Spahn HTT, 1.2, the language of higher category theory (Rev #10, changes)

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1.2.1 the opposite of an \infty-category

For topological and simplicial categories the definition of the opposite category is the same as the notion from classical category theory.

For a simplicial set we obtain the opposite simplicial set by component- wise reversing the order of the ordinal.

Definition

Let SS be a simplicial set. Let JJ be a linear ordered set. Then the face and degeneracy maps on S opS^{op} are given by.

(d i:S n opS n1 op)=(d ni:S nS n1)(d_i:S_n^{op}\to S_{n-1}^{op})=(d_{n-i}:S_n\to S_{n-1})
(s i:S n opS n+1 op)=(s ni:S nS n+1)(s_i:S_n^{op}\to S_{n+1}^{op})=(s_{n-i}:S_n\to S_{n+1})

1.2.2 mapping spaces in higher category theory

Definition

Let SS be a simplicial set. Let x,ySx,y\in S be vertices. Then the simplicial mapping space is defined by

Map S(x,y):=Map |S|(x,y)Map_S (x,y):=Map_{|S|} (x,y)

where ||:sSetCatsSet|-|:sSet Cat\to s Set denotes the adjoint of the homotopy coherent nerve: the homotopy coherent realization?. We have

||=Lan y|-|=Lan_y \mathfrak{C}

where y:Δ[Δ op,Set]y:\Delta\hookrightarrow [\Delta^{op},Set] denotes the Yoneda embedding and :ΔsSetCat\mathfrak{C}: \Delta\to sSet Cat denotes the cosimplicial-thickening functor. We think of \mathfrak{C} as assigning to an ordinal [n][n] (considered as a category) a simplicially-enriched category which is thickened.

Proposition 1.2.3.5

Let CC be an \infty-category. Two parallel edges of SS are called homotopic if there is a 22-simplex joining them.

Homotopy is an equivalence relation on hShS.

1.2.3 the homotopy category

Remark and Definition

Let CC be a classical category. Then

(hN):CatNsSet(h\dashv N):Cat\stackrel{N}{\to}sSet

exhibits CatCat as a full reflective subcategory of sSetsSet. Here NN denotes the (classical) nerve functor an hh assigns to a simplicial set SS its homotopy category. Joyal calls hShS the fundamental category of SS since if SS is a Kan complex hShS is the fundamental groupoid of SS.

Moreover NN can be written as a composition

CatisSetCatN sSetCat\xhookrightarrow{i}sSet Cat\stackrel{N^\prime}{\to}sSet

where N N^\prime denotes the simplicial nerve functor and ii denotes inclusion.

(π 0ι):SetιsSet(\pi_0\dashv \iota):Set\stackrel{\iota}{\to}sSet

is a reflective subcategory.

Remark

(presentation of the homotopy category by generators and relations) Let SS be a simplicial set.

  • We have Ob(hS)=Ob(S)Ob(hS)=Ob(S)

  • For each σ:Δ 1S\sigma:\Delta^1\to S, there is a morphism ϕ¯:ϕ(0)ϕ(1)\overline \phi:\phi(0)\to \phi(1).

  • For each σ:Δ 2S\sigma:\Delta^2\to S, we have d 0(σ)¯d 2(σ)¯=d 1(σ)¯\overline{d_0(\sigma)}\circ\overline{d_2(\sigma)}=\overline{d_1(\sigma)}

  • For each vertex xx of SS, the morphism s 0s¯\overline{s_0 s} is the identity id xid_x.

1.2.4 objects, morphisms and equivalences

Remark

Let SS be a simplicial set.

  • Vertices Δ 0S\Delta^0\to S of SS are called objects of SS.

  • Edges Δ 1S\Delta^1\to S are called morphisms of SS.

  • A morphism in SS is called an equivalence if it is an isomorphism in the homotopy category hShS.

  • Two parallel edges of SS are called equivalent if there is a 22-simplex between them which is an equivalence.

1.2.5 groupoids and classical homotopy theory

Proposition 1.2.5.1

Let CC be a simplicial set. The the following conditions are equivalent:

  1. CC is an \infty-category and hChC is a groupoid.

  2. CC satisfies the horn-filling condition.

  3. CC satisfies the horn-filling condition for all horns except the left outer horn.

  4. CC satisfies the horn-filling condition for all horns except the right outer horn.

1.2.6 homotopy commutativity versus homotopy coherence

Let F:JHF:J\to H be a diagram. If f,gMor(J)\f,g\in Mor(J) are Morphisms we will in general only have an equivalence

F(fg)F(f)F(g)F(f\circ g)\simeq F(f)\circ F(g)

and no equality. If for all morphism these equivalences can be chosen in a “coherent” way, FF is called a coherent diagram.

If JJ is a classical category and CC is a quasi-category then a homotopy coherent diagram can be defined to be a map of simplicial sets f:JCf: J\to C. This encodes the coherence data.

1.2.7 functors between higher categories

1.2.8 joins of \infty-categories

Definition

Let CC, DD be simplicial sets. A functor from CDC\to D is defined to be a morphism of simplicial sets; i.e. a natural transformation.

We consider only functors f:CDf:C\to D from a simplicial set to an \infty-category. By Fun(C,D)Fun(C,D) we denote the collection of functors fro, CDC\to D.

Definition Proposition and 1.2.7.3 remark

The Let category sSet K sSet K is be a monoidal simplicial category set. where the monoidal structure is induced by theordinal sum; i.e. the join of simplicial sets S,TS,T is defined by

(ST):= J=II S(I)×T(I )(S\star T):=\coprod_{J=I\cup I^\prime}S(I)\times T(I^\prime)
  1. For every \infty-category CC, the simplicial set Fun(K,C)Fun(K,C) is an \infty-category.

  2. Let CDC\to D be a categorical equivalence of \infty-ctegories. Then the induced map Fun(K,C)Fun(K,D)Fun(K,C)\to Fun(K,D) is a categorical equivalence.

  3. Let CC be an \infty-category. Let KK K\to K^\prime be a categorical equivalence of simplicial sets. The the induced map Fun(K ,C)Fun(K,C)Fun(K^\prime,C)\to Fun(K,C) is a categorical equivalence.

The empty simplicial set Δ 1\Delta^{-1} is the monoidal unit. Moreover we have natural isomorphisms

Φ ij:Δ i1Δ j1Δ (i+j)1\Phi_{ij}:\Delta^{i-1}\star \Delta^{j-1}\simeq\Delta^{(i+j)-1}

for all i,j0i,j\ge 0.

An important special case of this definition is that of a cone:

Let KK be a simplicial set. Then K :=Δ 0KK^{\triangleleft}:=\Delta^0\star K is called left cone of KK and K :=KΔ 0K^{\triangleright}:=K\star\Delta^0 is called right cone of KK.

If SS and TT are quasi-categories, so is STS\star T.

1.2.8 joins of \infty-categories

Definition and remark

The category sSetsSet is a monoidal category where the monoidal structure is induced by the ordinal sum; i.e. the join of simplicial sets S,TS,T is defined by

(ST):= J=II S(I)×T(I )(S\star T):=\coprod_{J=I\cup I^\prime}S(I)\times T(I^\prime)

The empty simplicial set Δ 1\Delta^{-1} is the monoidal unit. Moreover we have natural isomorphisms

Φ ij:Δ i1Δ j1Δ (i+j)1\Phi_{ij}:\Delta^{i-1}\star \Delta^{j-1}\simeq\Delta^{(i+j)-1}

for all i,j0i,j\ge 0.

An important special case of this definition is that of a cone:

Let KK be a simplicial set. Then K :=Δ 0KK^{\triangleleft}:=\Delta^0\star K is called left cone of KK and K :=KΔ 0K^{\triangleright}:=K\star\Delta^0 is called right cone of KK.

If SS and TT are quasi-categories, so is STS\star T.

Compare this notion of cone with the one from classical category theory:

Let JJ and CC be categories, let xC 0x\in C_0, let [x]:*x[x]:* \mapsto x denote the global element ‘’in xx’’, let !:I*!:I\to * and κ x:=[x]!:JC\kappa_x:=[x]\circ !:J\to C be the constant functor in xx. It is the terminal object in the functor category of its shape.

A natural transformation η x:κ xF\eta^x:\kappa_x\to F from the terminal diagram to F:JCF:J\to C is called cone for FF.

These consideration have an application in limits and colimits.

1.2.9 overcategories and undercategories

Definition and Remark

(over-simplicial-set, under-simplicial-set, over-quasi-category, under-quasi-category)

Let SS, KK be simplicial sets, let p:KSp:K\to S be an arbitrary map. Then there exists a simplicial set S /pS_{/p} satisfying

hom sSet(Y,S /p)=hom p(YK,S)hom_{sSet}(Y, S_{/p})=hom_p (Y\star K,S)

where the subscript pp on the right hand side indicates that we only consider those morphisms which restricted to KK coincide with pp. We can define S /pS_{/p} by

(S /p) n:=hom p(Δ nK,S)(S_{/p})_n:=hom_p (\Delta^n\star K,S)

If CC is an \infty-category, so is S/pS/p. In this case S/pS/p is called over-\infty-category

Dually the under \infty-category is defined analogously by replacing $YKY\star K with KYK\star Y.

Remark 1.2.9.6

If CC is a classical category, then there is a canonical equivalence

N(C)/XN(C/X)N( C )/X\simeq N(C/X)

1.2.10 fully faithful and essentially surjective functors

A functor between simplicial sets / simplicially enriched categories / topologically enriched categories is called an essentially surjective functor reps. fully faithful functor if the induced functor hFh F between the homotopy categories is.

1.2.11 subcategories of \infty-categories

Let CC be an \infty-category, let (hC) hC(hC)^\prime\subseteq hC be a subcategory of its homotopy category. Then there is a pullback diagram of simplicial sets

C C N(hC) N(hC)\array{ C^\prime&\to&C \\ \downarrow&&\downarrow \\ N(hC)^\prime&\to &N(hC) }

C C^\prime is called a sub-\infty-category of CC spanned by (hC) (hC)^\prime.

1.2.12 initial and final objects

Definition 1.2.12.1

(initial object, final object) An object of a simplicial set / a simplicial category / a topological category SS is called final reps. initial if it is final resp initial in the homotopy category hShS.

Definition 1.2.12.3

(strongly final object) Let CC be a simplicial set. An object XX of CC is called strongly final object if the projection C/XCC/X\to C is an acyclic fibration of simplicial sets.

Proposition 1.2.12.9 (Joyal)

Let CC be an \infty-category. Let DD be the full subcategory of CC spanned by the final vertices of CC. Then CC is either empty or a contractible Kan complex.

1.2.13 limits and colimits

The following definition says that just as in classical category theory a limit is a terminal cone and a colimit is an initial cocone:

Definition 1.2.13.4 (Joyal)

Let CC be an \infty-category, let p:KCp:K\to C be an arbitrary map of simplicial sets.

A colimit for pp is defined to be an initial object of p/Cp/C.

A limit for pp is defined to be an final object of C/pC/p.

By definition and the formula

hom sSet(K,p/C)=hom p(K ,C)hom_sSet(K, p/C)=hom_p (K^{\triangleright},C)

a colimit for pp is equivalently a map p¯:K C\overline p:K^\triangleleft\to C extending pp. We call such a p¯\overline p colimit diagram.

An example for a colimit preserving functor is the following: If a functor possessing a colimit factors into another functor pp followed by a projection out of an over category, then pp has a colimit and the projection preserves this colimit.

1.2.14 presentations of \infty-categories

By presentation is meant here (somehow unconcrete) a fibrant replacement of a simplicial set.

If this simplicial set has only finitely many non-degenerate cells this presentation is called finite.

Note that in the Joyal model structure precisely \infty-categories are the fibrant objects and consequently by the axioms of the notion of model category every simplicial set is categorical equivalent to an \infty-category. One such fibrant replacement of a simplicial set is obtained by taking the the nerve of its realization.

1.2.15 Set-theoretic technicalties

1.2.16 the \infty-category of spaces

For every cardinal κ\kappa we will assume the existence of a strongly inaccessible cardinal κ>κ 0\kappa\gt \kappa_0. By 𝔘(κ)\mathfrak{U}(\kappa) we denote the collection of sets with cardinality <κ\lt\kappa. 𝔘(κ)\mathfrak{U}(\kappa) is a Grothendieck universe.

1.2.16 the \infty-category of spaces

Definition 1.2.16.1

Let KanKan denote the full \infty-category of sSetsSet spanned by the collection of Kan complexes. We regard KanKan as a simplicial category. We call the homotopy coherent nerve

S:=N(Kan)S:=N(Kan)

the \infty-category of spaces.

Every \infty-category is enriched in SS.

Revision on June 23, 2012 at 01:13:08 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.