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 Grothendieck also continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs (Grothendieck (1991)).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
K
be an infinite regular cardinal and let
C
be a category. An object
X
of
C
is called
K
-presentable if the Hom functor
Hom
(
X
,
−
)
preserves
K
-directed colimits. The category
C
is called
K
-accessible provided that :
C
has
K
-directed colimits
C
has a set
P
of
K
-presentable objects such that every object of
C
is a
K
-directed colimit of objects of
P
A category
C
is called accessible if
C
is
K
-accessible for some infinite regular cardinal
K
.
A
ℵ
0
-presentable object is usually called finitely presentable, and an
ℵ
0
-accessible category is often called finitely accessible.
The category
R
-Mod of (left)
R
-modules is finitely accessible for any ring
R
. The objects that are finitely presentable in the above sense are precisely the finitely presented modules (which are not necessarily the same as the finitely generated modules unless
R
is noetherian).
The category of simplicial sets is finitely-accessible.
The category Mod(T) of models of some first-order theory T with countable signature is
ℵ
1
-accessible.
ℵ
1
-presentable objects are models with a countable number of elements.
When an accessible category
C
is also cocomplete,
C
is called locally presentable. Locally presentable categories are also complete.
