Accessible category

The theory of accessible categories originates from the work of Grothendieck completed by 1969 (Grothendieck (1972)) and Gabriel-Ulmer (1971). It has been further developed in 1989 by Michael Makkai and Robert Paré, with motivation coming from model theory, a branch of mathematical logic. Accessible categories have also applications in homotopy theory. Grothendieck also continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs (Grothendieck (1991)). Some properties of accessible categories depend on the set universe in use, particularly on the cardinal properties.

Let ${\displaystyle K}$ be an infinite regular cardinal and let ${\displaystyle C}$ be a category. An object ${\displaystyle X}$ of ${\displaystyle C}$ is called ${\displaystyle K}$-presentable if the Hom functor ${\displaystyle \operatorname {Hom} (X,-)}$ preserves ${\displaystyle K}$-directed colimits. The category ${\displaystyle C}$ is called ${\displaystyle K}$-accessible provided that :

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