# Contents * Automatic table of contents {: toc} ## 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. +-- {: .num_defn} ###### 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 +-- {: .num_defn} ###### 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. =-- +-- {: .un_prop} ###### 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 +-- {: .num_remark} ###### 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. =-- +-- {: .num_remark} ###### 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 +-- {: .num_remark} ###### 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 +-- {: .num_prop} ###### 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. 1. $C$ satisfies the horn-filling condition. 1. $C$ satisfies the horn-filling condition for all horns except the left outer horn. 1. $C$ 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 +-- {: .num_defn} ###### 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, $C\to D$. =-- +-- {: .un_prop #prop1.2.7.3} ###### 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. 1. 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. 1. 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 +-- {: .num_defn} ###### 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](#limits). ## 1.2.9 overcategories and undercategories +-- {: .num_defn} ###### 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 $$Y\star K$ with $K\star Y$. =-- +-- {: .un_remark} ###### Remark 1.2.9.6 If $C$ is a classical category, then there is a canonical equivalence $$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) }$$ $C^\prime$ is called a *sub-$\infty$-category of $C$ spanned by $(hC)^\prime$*. ## 1.2.12 initial and final objects +-- {: .un_defn} ###### Definition 1.2.12.1 (initial object, final object) An object of a simplicial set / a simplicial category / a topological category $S$ is called final reps. initial if it is final resp initial in the homotopy category $hS$. =-- +-- {: .un_defn} ###### Definition 1.2.12.3 (strongly final object) Let $C$ be a simplicial set. An object $X$ of $C$ is called *strongly final object* if the projection $C/X\to C$ is an acyclic fibration of simplicial sets. =-- +-- {: .un_prop} ###### Proposition 1.2.12.9 (Joyal) Let $C$ be an $\infty$-category. Let $D$ be the full subcategory of $C$ spanned by the final vertices of $C$. Then $C$ is either empty or a contractible Kan complex. =-- ## 1.2.13 limits and colimits {#limits} The following definition says that just as in classical category theory a *limit* is a terminal cone and a *colimit* is an initial cocone: +-- {: .un_defn} ###### Definition 1.2.13.4 (Joyal) Let $C$ be an $\infty$-category, let $p:K\to C$ be an arbitrary map of simplicial sets. A *colimit for $p$* is defined to be an initial object of $p/C$. A *limit for $p$* is defined to be an final object of $C/p$. =-- By definition and the formula $$hom_sSet(K, p/C)=hom_p (K^{\triangleright},C)$$ a colimit for $p$ is equivalently a map $\overline p:K^\triangleleft\to C$ extending $p$. We call such a $\overline p$ *colimit diagram*. An example for a colimit preserving functor is the following: If a functor possessing a colimit factors into another functor $p$ followed by a projection out of an over category, then $p$ has a colimit and the projection preserves this colimit. ## 1.2.14 presentations of $\infty$-categories By *presentation* is meant here (somehow unconcrete) a fibrant replacement of a simplicial set. If this simplicial set has only finitely many non-degenerate cells this presentation is called finite. Note that in the [[nLab:Joyal model structure]] precisely $\infty$-categories are the fibrant objects and consequently by the axioms of the notion of [[nLab:model category]] every simplicial set is categorical equivalent to an $\infty$-category. One such fibrant replacement of a simplicial set is obtained by taking the the nerve of its realization. ## 1.2.15 Set-theoretic technicalties For every [[nLab:cardinal]] $\kappa$ we will assume the existence of a strongly inaccessible cardinal $\kappa\gt \kappa_0$. By $\mathfrak{U}(\kappa)$ we denote the collection of sets with cardinality $\lt\kappa$. $\mathfrak{U}(\kappa)$ is a [[nLab:Grothendieck universe]]. ## 1.2.16 the $\infty$-category of spaces +-- {: .un_defn} ###### Definition 1.2.16.1 Let $Kan$ denote the full $\infty$-category of $sSet$ spanned by the collection of Kan complexes. We regard $Kan$ as a simplicial category. We call the [[nLab:homotopy coherent nerve]] $$S:=N(Kan)$$ the *$\infty$-category of spaces*. =-- Every $\infty$-category is enriched in $S$.