Enrichments for Higher Categories

Enrichments for higher categories

David Kern

Laboratoire Angevin de REcherche en MAthématiques — Séminaire doctoral

12th October 2018

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 1 / 22 Sommaire - Section 1: The algebraic structure of free objects

1 The algebraic structure of free objects

2 Monoids and enrichments Monoidal and enriched categories Monads

3 Categories up to homotopy

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 1 / 22 I Free category FQ with objects the vertices of Q morphisms homFQ (V0, V1) = {strings of composable arrows joining V0 to V1} Composition given by concatenation (and get identities as empty string)

Adjunction property Let C be a category. Any functor FQ C is determined by its action on Q, and any graph morphism Q C extends uniquely (by concatenation): → (F , C) ' ( , UC) U homCat→ Q homQuiver Q (with the “underlying quiver” functor)

The free category functor

Let Q = ({vertices}, {arrows}) be a quiver. What is the “best” way to turn it into a category? I Need to freely add a composition operation, respecting associativity (and units)

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 2 / 22 Adjunction property Let C be a category. Any functor FQ C is determined by its action on Q, and any graph morphism Q C extends uniquely (by concatenation): → (F , C) ' ( , UC) U homCat→ Q homQuiver Q (with the “underlying quiver” functor)

The free category functor

Let Q = ({vertices}, {arrows}) be a quiver. What is the “best” way to turn it into a category? I Need to freely add a composition operation, respecting associativity (and units) I Free category FQ with objects the vertices of Q morphisms homFQ (V0, V1) = {strings of composable arrows joining V0 to V1} Composition given by concatenation (and get identities as empty string)

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 2 / 22 The free category functor

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 2 / 22 Examples

I U : Vectk Set the forgetful “underlying set” functor; the free vector space k[S] on a set S is the k-linear span of its elements

I U : CAlgk → Vectk ; the free commutative k-algebra on V is the symmetric tensor • ∼ ⊗2 ⊗3 algebra: U (Sym V ) = k ⊕ V ⊕ (V )S2 ⊕ (V )S3 ⊕ · · · → Remark: FX is functorial in X, giving F :C D

→

Free objects in general

Let U :D C be a “forgetful” functor. For any object X ∈ C, the free object of D on X is FX ∈ D with ηX : X UFX (in C) such that: → ηX for any Y ∈ D and any morphism f : X UY in X UFX→ C, there is a unique f : FX Y (in D) making the Uf f diagram commute (in C) → UY →

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 3 / 22 Free objects in general

I U : Vectk Set the forgetful “underlying set” functor; the free vector space k[S] on a set S is the k-linear span of its elements

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 3 / 22 → Adjoint functors

Definition An adjunction F a G between categories C and D is F :C D and G :D C with bijections φX,Y : homD(FX, Y ) ' homC(X→, GY ) → natural in X ∈ C and Y ∈ D.

U :D C forgetful functor

→η φX,Y : hom(FX, Y ) 3 g 7 Ug ◦ ηX ∈ hom(X, UY ) X X UFX

∃! Uf bijective by free property f → UY = F : C ⊥ D : U

David Kern (LAREMA — SemDoc) Enrichments for higher categories⇒ 12th October 2018 4 / 22 Notation: η and ε are called the unit and counit of the adjunction

ε and η determine the adjunction φ, that is:

F and G are adjoint iff there are ε: FG idD and η: idC GF with the compatibility condition

F ⇒ ⇒G G Fη F ηG

C FGF D = C idF D and D GFG C = D idG C εF Gε F G F G

The algebra of adjunctions ' φX,FX : homD(FX, FX) − homC(X, GFX) maps idFX 7 ηX : X GFX I By functoriality of φ = natural transformation η: idC GF −1 → ' → → Similarly, by φGY ,Y : hom(GY , GY ) − hom(FGY , Y ), get ε: FG idD ⇒ ⇒ → ⇒

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 5 / 22 The algebra of adjunctions ' φX,FX : homD(FX, FX) − homC(X, GFX) maps idFX 7 ηX : X GFX I By functoriality of φ = natural transformation η: idC GF −1 → ' → → Similarly, by φGY ,Y : hom(GY , GY ) − hom(FGY , Y ), get ε: FG idD Notation: η and ε are called⇒ the unit and counit of the adjunction⇒ → ⇒ ε and η determine the adjunction φ, that is:

F and G are adjoint iff there are ε: FG idD and η: idC GF with the compatibility condition

F ⇒ ⇒G G Fη F ηG

C FGF D = C idF D and D GFG C = D idG C εF Gε F G F G

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 5 / 22 Similarly, FηG is a comultiplication FG (FG)◦2 on the counital FG on D

= GF and FG are an algebra and a cogebra for the composition products on ⇒ EndoFun(C) and EndoFun(D) ⇒

From adjunctions to monads

Using the counit, we get a natural transformation

µ := GεF :(GF) ◦ (GF) = G ◦ (FG) ◦ F GF.

⇒ I The endofunctor GF on C has a “multiplication” µ:(GF)◦2 GF and a “unit” η: idC GF (associative and unital by compatibility condition) ⇒ U :D C Example: ⇒ forgetful functor UF creates a free object and forgets the added structure µ remembers reduction→ along this structure (e.g. F = k[−]; µ is linear dependence)

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 6 / 22 From adjunctions to monads

Using the counit, we get a natural transformation

µ := GεF :(GF) ◦ (GF) = G ◦ (FG) ◦ F GF.

= GF and FG are an algebra and a cogebra for the composition products on ⇒ EndoFun(C) and EndoFun(D)

David Kern⇒(LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 6 / 22 Sommaire - Section 2: Monoids and enrichments

1 The algebraic structure of free objects

2 Monoids and enrichments Monoidal and enriched categories Monads

3 Categories up to homotopy

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 7 / 22 Sommaire - Section 2: Monoids and enrichments

1 The algebraic structure of free objects

2 Monoids and enrichments Monoidal and enriched categories Monads

3 Categories up to homotopy

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 8 / 22 Examples

I (ModR , ⊗R , R) I (EndoFun(C), ◦, idC) is strict monoidal (strictly associative) I Any category with finite products (1 is the final object)

Monoidal categories

Definition A monoidal structure on a category V is a bifunctor ⊗:V × V V and a choice of object 1 ∈ V, with natural transformations expressing associativity and unitality → ' I A braiding is a natural isomorphism τX,Y : X ⊗ Y − Y ⊗ X −1 I Symmetric structure if τX,Y = τY ,X → ' A monoidal functor (V, ⊗, 1) (N, , I) is F :V N with F(X ⊗ Y ) − FX FY ' (natural) and F(1) − I → → → →

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 9 / 22 I (EndoFun(C), ◦, idC) is strict monoidal (strictly associative) I Any category with finite products (1 is the final object)

Monoidal categories

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 9 / 22 I Any category with finite products (1 is the final object)

Monoidal categories

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 9 / 22 Monoidal categories

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 9 / 22 Enriched categories

Definition A (V, ⊗, 1)-enriched category C has a set of objects Ob(C) and, for any pair (A, B) of objects, an object hom(A, B) of V, along with hom(A, B) ⊗ hom(B, C) hom(A, C), “(f , g) 7 g ◦ f ” and idA : 1 hom(A, A), all following the category axioms. → → A V-functor F :C D is a map of sets Ob(C) Ob(D) and morphisms → homC(A, B) homD(FA, FB) in V with functoriality. → → Examples → I A category is a (Set, ×)-enriched category

I A ModR -enriched category is called an R-linear category. An R-algebra is an R-category with one object

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 10 / 22 Proposition A closed monoidal structure gives a self-enrichment of V with hom(−, −) = [−, −]

L I (ModR , ⊗R , R), and the dg-derived category (Ho(dgModR ), ⊗R , R[0]) I (Top, ×, pt), and Ho(Top) I (Cat, ×, pt) cartesian closed = Cat-enriched category, i.e. (strict) 2-category

⇒

Closure and self-enrichment

A closed symmetric monoidal category is (V, ⊗, 1) such that, ∀Y ∈ V, the (endo)functor − ⊗ Y has a right adjoint [Y , −]:

hom(X ⊗ Y , Z) = hom(X, [Y , Z])

Notation: the object [Y , Z] is called the internal hom from Y to Z

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 11 / 22 Closure and self-enrichment

A closed symmetric monoidal category is (V, ⊗, 1) such that, ∀Y ∈ V, the (endo)functor − ⊗ Y has a right adjoint [Y , −]:

hom(X ⊗ Y , Z) = hom(X, [Y , Z])

Notation: the object [Y , Z] is called the internal hom from Y to Z

Proposition A closed monoidal structure gives a self-enrichment of V with hom(−, −) = [−, −]

L I (ModR , ⊗R , R), and the dg-derived category (Ho(dgModR ), ⊗R , R[0]) I (Top, ×, pt), and Ho(Top) I (Cat, ×, pt) cartesian closed = Cat-enriched category, i.e. (strict) 2-category

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 11 / 22 ⇒ Sommaire - Section 2: Monoids and enrichments

1 The algebraic structure of free objects

2 Monoids and enrichments Monoidal and enriched categories Monads

3 Categories up to homotopy

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 12 / 22 Examples I Monoids in (Set, ×) are monoids. Monoids in monoids are commutative monoids.

I Monoids in ModR are R-algebras. (Commutative) monoids in dgModR are R-(c)dgas I Monoids in (EndoFun(C), ◦, idC) are monads on C (comonoids are comonads)

Remark: Define similarly modules and algebras over a monoid

Monoids in a monoidal category

Definition A monoid in (V, ⊗, 1) is M ∈ V with µ: M ⊗ M M and a generalised element η: 1 M, satisfying associativity and unitality → → µ⊗id id ⊗η η⊗id M ⊗ M ⊗ M M M ⊗ M M ⊗ 1 M M ⊗ M M 1 ⊗ M

idM ⊗µ µ and µ ' '

M ⊗ M µ M M Comonoid = monoid in Vop: comultiplication δ: M M ⊗ M, counit ε: M 1, coassociative and counital → →

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 13 / 22 Monoids in a monoidal category

Definition A monoid in (V, ⊗, 1) is M ∈ V with µ: M ⊗ M M and a generalised element η: 1 M, satisfying associativity and unitality → → Comonoid = monoid in Vop: comultiplication δ: M M ⊗ M, counit ε: M 1, coassociative and counital Examples → → I Monoids in (Set, ×) are monoids. Monoids in monoids are commutative monoids.

I Monoids in ModR are R-algebras. (Commutative) monoids in dgModR are R-(c)dgas I Monoids in (EndoFun(C), ◦, idC) are monads on C (comonoids are comonads)

Remark: Define similarly modules and algebras over a monoid

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 13 / 22 The forgetful functor AlgT C, (A, ρ) 7 A has a “free” left adjoint C 7 (T C, µC ). The monad for the adjunction is (universally) T . → → → Remark: An adjunction F : C ⊥ D : G is said monadic if AlgGF ' D

= AlgGF measures how free F is

⇒

Monads and adjunctions

Eilenberg–Moore category An algebra over (T , µ, η) is a functor A: pt C (equivalently, A ∈ C) with ρ: T A A such that µA ηA TT A T A → A T A → T ρ ρ and ρ commute. idA T A ρ A A A morphism of T -algebras (A, ρ) (A0, ρ0) is f : A A0 compatible with ρ, ρ0

→ →

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 14 / 22 Monads and adjunctions

The forgetful functor Alg C, ( , ρ) 7 has a “free” left adjoint 7 (T , µ ). The T A→ A → C C C monad for the adjunction is (universally) T . → → → Remark: An adjunction F : C ⊥ D : G is said monadic if AlgGF ' D

= AlgGF measures how free F is

David Kern (LAREMA⇒ — SemDoc) Enrichments for higher categories 12th October 2018 14 / 22 Algebras over β Let ρ: βX X be a β-algebra. Topologise X with U ⊂ X open if

∀ ∈ , ∀ ∈ β , ρ( ) = = ∈ → x U F X F x U F

I The space X is a compactum (quasicompact–Hausdorff)⇒

Example: the ultrafilter monad I

Ultrafilters form a monad

β: Set Set, X 7 {ultrafilters on X} ' homBool(P(X), 2)

I ηX (x) is the principal ultrafilter $x = {U ⊂ X | x ∈ U} → → I µX : ββX 3 F 7 U ⊂ X | {G ∈ βX | U ∈ G} ∈ F ∈ βX →

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 15 / 22 Example: the ultrafilter monad I

Ultrafilters form a monad

β: Set Set, X 7 {ultrafilters on X} ' homBool(P(X), 2)

I ηX (x) is the principal ultrafilter $x = {U ⊂ X | x ∈ U} → → I µX : ββX 3 F 7 U ⊂ X | {G ∈ βX | U ∈ G} ∈ F ∈ βX Algebras over β → Let ρ: βX X be a β-algebra. Topologise X with U ⊂ X open if

∀ ∈ , ∀ ∈ β , ρ( ) = = ∈ → x U F X F x U F

I The space X is a compactum (quasicompact–Hausdorff)⇒

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 15 / 22 Example: the ultrafilter monad II

Properties (X a topological space)

A net in X is ν: I X, I directed set. Converges to x if ∀U 3 x, ∃i, ∀j ≥ i, νj ∈ U I X is quasicompact iff every net has a convergent subnet I X is Hausdorff→ iff no net can converge to two distinct limits

For X a compactum, define β-algebra structure ρ: βX X by

ρ( ) = unique limit point of (i.e. filter {neighbourhoods of } ⊂ ) F x F → x F

In fact Algβ ' CHaus Remarks (β regularises infinite sets) ' I On a finite set every ultrafilter is principal, so thefree β-algebra β[n] − [n]: η[n]

I For any set X, βX is the (Stone–Čech) compactification of (X, τdiscrete) ← David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 16 / 22 Sommaire - Section 3: Categories up to homotopy

1 The algebraic structure of free objects

2 Monoids and enrichments Monoidal and enriched categories Monads

3 Categories up to homotopy

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 17 / 22 Monoidal structure: [n] [m] := [n + m + 1] (strict, but not symmetric!) 0,17 0 [0] is a monoid: [0] [0] = [1] −−−− [0] Universal property: cobar construction→ → Let (C, ⊗) be a strict monoidal category with a monoid M. There is a unique monoidal ⊗(n+1) functor BC(M)• : ∆+ C sending [0] to M (and generally [n] to M )

→

The augmented simplex category

Category ∆+ with objects [n] = {0 ≤ 1 ≤ · · · ≤ n} (seen as a category), with [−1] = ∅ arrows order-preserving, i.e. non-decreasing, functions (or simply functors) Note [−1] is initial

Also (non augmented) simplex catgory ∆ without [−1]

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 18 / 22 Universal property: cobar construction Let (C, ⊗) be a strict monoidal category with a monoid M. There is a unique monoidal ⊗(n+1) functor BC(M)• : ∆+ C sending [0] to M (and generally [n] to M )

→

The augmented simplex category

Also (non augmented) simplex catgory ∆ without [−1]

Monoidal structure: [n] [m] := [n + m + 1] (strict, but not symmetric!) 0,17 0 [0] is a monoid: [0] [0] = [1] −−−− [0] → →

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 18 / 22 The augmented simplex category

Also (non augmented) simplex catgory ∆ without [−1]

Monoidal structure: [n] [m] := [n + m + 1] (strict, but not symmetric!) 0,17 0 [0] is a monoid: [0] [0] = [1] −−−− [0] Universal property: cobar construction→ → Let (C, ⊗) be a strict monoidal category with a monoid M. There is a unique monoidal ⊗(n+1) functor BC(M)• : ∆+ C sending [0] to M (and generally [n] to M )

→ David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 18 / 22 n Given by diagram of Xn = X•([n]) with faces d = X•(δn) (resp. cofaces dn) and n degeneracies s = X•(σn) (resp. codegeneracies sn)

Cobar resolution id ⊗ id ⊗η ⊗η id id ⊗µ η ⊗2 BC(M)•(∆) = 1 M µ M id ⊗η⊗id ··· µ⊗ η⊗id id η⊗id ⊗ id

(Co)simplicial objects

Face maps δi :[n] [n + 1], {0 ≤ · · · ≤ n} 7 {0 ≤ · · · ≤ i − 1 ≤ i + 1 ≤ · · · ≤ n + 1} Degeneracy maps σj :[n] [n − 1], {0 ≤ · · · ≤ n} 7 {0 ≤ · · · ≤ j ≤ j ≤ · · · ≤ n − 1} → Every morphism in ∆ factors uniquely as degeneracies followed by faces → op An augmented (co)simplicial object in C is a functor X• : ∆+ C (or ∆+ C):

→ →

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 19 / 22 (Co)simplicial objects

Cobar resolution id ⊗ id ⊗η ⊗η id id ⊗µ η ⊗2 BC(M)•(∆) = 1 M µ M id ⊗η⊗id ··· µ⊗ η⊗id id η⊗id ⊗ id

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 19 / 22 The nerve problem

C a category. Its nerve N•(C) ∈ sSet has Ni (C) = homCat([i], C): strings of i composable unit. morphisms. It is 2-coskeletal: determined by N≤1 (as Qvr = s≤1Set)

Set E.g.: M a monoid, BM its 1-object category. Then N•(BM) = B (M)• is the bar resolution

The left adjoint of N• : Cat sSet (“fundamental category”) only depends on 0-, 1- and 2-simplices = How to define→ the nerve of an sSet-category, or the “correct” fundamental category of a simplicial set? ⇒

Simplicial categories and simplicial enrichments

I C a simplicially enriched category. op = Define simplicial object C• : ∆ Cat with Ob(Ci ) = Ob(C) and

homCi (X, Y ) = homC(X, Y )i op I Conversely,⇒ ∆ Cat with constant objects→ gives an sSet-category

→

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 20 / 22 The left adjoint of N• : Cat sSet (“fundamental category”) only depends on 0-, 1- and 2-simplices = How to define→ the nerve of an sSet-category, or the “correct” fundamental category of a simplicial set? ⇒

Simplicial categories and simplicial enrichments

I C a simplicially enriched category. op = Define simplicial object C• : ∆ Cat with Ob(Ci ) = Ob(C) and

homCi (X, Y ) = homC(X, Y )i op I Conversely,⇒ ∆ Cat with constant objects→ gives an sSet-category The nerve problem → C a category. Its nerve N•(C) ∈ sSet has Ni (C) = homCat([i], C): strings of i composable unit. morphisms. It is 2-coskeletal: determined by N≤1 (as Qvr = s≤1Set)

Set E.g.: M a monoid, BM its 1-object category. Then N•(BM) = B (M)• is the bar resolution

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 20 / 22 Simplicial categories and simplicial enrichments

I C a simplicially enriched category. op = Define simplicial object C• : ∆ Cat with Ob(Ci ) = Ob(C) and

Set E.g.: M a monoid, BM its 1-object category. Then N•(BM) = B (M)• is the bar resolution

David Kern⇒(LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 20 / 22  <  0 0 2 2 + +   and [2] = [2] =  < · <  • ≤1  “1 2” “0 1”  < 0<1 1 1 2

Homotopy coherent diagrams

Back to the free category construction F : Qvr ⊥ Cat : U

EndoFun(Cat) op I Comonad FU on Cat, with bar resolution B (FU)• : ∆ EndoFun(Cat) I Both U and F are identity on objects = simplicial categories + EndoFun(Cat) C• := B (FU)•(C) → ⇒ Definition I a category, C an sSet-category. + A homotopy coherent diagram of shape I in C is an sSet-functor I• C

(0<2)=(1<2)◦(0<1) → [2] = 0 1 2 0<1 1<2

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 21 / 22 Homotopy coherent diagrams

Back to the free category construction F : Qvr ⊥ Cat : U

(0<2)=(1<2)◦(0<1)  →<  0 0 2 2 + +   [2] = 0 1 2 and [2] = [2] =  < · <  0<1 1<2 • ≤1  “1 2” “0 1”  < 0<1 1 1 2

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 21 / 22 n I By the Yoneda lemma, Xn = hom(∆•, X•) n n I To have $ a N∆, we set N∆(C)n = hom(∆•, N∆(C)•) := hom($ (∆•), C)

= an n-simplex∞ in N∆(C) is a string of n homotopy composable morphisms∞ of C Interpretation ⇒ N∆C is a combinatorial representation of C exhibiting its structure of -category

∞

The homotopy coherent nerve

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 22 / 22 The homotopy coherent nerve

∈ sSet ∆n I By abstract nonsense (density theorem), any X• is lim− •, where n ∆• X• n op ∆• : ∆ Set is the representable hom(−, [n]) sSet $ ( ) := [→]+ I We define the fundamental -category X• −lim n • n → ∆• X• ∞ n → I By the Yoneda lemma, Xn = hom(∆•, X•) n n I To have $ a N∆, we set N∆(C)n = hom(∆•, N∆(C)•) := hom($ (∆•), C)

= an n-simplex∞ in N∆(C) is a string of n homotopy composable morphisms∞ of C Interpretation ⇒ N∆C is a combinatorial representation of C exhibiting its structure of -category

∞ David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 22 / 22 Bonus

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 23 / 22 Sommaire - Section 4: More on monadic algebra

4 More on monadic algebra

5 Comparison of models

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 24 / 22 Nerve and realisation

op I :D C a functor. Define an adjoint pair |−| : Fun(D , Set) ⊥ C : NI with:

op → op I op homC(−,C) NI (C):D −− C −−−−−− Set

→ → Realisations of simplicial sets: D = ∆

I I : ∆ Cat the inclusion: NI is the nerve n I I : ∆ Top, [n] 7 |∆ |: NI is the singular complex, |−| geometric realisation → Remark→ |−| is the→ left Kan extension of I along the Yoneda embedding: D I C univ. Fun(Dop, Set) |−| David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 25 / 22 F op F I There is a functor N :D Fun(C, Set) with N (D): C 7 homD(D, FC) I Right adjoint − t F : Fun(C, Set)op D → → Definition → F The codensity monad of F is the monad N ◦ (− t F) of the adjunction

The codensity monad I Dense and codense functors

I A functor F :C D is dense if every object D of D is canonically a colimit lim FC, with F(C)/D the comma category of morphisms FC D, C ∈ C − F(C)/D → op I Equivalently, NF :D Fun(C , Set) is fully faithful → → Density theorem: the Yoneda→ embedding is dense

F is codense if every is lim F I D −D/F(C) C

←

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 26 / 22 The codensity monad I Dense and codense functors

F is codense if every is lim F I D −D/F(C) C F op F I There is a functor N :D Fun(C, Set) with N (D): C 7 homD(D, FC) I Right adjoint − t F : Fun(C←, Set)op D → → Definition → F The codensity monad of F is the monad N ◦ (− t F) of the adjunction

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 26 / 22 op I The functor Cat/D Monad(D) is right adjoint to the Eilenberg–Moore algebra functor monadic w/ cod. mon. I The forgetful inclusion Cat/D , Cat/D has a left adjoint mapping F :C D → F to the forgetful functor Alg(N ◦ (− t F)) D → → I The codensity monad of FinSet , Set is the→ ultrafilter monad β fin.dim. ∨∨ I The codensity monad of Vectk , Vectk is the double dualisation monad (−) → →

The codensity monad II Properties and examples

Main properties I F is codense iff its codensity monad is the identity I The codensity monad is the right Kan extension of F along itself I If F has a left adjoint L, its codensity monad is FL

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 27 / 22 I The codensity monad of FinSet , Set is the ultrafilter monad β fin.dim. ∨∨ I The codensity monad of Vectk , Vectk is the double dualisation monad (−) → →

The codensity monad II Properties and examples

Main properties I F is codense iff its codensity monad is the identity I The codensity monad is the right Kan extension of F along itself I If F has a left adjoint L, its codensity monad is FL op I The functor Cat/D Monad(D) is right adjoint to the Eilenberg–Moore algebra functor monadic w/ cod. mon. I The forgetful inclusion Cat/D , Cat/D has a left adjoint mapping F :C D → F to the forgetful functor Alg(N ◦ (− t F)) D → → →

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 27 / 22 The codensity monad II Properties and examples

David Kern (LAREMA — SemDoc) Enrichments→ for higher categories 12th October 2018 27 / 22 Sommaire - Section 5: Comparison of models

4 More on monadic algebra

5 Comparison of models

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 28 / 22 Homotopy categories and derived functors

Goal ∼ M a category with W a wide class of morphisms (“weak equivalences”, written − ) ` The localisation M − M[W−1] is the universal (initial) functor sending the weak equivalences to isomorphisms → − Want to understand→ the homotopy category Ho(M, W) := M[W 1] Also, for F : (M, W) N, compute the derived functors

RF = Lan` F : Ho(M, W) N or LF = Ran` F : Ho(M, W) N → → → General construction 0 ∈W ∈W ∈W 0 homHo(M,W)(M, M ) = M −− M1 M2 −−· · · −− M Very unwieldy model (possibly not even locally small) ← → ← ← David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 29 / 22 Provides a left deformation Q:M Mcof ⊂ M and a right deformation R:M Mfib ⊂ M, ∼ ∼ with Q = idM and idM = R → → Theorem ⇒ ⇒ I Ho(M, W) ' “homotopy” classes of maps in Mcf full subcat. on fibrant–cofibrant I LF = F ◦ Q and RF = F ◦ R

Model structures

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 30 / 22 Model structures

A model structure on (M, W) consists of two classes C (cofibrations) and F (fibrations) satisfying conditions ((C, W ∩ F) and (W ∩ C, F) must be weak factorisation systems) In particular I Say M ∈ M is cofibrant if ∅ M is a cofibration, fibrant if M ∗ is a fibration ∼ ∼ I For every M there is a cofibrant QM with QM − M and a fibrant RM with M − RM, both functorially in M → → → → Provides a left deformation Q:M Mcof ⊂ M and a right deformation R:M Mfib ⊂ M, ∼ ∼ with Q = idM and idM = R → → Theorem ⇒ ⇒ I Ho(M, W) ' “homotopy” classes of maps in Mcf full subcat. on fibrant–cofibrant I LF = F ◦ Q and RF = F ◦ R

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 30 / 22 Joyal There is a model structure on sSet whose fibrant objects are quasi-categories (weak Kan complexes)

Bergner Model structure on CatsSet with fibrant objects locally Kan sSet-categories, W functors inducing equivalences of Ho(sSet, WKan)-categories Lurie The adjunction $ a N∆ induces an equivalence of homotopy categories

∞

Models for higher categories

Spaces as -groupoids I Model structure on Top with W the weak homotopy equivalences I Similar∞ model structure on sSet whose fibrant objects are the Kan complexes Nerve and realisation induce an equivalence of homotopy theories

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 31 / 22 Lurie The adjunction $ a N∆ induces an equivalence of homotopy categories

∞

Models for higher categories

Joyal There is a model structure on sSet whose fibrant objects are quasi-categories (weak Kan complexes)

Bergner Model structure on CatsSet with fibrant objects locally Kan sSet-categories, W functors inducing equivalences of Ho(sSet, WKan)-categories

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 31 / 22 Models for higher categories

Joyal There is a model structure on sSet whose fibrant objects are quasi-categories (weak Kan complexes)

∞

David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 31 / 22 Model-independence in -cosmoi

An -cosmos K is (roughly) a QCat∞ -enriched category of fibrant objects Much of -category theory can be done in the homotopy 2-category of K ∞ Virtual profunctor equipment ∞ -cosmoi admit a calculus of “bimodules” C 9 D, called profunctors, which are (fibrations defining) -functors “Dop × C Grpd ” ∞I Abstracts the internal hom functors ∞ → ∞ Model-independence Any notion that can be encoded in terms of the virtual equipment can be transported along cosmological 2-equivalences = Constructions appropriately performed in any -cosmos equivalent to QCat are valid for all models of -categories ⇒ ∞ ∞ David Kern (LAREMA — SemDoc) Enrichments for higher categories 12th October 2018 32 / 22