Questions tagged [abelian-categories]

Abelian categories are categories that possess most of properties of categories of modules over a ring, e.g. abelian group structure on morphisms, existence of kernels and cokernels of morphisms, existence of direct products and directs sums, etc.

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Reconstruction of monoidal categories

Both this post on mathoverflow and this wikipedia page claim that you can reconstruct a monoidal category from its Grothendieck ring and $6j$-symbols (or equivalently the associator). Bruce Westbury ...
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Mistake in contradiction argument to show that sheafification commutes with cokernel

The sheafification functor is a left-adjoint to the forgetful functor. Hence it commutes with colimits. The cokernel is a colimit. Hence the cokernel of a sheafified morphism is the same as the ...
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How is functor with “image” unique up to a unique isomorphism defined exactly?

In an abelian category $\mathscr A$ we encounters the notions of kernel, cokernel, chain homology, derived functors, etc. These notions are frequently referred to as functors, and yes, they actually ...
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Noetherian Objects in Quotient Category

I've recently been learning about the quotient of an abelian category by a Serre (thick) subcategory, specifically in the context of module categories. A thought occurred to me today about ...
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Direct sum of injective modules is injective.

By the Bass-Papp Theorem, for a unital ring $R$, any direct sum of injective left $R$-modules is injective if and only if $R$ is left Noetherian. I would like to restrict my consideration to an ...
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A chain complex is exact iff it is split

Let $C.$ be a chain complex in an abelian category. Suppose that the identity $Id_{C.}$ is null homotopic. Then there are morphisms $s_n:C_n\to C_{n+1}$ such that for every $n$ the following identity ...
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Short proof or wrong proof?

I found in Borceux' Categorical Algebra the following proposition: in an abelian category, the following are equivalent: 1) $f:A\longrightarrow B$ is a mono 2) $\operatorname{Ker} f=0$ 3) for ...
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When modular tensor categories are equivalent?

I would like to know when we say that two modular tensor categories are equivalent. Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ...
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A monoidal category that preserves subobjects

Let $X$, $Y$ be objects in a monoidal category $\mathcal{C}$, s.t. the functors $X \otimes \_$ and $\_\otimes Y$ preserve monomorphisms. Moreover, let $A \hookrightarrow X$, $B \hookrightarrow Y$ be ...
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Uniquenes of decompositions in abelian semisimple categories

Basing my intuition on the semisimple Lie algebra case, I have a question about semisimple abelian categories. Let $\mathcal{C}$ be such a category, and let $X$ be an object in $\mathcal{C}$. By ...
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extending functors

In "Functors on locally finitely presented additive categories", by H. Krause, one can read (in page 108): Proposition 2.3. There is, up to equivalence, a bijective correspondence between (skeletally ...
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Abelian categories linear over a field

A category $\mathcal{A}$ is abelian if it has the following properties: There is a zero object (both initial and terminal). Every pair of objects has a product and a coproduct (which will be ...
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What does an additive, mono-preserving functor have to do to become left exact?

Consider an additive functor $F \colon C → D$ of abelian categories preserving monos. As Martin has demonstrated here, $F$ may not be left exact. If $F$ even preserves kernels, $F$ is left exact. But ...
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Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
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Projective Module as a Direct Sum of Left Ideals

I wonder if the following statement is true: Every projective $R$-module is a direct sum of projective left ideals of $R$. Most examples of non-free projective modules I have seen are all left ...
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Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...
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Decomposing Semisimple Perverse Sheaves

Assume $\mathbf{G}$ is an algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ for some prime $p>0$. Let $\mathscr{M}\mathbf{G}$ be the category of all $\overline{\mathbb{Q}_{\ell}}$-...
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A finite diagram in an abelian category which may not be locally small

This question is motivated by this. I will use the notations of my answer to this. We say a category $\mathcal C$ is locally small if Hom($X, Y$) is small for any $X, Y \in$ Ob($C$). Let $\mathcal A$...
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Image of projective objects.

Let $A$ and $B$ be two abelian categories. Assume that there exist a functor $F$ between them which is exact, full and essencially surjective. If $x$ is a projective object in $A$, then $F(x)$ is a ...
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How much can the Freyd-Mitchell embedding theorem be improved?

Can the embedding theorem be improved to preserve injective objects or even better be part of an adjoint pair? Or if that's not possible are there stronger conditions (AB5, enough injectives, etc) on ...
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Products, coproducts and morphisms

The universal properties of products and coproducts "amount" to the statements $$\hom(\coprod_i X_i , Y) = \prod_i \hom(X_i, Y) \quad \text{and} \quad \hom(X,\prod_i Y_i) = \prod_i \hom(X,Y_i)$$ ...
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Derived functors commute with filtered colimits?

I have some trouble regarding the answer to this question. My problem with it has been mentioned in the comments below it, and I think adressed in an answer, but I can't understand this second answer. ...
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Intuition behind exact sequences as extensions in algebraic geometry

Introduction: There are certain categories of group schemes or group varieties in which one can define exact sequences. For instance, the categories of commutative group schemes of finite type over a ...
Once one studies algebra, one finds categories such as $R-\textbf{Mod}$, abelian groups, sheaves over abelian groups, $\mathcal R-\textbf{Mod}$ and the like. They are all abelian. On the other hand, ...
In "Deriving Auslander's Formula (Theorem 2.2)", I'm trying to construct a right adjoint to inclusion functor $\mathsf{i:eff~C \to mod~C}$. I constructed it as follows: The functor \$\mathsf{j:mod~C \...