nLab
accessible (infinity,1)-category

Idea

Beiing the (∞,1)-categorical generalization of accessible category, the notion of an accessible (∞,1)-category is a means to handle large $(\mathrm{\infty }\mathrm{,1})$-categories.

An accessible $(\mathrm{\infty }\mathrm{,1})$-category is one which may be large, but can entirely be accessed as an $(\mathrm{\infty }\mathrm{,1})$-category of “conglomerates of objects” in a small $(\mathrm{\infty }\mathrm{,1})$-category – precisely: that it is a category of $\kappa$-small ind-objects in the some small $(\mathrm{\infty }\mathrm{,1})$-category $C$.

Definition

Let $\kappa$ be a regular cardinal. An (∞,1)-category $C$ is accessible if it satisfies the following equivalent conditions

• $C$ is equivalent to the $(\mathrm{\infty }\mathrm{,1})$-category ${\mathrm{Ind}}_{\kappa }(C^0)$ of ind-objects for small $C^0$;

• $C$ admits small $\kappa$-filtered colimits and contains an essentially small full subcategory which consists of $\kappa$-compact objects and generates $C$ under small $\kappa$-filtered colimits.

References

The theory of accessible 1-categories is described in

• Adamek and Rosicky, Locally presentable and accessible categories, Cambridge University Press (1994)

The theory of accessible $(\mathrm{\infty }\mathrm{,1})$-categories is the topic of section 5.4 of

• Jacob Lurie, Higher Topos Theory