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 κ {\displaystyle \kappa } -accessible if it is equivalent to the ∞-category of κ {\displaystyle \kappa } -ind objects on some small ∞-category for some regular cardinal κ {\displaystyle \kappa } .^[2]

Facts

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

References

• 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

Further reading

• https://ncatlab.org/nlab/show/accessible+%28infinity%2C1%29-category