nLab
locally small (infinity,1)-category

Idea

The notion of locally small $(\mathrm{\infty },1)$-category is the generalization of the notion of locally small category from category theory to (∞,1)-category theory.

Definitions

Definition

A quasi-category $C$ is locally small if for all objects $x,y\in C$ the hom ∞-groupoid ${\mathrm{Hom}}_C(x,y)$ is essentially small.

This appears as HTT, below prop. 5.4.1.7.

Properties

Proposition

A quasi-category $C$ is locally small precisely if the following equivalent condition holds:

for every small set $S$ of objects in $C$, the full sub-quasi-category on $S$ is essentially small.

References

This is the topic of section 5.4.1 of

• Jacob Lurie, Higher Topos Theory .