Accessible ∞-category

In mathematics, especially category theory, an accessible quasi-category is a quasi-category in which each object is an ind-object on some small quasi-category. In particular, an accessible quasi-category is typically large (not small). The notion is a generalization of an earlier 1-category version of it, an accessible category introduced by Adámek and Rosický.^[1]

Definition

An ∞-category is called accessible or more precisely $\kappa$ -accessible if it is equivalent to the ∞-category of $\kappa$ -ind objects on some small ∞-category for some regular cardinal $\kappa$ .^[2]

Facts

A small ∞-category is accessible if and only if it is idempotent-complete.^[3]

References

^ Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)
^ Lurie 2009, Definition 5.4.2.1.
^ Lurie 2009, Corollary 5.4.3.6.

Lurie, Jacob (2009). Higher Topos Theory. Princeton University Press. arXiv:math/0608040. ISBN 978-0-691-14048-3.
Charles Rezk, Generalizing accessible ∞-categories, 2021 draft

Definition

Facts

References

Further reading