Tagged Questions

48 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
42 views

Commutativity of direct and inverse limits

In exercise 5.34(iv) of Homological Algebra book by Rotman one is asked to prove that direct limits and inverse limits do not necessarily commute. I have two questions : 1.) Is it true that ...
67 views

The splitting lemma and uniqueness

For the sake of concreteness, let's restrict discussion to the category of abelian groups. Throughout, $$0 \to A \overset{q}{\to} B \overset{r}{\to}C \to 0$$ is a short exact sequence. One part of ...
61 views

101 views

When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
78 views

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 ...
144 views

Cocartesian squares in the category of abelian groups.

Recently, I've been doing a recap of (basic) category theory and found an old exercise I seem to be unable to solve. The question is as follows. Let $A, B$ be abelian groups, $A'<A$ and $B'<B$ ...
472 views

Equivalences and isomorphisms of short exact sequences

In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
122 views

Let $A$ be an additive category. Namely $A$ has a zero object, $A$ has finite products and coproducts, and Every Hom-set is an Abelian group such that composition of morphisms is bilinear. ...
135 views

The forgetful functor $U : \mathsf{Ab} \to \mathsf{Set}$ is monadic, this follows from Beck's monadicity theorem or some other general result. Anyway, I would like to prove this directly, thereby ...
66 views

Inductive vs projective limit of sequence of split surjections II

This question is a follow-up of this earlier question I asked. Let $$A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots$$ be an inductive sequence of ...
88 views

Inductive vs projective limit of sequence of split surjections

Let $$A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots$$ be an inductive sequence of abelian groups, the connecting homomorphisms of which are ...
388 views

Coproducts in $\text{Ab}$

I am currently trying to understand why finite products and coproducts in the category $\text{Ab}$ coincide. In fact, I'm not even sure I can show it. My question is the following: Is there an ...
254 views

Adjoint of forgetful functor between category of vector spaces and category of abelian groups

I've just found out about the forgetful functor between the category of vector spaces and the category of abelian groups. It maps a vector space to it's additive abelian group. My question is, is ...