# Tagged Questions

108 views

### Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
78 views

### Tensor product of arbitrary categories

I would like to consider a definition of tensor product in the category of (small, finite, whatever is needed) categories, analogous to the tensor product of vector spaces. I will first rewrite ...
45 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 ...
48 views

### Monoidal product is coproduct in category of commutative monoids

If $V$ is a symmetric monoidal category, the category $\text{CMon}(V)$ of commutative monoids in $V$ has binary coproducts given by $\otimes$, the monoidal product of $V$. See for example Johnstone’s ...
125 views

### Natural Isomorphism: how can $A \otimes B \simeq B \otimes A$ and yet $A \otimes B \neq B \otimes A$

I am reading Braided Monoidal Categories by Joyal and Street. They say cateogories with tensor product arise naturally such as the category of Abelian Groups and that of Banach Spaces. Is there any ...
72 views

### Algeraic theory and definition of multiplication: tensor product v.s. Cartesian product

In a monoidal category, one can define multiplication on a object $M$ as a morphism $$M\otimes M\longrightarrow M,$$ or a morphism M\times ...
55 views

### Tensor Algebra = Universal Property of FORGETFUL FUNCTOR?

Hi there in wiki the tensor algebra is defined w.r.t. the adjoint of the forgetful functor rather than the forgetful functor itself - why so? Besides, does the existence of such algebras for every ...
48 views

### Characteristic Property = Universal Property?

Problem They seem to be the same -almost! But are they really or is it just unlucky accident that they look so similar however describe totally different notions? Example I was trying to set the ...
55 views

### Is it true to say that every tensor is an element of a monoid?

If we consider that: by definition, a tensor is an element of the tensor product of two algebraic structures the most abstract algrebraic structure on which the tensor product is defined are ...
29 views

### Adjoints to cofree modules tensor?

If $M$ is a cofree $R$-module and $A,B$ are arbitrary $R$-modules then, is there a left adjoint to the functor $M\otimes_R -$, i.e. is there an endofunctor $F$ on $_R \mathrm{Mod}$ such that ...
18 views

### How to define the tensor product via an initial morphism? [duplicate]

I am just getting to know some category theory. My understanding of the "universal property" (based on this Wikipedia article) is that it is characterized in terms of an "initial morphism" or ...
69 views

### Monoidal categories and tensor products

Does the multiplication $-\square-$ biendofunctor in a Monoidal category, $\mathfrak{C}$ necessarily commute with coproducts? This is true in some familiar categories, such as $_RMod$, $Grp$ $CRings$ ...
96 views

### $0$-th exterior power, empty product of modules and their tensor product

$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object in the category of multilinear maps from $\finiteprod En$ where ...
92 views

122 views

### Which graph products are categorical products?

There is a whole bunch of definitions of graph products, but only one of them - the tensor product - is the categorical product in the (standard) category of graphs with graph homomorphisms. I'd ...
253 views

### Categorical definition of tensor product

It is standard to define the tensor product $M\otimes_R N$ of $R$-modules as a universal object of bilinear maps from $M\times N$. Now, suppose that $\mathscr{F}$, $\mathscr{G}$ are sheaves of ...
281 views

### Tensor product of sets

The cartesian product of two sets $A$ and $B$ can be seen as a tensor product. Are there examples for the tensor product of two sets $A$ and $B$ other than the usual cartesian product ? The context ...
229 views

### Algebras over a field are flat - category theoretic proof?

Let $k$ be a field. Assume that you already know that the category $\mathrm{Alg}(k)$ of $k$-algebras (everything here is commutative and unital) has a coproduct $\sqcup$. But you don't know that this ...
723 views

### What is the categorical diagram for the tensor product?

The title says it all: what is the diagram that defines a tensor product? (I'm using the term diagram here in the technical sense it has in category theory.) Edit: This question was motivated by the ...
640 views

### Tensor product as a colimit

I've been dealing with category theory for three weeks now and we just started covering limits and colimits, meanwhile in my geometry class we defined the tensor product of vector spaces. Then I ...