equivalences in/of $(\infty,1)$-categories
Let $C$ be an (∞,1)-category and $X \in C$ an object.
A subobject of $X$ is a 1-monomorphism $K \hookrightarrow X$ into $X$.
The (∞,1)-category $Sub(X)$ of subobjects of $X$ is the $(-1)$-truncation of the slice-(∞,1)-category of $C$ over $X$
This is the category whose objects are monomorphisms $U \hookrightarrow X$ in $C$ and whose morphisms are 2-morphisms
in $C$.
$Sub(X)$ is a (0,1)-category (a poset).
This appears for instance in (Lurie, section 6.2).
If $C$ is a locally presentable (∞,1)-category then $Sub(X)$ is a small category.
subobject in an $(\infty,1)$-category