0
votes
1answer
46 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 ...
0
votes
1answer
42 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 ...
0
votes
1answer
41 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 ...
2
votes
1answer
26 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 ...
1
vote
0answers
17 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 ...
5
votes
1answer
50 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 $_RMod$, $Grp$ $CRings$ ...
2
votes
3answers
74 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 ...
3
votes
1answer
92 views

$(\beta_1 \otimes \beta_2)(\alpha_1 \otimes \alpha_2)=(\beta_1 \alpha_1)\otimes(\beta_2 \alpha_2)$

Let $V_1, V_2, W_1, W_2, U_1, U_2 \in$ K-Vect, $V_1 \xrightarrow{\;\; \alpha_1 \;\; }W_1 \xrightarrow{\;\; \beta_1 \;\; }U_1, V_2 \xrightarrow{\;\; \alpha_2 \;\; }W_2 \xrightarrow{\;\; \beta_2 \;\; ...
7
votes
1answer
136 views

Understanding the right-exactness of the tensor product using *only* its universal property and the Yoneda lemma

I would like to get an intuition for why $(-)\otimes N$ is right-exact using its universal property involving bilinear maps, not by appealing to higher-level observations such as "left-adjoints ...
0
votes
2answers
53 views

Do $\operatorname{Hom}( - , R)$ and $ - \otimes_R B$ commute when applied to $A\cong R^d?$

Let $R$ be a commutative ring with identity. Let $A$ and $B$ are $R$-modules, and further suppose that $A$ is free with finite rank. Is it true that $$ \operatorname{Hom} (A \otimes_R B , R) \cong ...
4
votes
1answer
154 views

Could I see that the tensor product is right-exact using its universal property and the Yoneda lemma?

I have been doing some review with the goal of trying to understand as much as I can via universal properties and category theory (already feeling comfortable with the mundane way of doing things). ...
2
votes
1answer
116 views

“Tensor product” $\otimes$ of monoids

Referring to the top answer in this post: http://mathoverflow.net/questions/19004/is-the-category-commutative-monoids-cartesian-closed Would it be reasonable to explain what is a tensor product in ...
5
votes
0answers
138 views

Mnemonic device for relationships between Hom and Tensor

Probably this is a stupid question, but nevertheless... Let $A$, $B$, $C$ and $D$ be rings, and $M$, $N$ and $K$ be appropriate bimodules between them. There are extremely well-known canonical ...
8
votes
2answers
280 views

Tensor product of monoids and arbitrary algebraic structures

Let $C$ be the category of algebraic structures of a certain type and let us denote by $|~|$ the underlying functor $C \to \mathsf{Set}$. For $M,N \in C$ we have a functor $\mathrm{BiHom}(M,N;-) : C ...
2
votes
2answers
110 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 ...
4
votes
1answer
219 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 ...
0
votes
1answer
238 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 ...
5
votes
2answers
207 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 ...
3
votes
2answers
614 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 ...
2
votes
2answers
560 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 ...
35
votes
1answer
2k views

Was Grothendieck familiar with Stone's work on Boolean algebras?

In short, my question is: Was Grothendieck familiar with Stone's work on Boolean algebras? Background: In an answer to Pierre-Yves Gaillard's question Did Zariski really define the Zariski ...
8
votes
1answer
323 views

Adjointness of Hom and Tensor

Could someone provide me a link to the proof of the adjointness of Hom and Tensor. I did an extensive google search but could not find anything self contained that presented the proof in full ...