nLab
wide subcategory

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Definition

Of a general category

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

Equivalently, it is a subcategory through which the canonical functor disc(Obj(C))Cdisc(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 𝒲𝒜\mathcal{W} \hookrightarrow \mathcal{A} of an abelian category 𝒜\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 𝒲\mathcal{W} in this sense, one can consider the minimal torsion class T(𝒲)T(\mathcal{W}) containing it. Conversely, if 𝒯\mathcal{T} is a torsion class, define W(𝒯)W(\mathcal{T}) to be the full subcategory on those objects XOb(𝒯)X \in \mathrm{Ob}(\mathcal{T}) such that, for any YOb(𝒯)Y \in \mathrm{Ob}(\mathcal{T}) and any g:YXg: Y \to X, the kernel of gg is in 𝒯\mathcal{T}. The composition WTW \circ T is the identity, thus exhibiting the poset of wide subcategories as a retract of the poset of torsion classes.

The surjection WW 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.

References

General

For abelian categories

Last revised on April 28, 2019 at 05:27:03. See the history of this page for a list of all contributions to it.