nLab thick subcategory

Definition in (pre)triangulated categories

A non-empty full (pre-)triangulated subcategory $T$ 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 $M\oplus N$ is isomorphic to an object in $T$ then both $M$ and $N$ are isomorphic to objects in $T$.

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 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 $A$ 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 $X/X' \in Ob T$, $Y'\in Ob T$.

It remains to define bilinear composition laws:

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

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

Let $h$ be the composition of $f'$, of the canonical isomorphism of $(N''+ N')/N'$ on $N''/(N''\cap N')$ and $g'$,

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

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

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

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 Gabriel and Zisman, Calculus of fractions and homotopy theory I.2.5.d) and 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 (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:

• 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