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.Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)
Definition An ∞-category is called accessible or more precisely -accessible if it is equivalent to the ∞-category of -ind objects on some small ∞-category for some regular cardinal .
Facts A small ∞-category is accessible if and only if it is idempotent-complete.
References
Charles Rezk, Generalizing accessible ∞-categories, 2021 draft
Further reading https://ncatlab.org/nlab/show/accessible+%28infinity%2C1%29-category