Corollary 4.1.2.12 (01CR)

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. $\square$