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7
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0answers
55 views

Is the double orthogonal of a localizing Serre subcategory still a localizing Serre subcategory?

Let $R$ be an unital ring and consider the category of left $R$-modules $R$-Mod. We call a full subcategory $\mathcal{W}\subset R$-Mod a localizing Serre subcategory if $\mathcal{W}$ is closed under ...
4
votes
1answer
131 views

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 ...
3
votes
1answer
105 views

Tannaka-Krein reconstruction and rigidity

Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have ...
2
votes
1answer
105 views

Construction of differentials in the spectral sequence for double complexes

I was reading through Ravi Vakil's book/lecture notes on spectral sequences, but I came to an impasse. He leaves as an exercise the construction of the $d_2$ differentials of the spectral sequence (...
11
votes
0answers
169 views

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(...
2
votes
1answer
71 views

An immersion of $\text{Sub}(X) \to \text{EqRel}(X)$ in a Malcev category

In an abelian category, each subobject $A \stackrel{f}{\to} X$ individuate an equivalence relation $R(f) \to X^2$ which is given by the equalizer of $$X^2 \rightrightarrows X \to \text{Coker}(f). $$ ...
0
votes
1answer
83 views

Misunderstanding of the proof of the Embedding Theorem in Borceux

In the proof of the full exact embedding theorem of the "Handbook of Categorical Algebra 2" of F. Borceux at the step 2, there is something I do not understand. It is at page 82 just after the ...
14
votes
1answer
646 views

Is the category of left exact functors abelian?

Recently I asked on Math Stack Exchange here, if the category $\mathbf{Lex(\mathcal{A,B})}$ of left exact functors between to abelian categories $\mathcal{A,B}$ is abelian? In his thesis Des ...
3
votes
1answer
88 views

Brauer group classifying some splitting categories

Notation: $k$ - field. "$k$-category" = $k$-linear abelian category. $Vect_k$ - the $k$-category of $k$-vector spaces. For a field extension $K/k$ and a $k$-category $\mathcal{A}$, denote by $\mathcal{...
8
votes
1answer
423 views

Example of an abelian category with enough projectives and injectives which are not dual

For trying to understand how general a certain theorem is, I'm looking for an example of an essentially small abelian category which has enough projectives and enough injectives, but whose category of ...
29
votes
1answer
1k views

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 ...
4
votes
0answers
147 views

On representable abelian sheaves vs abelian sheaves

Let $S$ be a scheme, $(\text{Sch}/S)_{\rm Ét}$ a big étale site, and $A$ a representable (either in schemes or algebraic spaces over $S$) abelian sheaf on $(\text{Sch}/S)_{\rm Ét}$. Suppose there is ...
10
votes
2answers
930 views

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 ...
23
votes
0answers
1k views

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 ...
1
vote
1answer
152 views

Abelian category which is nice and its dual is also

Is there a nontrivial abelian category $C$ such that both $C$ and $C^{op}$ satisfy AB3-AB5? ("nontrivial" means that there are nonzero objects in $C$)
37
votes
4answers
4k views

What's there to do in category theory?

Disclaimer: I posted this question on MSE only a few days ago; and received very few comments. I know that the etiquette is to wait a bit more than that before moving a post from MSE to MO, but I ...
4
votes
0answers
106 views

Category of (co)commutative Hopf monoids in an exact category

I'm transferring this question over from SE, since it didn't get much attention over there. Let $(C, \otimes)$ be an exact monoidal category, and let $H(C)$ be the category of cocommutative and ...
8
votes
0answers
281 views

Abelian categories have become the language of homological algebra. Why haven't Zariski categories become the language of commutative algebra?

I'm not seeing much mention of Zariski categories in the literature. There is no article on Zariski categories in nLab, which would seem to be an obvious place to have such an article. What has ...
3
votes
0answers
123 views

About functor categories and enough injectives [duplicate]

Let $\mathcal{I}$ be a small category and $\mathcal{A}$ an abelian category. If $\mathcal{A}$ is complete (that is, the product of any set of objects exists) and has enough injectives, how can I prove ...
7
votes
0answers
128 views

Quotifiable Subcategories

I asked this on math.stackexchange but haven't got a response. If this question is unsuitable for this website please let me know. In Popescu's textbook Abelian Categories with Applications to Rings ...
2
votes
0answers
127 views

Are finitely generated objects in a locally Noetherian category also finitely presented?

Let $\mathcal C$ be a locally Noetherian category. Then, we know that every finitely generated object is Noetherian. I cannot seem to be able to prove that Noetherian objects are finitely presented. ...
8
votes
0answers
236 views

Is every $R^iF(M)$ isomorphic to some $F(N)$?

Let $A$ and $B$ be abelian categories. Assume that $A$ has enough injectives. Is there a "useful" (take that to mean what you will) condition on $A$ and $B$ such that the following is true? For all ...
3
votes
1answer
203 views

A simple colimit in the derived category?

I have recently come across the following question : Let $X$ be a (bounded below)chain complex in an arbitrary abelian category, and denote ${\sigma_{\leq n}}$ the stupid truncations functors (i.e. ...
5
votes
2answers
425 views

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 ...
3
votes
1answer
146 views

Exactness of $j_!$ in abelian category recollement

Consider a recollement situation, with notation the same as on the nLab page. That is, we have adjunctions $i^* \dashv i_* \dashv i^!$ and $j_! \dashv j^* \dashv j_*$ between the abelian categories $\...
8
votes
0answers
332 views

How to Compute Ext-Groups for Categories without Enough Injectives/Projectives?

I am studying a category of representations of an infinite-dimensional Lie algebra, which is an abelian category. Unfortunately, the category does not have enough injectives/projectives. I would ...
4
votes
0answers
168 views

Checking a monad is idempotent

I have a monad $T: \mathcal{C} \to \mathcal{C}$ on a (Grothendieck) abelian category which preserves filtered colimits and direct sums (but is not exact). There is a finite collection $G$ of compact, ...
2
votes
0answers
158 views

Why holim and not Rlim?

Let $\mathcal{A}$ be a Grothendieck category (I care mostly about modules over a ring). Let $\operatorname{Ch}^+(\mathcal{A})$ the category of bounded below cochain complexes, $\operatorname{D}^+(\...
2
votes
1answer
338 views

Let $F:\mathscr{A}\to\mathscr{B}$ be an equivalence of Abelian categories. Must $F$ be additive?

What if we simply require $\mathscr{A}$ to be pre-additive, or additive? I have seen it stated without proof that if $\mathscr{A}$ and $\mathscr{B}$ are abelian categories, then any equivalence $F:\...
3
votes
1answer
458 views

On an exercise from Weibel's book on homological algebra

I was revising some old postgraduate notes of mine in homological algebra (written during a postgrad course on the topic, I had taken more than ten ;) years ago) and I came accross the following ...
1
vote
1answer
146 views

Is the minimal Serre subcategory containing a set also a set?

Let $\mathscr{A}$ be an abelian category, and $S\subset\mathscr{A}$ some set of objects in $\mathscr{A}$ (i.e. $S$ is not a proper class), so we can fairly easily define the category $\mathscr{C}_S$ ...
4
votes
1answer
281 views

Colimits in colimits of categories

I was considering the following situation. I have a directed/filtering system of categories $C_i$. I understand how take its direct limit (aka colimit) $C$ in the category of categories. My question ...
1
vote
0answers
282 views

Is every Artinian abelian category necessarily Noetherian?

Let $\mathcal{A}$ be a small abelian category (additive category with AB1) and AB2)). We say $\mathcal{A}$ is artinian if for every $A\in\mathcal{A}$, every descending chain of subobjects of $A$ ...
28
votes
3answers
3k views

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 ...
3
votes
1answer
255 views

Does the existence of an injective cogenerator “help” in finding generators of an abelian category?

I have an abelian category $A$ that is AB4, AB3* and has an injective cogenerator. Do these conditions "help" in checking whether a given family $a_i$ of (compact) objects of $A$ is generating in it? ...
17
votes
3answers
1k views

Can $\mathcal O_X$ be recognized abstract-nonsensically?

This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't. In the ...
5
votes
0answers
128 views

Names for $\operatorname{Coker}\operatorname{ker}\cong \operatorname{Ker}\operatorname{coker}$ and $\operatorname{Coim}\cong \operatorname{Im}$?

Let $\operatorname{Coim}f$ be the cokernel of the kernel pair of an arrow. Let $\operatorname{Im}f$ be the kernel of the cokernel pair. An interesting property of a category is the isomorphy of the ...
9
votes
1answer
459 views

Objects of which Grothendieck abelian categories have elements?

The Freyd-Mitchell embedding theorem is a very useful tool for dealing with small abelian categories. However, it does not allow to use "elements" of objects of an abelian category $A$ in those ...
4
votes
2answers
442 views

On various relations between “additional axioms” for AB4 and Grothendieck abelian categories

Let $A$ be an abelian category that has a generator and satisfies the AB4 axiom. I would like to understand (better) the relations between various additional "restrictions" on $A$. So here is my list ...
4
votes
0answers
145 views

Relationship between pointed protomodularity and $\operatorname{Coker}\ker f\overset{\cong}{\longrightarrow}\operatorname{Ker}\operatorname{coker}f$

In hopes of understanding algebra better, I've been reading here and there about protomodular categories and the like. Among other things, the theory surrounding these (and some other) categories ...
19
votes
0answers
268 views

Homotopic version of Freyd's AT category observations

Freyd was the first to formalize a striking comparison between abelian categories and topoi, showing that their exactness properties can be jointly captured by the axioms of AT categories, and the ...
2
votes
0answers
196 views

On the category of $D$-modules

Let $X$ be a smooth variety over an algebraically closed field $k$ of char. $0$. 1) Is the abelian category $M(X)$ of $D$-modules on $X$, which are quasi-coherent as $O$-modules, a Grothendieck ...
2
votes
1answer
174 views

Does a binormal category always admit an additive structure?

Let $\mathsf{C}$ be a category. We call $\mathsf{C}$ binormal if it has a null object, has all equilizers and coequilizers, all monomorphisms are kernels and all epimorphisms are cokernels (whereby a ...
4
votes
1answer
127 views

Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker (f))\...
3
votes
1answer
317 views

Is a left-exact limit-preserving functor $Ab \to Ab$ necessarily representable?

Let $Ab$ be the category of abelian groups, and let $F: Ab \to Ab$ be a covariant functor which is left-exact and limit-preserving. Is $F$ necessarily naturally equivalent to a functor of the form $\...
3
votes
1answer
115 views

Direct limit closure of Serre subcategories

Let $C$ be a Grothendieck category and $T$ a Serre subcategory of $C$. Let $\tilde{T}$ be the full subcategory of $C$ consisting of all direct limits of objects in $T$. Is $\tilde{T}$ a Serre ...
4
votes
0answers
183 views

Is an abelian category a Serre subcategory of its ind-category?

Let $\mathcal C$ be an abelian category and consider its ind-category $Ind(\mathcal C)$: (1) Is $Ind(\mathcal C)$ always abelian? (If not, what conditions are needed?) (2) Is $\mathcal C\subseteq ...
2
votes
1answer
166 views

Additive functors to abelian groups: “additional structure” and functors induced by “additive pseudo-functors”; references?

Let $\mathcal{A}$ be a small additive category. Consider the category $PreSh(\mathcal{A})$ of all additive functors from $\mathcal{A}^{op}$ into abelian groups; note that this category is abelian and ...
3
votes
1answer
131 views

Semisimple monoidal category with duals

We say that an abelian category is semisimple if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects. Let $({\cal C},\otimes,*)$ be a semisimple ...
0
votes
1answer
118 views

Adjointable Abelian Monoidal Functor

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?