nLab
subobject in an (infinity,1)-category

Definition

Let $C$ be an (∞,1)-category and $X\in C$ an object.

Definition

A subobject of $X$ is a 1-monomorphism $K\hookrightarrow X$ into $X$.

The (∞,1)-category $\mathrm{Sub}(X)$ of subobjects of $X$ is the $(-1)$-truncation of the slice-(∞,1)-category of $C$ over $X$

$$\mathrm{Sub}(X):={\tau }_{-1}{C}_{/X}.$$Sub(X) := \tau_{-1} C_{/X} \,.

This is the category whose objects are monomorphisms $U\hookrightarrow X$ in $C$ and whose morphisms are 2-morphisms

$$\begin{array}{ccccc}{U}_1& & \to & & U_2\\ & \searrow & ⇙& \swarrow \\ & & X\end{array}$$\array{ U_1 &&\to&& U_2 \\ & \searrow &\swArrow& \swarrow \\ && X }

in $C$.

Properties

Proposition

$\mathrm{Sub}(X)$ is a (0,1)-category (a poset).

This appears for instance in (Lurie, section 6.2).

Proposition

If $C$ is a locally presentable (∞,1)-category then $\mathrm{Sub}(X)$ is a small category.

Related concepts

• subobject

• subobject in an $(\mathrm{\infty },1)$-category

• n-image, n-truncated object in an (∞,1)-category

References

• Jacob Lurie, Higher Topos Theory