Spahn HTT, 1.2, the language of higher category theory

This is a subentry of a reading guide to HTT.

Nerve and realization

We are dealing here only with the fragment of the more general theory of nerve and realization; namely the one based on the simplex category? $\Delta$ as category of geometric shapes.

Definition

(nerve)

Let $C$ be a locally small category. Let $\Delta_C:\Delta\to C$ be a cosimplicial object of $C$. Let $c\in C$ be an object

The nerve of $c$ wrt. $\Delta_C$ is defined to be the simplicial set $N_C:\Delta^{op}\to Set$ which is given in degree $n$ by

$$N_C(c)_n:=hom_C(\Delta_C[n],c)$$

Let now $C:=Cat$ The ordinary nerve $N(c) of the category$c$* is defined to be the nerve of$C$wrt. the inclusion$\Delta_i:\Delta\hookrightarrow Cat$ sending an ordinal to free [[nLab:quiver]] on it.

This means that we have

$$N(c)_n = \underbrace{ Mor(c) {}_t \times_s Mor(c) {}_t \times_s \cdots {}_t \times_s Mor(c)}_{n factors}$$

Definition

(nerve and realization)

(…)

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

$$(d_i:S_n^{op}\to S_{n-1}^{op})=(d_{n-i}:S_n\to S_{n-1})$$

$$(s_i:S_n^{op}\to S_{n+1}^{op})=(s_{n-i}:S_n\to S_{n+1})$$

1.2.2 mapping spaces in higher category theory

Definition

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

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

where $|-|:sSet Cat\to s Set$ denotes the adjoint of the [[nLab:homotopy coherent nerve]]: the [[homotopy coherent realization]]. We have

$$|-|=Lan_y \mathfrak{C}$$

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

Proposition 1.2.3.5

Let $C$ be an $\infty$-category. Two parallel edges of $S$ are called homotopic if there is a $2$-simplex joining them.

Homotopy is an equivalence relation on $hS$.

1.2.3 the homotopy category

Remark and Definition

Let $C$ be a classical category. Then

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

exhibits $Cat$ as a full reflective subcategory of $sSet$. Here $N$ denotes the (classical) nerve functor an $h$ assigns to a simplicial set $S$ its homotopy category. Joyal calls $hS$ the fundamental category of $S$ since if $S$ is a Kan complex $hS$ is the fundamental groupoid of $S$.

Moreover $N$ can be written as a composition

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

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

$$(\pi_0\dashv \iota):Set\stackrel{\iota}{\to}sSet$$

is a reflective subcategory.

Remark

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

• We have $Ob(hS)=Ob(S)$

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

• For each $\sigma:\Delta^2\to S$, we have $\overline{d_0(\sigma)}\circ\overline{d_2(\sigma)}=\overline{d_1(\sigma)}$

• For each vertex $x$ of $S$, the morphism $\overline{s_0 s}$ is the identity $id_x$.

1.2.4 objects, morphisms and equivalences

Remark

Let $S$ be a simplicial set.

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

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

• A morphism in $S$ is called an equivalence if it is an isomorphism in the homotopy category $hS$.

• Two parallel edges of $S$ are called equivalent if there is a $2$-simplex between them which is an equivalence.

1.2.5 groupoids and classical homotopy theory

Proposition 1.2.5.1

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

1. $C$ is an $\infty$-category and $hC$ is a groupoid.

2. $C$ satisfies the horn-filling condition.

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

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

1.2.6 homotopy commutativity versus homotopy coherence

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

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

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

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

1.2.7 functors between higher categories

Definition

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

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

Proposition 1.2.7.3

Let $K$ be a simplicial set.

1. For every $\infty$-category $C$, the simplicial set $Fun(K,C)$ is an $\infty$-category.

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

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

1.2.8 joins of $\infty$-categories

Definition and remark

The category $sSet$ is a monoidal category where the monoidal structure is induced by the [[nLab:ordinal sum]]; i.e. the join of simplicial sets $S,T$ is defined by

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

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

$$\Phi_{ij}:\Delta^{i-1}\star \Delta^{j-1}\simeq\Delta^{(i+j)-1}$$

for all $i,j\ge 0$.

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

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

If $S$ and $T$ are quasi-categories, so is $S\star T$.

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

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

A natural transformation $\eta^x:\kappa_x\to F$ from the terminal diagram to $F:J\to C$ is called cone for $F$.

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 $S$, $K$ be simplicial sets, let $p:K\to S$ be an arbitrary map. Then there exists a simplicial set $S_{/p}$ satisfying

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

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

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

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

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

Remark 1.2.9.6

If $C$ is a classical category, then there is a canonical equivalence

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

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 $h F$ between the homotopy categories is.

1.2.11 subcategories of $\infty$-categories

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

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

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

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 $S$ is called final reps. initial if it is final resp initial in the homotopy category $hS$.

Definition 1.2.12.3

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

Proposition 1.2.12.9 (Joyal)

Let $C$ be an $\infty$-category. Let $D$ be the full subcategory of $C$ spanned by the final vertices of $C$. Then $C$ 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 $C$ be an $\infty$-category, let $p:K\to C$ be an arbitrary map of simplicial sets.

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

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

By definition and the formula

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

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

An example for a colimit preserving functor is the following: If a functor possessing a colimit factors into another functor $p$ followed by a projection out of an over category, then $p$ 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

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

1.2.16 the $\infty$-category of spaces

Definition 1.2.16.1

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

$$S:=N(Kan)$$

the $\infty$-category of spaces.

Every $\infty$-category is enriched in $S$.