nLab
compact object in an (infinity,1)-category

Idea

The notion of compact object in an ordinary category is motivated by the example of Top. But as $\mathrm{Top}$ is an $(\mathrm{\infty }\mathrm{,1})$-category, it should also make sense in that context.

Definition

The general definition has a bit of cumbersome fine print, see definition 5.3.4.5 in

• Jacob Lurie, Higher Topos Theory

It becomes simpler in stable (infinity,1)-categoris:

An object $X$ in a stable (infinity,1)-category is compact if the $(\mathrm{\infty }\mathrm{,1})$-functor

$$C(X,-):C\to C$$C(X,-) : C \to C

commutes with all colimits.

Remarks

• For more discussion of the relevance of compact objects, see also geometric infinity-function theory.