functorially finite subcategory

Functorially finite subcategories of abelian categories

Functorially finite subcategories of abelian categories

Let 𝒜\mathcal{A} be an abelian category and 𝒞\mathcal{C} a full subcategory. Given an object XX of 𝒜\mathcal{A}, a morphism f:CXf: C \to X from COb(𝒞)C \in \mathrm{Ob}(\mathcal{C}) is called a right 𝒞\mathcal{C} approximation of X if, for any COb(𝒞)C' \in \mathrm{Ob}(\mathcal{C}) and any morphism f:CXf': C' \to X, we can factor ff' and fgf \circ g.

For example, suppose that we have finitely many objects C 1C_1, C 2C_2, …, C nC_n and 𝒞\mathcal{C} is the full subcategory on direct sums of C jC_j‘s. Suppose also that our category is kk linear for some field kk and that Hom(C j,X)\mathrm{Hom}(C_j, X) is finite dimensional over kk. Then jHom(C j,X) kC jX\bigoplus_j \Hom(C_j, X) \otimes_k C_j \longrightarrow X will be a right approximation of XX.

As an example where we do NOT have a left approximation, let 𝒜\mathcal{A} be finitely generated abelian groups and let 𝒞\mathcal{C} be finite abelian groups. Take X=X = \mathbb{Z}. For any finite abelian group CC and map C\mathbb{Z} \to C, we can choose a prime pp not dividing |C||C|, then /p\mathbb{Z} \to \mathbb{Z}/p \mathbb{Z} will not factor through C\mathbb{Z} \to C. (On the other hand, all finitely generated abelian groups do have right approximations in this setting – the inclusion of the torsion subgroup is a right approximation.)

The full subcategory 𝒞\mathcal{C} is called right functorially finite if every object in 𝒜\mathcal{A} has a right 𝒞\mathcal{C} approximation. We define left 𝒞\mathcal{C} approximation and left functorially finite in a dual manner. A full subcategory which is both right and left functorially finite is called functorially finite.

Last revised on April 27, 2019 at 18:45:17. See the history of this page for a list of all contributions to it.