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11
votes
2answers
439 views

Tensor product of monoids and arbitrary algebraic structures

Question. Do you know a specific example which demonstrates that the tensor product of monoids (as defined below) is not associative? Let $C$ be the category of algebraic structures of a fixed type,...
8
votes
2answers
146 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
13
votes
2answers
225 views

Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...
5
votes
1answer
118 views

universal property in quotient topology

The following is a theorem in topology: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/\sim$ be the canonical projection. If $g : X → Z$ is a continuous ...
7
votes
2answers
180 views

How to get used to commutative diagrams? (the case of products).

I've returned to Aluffi's book (after getting the basics of groups from Herstein's) and I hit the same brick wall that made me put it aside. My general problem is this: I can't seem to get used to ...
5
votes
1answer
167 views

Universal Property: do people study terminal objects in $(X\downarrow U)$?

Suppose that $U:D\to C$ is a functor and $X$ is an object of $C$. In defining Universal property, Wikipedia writes about terminal objects in $(U\downarrow X)$ and initial objects in the category $(X\...
5
votes
1answer
330 views

The Kahler differential is zero

While studying for my commutative algebra exam, I have come across this problem. Let $K$ be a field. Let $A$ be a $K-$algebra, finitely generated as a $K$-vector space. Prove that $\Omega^1_{A/K} =...
6
votes
1answer
116 views

Does every continuous map between compact metrizable spaces lift to the Cantor set?

I'm interested in the universal properties of the Cantor set. It is well-know that the Cantor set $2^\mathbb{N}$ is "universal" in the category of metrizable compact spaces, in the sense that every ...
1
vote
1answer
498 views

Direct (Inductive) limit of (locally convex) TVEs and universal property

This is not really a question, I'd just like to discuss a little about universal properties (more specifically, the direct limit) in TVEs. I'm trying to work with universal properties in Topological ...