# A classification of accessible categories

### Jiri Adámek, Francis Borceux, Stephen Lack, and
Jiri Rosický

This appeared in Journal of Pure and
Applied Algebra 175:7-30, 2002.
For a suitable collection **D** of small categories, we define
the **D**-accessible categories, generalizing the *k*-accessible
categories (for a regular cardinal *k*) of Lair, Makkai,
and Paré here the *k*-accessible categories are seen as the
**D**-accessible ones where **D** consists of the *k*-small
categories. A small category *C* is called **D**-filtered when
*C*-colimits commute with **D**-limits in the category of sets.
An object of a category is called **D**-presentable when the corresponding
representable functor preserves **D**-filtered colimits. The
**D**-accessible categories are then the categories with **D**-filtered
colimits and a small set of **D**-presentable objects which is
``dense with respect to **D**-filtered colimits''.

We suppose always that **D** satisfies a technical condition called
``soundness'': this is the ``suitable'' case mentioned above. Every
**D**-accessible category is accessible; thus the choice of different
sound **D** provides a classification of accessible categories, as
referred to in the title. A surprising number of the main results from
the theory of accessible categories remain valid in the **D**-accessible
context.

The locally **D**-presentable categories are defined as the cocomplete
**D**-accessible ones. When **D** consists of the finite categories,
these are precisely the locally finitely presentable categories are Gabriel
and Ulmer. When **D** consists of the finite discrete categories, these
are the finitary (many-sorted) varieties.

As a by-product of this theory, we prove that the free completion under
**D**-filtered colimits distributes over the free completion under limits.
This result is new, even in the case where **D** is empty and
**D**-filtered colimits are just arbitrary (small) colimits.

Click here to download a gzipped postscript file of
the complete paper.

Steve Lack
Last modified: Fri Aug 30 09:09:56 EST 2002