Questions tagged [abelian-categories]
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Projectives and Injectives in Functor Categories
Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories?
Here is a precise question. Let $C$ be a small category, whose ...
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What was the error in the proof of Roos' theorem?
Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse ...
- 61.5k
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Locally presentable abelian categories with enough injective objects
I came to the following question when thinking about the (infinitely generated) tilting-cotilting correspondence, where it appears to be relevant.
Does there exist a locally presentable abelian ...
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Is every Grothendieck category with a generator a category of sheaves?
The Gabriel-Popescu theorem tells us that every Grothendieck category with a generator is a left exact localization of a module category. I'm interested in a slightly different way of "representing" ...
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Is every "nice" abelian category with enough projectives an additive presheaf category?
A "nice" category $\mathcal{C}$ should be (for the purposes of this question) locally presentable at a minimum, and maybe a bit more. One might require $\mathcal{C}$ to be (in roughly order of ...
- 61.5k
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Cocomplete but not complete abelian category
This is a duplicate of the following question to which I did not receive any answer: https://math.stackexchange.com/questions/238247/complete-but-not-cocomplete-category
Let $\mathfrak C$ be an ...
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What is a triangle?
So I've been reading about derived categories recently (mostly via Hartshorne's Residues and Duality and some online notes), and while talking with some other people, I've realized that I'm finding it ...
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Are there (enough) injectives in condensed abelian groups?
The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ?
Does it, in fact, ...
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Can a topos ever be an abelian category?
Can a topos ever be a nontrivial abelian category? If not, where does the contradiction lie? If a topos can be an abelian category, can you give a (notrivial!) example?
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Name for abelian category in which every short exact sequence splits
What is the name of the class of abelian categories defined by the following property: every short exact sequence splits?
user138292
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Characterization of geometric morphisms without referring explicitly to the left adjoint?
Recall that a functor $f_\ast : \mathcal E \to \mathcal F$ between toposes is called a geometric morphism if it has a left exact left adjoint $f^\ast$. Is there an intrinsic characterization of such ...
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Abelian categories that are not monoidal
Forgive me if this turns out to be a naive question. I'm quite convinced that not all abelian categories admit (symmetric?) monoidal structure (of course, with the tensor product being additive, ...
- 1,054
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Mitchell's embedding theorem
Mitchell's embedding theorem http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem tells us that every small abelian category ${\cal A}$ has a full, faithful and exact embedding $V : {\cal A} \...
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Are the Alexander-Whitney and Eilenberg-Zilber maps homotopy inverse in arbitrary abelian categories?
Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let
$$AW_{A,B}\colon N_\ast(...
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Example of a Grothendieck category which is not Gabriel?
Following Gabriel, for $\mathcal C$ a Grothendieck category, set $\mathcal T(\mathcal C)$ to be the localizing subcategory generated by the objects of finite length, and $\mathcal C' = \mathcal C / \...
- 61.5k
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Examples of applications of the Freyd-Mitchell embedding theorem.
The Freyd-Mitchell embedding theorem states the following:
Let $\mathcal{A}$ be a small abelian category. There exists a unital ring $R$ and a full, faithful and exact functor $F\colon\mathcal{A}\...
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Abelian category which is not well-powered
Can you give an example of an abelian category which is not well-powered? If not, maybe you can give any reason why there are such abelian categories?
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Does every monoidal abelian category admit an exact, lax monoidal functor to abelian groups?
Let $\mathcal A$ be a (small) non-zero abelian category equipped with a monoidal structure $\otimes$ which is right-exact in each variable. (Maybe feel free to assume more if that makes things easier -...
- 61.5k
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Abelian category from the category of Hopf algebras
The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf
sub-algebra of $H_1$. Is there some replacement or alteration of the notion
of a kernel in the Hopf algebra setting. Same ...
- 1,144
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Categorical presentation of direct sums of vector spaces, versus tensor products
My apologies in advance if this question is to vague, but here goes.... In the category of vector spaces, products are given by direct sums. In general category theory, the existence of products is a ...
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Existence of projective resolutions in abelian categories
It is a standard result of elementary homological algebra that to every R-module $A$ there exists a projective resolution. It is often said that the category of R-modules has "enough projectives." ...
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DGA for a general abelian category
Differential graded algebras dga-s are fundamental objects of study in homological algebra and category theory. On the nlab webpage, they are defined as follows:
a dga is a monoid in the symmetric ...
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Coreflective subcategories in Grothendieck/locally presentable categories
This question is a reference request for the following result or two results, which I believe are rather easy to prove.
Lemma. Let $\mathcal K$ be a locally presentable category and $\mathcal A\...
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Why does tensor product in Ab(V) require colimits in V?
In Tom Leinster's book on operads, he gives Ab(V), the category of abelian groups in a symmetric monoidal category V, as an example of a multicategory that doesn't arise from a monoidal category, ...
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Holomorphic vector bundles over a Riemann surface does not satisfy $\mathbf{AB2}$ but satisfies $\mathbf{AB1}$
in Grothendieck's Tohoku paper (page 127), Grothendieck asserts that the category of holomorphic vector bundles $\mathbf{Bund}(X)$ over a fixed Riemann surface $X$ does not satisfy $\mathbf{AB2}$, but ...
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Locally noetherian (AB4*) Grothendieck categories
Let A be a Grothendieck category satisfying (AB4*) (that is, with exact products).
Assuming that, moreover, A is locally noetherian, must this category have enough small projective objects (so, to be ...
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