nLab thick subcategory



Definition in (pre)triangulated categories

A non-empty full (pre-)triangulated subcategory TT 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 MNM\oplus N is isomorphic to an object in TT then both MM and NN are isomorphic to objects in TT.

In abelian categories

A nonempty full subcategory TT of an abelian category AA is thick (in the strong sense) if for every exact sequence

0MMM0 0 \longrightarrow M\longrightarrow M''\longrightarrow M'\longrightarrow 0

in AA, the object MM'' is in TT iff MM and MM' are in TT.

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.

Remarks on terminology in abelian context

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 AA one denotes by T¯\bar{T} the full subcategory of AA generated by all objects NN for which any (nonzero) subquotient of NN in TT has a (nonzero) subobject from TT. This becomes an idempotent operation on the class of subcategories of AA where TT¯T\subset \bar{T} iff TT is topologizing. Moreover T¯\bar{T} is always thick in the stronger sense. Serre subcategories in the strong sense are those (nonempty) subcategories which are stable under the operation TT¯T\mapsto\bar{T}. 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.

Serre quotient category

Following the extensions of an early work of Serre by Grothendieck and Gabriel, for any thick subcategory TT in an abelian category AA, one defines the (Serre) quotient category A/TA/T as the one having the same objects as AA and hom-sets given by

(A/T)(X,Y):=colimA(X,Y/Y) (A/T)(X,Y) := colim A(X',Y/Y')

where the colimit runs through all subobjects XXX'\subset X, YYY'\subset Y such that X/XObTX/X' \in Ob T, YObTY'\in Ob T.

It remains to define bilinear composition laws:

Hom A/T(M,N)×Hom A/T(N,P)Hom A/T(M,P). Hom_{A/T}(M, N)\times Hom_{A/T}(N, P)\longrightarrow Hom_{A/T}(M,P).

For this, let f¯\overline{f} be an element of Hom A/T(M,N)Hom_{A/T}(M, N) and let g¯\overline{g} be an element of Hom A/T(N,P)Hom_{A/T}(N, P). The element f¯\overline{f} is the image of a morphism f:MN/Nf: M'\to N/N' where M/MM/M' and NN' are objects of TT. Similarly, g¯\overline{g} is the image of a morphism g:NP/Pg:N'\to P/P', where N/NN/N' and PP' are objects of TT. If MM'' designates the inverse image f 1((N+N)/N)f^{-1}((N''+N')/N'), it is easy to see that M/MM/M'' belongs to TT; we denote by ff' the morphism from MM'' to (N+N)/N(N''+N')/N' which is induced by ff. Likewise, g(NN)g(N''\cap N') is an object of TT; if PP'' denotes the sum P+g(NN)P'+g(N''\cap N'), it is easy to see that PP'' belongs to TT; we denote gg' the morphism of N/(NN)N''/(N''\cap N') in P/PP/P'' which is induced by gg.

Let hh be the composition of ff', of the canonical isomorphism of (N+N)/N(N''+ N')/N' on N/(NN)N''/(N''\cap N') and gg',

Mf (N+N)/NcanN/(NN)g P/P. M'' \overset{f^'}\longrightarrow (N''+N')/N' \overset{can}\longrightarrow N''/(N''\cap N')\overset{g^'}\longrightarrow P/P''.

The image h¯\overline{h} of hh in Hom A/T(M,P)Hom_{A/T}(M, P) depends only on f¯\overline{f} and g¯\overline{g} and not on ff and gg. It is therefore justified to define the composition in A/TA/T by equality g¯f¯=h¯\overline{g}\circ\overline{f} = \overline{h}. These composition laws are bilinear; they make A/TA/T a category.

(adapted from Gabriel, Des Catégories Abéliennes, III.1)

The quotient functor Q:AA/TQ \colon A\to A/T 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 Σ\Sigma of all morphisms whose kernel and cokernel are in TT. Although Σ\Sigma 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 \mathbb{Q}-vector spaces, by the functor which maps an abelian group MM to the scalar extension M M \otimes_{\mathbb{Z}} \mathbb{Q}. (See the Stacks Project, Tag 0B0J for a proof.)

Localizing subcategories

A thick subcategory (here always in the strong sense) is said to be localizing if TT is thick and the Serre quotient functor QQ admits a right adjoint A/TAA/T\to A, often called the section functor. In other words A/TA/T is then a reflective subcategory of AA. Every coreflective thick subcategory TT admits a section functor, and the converse holds if AA has injective envelopes. A thick subcategory TAT\subset A is a coreflective iff (T,F)(T,F) is a torsion theory where

F{XObA|A(T,X)=0} F \coloneqq \{X\in Ob A\,|\,A(T,X) = 0\}

Thick subcategories and saturation

Recall that a class Σ\Sigma of morphisms with category of fractions 𝒞 Σ\mathcal{C}_\Sigma is called saturated if Σ\Sigma coincides with the class of morphisms inverted by the canonical functor 𝒞𝒞 Σ\mathcal{C}\to \mathcal{C}_\Sigma.

Let AA be an abelian category, TT be a thick subcategory in the strong sense and let Σ T\Sigma_T be the class of morphisms in AA such that their kernel and cokernel are in TT. Then Σ T\Sigma_T has a right and a left calculus of fractions and is saturated for the canonical functor p:AA/Tp:A\to A/T. Furthermore, pp maps to zero objects precisely the objects in TT.

Conversely, let Σ\Sigma be a saturated class of morphisms of AA 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 p:AA Σp:A\to A_\Sigma 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).


Abelian context

  • 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:

Discussion for quiver representations:

  • Nikolay Dimitrov Dichev, Thick subcategories for quiver representations 2009 (pdf)

Triangulated context

Last revised on July 2, 2024 at 23:37:55. See the history of this page for a list of all contributions to it.