The Grothendieck Constructionfor Enriched, Internal and ∞-Categories

Liang Ze Wong

Final Exam

26 Feb 2019

Publications

BW1 Jonathan Beardsley and Liang Ze Wong. The enrichedGrothendieck construction. Advances in Math, 2019.

BW2 . The operadic nerve, relative nerve, and theGrothendieck construction. arXiv:1808.08020, 2018.

W Liang Ze Wong. Smash products for Non-cartesian InternalPrestacks, 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. Aco-reflection of cubical sets into simplicial sets withapplications to model structures, 2019.

Simon Cho, Cory Knapp, Clive Newstead and Liang Ze Wong.Weak equivalences between categories of models of typetheory. (in preparation)

Semi-direct Products

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

G × N → N.

Can form N o G = the set N × G with multiplication

(n, g)(m, f ) = (n (g ·m), gf ).

Also have a split surjection:

N = ker π

N o G Gπ

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

Semi-direct Products

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

G × N → N.

Can form N o G

= the set N × G with multiplication

(n, g)(m, f ) = (n (g ·m), gf ).

Also have a split surjection:

N = ker π

N o G Gπ

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

Semi-direct Products

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

G × N → N.

Can form N o G = the set N × G with multiplication

(n, g)(m, f ) = (n (g ·m), gf ).

Also have a split surjection:

N = ker π

N o G Gπ

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

Semi-direct Products

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

G × N → N.

Can form N o G = the set N × G with multiplication

(n, g)(m, f ) = (n (g ·m), gf ).

Also have a split surjection:

N = ker π

N o G Gπ

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

Semi-direct Products

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

G × N → N.

Can form N o G = the set N × G with multiplication

(n, g)(m, f ) = (n (g ·m), gf ).

Also have a split surjection:

N = ker π N o G Gπ

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

∼=o

ker

Split surjections

N o G � G

Today, we’ll see that G and N don’t have to be groups:They can be algebras, categories, ∞-categories, and more!

Semi-direct Products

Splitting Lemma (Classical)

There is a bijective correspondence: G -actions

G × N → N

∼=o

ker

Split surjections

N o G � G

Today, we’ll see that G and N don’t have to be groups:

They can be algebras, categories, ∞-categories, and more!

Semi-direct Products

Splitting Lemma (Classical)

There is a bijective correspondence: G -actions

G × N → N

∼=o

ker

Split surjections

N o 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 .

A group action G × N → N can be treated as a group homG → Aut(N), or a functor

C → Grp, ∗ 7→ 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 7→ Nc , (cg−→ d) 7→ (Nc

g∗−→ Nd).

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 homG → Aut(N)

, or a functor

C → Grp, ∗ 7→ 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 7→ Nc , (cg−→ d) 7→ (Nc

g∗−→ Nd).

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 homG → Aut(N), or a functor

C → Grp, ∗ 7→ 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 7→ Nc , (cg−→ d) 7→ (Nc

g∗−→ Nd).

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 homG → Aut(N), or a functor

C → Grp, ∗ 7→ 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 7→ Nc , (cg−→ d) 7→ (Nc

g∗−→ Nd).

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 homG → Aut(N), or a functor

C → Grp, ∗ 7→ 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 7→ Nc , (cg−→ d) 7→ (Nc

g∗−→ Nd).

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 homG → Aut(N), or a functor

C → Grp, ∗ 7→ 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 7→ Nc , (cg−→ d) 7→ (Nc

g∗−→ Nd).

The Grothendieck Construction

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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

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

objects are (x , c) where x ∈ Nc

arrows are (g∗xn−→ 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)

∼=o

ker

Split surjections

N o G � G

Theorem (Grothendieck 1959)

There is an isomorphism of categories: Functors

N• : C → Cat

∼=o

fibers

Split opfibrations

N• o C → 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/cg◦−−−−−−−→ C/d .

(C/•) o C has objects (x → c , c) and morphisms:

x y

c dg

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

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/cg◦−−−−−−−→ C/d .

(C/•) o C has objects (x → c , c) and morphisms:

x y

c dg

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

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/cg◦−−−−−−−→ C/d .

(C/•) o C has objects (x → c , c) and morphisms:

x y

c dg

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

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/cg◦−−−−−−−→ C/d .

(C/•) o C has objects (x → c , c) and morphisms:

x y

c dg

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

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/cg◦−−−−−−−→ C/d .

(C/•) o C has objects (x → c , c) and morphisms:

x y

c dg

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

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/cg◦−−−−−−−→ C/d .

(C/•) o C has objects (x → c , c) and morphisms:

x y

c dg

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

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/cg◦−−−−−−−→ C/d .

(C/•) o C has objects (x → c , c) and morphisms:

x y

c dg

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

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/cg◦−−−−−−−→ C/d .

(C/•) o C has objects (x → c , c) and morphisms:

x y

c dg

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

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/cg◦−−−−−−−→ C/d .

(C/•) o C has objects (x → c , c) and morphisms:

x y

c dg

(C/•) o 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 .

Mod• o Ringop has objects (M,R) and morphisms:

(M,R)(

M→f ∗N

,

Rf−→S

)−−−−−−−−−−−→ (N,S)

This is the global module category Mod.

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to

f ∗ : ModS →ModR .

Mod• o Ringop has objects (M,R) and morphisms:

(M,R)(

M→f ∗N

,

Rf−→S

)−−−−−−−−−−−→ (N,S)

This is the global module category Mod.

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to

f ∗ : ModS →ModR .

Mod• o Ringop has objects (M,R) and morphisms:

(M,R)(

M→f ∗N

,

Rf−→S

)−−−−−−−−−−−→ (N, S)

This is the global module category Mod.

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to

f ∗ : ModS →ModR .

Mod• o Ringop has objects (M,R) and morphisms:

(M,R)(

M→f ∗N

, Rf−→S )−−−−−−−−−−−→ (N, S)

This is the global module category Mod.

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to

f ∗ : ModS →ModR .

Mod• o Ringop has objects (M,R) and morphisms:

(M,R)( M→f ∗N, R

f−→S )−−−−−−−−−−−→ (N, S)

This is the global module category Mod.

Examples

We have a functor Mod• : Ringop → Cat sending f : R → S to

f ∗ : ModS →ModR .

Mod• o 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.

Instead of G acting on another group, supposeit acts on a k-algebra A

G × A→ A.

Can form the skew group ring Ao G =⊕

g∈G A where

(a, g) · (b, h) = (a (g · b), gh).

But we don’t have an algebra map Ao G → kG . . .

The Skew Group Ring and Smash Products

Let G be a group. Instead of G acting on another group, supposeit acts on a k-algebra A

G × A→ A.

Can form the skew group ring Ao G =⊕

g∈G A where

(a, g) · (b, h) = (a (g · b), gh).

But we don’t have an algebra map Ao G → kG . . .

The Skew Group Ring and Smash Products

Let G be a group. Instead of G acting on another group, supposeit acts on a k-algebra A

G × A→ A.

Can form the skew group ring Ao G

=⊕

g∈G A where

(a, g) · (b, h) = (a (g · b), gh).

But we don’t have an algebra map Ao G → kG . . .

The Skew Group Ring and Smash Products

Let G be a group. Instead of G acting on another group, supposeit acts on a k-algebra A

G × A→ A.

Can form the skew group ring Ao G =⊕

g∈G A where

(a, g) · (b, h) = (a (g · b), gh).

But we don’t have an algebra map Ao G → kG . . .

The Skew Group Ring and Smash Products

Let G be a group. Instead of G acting on another group, supposeit acts on a k-algebra A

G × A→ A.

Can form the skew group ring Ao G =⊕

g∈G A where

(a, g) · (b, h) = (a (g · b), gh).

But we don’t have an algebra map Ao G → kG . . .

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 anymonoidal 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.

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 anymonoidal 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.

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 anymonoidal 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.

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 anymonoidal 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.

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 anymonoidal 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.

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 anymonoidal 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.

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 anymonoidal 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.

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 anymonoidal 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.

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 anymonoidal 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 Ao G =⊕

g∈G A to kG .

But we do have a G -grading on Ao G , or equivalently, akG -coaction on Ao G

(a, g) 7→ (a, g)⊗ g .

The coaction perspective allows us to replace kG with anybialgebra or Hopf algebra H.

The Skew Group Ring and Smash Products

We don’t have an algebra map from Ao G =⊕

g∈G A to kG .

But we do have a G -grading on Ao G ,

or equivalently, akG -coaction on Ao G

(a, g) 7→ (a, g)⊗ g .

The coaction perspective allows us to replace kG with anybialgebra or Hopf algebra H.

The Skew Group Ring and Smash Products

We don’t have an algebra map from Ao G =⊕

g∈G A to kG .

But we do have a G -grading on Ao G , or equivalently, akG -coaction on Ao G

(a, g) 7→ (a, g)⊗ g .

The coaction perspective allows us to replace kG with anybialgebra or Hopf algebra H.

The Skew Group Ring and Smash Products

We don’t have an algebra map from Ao G =⊕

g∈G A to kG .

But we do have a G -grading on Ao G , or equivalently, akG -coaction on Ao G

(a, g) 7→ (a, g)⊗ g .

The coaction perspective allows us to replace kG with anybialgebra 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

∼=o

fibers

G -graded algebras

Ao 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

∼=o

coinv

H-comodule algebras

Ao 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

∼=o

fibers

G -graded algebras

Ao 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

∼=o

coinv

H-comodule algebras

Ao H

?

SmashProductAoH

GrothendieckConstruction

N•oC

Semi-directProductNoG

k-linear many objects

?

SmashProductAoH

GrothendieckConstruction

N•oC

Semi-directProductNoG

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)

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:

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:

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:

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:

objects C0 ∈ Varrows 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:

objects C0 ∈ V objects C0 ∈ Comon(V)

arrows C1 ∈ V arrows C1 ∈ C0ComodC0

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:

objects 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)

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:

objects 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)

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:

objects 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)

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:

objects 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)

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:

objects 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)

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:

objects 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)

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:

objects 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)

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:

objects 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)

Internalversion

SmashProductAoH

Enrichedversion

SmashProductAoH

GrothendieckConstruction

N•oC

Semi-directProductNoG

cocomm. comon.of objects k-linear objects

k-linear Homs

k-linear many objects

Internalversion

SmashProductAoH

Enrichedversion

SmashProductAoH

GrothendieckConstruction

N•oC

Semi-directProductNoG

cocomm. comon.of objects k-linear objects

k-linear Homs

k-linear many objects

Internalversion

SmashProductAoH

Enrichedversion

SmashProductAoH

GrothendieckConstruction

N•oC

Semi-directProductNoG

cocomm. comon.of objects k-linear objects

k-linear Homs

k-linear many objects

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them.

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o 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

∼=o

coinv

C-comodule V-cats

Ao C

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them.

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o 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

∼=o

coinv

C-comodule V-cats

Ao C

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them.

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o 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

∼=o

coinv

C-comodule V-cats

Ao C

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them.

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o 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

∼=o

coinv

C-comodule V-cats

Ao C

Enriched Versions

Suppose V has coproducts, and ⊗ preserves them.

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009) Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o 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

∼=o

coinv

C-comodule V-cats

Ao 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

∼=o

coinv

C-comod int cats

Ao C

Let C = (C0,C1) be comonoidal internal category, andA = (A0,A1) be a C-module category.

Can form Ao 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

Theorem (W)

Suppose V has equalizers, and ⊗ preserves them.Let C be a comonoidal internal category. Then C-module int cats

C ⊗ A → A

∼=o

coinv

C-comod int cats

Ao C

Let C = (C0,C1) be comonoidal internal category, andA = (A0,A1) be a C-module category.

Can form Ao 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

Theorem (W)

Suppose V has equalizers, and ⊗ preserves them.Let C be a comonoidal internal category. Then C-module int cats

C ⊗ A → A

∼=o

coinv

C-comod int cats

Ao C

Let C = (C0,C1) be comonoidal internal category, andA = (A0,A1) be a C-module category.

Can form Ao 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

Theorem (W)

Suppose V has equalizers, and ⊗ preserves them.Let C be a comonoidal internal category. Then C-module int cats

C ⊗ A → A

∼=o

coinv

C-comod int cats

Ao C

Let C = (C0,C1) be comonoidal internal category, andA = (A0,A1) be a C-module category.

Can form Ao 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

Theorem (W)

Suppose V has equalizers, and ⊗ preserves them.Let C be a comonoidal internal category. Then C-module int cats

C ⊗ A → A

∼=o

coinv

C-comod int cats

Ao C

Let C = (C0,C1) be comonoidal internal category, andA = (A0,A1) be a C-module category.

Can form Ao 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

Theorem (W)

Suppose V has equalizers, and ⊗ preserves them.Let C be a comonoidal internal category. Then C-module int cats

C ⊗ A → A

∼=o

coinv

C-comod int cats

Ao C

Let C = (C0,C1) be comonoidal internal category, andA = (A0,A1) be a C-module category.

Can form Ao 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

Internalversion

SmashProductAoH

Enrichedversion

SmashProductAoH

GrothendieckConstruction

N•oC

Semi-directProductNoG

cocomm. comon.of objects k-linear objects

k-linear Homs

k-linear N,G many objects

Internalversion

SmashProductAoH

Enrichedversion

SmashProductAoH

GrothendieckConstruction

N•oC

Semi-directProductNoG

cocomm. comon.of 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

∼=o

fibers

C -graded V-cats

A• o C

When do we get an actual functor A• o C → C

V

?

Theorem (BW1)

Suppose further that 1 is terminal, V has pullbacks, and pullbacksand HomV(1,−) preserve coproducts. Then: Functors

A• : C → V-Cat

∼=o

fibers

Split opfibrations

A• o C → CV

e.g. V = sSet

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)

Suppose V has coproducts, and ⊗ preserves them. Then: Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o C

When do we get an actual functor A• o C → C

V

?

Theorem (BW1)

Suppose further that 1 is terminal, V has pullbacks, and pullbacksand HomV(1,−) preserve coproducts. Then: Functors

A• : C → V-Cat

∼=o

fibers

Split opfibrations

A• o C → CV

e.g. V = sSet

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)

Suppose V has coproducts, and ⊗ preserves them. Then: Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o C

When do we get an actual functor A• o C → CV?

Theorem (BW1)

Suppose further that 1 is terminal, V has pullbacks, and pullbacksand HomV(1,−) preserve coproducts. Then: Functors

A• : C → V-Cat

∼=o

fibers

Split opfibrations

A• o C → CV

e.g. V = sSet

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)

Suppose V has coproducts, and ⊗ preserves them. Then: Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o C

When do we get an actual functor A• o C → CV?

Theorem (BW1)

Suppose further that 1 is terminal, V has pullbacks, and pullbacksand HomV(1,−) preserve coproducts.

Then: Functors

A• : C → V-Cat

∼=o

fibers

Split opfibrations

A• o C → CV

e.g. V = sSet

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)

Suppose V has coproducts, and ⊗ preserves them. Then: Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o C

When do we get an actual functor A• o C → CV?

Theorem (BW1)

Suppose further that 1 is terminal, V has pullbacks, and pullbacksand HomV(1,−) preserve coproducts. Then: Functors

A• : C → V-Cat

∼=o

fibers

Split opfibrations

A• o C → CV

e.g. V = sSet

Enriched Results

Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)

Suppose V has coproducts, and ⊗ preserves them. Then: Functors

A• : C → V-Cat

∼=o

fibers

C -graded V-cats

A• o C

When do we get an actual functor A• o C → CV?

Theorem (BW1)

Suppose further that 1 is terminal, V has pullbacks, and pullbacksand HomV(1,−) preserve coproducts. Then: Functors

A• : C → V-Cat

∼=o

fibers

Split opfibrations

A• o C → CV

e.g. V = sSet

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 . . .

objects arrows homotopies

i.e. an ∞-category!

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 . . .

objects arrows homotopies

i.e. an ∞-category!

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 . . .

objects arrows homotopies

i.e. an ∞-category!

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 . . .

objects arrows homotopies

i.e. an ∞-category!

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 . . .

objects arrows homotopies

i.e. an ∞-category!

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 . . .

objects arrows homotopies

i.e. an ∞-category!

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 . . .

objects arrows homotopies

i.e. an ∞-category!

Simplicial sets and ∞-categories

But simplicial sets themselves model ∞-categories:

X0 X1 X2 X3 . . .

objects arrows homotopies

And both models are related:

sSet sCat

C

N

a

Simplicial sets and ∞-categories

But simplicial sets themselves model ∞-categories:

X0 X1 X2 X3 . . .

objects arrows homotopies

And both models are related:

sSet sCat

C

N

a

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets:

Theorem (Lurie 2009)

Simplical maps

A• : S → Cat∞

'o

Cocartesian fibrations

A• o 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• : CA•−−−−−→ sCat

N−−−−→ sSet. Then

N(A•) o N(C ) ∼= N (A• o C ) .

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets:

Theorem (Lurie 2009)

Simplical maps

A• : S → Cat∞

'o

Cocartesian fibrations

A• o 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• : CA•−−−−−→ sCat

N−−−−→ sSet. Then

N(A•) o N(C ) ∼= N (A• o C ) .

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets:

Theorem (Lurie 2009)

Simplical maps

A• : S → Cat∞

'o

Cocartesian fibrations

A• o 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• : CA•−−−−−→ sCat

N−−−−→ sSet. Then

N(A•) o N(C ) ∼= N (A• o C ) .

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets:

Theorem (Lurie 2009)

Simplical maps

A• : S → Cat∞

'o

Cocartesian fibrations

A• o 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• : CA•−−−−−→ sCat

N−−−−→ sSet.

Then

N(A•) o N(C ) ∼= N (A• o C ) .

∞-categorical Grothendieck construction

Have an ∞-categorical version in terms of (marked) simplicial sets:

Theorem (Lurie 2009)

Simplical maps

A• : S → Cat∞

'o

Cocartesian fibrations

A• o 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• : CA•−−−−−→ sCat

N−−−−→ sSet. Then

N(A•) o N(C ) ∼= N (A• o C ) .

Internalversion ∞-version

SmashProductAoH

Enrichedversion

SmashProductAoH

GrothendieckConstruction

N•oC

Semi-directProductNoG

cocomm. comon.of objects

many objects

Thank you!

Questions?

Application: Monoidal ∞-categories

Recall the simplex category ∆:

objects 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] 7→ Cn.

∗ C C 2 . . .

∗ 1

c ⊗ d (c , d)

C⊗ := C • o ∆op has an opfibration down to ∆op. In fact, we candefine monoidal categories in terms of opfibrations M → ∆op.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

objects 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] 7→ Cn.

∗ C C 2 . . .

∗ 1

c ⊗ d (c , d)

C⊗ := C • o ∆op has an opfibration down to ∆op. In fact, we candefine monoidal categories in terms of opfibrations M → ∆op.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

objects 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] 7→ Cn.

∗ C C 2 . . .

∗ 1

c ⊗ d (c , d)

C⊗ := C • o ∆op has an opfibration down to ∆op. In fact, we candefine monoidal categories in terms of opfibrations M → ∆op.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

objects 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] 7→ Cn.

∗ C C 2 . . .

∗ 1

c ⊗ d (c , d)

C⊗ := C • o ∆op has an opfibration down to ∆op. In fact, we candefine monoidal categories in terms of opfibrations M → ∆op.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

objects 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] 7→ Cn.

∗ C C 2 . . .

∗ 1

c ⊗ d (c , d)

C⊗ := C • o ∆op has an opfibration down to ∆op. In fact, we candefine monoidal categories in terms of opfibrations M → ∆op.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

objects 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] 7→ Cn.

∗ C C 2 . . .

∗ 1

c ⊗ d (c , d)

C⊗ := C • o ∆op has an opfibration down to ∆op. In fact, we candefine monoidal categories in terms of opfibrations M → ∆op.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

objects 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] 7→ Cn.

∗ C C 2 . . .

∗ 1

c ⊗ d (c , d)

C⊗ := C • o ∆op

has an opfibration down to ∆op. In fact, we candefine monoidal categories in terms of opfibrations M → ∆op.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

objects 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] 7→ Cn.

∗ C C 2 . . .

∗ 1

c ⊗ d (c , d)

C⊗ := C • o ∆op has an opfibration down to ∆op.

In fact, we candefine monoidal categories in terms of opfibrations M → ∆op.

Application: Monoidal ∞-categories

Recall the simplex category ∆:

objects 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] 7→ Cn.

∗ C C 2 . . .

∗ 1

c ⊗ d (c , d)

C⊗ := C • o ∆op has an opfibration down to ∆op. In fact, we candefine 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⊗).

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 ∞-categoriesarising from simplicial 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 ∞-categoriesarising from simplicial 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 ∞-categoriesarising from simplicial 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 ∞-categoriesarising from simplicial monoidal categories.