A non-empty full (pre-)triangulated subcategory is called thick (or épaisse) if it is “closed under direct summands” (e.g. Murayama, Def. 11, see Stacks Project, Def. 13.6.1). In other words, if is isomorphic to an object in then both and are isomorphic to objects in .
A nonempty full subcategory of an abelian category is thick (in the strong sense) if for every exact sequence
in , the object is in iff and are in .
In words, it is an additive full subcategory closed under subobjects, quotients and extensions (e.g. Dichev 2009 and Pierre Gabriel in Des Catégories Abéliennes).
In any abelian category all subobjects are kernels of epimorphisms; and all quotients are cokernels of monomorphisms.
Another way to pack the same definition is that thick subcategory of an abelian category is a topologizing subcategory closed under extensions.
For some authors the thick subcategory is also called a Serre subcategory (in a weak sense), the term which we reserve for (generally) a stronger notion.
For any subcategory of an abelian category one denotes by the full subcategory of generated by all objects for which any (nonzero) subquotient of in has a (nonzero) subobject from . This becomes an idempotent operation on the class of subcategories of where iff is topologizing. Moreover is always thick in the stronger sense. Serre subcategories in the strong sense are those (nonempty) subcategories which are stable under the operation . In full generality this is a bit stronger than thick subcategory and the two senses agree if the ambient category is a full category of modules.
Following the extensions of an early work of Serre by Grothendieck and Gabriel, for any thick subcategory in an abelian category , one defines the (Serre) quotient category as the one having the same objects as and hom-sets given by
where the colimit runs through all subobjects , such that , .
It remains to define bilinear composition laws:
For this, let be an element of and let be an element of . The element is the image of a morphism where and are objects of . Similarly, is the image of a morphism , where and are objects of . If designates the inverse image , it is easy to see that belongs to ; we denote by the morphism from to which is induced by . Likewise, is an object of ; if denotes the sum , it is easy to see that belongs to ; we denote the morphism of in which is induced by .
Let be the composition of , of the canonical isomorphism of on and ,
The image of in depends only on and and not on and . It is therefore justified to define the composition in by equality . These composition laws are bilinear; they make a category.
(adapted from Gabriel, Des Catégories Abéliennes, III.1)
The quotient functor is obvious.
Notice that the set of morphisms is indeed small, so that the Serre quotient category exists as a locally small category. On the other hand, one can construct an equivalent localization by the Gabriel-Zisman localizing at the class of all morphisms whose kernel and cokernel are in . Although admits the calculus of fractions, this method does not guarantee the existence in general.
A basic example is the quotient of the category of abelian groups modulo the torsion groups. This category is equivalent to the category of -vector spaces, by the functor which maps an abelian group to the scalar extension . (See the Stacks Project, Tag 0B0J for a proof.)
A thick subcategory (here always in the strong sense) is said to be localizing if is thick and the Serre quotient functor admits a right adjoint , often called the section functor. In other words is then a reflective subcategory of . Every coreflective thick subcategory admits a section functor, and the converse holds if has injective envelopes. A thick subcategory is a coreflective iff is a torsion theory where
Recall that a class of morphisms with category of fractions is called saturated if coincides with the class of morphisms inverted by the canonical functor .
Let be an abelian category, be a thick subcategory in the strong sense and let be the class of morphisms in such that their kernel and cokernel are in . Then has a right and a left calculus of fractions and is saturated for the canonical functor . Furthermore, maps to zero objects precisely the objects in .
Conversely, let be a saturated class of morphisms of with a calculus of fractions on the right and on the left. Then the full subcategory on the objects that are mapped to zero by the canonical is thick. In other words, there is a bijection between thick subcategories in the strong sense and saturated classes of morphisms with a calculus of fractions on the right and on the left.
(For this material see Gabriel and Zisman, Calculus of fractions and homotopy theory I.2.5.d) and Schubert 1970, pp. 105–107).
Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448 (numdam)
eom, Localization of categories (for Serre quotient category)
A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9
It is called “dense subcategory” (see page 165) in Popescu:
N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp
H. Schubert, Kategorien II , Springer Heidelberg 1970.
Discussion for quiver representations:
Takumi Murayama, The classification of thick subcategories and Balmer’s reconstruction theorem (pdf)
Jacob Lurie, Chromatic Homotopy Theory, lecture 26, Thick subcategories (pdf)
Last revised on July 2, 2024 at 23:37:55. See the history of this page for a list of all contributions to it.