nLab thick subcategory

Definition in (pre)triangulated categories

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 abelian categories

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

$$0 \longrightarrow M\longrightarrow M''\longrightarrow M'\longrightarrow 0$$

in $A$, the object $M''$ is in $T$ iff $M$ and $M'$ are in $T$.

In words, it is an additive full subcategory closed under subobjects, quotients and extensions (e.g. Dichev 2009 and Pierre Gabriel in Des Categories Abeliennes).

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 $A$ one denotes by $\bar{T}$ the full subcategory of $A$ generated by all objects $N$ for which any (nonzero) subquotient of $N$ in $T$ has a (nonzero) subobject from $T$. This becomes an idempotent operation on the class of subcategories of $A$ where $T\subset \bar{T}$ iff $T$ is topologizing. Moreover $\bar{T}$ is always thick in the stronger sense. Serre subcategories in the strong sense are those (nonempty) subcategories which are stable under the operation $T\mapsto\bar{T}$. 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 $T$ in an abelian category $A$, one defines the (Serre) quotient category $A/T$ as the one having the same objects as $A$ and hom-sets given by

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

where the colimit runs through all subobjects $X'\subset X$, $Y'\subset Y$ such that $X/X' \in Ob T$, $Y'\in Ob T$. The quotient functor $Q \colon A\to A/T$ 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 $\Sigma$ of all morphisms whose kernel and cokernel are in $T$. Although $\Sigma$ 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 $M$ to the scalar extension $M \otimes_{\mathbb{Z}} \mathbb{Q}$. (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 $T$ is thick and the Serre quotient functor $Q$ admits a right adjoint $A/T\to A$, often called the section functor. In other words $A/T$ is then a reflective subcategory of $A$. Every coreflective thick subcategory $T$ admits a section functor, and the converse holds if $A$ has injective envelopes. A thick subcategory $T\subset A$ is a coreflective iff $(T,F)$ is a torsion theory where

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

Thick subcategories and saturation

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

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

Conversely, let $\Sigma$ be a saturated class of morphisms of $A$ 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 $p:A\to A_\Sigma$ 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 Schubert 1970, pp.105-107).

Related concepts

• thick subcategory theorem

References

Abelian context

• Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448 (numdam)

• eom, Localization of categories

• 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:

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

Triangulated context

• Takumi Murayama, The classification of thick subcategories and Balmer’s reconstruction theorem (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, lecture 26, Thick subcategories (pdf)

• The Stacks Project, Quotients of triangulated categories