The theory of accessible categories is a part of
mathematics, specifically of
category theory. It attempts to describe categories in terms of the "size" (a
cardinal number) of the operations needed to generate their objects.
The theory originates in the work of
Grothendieck completed by 1969,[1] and Gabriel and Ulmer (1971).[2] It has been further developed in 1989 by
Michael Makkai and Robert Paré, with motivation coming from
model theory, a branch of
mathematical logic.[3]
A standard text book by Adámek and Rosický appeared in 1994.[4]
Accessible categories also have applications in
homotopy theory.[5][6] Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs.[7]
Some properties of accessible categories depend on the
set universe in use, particularly on the
cardinal properties and
Vopěnka's principle.[8]
κ-directed colimits and κ-presentable objects
Let be an infinite
regular cardinal, i.e. a
cardinal number that is not the sum of a smaller number of smaller cardinals; examples are (
aleph-0), the first infinite cardinal number, and , the first uncountable cardinal). A
partially ordered set is called -directed if every subset of of cardinality less than has an upper bound in . In particular, the ordinary
directed sets are precisely the -directed sets.
Now let be a
category. A
direct limit (also known as a directed colimit) over a -directed set is called a -directed colimit. An object of is called -presentable if the
Hom functor preserves all -directed colimits in . It is clear that every -presentable object is also -presentable whenever , since every -directed colimit is also a -directed colimit in that case. A -presentable object is called finitely presentable.
Examples
In the category Set of all sets, the finitely presentable objects coincide with the finite sets. The -presentable objects are the sets of cardinality smaller than .
In the
category of all groups, an object is finitely presentable if and only if it is a
finitely presented group, i.e. if it has a presentation with finitely many generators and finitely many relations. For uncountable regular , the -presentable objects are precisely the groups with cardinality smaller than .
contains a set of -presentable objects such that every object of is a -directed colimit of objects of .
An -accessible category is called finitely accessible.
A category is called accessible if it is -accessible for some infinite regular cardinal .
When an accessible category is also
cocomplete, it is called locally presentable.
A functor between -accessible categories is called -accessible provided that preserves -directed colimits.
Examples
The category Set of all sets and functions is locally finitely presentable, since every set is the direct limit of its finite subsets, and finite sets are finitely presentable.
The category -Mod of (left) -modules is locally finitely presentable for any ring .
The category Mod(T) of models of some
first-order theory T with countable signature is -accessible. -presentable objects are models with a countable number of elements.
One can show that every locally presentable category is also
complete.[9] Furthermore, a category is locally presentable if and only if it is equivalent to the category of models of a limit
sketch.[10]
Adjoint functors between locally presentable categories have a particularly simple characterization. A functor between locally presentable categories:
is a left adjoint if and only if it preserves small colimits,
is a right adjoint if and only if it preserves small limits and is accessible.
Notes
^
Grothendieck, Alexander; et al. (1972), Théorie des Topos et Cohomologie Étale des Schémas, Lecture Notes in Mathematics 269, Springer
^
Gabriel, P; Ulmer, F (1971), Lokal Präsentierbare Kategorien, Lecture Notes in Mathematics 221, Springer
^
Makkai, Michael; Paré, Robert (1989), Accessible categories: The foundation of Categorical Model Theory, Contemporary Mathematics, AMS,
ISBN0-8218-5111-X