The tag has no usage guidance, but it has a tag wiki.

learn more… | top users | synonyms

3
votes
1answer
48 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
82 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 ...
1
vote
1answer
36 views

Finite graphs forms Fraïssé Class with limit Rado/random graph

I was reading Peter Cameron's explanation of Fraïssé's Theorem in his book Permutation Groups (Chapter 5). He states the Fraïssé class of finite graphs has limit being the countable random/Rado graph, ...
0
votes
1answer
19 views

What does “4-universal hash function” mean?

I encountered the notion of 4-universal hash function and I cannot understand what exactly it means. This article https://en.wikipedia.org/wiki/Universal_hashing did not really help to clarify it. ...
0
votes
1answer
35 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
148 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 ...
4
votes
0answers
52 views

Monomorphisms of monoids are stable under coproducts

Let $M,N,K$ be three monoids (or even groups, if you like) and let $N \to K$ be an injective homomorphism. Then, the induced morphism $M \sqcup N \to M \sqcup K$ is also injective. This is easy to ...
3
votes
0answers
78 views

Universal property of l^p-spaces

The category $\mathsf{Ban_1}$ of Banach spaces together with short linear maps (i.e. those of norm $\leq 1$) seems to have a natural construction which interpolates between coproduct and product: Let ...
3
votes
0answers
60 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
61 views

Grothendieck's definition of a universal problem

In EGA I, Grothendieck writes (I'm paraphrasing): Let $\mathbf{K}$ be a category, $(A_{\alpha})_{\alpha \in I}, (A_{\alpha \beta})_{(\alpha,\beta) \in I \times I}$ two families of objects of ...
2
votes
0answers
100 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
40 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
136 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
63 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
46 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
20 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
29 views

Proving “up to unique isomorphism”: Universal property of cokernel

This is a homework question. I have already searched the web, but the proofs I have found are very generalized using category theory. Problem is the following: Suppose you have homomorphism $f: M ...
0
votes
0answers
39 views

Universal property of free products.

In my textbook, the Van der Waerden trick is used to prove that every word is equivalent to exactly one reduced word. For this, the author used the universal property of free products, so I tried to ...
0
votes
0answers
29 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 ...