nLab
subobject in an (infinity,1)-category

Contents

Definition

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

Definition

A subobject of XX is a 1-monomorphism KXK \hookrightarrow X into XX.

The (∞,1)-category Sub(X)Sub(X) of subobjects of XX is the (1)(-1)-truncation of the slice-(∞,1)-category of CC over XX

Sub(X):=τ 1C /X. Sub(X) := \tau_{-1} C_{/X} \,.

This is the category whose objects are monomorphisms UXU \hookrightarrow X in CC and whose morphisms are 2-morphisms

U 1 U 2 X \array{ U_1 &&\to&& U_2 \\ & \searrow &\swArrow& \swarrow \\ && X }

in CC.

Properties

Proposition

Sub(X)Sub(X) is a (0,1)-category (a poset).

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

Proposition

If CC is a locally presentable (∞,1)-category then Sub(X)Sub(X) is a small category.

References

Revised on December 4, 2012 00:57:53 by Urs Schreiber (82.169.65.155)