4.1.2 Subcategories of $\infty $-Categories
Let $\operatorname{\mathcal{C}}$ be a category, and let $\operatorname{Ob}(\operatorname{\mathcal{C}})$ be the set of objects of $\operatorname{\mathcal{C}}$. Suppose that we are given a subset $\operatorname{Ob}'(\operatorname{\mathcal{C}}) \subseteq \operatorname{Ob}(\operatorname{\mathcal{C}})$ and, for every pair of objects $X,Y \in \operatorname{Ob}'(\operatorname{\mathcal{C}})$, a subset $\operatorname{Hom}'_{\operatorname{\mathcal{C}}}(X,Y) \subseteq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ satisfying the following conditions:
For every object $X \in \operatorname{Ob}'(\operatorname{\mathcal{C}})$, the identity morphism $\operatorname{id}_{X}$ belongs to $\operatorname{Hom}'_{\operatorname{\mathcal{C}}}(X,X)$.
For every triple of objects $X,Y,Z \in \operatorname{Ob}'(\operatorname{\mathcal{C}})$ and every pair of morphisms $f \in \operatorname{Hom}'_{\operatorname{\mathcal{C}}}(X,Y)$, $g \in \operatorname{Hom}'_{\operatorname{\mathcal{C}}}(Y,Z)$, the composition $g \circ f$ belongs to $\operatorname{Hom}'_{\operatorname{\mathcal{C}}}(X,Z)$.
In this case, we can construct a category $\operatorname{\mathcal{C}}'$ by setting $\operatorname{Ob}(\operatorname{\mathcal{C}}') = \operatorname{Ob}'(\operatorname{\mathcal{C}})$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) = \operatorname{Hom}'_{\operatorname{\mathcal{C}}}(X,Y)$ for every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}')$ (where the composition of morphisms in $\operatorname{\mathcal{C}}'$ agrees with their composition in $\operatorname{\mathcal{C}}$). In this situation, we refer to $\operatorname{\mathcal{C}}'$ as the subcategory of $\operatorname{\mathcal{C}}$ spanned by the objects $\operatorname{Ob}'(\operatorname{\mathcal{C}})$ and the morphisms $\{ \operatorname{Hom}'_{\operatorname{\mathcal{C}}}(X,Y) \} _{X,Y \in \operatorname{Ob}'(\operatorname{\mathcal{C}})}$.
We write $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ to indicate that $\operatorname{\mathcal{C}}'$ is a subcategory of $\operatorname{\mathcal{C}}$.
We now generalize the notion of subcategory to the setting of $\infty $-categories.
Definition 4.1.2.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A subcategory of $\operatorname{\mathcal{C}}$ is a simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ for which the inclusion map $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is an inner fibration.
Example 4.1.2.4. Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be its nerve. For every subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$, the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}')$ can be viewed as a subcategory of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (the inclusion map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is automatically an inner fibration, by virtue of Proposition 4.1.1.10). We will see in a moment that every subcategory of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ arises in this way (Corollary 4.1.2.11). In other words, when restricted to (the nerves of) ordinary categories, Definition 4.1.2.2 reduces to the classical notion of subcategory.
Warning 4.1.2.5. The terminology of Definition 4.1.2.2 has the potential to cause confusion. If $\operatorname{\mathcal{C}}$ is an $\infty $-category and $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is a subcategory, then $\operatorname{\mathcal{C}}'$ need not be (isomorphic to the nerve of) an ordinary category. Our use of the term “subcategory” (rather than the more technically correct “sub-$\infty $-category”) is intended to avoid awkward language.
The implications $(1) \Rightarrow (2) \Rightarrow (3)$ are immediate from the definitions, and the implication $(3) \Rightarrow (1)$ follows from fact that the inclusion $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is weakly right orthogonal to the inner anodyne morphism $\operatorname{Spine}[n] \hookrightarrow \Delta ^{n}$ (see Example 1.5.7.7 and Proposition 4.1.3.1).
Proposition 4.1.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ be its homotopy category, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ be the canonical map. Then the construction $(\operatorname{\mathcal{D}}\subseteq \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) \mapsto ( F^{-1}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})) \subseteq \operatorname{\mathcal{C}})$ induces a bijection
\[ \{ \textnormal{Subcategories of the ordinary category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$} \} \simeq \{ \textnormal{Subcategories of the $\infty $-category $\operatorname{\mathcal{C}}$} \} \]
Proof.
We first observe that if $\operatorname{\mathcal{D}}$ is a subcategory of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a subcategory of the $\infty $-category $\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ (Example 4.1.2.4), so that $F^{-1}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) )$ is a subcategory of the $\infty $-category $\operatorname{\mathcal{C}}$ (Remark 4.1.2.6). Moreover, the subcategory $\operatorname{\mathcal{D}}$ is uniquely determined by its inverse image $F^{-1}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$): this follows from the fact that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is an epimorphism of simplicial sets (Remark 1.5.7.10). To complete the proof, it will suffice to show that every subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ arises in this way. Note that the inclusion map $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ induces a functor of homotopy categories $G: \mathrm{h} \mathit{\operatorname{\mathcal{C}}'} \hookrightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, which is obviously injective at the level of objects. In addition, for every pair of objects $X,Y \in \mathrm{h} \mathit{\operatorname{\mathcal{C}}'}$, the functor $G$ induces a monomorphism $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}'}}( X,Y) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$: this follows from the observation that a pair of morphisms $f,g: X \rightarrow Y$ are homotopic in the $\infty $-category $\operatorname{\mathcal{C}}'$ if and only if they are homotopic in the $\infty $-category $\operatorname{\mathcal{C}}$ (Remark 4.1.2.8). It follows that the functor $G$ induces an isomorphism from $\mathrm{h} \mathit{\operatorname{\mathcal{C}}'}$ onto a subcategory $\operatorname{\mathcal{D}}\subseteq \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. We therefore have an inclusion $\operatorname{\mathcal{C}}' \subseteq F^{-1}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) )$. To complete the proof, it will suffice to show that this inclusion is an equality. In other words, we must show that an $n$-simplex $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{C}}$ is contained in $\operatorname{\mathcal{C}}'$ if and only if the induced map $[n] \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ factors through the subcategory $\operatorname{\mathcal{D}}\subseteq \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Without loss of generality, we may assume that $n > 0$ (the case $n=0$ is trivial). Using Remark 4.1.2.9, we can reduce to the case where $n=1$, so that $\sigma $ can be identified with a morphism $g: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Our assumption that $F( \sigma )$ belongs to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ guarantees that $g$ is homotopic to a morphism $f: X \rightarrow Y$ which belongs to the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ (and, in particular, that the objects $X$ and $Y$ belong to $\operatorname{\mathcal{C}}'$). Invoking Remark 4.1.2.8, we conclude that $g$ also belongs to the subcategory $\operatorname{\mathcal{C}}'$, as desired.
$\square$
Corollary 4.1.2.11. Let $\operatorname{\mathcal{C}}$ be an ordinary category. Then the construction $\operatorname{\mathcal{C}}' \mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}')$ induces a bijection
\[ \{ \textnormal{Subcategories of the ordinary category $\operatorname{\mathcal{C}}$} \} \simeq \{ \textnormal{Subcategories of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$} \} \]
Proof.
Combine Proposition 4.1.2.10 with Example 1.4.5.4.
$\square$
Corollary 4.1.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $S$ be a collection of objects of $\operatorname{\mathcal{C}}$, and let $T$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
There exists a subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ whose objects are the elements of $S$ and whose morphisms are the elements of $T$.
The collections $S$ and $T$ satisfy the following conditions:
- $(1)$
For each object $X \in S$, the identity morphism $\operatorname{id}_{X}$ belongs to $T$.
- $(2)$
For each morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ which belongs to $T$, the objects $X$ and $Y$ belong to $S$.
- $(3)$
If $f: X \rightarrow Y$ is an morphism of $\operatorname{\mathcal{C}}$ which belongs to $T$ and $g: X \rightarrow Y$ is a morphism of $\operatorname{\mathcal{C}}$ which is homotopic to $f$, then $g$ also belongs to $T$.
- $(4)$
If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are morphisms of $\operatorname{\mathcal{C}}$ which belong to $T$, then some composition $(g \circ f): X \rightarrow Z$ also belongs to $T$.
Moreover, if these conditions are satisfied, then the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is uniquely determined by $S$ and $T$.
Proof.
The necessity of conditions $(1)$ and $(2)$ is immediate, and the necessity of $(3)$ and $(4)$ follow from Remark 4.1.2.8 and Remark 4.1.2.7. Conversely, suppose that conditions $(1)$ through $(4)$ are satisfied. Using $(1)$, $(2)$, and $(4)$, we deduce that there exists a subcategory $\operatorname{\mathcal{D}}\subseteq \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ whose objects are the elements of $S$ and whose morphisms are the homotopy classes of morphisms belonging to $T$. Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the inverse image of the subcategory $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \subseteq \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$. It follows immediately from the definition that an object of $\operatorname{\mathcal{C}}$ belongs to the subcategory $\operatorname{\mathcal{C}}'$ if and only if it is an element of $S$, and from $(3)$ that a morphism of $\operatorname{\mathcal{C}}$ belongs to the subcategory $\operatorname{\mathcal{C}}'$ if and only if it is an element of $T$. The uniqueness of the subcategory $\operatorname{\mathcal{C}}'$ follows from Proposition 4.1.2.10.
$\square$
Definition 4.1.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Suppose we are given a collection $S$ of objects of $\operatorname{\mathcal{C}}$ and a collection $T$ of morphisms of $\operatorname{\mathcal{C}}$ satisfying the assumptions of Corollary 4.1.2.12, so that there exists a unique subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ whose objects are the elements of $S$ and whose morphisms are the elements of $T$. In this case, we will refer to $\operatorname{\mathcal{C}}'$ as the subcategory of $\operatorname{\mathcal{C}}$ spanned by the objects of $S$ and the morphisms of $T$.
Let $\operatorname{\mathcal{C}}$ be an ordinary category. Recall that a subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is full if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}'$, the inclusion map $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is bijective. This definition has an obvious counterpart in the $\infty $-categorical setting.
Definition 4.1.2.15. Let $\operatorname{\mathcal{C}}$ be a simplicial set. We say that a simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is full if it satisfies the following condition:
Let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ be a simplex of $\operatorname{\mathcal{C}}$ having the property that, for each integer $0 \leq i \leq n$, the vertex $\sigma (i) \in \operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}'$. Then $\sigma $ belongs to $\operatorname{\mathcal{C}}'$.
If this condition is satisfied, then the inclusion map $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is an inner fibration. In particular, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then $\operatorname{\mathcal{C}}'$ is a subcategory of $\operatorname{\mathcal{C}}$; in this case, we will say that $\operatorname{\mathcal{C}}'$ is a full subcategory of $\operatorname{\mathcal{C}}$.
Proposition 4.1.2.16. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $S$ be a collection of vertices of $\operatorname{\mathcal{C}}$. Then there exists a unique full simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ having vertex set $S$.
Proof.
Take $\operatorname{\mathcal{C}}'$ to be the simplicial subset of $\operatorname{\mathcal{C}}$ consisting of those simplices $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ having the property that, for $0 \leq i \leq n$, the vertex $\sigma (i)$ belongs to $S$.
$\square$
Definition 4.1.2.17. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $S$ be a collection of vertices of $\operatorname{\mathcal{C}}$. By virtue of Proposition 4.1.2.16, there exists a unique full simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ having vertex set $S$. We will refer to $\operatorname{\mathcal{C}}'$ as the full simplicial subset of $\operatorname{\mathcal{C}}$ spanned by $S$. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, we will refer to $\operatorname{\mathcal{C}}'$ as the full subcategory of $\operatorname{\mathcal{C}}$ spanned by $S$.