Accessible category

Summary

An accessible category is a large category that is generated, under highly filtered colimits, by a small set of small objects. Here small objects are the $\lambda$-presentable ones, those whose hom functor preserves $\lambda$-filtered colimits for a regular cardinal $\lambda$. Accessible categories are exactly the categories of models of sketches, and they include the categories of models of many first-order theories. The cocomplete ones are the locally presentable categories.