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 $(\infty,1)$-category $D$ for $n \geq 1$ is an (∞,1)-functor

$$F : D \to C$$

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

$$F_{x,y} : Hom_D(x,y) \to Hom_C(F(x), F(y))$$

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 $sSet$-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 $(\infty,1)$-functor, choose in each preimage $F^{-1}(c)$ for each object $c \in C$ a representative, and let $C'$ be the full sSet-enriched subcategory on these representatives.

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

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

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

Reflective sub-$(\infty,1)$-categories

If the $(\infty,1)$-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

$$L \circ F \stackrel{}{\to} Id_D$$

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

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

2-Subcategory

For $sSet$-enriched and quasi-categories

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

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

Let $h D \to h C$ be a faithful functor. Then if we have a pullback in sSet-Cat

$$\array{ D &\to& C \\ \downarrow && \downarrow \\ h D &\to& h C }$$

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

• objects are those of $h D$;

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

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 $h D \to h C$ is an inclusion on equivalence classes of objects then this is the definition of subcategory of an $(\infty,1)$-category that appears in HTT, section 1.2.11.

As 2-subobjects

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

$$\array{ Q &\to& Set_* \\ \downarrow && \downarrow \\ h C &\to& Set }$$

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

$$\array{ D &\to& h D &\to& Set_* \\ \downarrow && \downarrow && \downarrow \\ C &\to& h C &\to& Set }$$

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

References

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

• Jacob Lurie, Higher Topos Theory