category. Then    C-module V-cats C ⊗ A → A    ∼ =

⋊ coinv

   C-comodule V-cats A ⋊ C   

Internal Version

Theorem (W) Suppose V has equalizers, and ⊗ preserves them.

Internal Version

Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then    C-module int cats C ⊗ A → A    ∼ =

⋊ coinv

   C-comod int cats A ⋊ C   

Internal Version

Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then    C-module int cats C ⊗ A → A    ∼ =

⋊ coinv

   C-comod int cats A ⋊ C    Let C = (C0, C1) be comonoidal internal category, and A = (A0, A1) be a C-module category.

Internal Version

Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then    C-module int cats C ⊗ A → A    ∼ =

⋊ coinv

   C-comod int cats A ⋊ C    Let C = (C0, C1) be comonoidal internal category, and A = (A0, A1) be a C-module category. Can form A ⋊ C with objects A0

Internal Version

Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then    C-module int cats C ⊗ A → A    ∼ =

⋊ coinv

   C-comod int cats A ⋊ C    Let C = (C0, C1) be comonoidal internal category, and A = (A0, A1) be a C-module category. Can form A ⋊ C with objects A0 and arrows A1 ⊠A0 (C1 ⊠C0 A0).

Internal Version

Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then    C-module int cats C ⊗ A → A    ∼ =

⋊ coinv

   C-comod int cats A ⋊ C    Let C = (C0, C1) be comonoidal internal category, and A = (A0, A1) be a C-module category. Can form A ⋊ C with objects A0 and arrows A1 ⊠A0 (C1 ⊠C0 A0). When C = (k, H), A = (k, A), this is just A ⊠k (H ⊠k k) ∼ = A ⊗ H.

Internal Version

A1 ⊠A0 C1 ⊠C0 A0 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A1 ⊠A0 C1 ⊠C0 C1 ⊠C0 A0 A1 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A0

Internal Version

A1 ⊠A0 C1 ⊠C0 A0 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A1 ⊠A0 C1 ⊠C0 C1 ⊠C0 A0 A1 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A0

Internal Version

A1 ⊠A0 C1 ⊠C0 A0 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A1 ⊠A0 C1 ⊠C0 C1 ⊠C0 A0 A1 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A0

Internal Version

A1 ⊠A0 C1 ⊠C0 A0 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A1 ⊠A0 C1 ⊠C0 C1 ⊠C0 A0 A1 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A0

Internal Version

A1 ⊠A0 C1 ⊠C0 A0 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A1 ⊠A0 C1 ⊠C0 C1 ⊠C0 A0 A1 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A0

Internal Version

A1 ⊠A0 C1 ⊠C0 A0 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A1 ⊠A0 C1 ⊠C0 C1 ⊠C0 A0 A1 ⊠A0 A1 ⊠A0 C1 ⊠C0 A0 A1 ⊠A0 C1 ⊠C0 A0

Internal version Smash Product A⋊H Enriched version Smash Product A⋊H Grothendieck Construction N•⋊C Semi-direct Product N⋊G

• cocomm. comon.
• f objects

k-linear objects k-linear Homs k-linear N,G many objects

Internal version Smash Product A⋊H Enriched version Smash Product A⋊H Grothendieck Construction N•⋊C Semi-direct Product N⋊G

• cocomm. comon.
• f objects

k-linear objects k-linear Homs k-linear N,G many objects

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Suppose V has coproducts, and ⊗ preserves them. Then:    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C   

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Suppose V has coproducts, and ⊗ preserves them. Then:    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C    When do we get an actual functor A• ⋊ C → C ?

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Suppose V has coproducts, and ⊗ preserves them. Then:    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C    When do we get an actual functor A• ⋊ C → C V?

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Suppose V has coproducts, and ⊗ preserves them. Then:    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C    When do we get an actual functor A• ⋊ C → C V? Theorem (BW1) Suppose further that 1 is terminal, V has pullbacks, and pullbacks and HomV(1, −) preserve coproducts.

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Suppose V has coproducts, and ⊗ preserves them. Then:    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C    When do we get an actual functor A• ⋊ C → C V? Theorem (BW1) Suppose further that 1 is terminal, V has pullbacks, and pullbacks and HomV(1, −) preserve coproducts. Then:    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   Split opfibrations A• ⋊ C → CV   

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Suppose V has coproducts, and ⊗ preserves them. Then:    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   C-graded V-cats A• ⋊ C    When do we get an actual functor A• ⋊ C → C V? Theorem (BW1) Suppose further that 1 is terminal, V has pullbacks, and pullbacks and HomV(1, −) preserve coproducts. Then:    Functors A• : C → V-Cat    ∼ =

⋊ fibers

   Split opfibrations A• ⋊ C → CV    e.g. V = sSet

Simplicial sets and ∞-categories

A simplicial set is a functor X• : ∆op → Set.

Simplicial sets and ∞-categories

A simplicial set is a functor X• : ∆op → Set. X0 X1 X2 X3 . . .

Simplicial sets and ∞-categories

A simplicial set is a functor X• : ∆op → Set. X0 X1 X2 X3 . . . points paths homotopies

Simplicial sets and ∞-categories

A simplicial set is a functor X• : ∆op → Set. X0 X1 X2 X3 . . . points paths homotopies Simplicial sets are thus combinatorial models of topological spaces.

Simplicial sets and ∞-categories

A simplicial set is a functor X• : ∆op → Set. X0 X1 X2 X3 . . . points paths homotopies Simplicial sets are thus combinatorial models of topological spaces. sSet-enriched categories are ‘categories enriched in spaces’:

Simplicial sets and ∞-categories

A simplicial set is a functor X• : ∆op → Set. X0 X1 X2 X3 . . . points paths homotopies Simplicial sets are thus combinatorial models of topological spaces. sSet-enriched categories are ‘categories enriched in spaces’: Ob(C) X0 X1 X2 . . .

• bjects

arrows homotopies

Simplicial sets and ∞-categories

A simplicial set is a functor X• : ∆op → Set. X0 X1 X2 X3 . . . points paths homotopies Simplicial sets are thus combinatorial models of topological spaces. sSet-enriched categories are ‘categories enriched in spaces’: Ob(C) X0 X1 X2 . . .

• bjects

arrows homotopies i.e. an ∞-category!

Simplicial sets and ∞-categories

But simplicial sets themselves model ∞-categories: X0 X1 X2 X3 . . .

• bjects

arrows homotopies

Simplicial sets and ∞-categories

But simplicial sets themselves model ∞-categories: X0 X1 X2 X3 . . .

• bjects

arrows homotopies And both models are related: sSet sCat

C N

⊣

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets: Theorem (Lurie 2009)      Simplical maps A• : S → Cat∞      ≃

⋊

     Cocartesian fibrations A• ⋊ S → S     

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets: Theorem (Lurie 2009)      Simplical maps A• : S → Cat∞      ≃

⋊

     Cocartesian fibrations A• ⋊ S → S      But applying the result of BW1 gives a sSet-enriched version.

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets: Theorem (Lurie 2009)      Simplical maps A• : S → Cat∞      ≃

⋊

     Cocartesian fibrations A• ⋊ S → S      But applying the result of BW1 gives a sSet-enriched version. How do these compare?

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets: Theorem (Lurie 2009)      Simplical maps A• : S → Cat∞      ≃

⋊

     Cocartesian fibrations A• ⋊ S → S      But applying the result of BW1 gives a sSet-enriched version. How do these compare? Theorem (BW2) Let A• : C → sCat and A• : C

A•

− − − − − → sCat

N

− − − − → sSet.

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets: Theorem (Lurie 2009)      Simplical maps A• : S → Cat∞      ≃

⋊

     Cocartesian fibrations A• ⋊ S → S      But applying the result of BW1 gives a sSet-enriched version. How do these compare? Theorem (BW2) Let A• : C → sCat and A• : C

A•

− − − − − → sCat

N

− − − − → sSet. Then N(A•) ⋊ N(C) ∼ = N (A• ⋊ C) .

Internal version ∞-version Smash Product A⋊H Enriched version Smash Product A⋊H Grothendieck Construction N•⋊C Semi-direct Product N⋊G

• cocomm. comon.
• f objects

many objects

Thank you!

Questions?

Application: Monoidal ∞-categories

Recall the simplex category ∆:

• bjects are [n] = {0 ≤ 1 ≤ · · · ≤ n}

morphisms are order-preserving maps

Application: Monoidal ∞-categories

Recall the simplex category ∆:

• bjects are [n] = {0 ≤ 1 ≤ · · · ≤ n}

morphisms are order-preserving maps Let (C, ⊗, 1) be a strict monoidal category.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

• bjects are [n] = {0 ≤ 1 ≤ · · · ≤ n}

morphisms are order-preserving maps Let (C, ⊗, 1) be a strict monoidal category. Then we have: C • : ∆op → Cat, [n] → C n.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

• bjects are [n] = {0 ≤ 1 ≤ · · · ≤ n}

morphisms are order-preserving maps Let (C, ⊗, 1) be a strict monoidal category. Then we have: C • : ∆op → Cat, [n] → C n. ∗ C C 2 . . .

Application: Monoidal ∞-categories

Recall the simplex category ∆:

• bjects are [n] = {0 ≤ 1 ≤ · · · ≤ n}

morphisms are order-preserving maps Let (C, ⊗, 1) be a strict monoidal category. Then we have: C • : ∆op → Cat, [n] → C n. ∗ C C 2 . . . ∗ 1

Application: Monoidal ∞-categories

Recall the simplex category ∆:

• bjects are [n] = {0 ≤ 1 ≤ · · · ≤ n}

morphisms are order-preserving maps Let (C, ⊗, 1) be a strict monoidal category. Then we have: C • : ∆op → Cat, [n] → C n. ∗ C C 2 . . . ∗ 1 c ⊗ d (c, d)

Application: Monoidal ∞-categories

Recall the simplex category ∆:

• bjects are [n] = {0 ≤ 1 ≤ · · · ≤ n}

morphisms are order-preserving maps Let (C, ⊗, 1) be a strict monoidal category. Then we have: C • : ∆op → Cat, [n] → C n. ∗ C C 2 . . . ∗ 1 c ⊗ d (c, d) C ⊗ := C • ⋊ ∆op

Application: Monoidal ∞-categories

Recall the simplex category ∆:

• bjects are [n] = {0 ≤ 1 ≤ · · · ≤ n}

morphisms are order-preserving maps Let (C, ⊗, 1) be a strict monoidal category. Then we have: C • : ∆op → Cat, [n] → C n. ∗ C C 2 . . . ∗ 1 c ⊗ d (c, d) C ⊗ := C • ⋊ ∆op has an opfibration down to ∆op.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

• bjects are [n] = {0 ≤ 1 ≤ · · · ≤ n}

morphisms are order-preserving maps Let (C, ⊗, 1) be a strict monoidal category. Then we have: C • : ∆op → Cat, [n] → C n. ∗ C C 2 . . . ∗ 1 c ⊗ d (c, d) C ⊗ := C • ⋊ ∆op has an opfibration down to ∆op. In fact, we can define monoidal categories in terms of opfibrations M → ∆op.

Application: Monoidal ∞-categories

Proposition (Lurie 2007) A simplicial monoidal category (C, ⊗, 1) gives rise to a monoidal ∞-category N(C ⊗).

Application: Monoidal ∞-categories

Proposition (Lurie 2007) A simplicial monoidal category (C, ⊗, 1) gives rise to a monoidal ∞-category N(C ⊗). Theorem (BW2) Let C be a strict simplicial monoidal category.

Application: Monoidal ∞-categories

Proposition (Lurie 2007) A simplicial monoidal category (C, ⊗, 1) gives rise to a monoidal ∞-category N(C ⊗). Theorem (BW2) Let C be a strict simplicial monoidal category. Then N(C op ⊗) and N(C ⊗)op are equivalent as monoidal ∞-categories.

Application: Monoidal ∞-categories

Proposition (Lurie 2007) A simplicial monoidal category (C, ⊗, 1) gives rise to a monoidal ∞-category N(C ⊗). Theorem (BW2) Let C be a strict simplicial monoidal category. Then N(C op ⊗) and N(C ⊗)op are equivalent as monoidal ∞-categories. This gives a better handle on coalgebras in monoidal ∞-categories arising from simplicial monoidal categories.