nLab
sub-(infinity,1)-category

Idea

The analog of the notion of subcategory for (∞,1)-categories.

Definition

Say that an equivalence of (∞,1)-categories $D\stackrel{\simeq }{\to }C$ exhibits $D$ as a 0-subcategory of $C$.

Then define recursively, for $n\in \mathbb{N}$:

an $n$-subcategory of an $(\mathrm{\infty },1)$-category $D$ for $n\ge 1$ is an (∞,1)-functor

$$F:D\to C$$F : D \to C

such that for all objects $x,y\in D$ the component-$(\mathrm{\infty },1)$-functor on the hom-objects

$$F_{x,y}:{\mathrm{Hom}}_D(x,y)\to {\mathrm{Hom}}_C(F(x),F(y))$$F_{x,y} : Hom_D(x,y) \to Hom_C(F(x), F(y))

exhibits an $(n-1)$-subcategory.

Special cases

1-Subcategory / full subcategory

A full subcategory is a 1-subcategory, exhibited by a full and faithful (∞,1)-functor.

For $\mathrm{sSet}$-enriched and quasi-categories

Let $C$ and $D$ be incarnated specifically as fibrant simplicially enriched categories. Then for $F:D\to C$ a full and faithful $(\mathrm{\infty },1)$-functor, choose in each preimage $F^{-1}(c)$ for each object $c\in C$ a representative, and let $C\prime$ be the full sSet-enriched subcategory on these representatives.

Then the evident projection functor $D\stackrel{\simeq }{\to }D\prime$ is manifestly an equivalence and the original $F:D\to C$ factors as

$$F:D\stackrel{\simeq }{\to }D\prime \hookrightarrow C,$$F : D \stackrel{\simeq}{\to} D' \hookrightarrow C \,,

where the second morphism is an ordinary inclusion of objects and hom-complexes.

Reflective sub-$(\mathrm{\infty },1)$-categories

If the $(\mathrm{\infty },1)$-functor $F:D\to C$ has a left adjoint (∞,1)-functor $L:C\to D$, then $F$ is full and faithful and hence exhibits a 1-subcategory precisely if the counit

$$L\circ F\stackrel{}{\to }{\mathrm{Id}}_D$$L \circ F \stackrel{}{\to} Id_D

is an equivalence of (∞,1)-functors. (See also HTT, p. 308).

In this case $D$ is a reflective (∞,1)-subcategory.

2-Subcategory

For $\mathrm{sSet}$-enriched and quasi-categories

Let the $(\mathrm{\infty },1)$-categories $C$ and $D$ concretely be incarnated as fibrant simplicially enriched categories.

Write $hC:=\mathrm{Ho}(C)$ and $hD:=\mathrm{Ho}(D)$ for the corresponding homotopy category of an (∞,1)-category (hom-wise the connected components of the corresponding simplicially enriched category).

Let $hD\to hC$ be a faithful functor. Then if we have a pullback in sSet-Cat

$$\begin{array}{ccc}D& \to & C\\ \downarrow & & \downarrow \\ hD& \to & hC\end{array}$$\array{ D &\to& C \\ \downarrow && \downarrow \\ h D &\to& h C }

$D$ is a 2-sub-$(\mathrm{\infty },1)$-category of $C$. This pullback manifestly produces the simplicially enriched category whose

• objects are those of $hD$;

• hom-complexes are precisely the unions of those connected components of the hom-complexes of $C$ whose equivalence class is present in $hD$.

Therefore the inclusion functor $D\to C$ is on each hom-complex a full and faithful (∞,1)-functor. Hence this identifies $D$ as a 2-subcategory of $C$.

If $hD\to hC$ is an inclusion on objects (which is a bit evil to say) then this is the definition of subcategory of an $(\mathrm{\infty },1)$-category that appears in HTT, section 1.2.11.

As 2-subobjects

Let $\mathrm{core}({\mathrm{Set}}_{*})\to \mathrm{core}(\mathrm{Set})$ be the 2-subobject classifier in the (∞,1)-topos ∞Grpd. Then for $C\in \mathrm{\infty }\mathrm{Grpd}$ a 1-subobject is classified by an $\mathrm{\infty }$-functor $C\to \mathrm{Set}$. This factors through the homotopy category of $C$ as $C\to hC\to \mathrm{Set}$. Since ${\mathrm{Set}}_{*}\to \mathrm{Set}$ is the universal faithful functor, the pullback

$$\begin{array}{ccc}Q& \to & {\mathrm{Set}}_{*}\\ \downarrow & & \downarrow \\ hC& \to & \mathrm{Set}\end{array}$$\array{ Q &\to& Set_* \\ \downarrow && \downarrow \\ h C &\to& Set }

gives an ordinary subcategory of $hC$. This means that the total pullback $D\to C$

$$\begin{array}{ccccc}D& \to & hD& \to & {\mathrm{Set}}_{*}\\ \downarrow & & \downarrow & & \downarrow \\ C& \to & hC& \to & \mathrm{Set}\end{array}$$\array{ D &\to& h D &\to& Set_* \\ \downarrow && \downarrow && \downarrow \\ C &\to& h C &\to& Set }

gives a 2-sub-$(\mathrm{\infty },1)$-category $K$ of $X$ (where both happen to be $\mathrm{\infty }$-groupoids) here.

References

What we call a 2-subcategory of an $(\mathrm{\infty },1)$-category appears in section 1.2.11 of

• Jacob Lurie, Higher Topos Theory