Spahn HTT, 1.2, the language of higher category theory (Rev #3, 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

1.2.4 objects morphisms and equivalences

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

1.2.7 functors between higher categories

1.2.8 joins of \infty-categories

1.2.9 overcategories and undercategories

1.2.10 fully faithful and essentially surjective functors

1.2.11 subcategories of \infty-categories

1.2.12 initial and final objects

1.2.13 limits and colimits

1.2.14 presentations of \infty-categories

1.2.15 Set-theoretic technicalties

1.2.16 the \infty-category of spaces

Revision on June 22, 2012 at 14:31:54 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.