nLab
over quasi-category

Idea

For $C$ an ordinary category and $c\in C$ an object of $C$, the ordinary over category $C\downarrow c$ satisfies the universal property that for any other category $C\prime$ there is a natural equivalence of categories

$$\mathrm{Hom}(C\prime ,C\downarrow c)\simeq {\mathrm{Hom}}_c(C\prime \star \{\top \},C),$$Hom(C',C\downarrow c) \simeq Hom_{c}(C' \star \{\top\}, C) \,,

where

• $C\prime \star \{\top \}$ denotes the category $C\prime$ with a freely adjoined terminal object $\top$;

• ${\mathrm{Hom}}_c(C\prime \star \{\top \},C)$ denotes the category of pairs $(F,\gamma )$, where $F:C\prime \star \{\top \}\to C$ is a functor and $\gamma :F(\top )\to c$ is an isomorphism in $C$.

The idea of the definition of over category in the context of quasi-categories is to mimic this universal property. This relies crucially on generalizing the construction $C\prime \star \{\top \}$ to the context of quasi-categories, in terms of the join of quasi-categories.

Definition

Let $C$ be a quasi-category. let $K$ be any simplicial set and let $F:K\to C$ be a morphism of simplicial sets (hence a morphism of quasi-categories if $K$ itself is a quasi-category).

The over-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

$${\mathrm{Hom}}_{\mathrm{SSet}}(S,C_{/F})\simeq {\mathrm{Hom}}_F(S\star K,C),$$Hom_{SSet}(S, C_{/F}) \simeq Hom_{F}( S \star K, C ) \,,

where

• $S\star K$ is the join of simplicial sets of $S$ with $K$;

• the hom-set on the right is the subset of morphism $f:S\star K\to C$ such that $f{\mid }_K=F$.

Concretely, the underlying simplicial set of $C_{/F}$is given by

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

where ${\mathrm{Hom}}_F(-)$ denotes the subset of morphisms of simplicial sets that restricts to $F$ on $K$.

Properties

• The simplicial set $C_{/F}$ is indeed again a quasi-category.

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

• For $C$ (the nerve of) an ordinary category this construction coincides with the ordinary notion of overcategory.

References

See proposition 1.2.9.2, p. 44 and the text leading to and including proposition 1.2.9.3 of

• Jacob Lurie, Higher Topos Theory