nLab
over-(infinity,1)-category

Idea

Definition

We give the definition in terms of the model of (∞,1)-categories in terms of quasi-categories.

Recall from join of quasi-categories that there are two different but quasi-catgorically equivalent definitions of join, denoted $\star$ and $♢$. Accordingly we have the following two different but quasi-categoricaly equivalent definitions of over/under quasi-categories.

Defnition/Proposition

Let $C$ be a quasi-category. let $K$ be any simplicial set and let

$$F:K\to C$$F : K \to C

be an (∞,1)-functor – a morphism of simplicial sets.

1. The under-quasi-category $C_{F/}$ is the simplicial set characterized by the property that for any other simplicial set $S$ there is a natural bijection of hom-sets

   $${\mathrm{Hom}}_{\mathrm{sSet}}(S,C_{F/})\cong ({\mathrm{Hom}}_{(K\downarrow \mathrm{SSet})})(i_{K,S},F),$$Hom_{sSet}(S, C_{F/}) \cong (Hom_{(K\downarrow SSet)})(i_{K,S} , F) \,,

   where $i_{K,S}:K\to K\star S$ is the canonical inclusion of $K$ into its join of simplicial sets with $S$.

   Similarly, the over quasi-category over $F$ is the simplicial set characterized by the property that

   $${\mathrm{Hom}}_{\mathrm{sSet}}(S,C_{/F})\simeq {\mathrm{Hom}}_{(K\downarrow \mathrm{SSet})}(j_{K,S},F)$$Hom_{sSet}(S, C_{/F}) \simeq Hom_{(K\downarrow SSet)}( j_{K,S} , F )

   naturally in $S$, where $j_{K,S}$ is the canonical inclusion map $K\to S\star K$.

2. The functor

   $$\mathrm{sSet}\to {\mathrm{sSet}}_{K/}$$sSet \to sSet_{K/}
   $$S\mapsto K♢S$$S \mapsto K \diamondsuit S

   with $♢$ denoting the other definition of join of quasi-categories (as described there)

   has a right adjoint

   $${\mathrm{sSet}}_{K/}\to \mathrm{sSet}$$sSet_{K/} \to sSet
   $$(F:K\to C)\mapsto C^{F/}$$(F : K \to C) \mapsto C^{F/}

   and its image $C^{F/}$ is another definition of the quasi-category under $F$.

The first definition in terms of the the mapping property is due to Andre Joyal. Together with the discussion of the concrete realization it appears as HTT, prop 1.2.9.2. The second definition appears in HTT above prop. 4.2.1.5.

Proposition

The simplicial sets $C_{/F}$ and $C_{F/}$ are indeed themselves again quasi-categories.

Proof

This appears as HTT, prop. 1.2.9.3

Proposition

The two definitions yield equivalent results in that the canonical morphism

$$C_{/F}\to C^{/F}.$$C_{/F} \to C^{/F} \,.

is an equivalence of quasi-categories.

Proof

This is HTT, prop. 4.2.1.5

From the formula

$$(C_{/F}{)}_n=({\mathrm{Hom}}_{\mathrm{sSet}}{)}_F({\Delta }^n\star K,C)$$(C_{/F})_n = (Hom_{sSet})_F(\Delta^n \star K , C)

we see that

• an object in the over quasi-category $C_{/F}$ is a cone over $F$;.

  For instance if $K=\Delta [1]$ then an object in $C_{/F}$ is a 2-cell

  $$\begin{array}{ccc}& & v\\ & \swarrow & ⇙& \searrow \\ F(0)& & \to & & F(1)\end{array}$$\array{ && v \\ & \swarrow &\swArrow& \searrow \\ F(0) &&\to&& F(1) }

  in $C$.

• a morphism in $C_{/F}$ is a morphism of cones,

• etc:.

So we may think of the overcategory $C_{/F}$ as the quasi-category of cones over $F$. Accordingly we have that

• the terminal object in $C_{/F}$ is (if it exists) the limit in $C$ over $F$;

• the initial object in $C_{F/}$ is (if it exists) the colimit of $F$ in $C$.

Properties

Proposition

For $C=N(𝒞)$ (the nerve of) an ordinary category $𝒞$ and $K=*$, this construction coincides with the ordinary notion of overcategory $𝒞/F$ in that there is a canonical isomorphism of simplicial sets

$$N(𝒞/F)\simeq N(𝒞)/F.$$N(\mathcal{C}/F) \simeq N(\mathcal{C})/F \,.

Proof

This appears as HTT, remark 1.2.9.6.

Proposition

If $q:C\to D$ is a categorical equivalence then so is the induced morphism $C_{/F}\to C_{/qF}$.

Proof

This appears as HTT, prop 1.2.9.3.

Proposition

For $C$ an (∞,1)-category and $X\in C$ an object in $C$ and $f:A\to X$ and $g:B\to X$ two objects in $C/X$, the hom-∞-groupoid $C/X(f,g)$ is equivalent to the homotopy fiber of $C(A,B)\stackrel{g_{*}}{\to }C(A,X)$ over the morphism $f$: we have an (∞,1)-pullback diagram

$$\begin{array}{ccc}C/X(f,g)& \to & C(A,B)\\ \downarrow & & {\downarrow }^{g_{*}}\\ *& \stackrel{f}{\to }& C(A,X)\end{array}.$$\array{ C/X(f,g) &\to& C(A,B) \\ \downarrow && \downarrow^{\mathrlap{g_*}} \\ {*} & \stackrel{f}{\to} & C(A,X) } \,.

This is HTT, prop. 5.5.5.12.

Related concepts

• over-category

  • slice category

  • under category

  • over topos

• over-(∞,1)-category

  • model structure on an over category

  • over-(∞,1)-topos

References

Section 1.2.9 of

• Jacob Lurie, Higher Topos Theory