Accessible ∞-category
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In mathematics, especially
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, an accessible quasi-category is a
quasi-category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. T ...
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.


Definition

An
∞-category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a Category (ma ...
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.


Facts

A small ∞-category is accessible if and only if it is idempotent-complete.


References

* * Charles Rezk, Generalizing accessible ∞-categories, 202
draft


Further reading

* https://ncatlab.org/nlab/show/accessible+%28infinity%2C1%29-category {{categorytheory-stub Category theory