objects such that commutes with certain colimits
An accessible category is a possibly large category which is however essentially determined by a small category. There are several ways to make this precise, all equivalent.
A locally small category is accessible if for some regular cardinal :
If satisfies these properties for some , we say that it is -accessible. Unlike for locally presentable categories, it does not follow that if is -presentable and then is also -presentable. It is true, however, that for any accessible category, there are arbitrarily large cardinals such that is -presentable.
Equivalent characterizations include that is accessible iff:
The important notion of functor between accessible categories is an accessible functor, meaning a functor such that there exists a such that and are both -accessible and preserves -filtered colimits.
If an accessible category in addition has all (small) colimits (or, equivalently, limits), then it is a locally presentable category.
A small category is accessible precisely when it is idempotent complete. Makkai-Paré say that this means accessibility is an “almost pure smallness condition.”
Let be a functor between locally presentable categories which preserves -filtered colimits, and let be an accessible subcategory. Then the inverse image is a -accessible subcategory.
This appears as HTT, corollary A.2.6.5.
Let be a combinatorial model category, its arrow category, the full subcategory on the weak equivalences and the full subcategory on the fibrations. Then , and are accessible subcategories of .
This appears as HTT, corollary A.2.6.6.
In addition:
The 2-category of accessible categories, accessible functors, and natural transformations has all small 2-limits.
Every accessible functor satisfies the solution-set condition?, and every left or right adjoint between accessible categories is accessible. Therefore, the adjoint functor theorem takes an especially pleasing form for accessible categories: a functor is a left (resp. right) adjoint iff it is accessible and preserves all small colimits (resp. limits).
The term accessible category is due to
The standard textbook on the theory of accessible categories is
A discussion of accessible (infinity,1)-categories is in section 5.4, p. 341 of