# Tag Info

Accepted

### How much data does a category contain?

It indeed looks like you need extra information to answer the question whether a given category $\mathcal{C}$ is an abelian category, but this is actually not true. An abelian category is an additive ...
• 5,987
Accepted

### Why is the category of finitely generated modules over a non-noetherian ring not abelian?

Claim: The inclusion $R\text{-mod}\hookrightarrow R\text{-Mod}$ preserves kernels. Once this is known, it follows that a kernel in $R\text{-mod}$, if it exists, must be isomorphic to the ...
• 19.6k
Accepted

### Coimage and Image in Abelian Categories

Notice firstly that $f(\operatorname{ker}f) = 0$ and $(\operatorname{coker}f)f = 0$. Use this to argue that $f = (\operatorname{im}f)f''(\operatorname{coim} f)$. Once you've done that, we want to ...
• 3,930
Accepted

### What's wrong with my understanding of the Freyd-Mitchell Embedding Theorem?

You don't show that $\mathcal{L}^{op}$ actually is a category of modules, just that any small subcategory of it fully and exactly embeds in one. Indeed, an abelian category with a projective ...
• 333k
Accepted

### Commutative ring category is not additive category

The ring $0$ is not the zero object. Indeed a zero object is an object which is both terminal and initial but in rings with unit the initial object is $\mathbb{Z}$ and the terminal one is $0$. ...
• 1,048
Accepted

### Convolution in Hopf algebras

You can try "Hopf algebras. An introduction" by Dascalescu, Nastasescu, Raianu. They have a detailed account with full proofs and further examples and exercises. Kassel's book "Quantum groups" and ...
• 7,354
Accepted

### Where have I used commutative (coproduct in $\mathbf{Ab}$ vs. $\mathbf{Grp}$)

The typical proof that your $A \times B$ satisfies the universal property of the coproduct in $\mathbf{Ab}$ is as follows: if $f: A \to C$ and $g: B \to C$ are two homomorphisms of abelian groups, ...
• 4,757

### Why do universal $\delta$-functors annihilate injectives?

In the 10 years since I asked this question, I learned some other theories of derived functors, and the fact that there has been no answer has been forthcoming leads me to believe that the question ...
• 90.6k
Accepted

### Opposite of a category of modules is not a category of modules

Thinking about Grothendieck categories is making things way more complicated than necessary to prove just that $\bf{R-Mod}^{op}$ is not a module category. You can prove this by simply thinking about ...
• 333k

### Homological algebra using nonabelian groups

A good setting for generalizing homological algebra is that of homological categories. There is a book on the subject due to Borceux and Bourn references at the linked article, as well as a very ...
• 52.7k
Accepted

### Writting objects in abelian category as limit of injective objects

This is false. To my mind it's a little easier to think about the dual question: in an abelian category with enough projectives, is every object a filtered colimit of projective objects? In the ...
• 425k
Accepted

### Definition of a presheaf on a topological space with values in an abelian category

No, that does not follow from the definition as a contravariant functor, and it is incorrect to add it to the definition of a presheaf. I've never seen a source that does this, and it would be ...
• 52.7k

Accepted

### Is taking (co)limits exact in an Abelian category?

In complete generality you only preserve one side of the short exact sequence. Taking colimits is a right exact operation while taking limits is a left exact operation and in general neither is two-...
• 1,594

### Is this categorical definition of homology correct and, furthermore, used to teach homology in some book?

This is exactly how Michael Barr develops homology theory in section 2.4 of his book Acyclic Models.
• 15.6k
Accepted

### In an abelian category, does every family of subobjects have an intersection?

Here is a counterexample. Let $k$ be your favorite field, and let $C$ be the category of "eventually constant" sequences of vector spaces over $k$. That is, an object of $C$ is a sequence $(V_n)$ of ...
• 333k

• 333k

### A chain complex is contractible iff it is split exact

A couple of housekeeping things: First, the title of your question is false; there exist chain complexes which are exact but not split. For instance, one can check that if $R=\mathbb{Z}$, then the ...
• 1,632

### Motivation for spectrum of an Abelian category

The basic motivation for this definition is that if $\mathbf{A}$ is the category of modules over a commutative ring $R$, then the $\sim$-equivalence classes of $\operatorname{Spec}(\mathbf{A})$ are ...
• 333k
Accepted

### Limits and $Hom(-,Y)$-functor in abelian categories

Yes, it is true in any abelian category. In fact, moreover, it's true in every category full stop that $$\hom(\text{colim}_i x_i,y)\cong \lim_i\hom(x_i,y).$$ (in category theory, a projective limit is ...
• 16.9k
Accepted

### Extension group and Baer sum

The Baer sum is not that surprising. Here is how I see it: The group structure on Hom-set may be recovered from the biproduct $\oplus$. Indeed, if $f:A\to B$ and $f':A\to B$ are two parallel arrows, ...
• 12.6k