# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 A\...
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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 $... 0answers 142 views ### Does zero-kernel imply monic in Abelian categories? I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the ... 0answers 64 views ### Relative topological tensor product I know that for a pair of topological rings$R$and$R'$(perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ... 0answers 46 views ### Show that$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)$. ... 0answers 42 views ### How do I get from the universal product of the tensor product to other definitions. I was wondering how you can "derive" the common (or "classical") definition of the tensor product before the universal property was established (I think there is no need to repeat it here), i.e. a ... 0answers 50 views ### Localization and the universal property I've been trying to get to grips with the universal property and was looking at the localization of a commutative ring$A$at some multiplicative set$S$, denoted$S^{-1}A$, to get an idea of what the ... 0answers 21 views ### “Universal solution” to factorization of arithmetic mean through a semigroup homomorphism. Headnote A: my apologies for the long winded exposition; I couldn't find a more compact way to convey what I'm looking for, why I'm looking for it, and what have I tried myself. Suggestions for ... 0answers 8 views ### How to prove pairwise independence of a family of hash functions? I want to prove pairwise independence of a family of hash functions, but I don't know where to start. Given the family of hash functions: H with h(x) = a * x + b (mod M). ( Say h: U -> V, then: M ... 0answers 32 views ### Proving “up to unique isomorphism”: Universal property of cokernel This is a homework question. I have already searched the web, but the proofs I have found are very generalized using category theory. Problem is the following: Suppose you have homomorphism$f: M \...
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 ...