nLab wide subcategory

Definition

Of a general category

Traditionally, a wide subcategory of a category $C$ is a subcategory containing all the objects of $C$.

Equivalently, it is a subcategory through which the canonical functor $disc(Obj(C)) \to C$ (from the discrete category on the collection of objects) factors, or whose inclusion functor is bijective on objects.

Notice that the condition to contain all the objects is not invariant under equivalence of categories and so the definition of wide subcategory above violates the principle of equivalence. A variant of the definition which fixes this is:

an essentially wide subcategory contains at least one object from each isomorphism class of objects; that is, its inclusion functor is essentially surjective on objects.

A wide subcategory is also called a lluf subcategory (“lluf” being “full” spelled backwards).

Of an abelian category

An unrelated definition of “wide subcategory” is commonly used in the study of derived categories and stability conditions.

In this context, a full subcategory $\mathcal{W} \hookrightarrow \mathcal{A}$ of an abelian category $\mathcal{A}$ is called wide if it is closed under kernels, cokernels and extensions.

See, for example, Hovey 01, Ingalls-Thomas 09, Marks-Stovicek 15.

Given a wide subcategory $\mathcal{W}$ in this sense, one can consider the minimal torsion class $T(\mathcal{W})$ containing it. Conversely, if $\mathcal{T}$ is a torsion class, define $W(\mathcal{T})$ to be the full subcategory on those objects $X \in \mathrm{Ob}(\mathcal{T})$ such that, for any $Y \in \mathrm{Ob}(\mathcal{T})$ and any $g: Y \to X$, the kernel of $g$ is in $\mathcal{T}$. The composition $W \circ T$ is the identity, thus exhibiting the poset of wide subcategories as a retract of the poset of torsion classes.

The surjection $W$ becomes an injection when restricted to functorially finite torsion classes, and is often a bijection between functorially finite torsion classes and functorially finite wide subcategories; see Marks-Stovicek 15.

Examples

General

Given any category $C$, the maximal sub-groupoid of $C$ is the subcategory consisting of all objects of $C$ but with morphisms only the isomorphisms of $C$. This is the core of a category, and it is a wide subcategory.

References

General

…

For abelian categories

• Mark Hovey, Classifying subcategories of modules, Trans. Amer. Math. Soc. 353 (2001), 3181-3191 (doi:10.1090/S0002-9947-01-02747-7)

• Colin Ingalls, Hugh Thomas, Noncrossing partitions and representations of quivers, Volume 145, Issue 6 November 2009 , pp. 1533-1562 (doi:10.1112/S0010437X09004023)

• Frederik Marks, Jan Stovicek, Torsion classes, wide subcategories and localisations arXiv:1503.04639