2
votes
2answers
88 views

Is the categorical product for projective spaces essentially the tensor product?

I wonder whether the categorical product of two projective spaces is essentially given by the tensor product of the underlying vector spaces. Is this at least true for projective Hilbert spaces? One ...
4
votes
1answer
100 views

iterated dual vector spaces

Let $K$ be a field and $\mathcal U$ a universe such that $K\in\mathcal U$. (Here, "universe" means "uncountable Grothendieck universe".) Let $\mathcal C$ be the category of $K$-vector spaces belonging ...
0
votes
0answers
39 views

$\mathbf{Vec}_k$ is not autodual

I am looking to prove that the category of vector spaces is not autodual, i.e., equivalent to its opposite category, in the simplest way. Here are my ideas. We have an equivalence between ...
1
vote
0answers
38 views

Categorical formulation of linear transformations between vector spaces

My question regards the formulation of linear transformations between vector spaces as morphisms in an appropriate category. I know that any biproduct category admits a calculus of matrix. What I'm ...
3
votes
2answers
59 views

Does first isomorphism theorem hold in the category of normed linear spaces?

Consider the category of normed linear spaces over $\mathbb{C}$ with bounded linear maps as morphisms. If $M\subset X$ is a subspace, then the quotient space $X/M$ has a map $\|x+M\|: = \inf_{y\in ...
1
vote
2answers
48 views

Universal property of tensor products / Categorial expression of bilinearity

Let $V$ and $W$ be linear spaces. According to Wikipedia, the universal definition of the tensor product $V \otimes W$ satisfies the following property: There is a bilinear map (i.e., linear in each ...
29
votes
1answer
356 views

In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
6
votes
0answers
71 views

Lie algebra of Derivations as a functor?

To an associative algebra $A$ one can associate a Lie algebra $\operatorname{Der} A$ of all derivations $D:A\to A$. To any morphism of associative algebras $\alpha:A\to B$, how can one associate a ...
2
votes
1answer
45 views

Choice of the right isomorphisms

The question makes sense in every abelian category, but for the moment let's work in the category of vector spaces over a field. PS: I previously posted a similar question which didn't make a lot of ...
4
votes
1answer
120 views

Dual and adjoint operator

Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual ...
2
votes
2answers
72 views

Category equivalence of sets and vector spaces

It seems to me that for a field $K$, the functor $\mathbf{Set}\to\mathbf{Vec}_K$, sending a set to the free $K$-vector space on it, is a category equivalence. It is full and faithful because a linear ...
1
vote
0answers
83 views

Epimorphisms and monomorphisms in the category of linear maps

Let $V,W$ be fixed $k$-vector spaces. Let $C$ be the category whose objects are linear maps $f:V\to W$ and morphisms from $f$ to $g$ are pairs of linear maps $(\alpha,\beta)$ where $\alpha:V\to ...
2
votes
2answers
268 views

Injectivity of the dual map

Suppose V and W are vector spaces of possibly finite and infinite dimension over a field K. Show that if a linear map $L : V → W$ is surjective the its dual is injective. Also prove the converse of ...
1
vote
1answer
70 views

Pushouts and Pullbacks in Category Theory

How would one prove existence of pushouts and pullbacks where the objects are vector spaces and the morphisms are linear transformations?
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 \;\; ...
6
votes
2answers
299 views

Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
2
votes
1answer
105 views

Sequence of vectors spaces of linear transformations of vector spaces.

Let $V_0$ be a vector space over some field k. Then the set of linear transformations $V_1 = \{T:V_0\rightarrow V_0\mid T\text{ is linear}\}$ is a vector space. Now, let $V_{n+1}= \{T:V_n\rightarrow ...
4
votes
1answer
116 views

Pullbacks and transpose map

Given maifolds $M,N$ and a smooth map $\phi:M \to N$, and a smooth function $f:N \to \mathbb{R}$, we have the pullback of $\phi$ by $f$ to be the function $\phi^* f = f \circ \phi : M \to \mathbb{R}$. ...
10
votes
1answer
145 views

Other Euler characteristics?

At the end of V.3.4 in Algebra: Chapter 0, Aluffi describes the construction of a Grothendieck group over the category of finite dimensional $\operatorname{k}$-vector spaces, ...
3
votes
1answer
92 views

Connection between dual space V* and negation P^c

Notice the following similarity between the vector space dual and negation in propositional logic: $$ V^* \equiv V \rightarrow F $$ $$ P^c \equiv P \rightarrow \bot $$ Is there some general notion ...
3
votes
0answers
100 views

Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
9
votes
2answers
449 views

natural isomorphism in linear algebra

Let $\mathsf{C}$ and $\mathsf{D}$ two categories and $\mathcal F,\mathcal G$ two functors $\mathsf{C}\rightarrow\mathsf{D}$. A natural isomorphism from $\mathcal F$ to $\mathcal G$ is the datum ...
3
votes
1answer
179 views

Sending vector space to dual is a functor

In the category of finite dimensional vector spaces over a field and linear maps between them, the map that sends each space to its dual and linear map to its transpose is a functor, right? But this ...
2
votes
0answers
98 views

Involutive Functors

Let us call a (co)functor $F:\mathcal{C}\to\mathcal{C}$ on a category $\mathcal{C}$ involutive, if $F^2$ is naturally isomorphic to $1_\mathcal{C}$. For example, if ...
2
votes
1answer
545 views

When is the pullback of a linear injection a surjection on dual space?

Due to the contravariance of the dual space functor on vector spaces, one might expect the pullback of an injection to be a surjection, and the pullback of a surjection to be an injection. Indeed, for ...
21
votes
3answers
1k views

Natural and coordinate free definition for the Riemannian volume form?

In linear algebra and differential geometry, there are various structures which we calculate with in a basis or local coordinates, but which we would like to have a meaning which is basis independent ...