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3
votes
1answer
36 views

Notions of free (and/or cofree) Hopf algebras?

I don't know if this somewhat vague question has a precise answer, but I'd appreciate thoughts and references. Let $k$ be a field. I am looking at Hopf algebras over $k$ by which I mean an algebra ...
3
votes
1answer
53 views

Universal property of natural number semi-ring

I asked a question similar to the one I am about to ask, and I think I got a satisfactory answer. However, this time I have some more specific question. Let a semiring $(R,+,\times)$ be an algebraic ...
0
votes
1answer
25 views

Transition functions of sheaf tensor product

Suppose $\mathcal{L},\mathcal{M}$ are invertible sheafs on a scheme $X$. I've seen an abstract construction of $\mathcal{L}\otimes_X \mathcal{M}$, but I'm having trouble connecting this with a more ...
5
votes
0answers
112 views

Universal Property for Tangent Space

As far as I know there are basically three different approaches to the tangent space -all of them coming with advantages and disadvantages: Though the oldest one precisely represents the derivative ...
3
votes
0answers
57 views

A mathematical object that can be characterised by axioms as well as by universal constructions

This question (more specifically one of the comments) prompted me to ask this question: Can someone give me an example of a mathematical object that one can characterize both via axioms and universal ...
2
votes
0answers
97 views

What should this definition be?

This is from Advanced Linear Algebra by S.Roman. Definition (on page 356). Referring to the figure below, let A be a set and let $\mathcal S$ be a family of sets. Let $$ \mathcal F = \{g\colon ...
2
votes
0answers
31 views

Characterization of epic morphisms in the category of rings.

Let $C$ denotes the category of rings (with identities). It's easy to prove that for any $R\in \text{ob }C$, any multiplicative set $S\subset R$ and any ideal $I\leqslant R$, both $R\to S^{-1}R$ and ...
2
votes
0answers
93 views

Does zero-kernel imply monic in Abelian categories?

I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the ...
2
votes
0answers
56 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
2
votes
0answers
43 views

Show that $H *_{G} (G/N) \cong H/N'$ where $f(N)=N' \subset H$.

I was doing some group theory and this exercise came up. Let $f: G \to H$ be a group morphism and $N$ a normal subgroup of $G$. Denote $\pi: G \to G/N$ and let $N'$ be the normal closure of $f(N)$. ...
1
vote
0answers
17 views

“Universal solution” to factorization of arithmetic mean through a semigroup homomorphism.

Headnote A: my apologies for the long winded exposition; I couldn't find a more compact way to convey what I'm looking for, why I'm looking for it, and what have I tried myself. Suggestions for ...
0
votes
0answers
24 views

Description of a universal regular space

tl;dr: Given a topological space $X$, is there a useful description of the universal continuous map from $X$ to a regular (Hausdorff) space? The Čech–Stone compactification $\beta X$, given a ...
0
votes
0answers
26 views

Subset/subgroups/etc as Equalizers using Pushouts

I am trying to show that in a given category $\mathcal{C}$ which has pushouts, we can show that any morphism $f:X\rightarrow Y$ is an equalizer of some pair of morphisms. My solution to this is by ...