Intereting Posts

calculate $\int_0^{\pi}\frac {x}{1+\cos^2x}dx$
Pointwise convergence of sequences of holomorphic functions to holomorphic functions
Who are the most inspiring communicators of math for a general audience?
A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic
Is the topology of the p-adic valuation to the unramfied extension discrete?
Two styles of semantics for a first-order language: what's to choose?
What can be said about the number of connected components of $G(n,p)$ random graphs?
Cofinality of $2^{\aleph_\omega}$
maximum eigenvalue of a diagonal plus rank-one matrix
Multiplicative Functions
Dulac's criterion and global stability connection
If $\left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right)$
GP 1.3.9(b) Every manifold is locally expressible as a graph.
basis for hermitian matrices
How would you find the exact roots of $y=x^3+x^2-2x-1$?

Suppose $V_\mathbb{F_V}$ and $W_\mathbb{F_W}$ are two vector spaces over fields $\mathbb{F}_V$ and $\mathbb{F}_W$. Then a homomorphism of these vector spaces consists of maps $f:V\rightarrow W$ and $f_\mathbb{F}:\mathbb{F}_V\rightarrow \mathbb{F}_W$ satisfying:

$$

f\left(a.v+b.u\right)= f_{\mathbb{F}}\left(a\right)f\left(v\right)+f_{\mathbb{F}}\left(b\right)f\left(u\right)

$$

for all $a,b \in \mathbb{F}_V$ and $v,u \in V$.

With such morphisms we can talk about the category of all vector spaces over arbitrary fields. But, I have never seen such examples. Why is that? Is it because the category of fields is not very *welcoming* of a place.

- dimension of a subspace spanned by two subspaces
- Linear independence of images by $A$ of vectors whose span trivially intersects $\ker(A)$
- What are the important properties that categories are really abstracting?
- Question about basis and finite dimensional vector space
- Utility of the 2-Categorical Structure of $\mathsf{Top}$?
- Dependency of linear map definition on basis
- Equivalence of categories ($c^*$ algebras <-> topological spaces)
- Computing the dimension of a vector space of matrices that commute with a given matrix B,
- Dot Product Intuition
- General Steinitz exchange lemma

There is indeed a category of *all* vector spaces with morphisms as you describe. It has many interesting properties – first of all, notice that it comes equipped with a projection to the category of all fields, $p : \textbf{Vect} \to \textbf{Fld}$. Let $\textbf{Vect}(K)$ be the non-full subcategory of $\textbf{Vect}$ of objects $V$ such that $p V = K$ and morphisms $f$ such that $p f = \textrm{id}_K$. This is easily seen to be isomorphic to the usual category of $K$-vector spaces. Given any field homomorphism $\phi : K \to L$, we get a functor $\phi^\sharp : \textbf{Vect}(L) \to \textbf{Vect}(K)$, and it is not hard to check that the operation $(-)^\sharp$ is strictly functorial. The category $\textbf{Vect}$ is then seen to be the Grothendieck construction applied to $(-)^\sharp$, and therefore $p : \textbf{Vect} \to \textbf{Fld}$ is a Grothendieck fibration.

Why is this interesting? Well, it gives us a way to consider vector spaces over all fields on equal grounds, and the universal property of some familiar constructions is best expressed in terms of this Grothendieck fibration. For example, if $W$ is a $L$-vector space, then $\phi^\sharp W$ is a $K$-vector space $V$ and a morphism $f : V \to W$ lying over $\phi : K \to L$ such that for all morphisms $h : U \to W$ in $\textbf{Vect}$ lying over $\chi : F \to L$ in $\textbf{Fld}$ and all factorisations $\chi = \phi \circ \psi$, there is a unique morphism $g : U \to W$ lying over $\psi : F \to K$ such that $h = g \circ f$. If you draw the diagram you will see this is basically the universal property of a pullback, but two different categories are involved here.

On the other hand, $\phi^\sharp : \textbf{Vect}(L) \to \textbf{Vect}(K)$ has a well-known left adjoint $\phi_\sharp : \textbf{Vect}(K) \to \textbf{Vect}(L)$, namely the tensor product $\phi_\sharp V = L \otimes_K V$. This makes $p : \textbf{Vect} \to \textbf{Fld}$ into a Grothendieck bifibration, and again this means $\phi_\sharp V$ can be described in terms of a universal property.

You are quite right that $\textbf{Fld}$ isn’t a category with particularly good properties – and unfortunately that means $\textbf{Vect}$ also lacks the same properties. For example, there is no terminal object in either $\textbf{Fld}$ or $\textbf{Vect}$. In this respect, the category $\textbf{Mod}$ of *all* modules over *all* commutative rings is more well-behaved. $\textbf{Mod}$ has some remarkable properties – in addition to being a Grothendieck bifibration, it (or rather $\textbf{Mod}^\textrm{op}$) is what is known as a stack for the faithfully flat topology on $\textbf{CRing}^\textrm{op}$. This is studied at length in SGA1 and is the motivating example behind the whole theory of fibred categories in general.

Your conditions imply that $F_V$ can be viewed as a subfield of $F_W$ and via that $W$ is a $F_V$-vector space and $f$ is $F_V$-linear.

For example nothing prevents you from considering the linear maps from the real vector space $C([0,1])$ of continuous function on the interval $[0,1]$ to the complex vector space $\mathbb C^{42}$. But what you get is the same as if you considered $\mathbb C^{42}$ just as $\mathbb R^{84}$.

Well, you will be able to talk about morphisms between vector spaces over different fields as long as you think up a category in which those vector spaces coexist as objects 🙂

There is something similar (and somewhat important) for semisimple rings. I’m doing this from a memory from Jacobson’s BA2 text, so I dearly hope I’m not too far off of the correct statement.

I think the question is: For finite dimensional vector spaces $V_F$ and $W_{F’}$, if $End(V_F)\cong End(W_{F’})$ as rings, what can you conclude about the relationship of $V_F$ to $W_{F’}$?

The answer is that $V$ and $W$ are semilinearly isomorphic, which is a generalization of a linear isomorphism. You might be interested in semilinear transformations 🙂

- Binomial expansion of $(1-x)^n$
- Assume $P$ is a prime ideal s.t. $K \subset P$, show $f(P)$ is a prime ideal
- Is this set compact?
- Check if a point is within an ellipse
- $A\subseteq B\subseteq C$ ring extensions, $A\subseteq C$ finite/finitely-generated $\Rightarrow$ $A\subseteq B$ finite/finitely-generated?
- Matrix characterization of surjective and injective linear functions
- An integral to prove that $\log(2n+1) \ge H_n$
- Definition of independence of infinite random variables
- Poincaré Inequality
- What is so special about triangles?!
- I have simple multiplication confusion, any help would be great
- Real Analysis: open and closed sets
- Multiplication by $m$ isogenies of elliptic curves in characteristic $p$
- Help with $\lim_{x, y\to(0, 0)} \frac{x^2y}{x^4+y^2}?$
- Extending partial sums of the Taylor series of $e^x$ to a smooth function on $\mathbb{R}^2$?