2
votes
0answers
159 views

Banach space interpolation theory in terms of categories

I have recently learned a bit about higher category theory. And there is a question I would like to ask. Can we enrich banach space interpolation theory by using higher category theory? Is it ...
1
vote
1answer
215 views

When are completely positive maps monic/epic?

In the category of C*-algebras and $*$-homomorphisms, a morphism is monic precisely when it is injective, and epic precisely when it is surjective (see Mono- and epi-morphisms for C*-algebras). Is ...
2
votes
0answers
103 views

Projective tensor powers of Banach spaces over a normed field

Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
2
votes
1answer
287 views

The point of view of semicats in functional analysis

I'm completing a paper about (Mitchell's) semicats (well, not exactly, but let's say so for simplicity), and as a motivational example I'd like to mention at some point that the monic/epic morphisms ...
11
votes
1answer
447 views

bornological vector spaces over a non-archimedean field

Let $k$ be a complete non-archimedean field. In definitions I have seen of bornological vector spaces over $k$ there are usually some extra assumptions on the non-archimedean field. For instance in ...
4
votes
0answers
207 views

Terminology for notion dual to “support”

If $X$ is a set (feel free to think of it as finite, but it doesn't have to be) and $f$ a real function on $X$, call the support $\operatorname{supp} f$ the subset of $X$ consisting of all elements ...
5
votes
0answers
183 views

Diffusion processes in wide generality

It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality. Hard question: What are the most general structures on which one may define something ...
15
votes
1answer
798 views

Convolution algebras for double groupoids?

There is a lot of work of course on convolution algebras of measured groupoids, and this gives "Noncommutative geometry". However there is a lot of interest in algebraically structured groupoids, for ...
16
votes
1answer
1k views

Theme of Isbell duality

Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop ...
33
votes
15answers
6k views

Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.] I've read looked at the examples in ...
2
votes
0answers
313 views

Quantum sheaves

Are the following definitions known? Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions: (a) {0} and H lie in Sigma (b) If ...
10
votes
3answers
2k views

What is an isomorphism of Banach spaces?

The nLab page on Banach spaces (http://ncatlab.org/nlab/show/Banach%20space) was recently criticised as being, in effect, too heavily biased to category theory (not of the Baire kind) and not enough ...
0
votes
1answer
292 views

products in the category of banach spaces

Let $\{X_{\alpha} \}_{\alpha \in A}$ be a collection of Banach spaces. It is easy to show that $ P = \{(x_{\alpha}) : {\rm sup}_{\alpha} \|x_{\alpha} \| < \infty \} $ with $\| (x_{\alpha} ) \| = ...
5
votes
1answer
561 views

Short five lemma in Banach spaces

Denote by $\mathbf{Ban}$ the category of Banach spaces and bounded linear maps and by $\mathbf{Banc}$ the subcategory of Banach spaces and linear contractions. The isomorphisms of $\mathbf{Ban}$ are ...
3
votes
2answers
492 views

Projective Banach spaces

Injective Banach spaces, with morphisms as contractive linear maps, have been classically studied (and are $C(K)$ spaces with $K$ Stonian). But what about projectives? So $P$ will be projective if ...
4
votes
2answers
276 views

Looking for substitutes for co-free modules in a topological setting

I should say that I'm not a category theorist or an abstract algebraist, so maybe this will be very pedestrian. I have the following, somewhat vague question: I have categories C and D, a ...
5
votes
1answer
495 views

Looking for references talking about category of topological vector spaces

It's known that category of topological vector spaces is not abelian but quasi-abelian or exact category. I am looking for the references playing with this category(category theory). All the related ...
7
votes
3answers
391 views

Compact Hausdorff and C^*-algebra “objects” in a category.

This is yet more on "algebraic objects in functional analysis". Since Compact Hausdorff spaces are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable ...
3
votes
2answers
354 views

Which Banach spaces have categorical duals?

I was looking carefully at all the definitions, trying to understand exactly what was going on in this question on categorical duals in Banach spaces. It seems that in the category of Banach spaces ...
6
votes
1answer
396 views

Categorical duals in Banach spaces

Near the bottom of the nlab page for Banach space I see "To be described: duals (p+q=pq)". Are $(\mathbb{R}^n)_p$ and $(\mathbb{R}^n)_q$ dual objects in the closed symmetric monoidal category of ...
3
votes
1answer
291 views

Request for reference: Banach-type spaces as algebraic theories.

Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight ...
8
votes
1answer
366 views

What's the nearest algebraic theory to inner product spaces?

Following the references to the accepted answer to Is the category of Banach spaces with contractions an algebraic theory? one discovers that there is an algebraic theory (infinitary) which is closely ...
4
votes
1answer
201 views

What functorial topologies are there on the space of linear maps between LCTVS?

Setup: we consider the category of locally convex topological vector spaces with morphisms as continuous linear maps. This time, I'm explicitly allowing the axiom of choice (or at least the ...
8
votes
4answers
900 views

Is the category of Banach spaces with contractions an algebraic theory?

Consider the category of Banach spaces with contractions as morphisms (weak, so $\|T\| \le 1$). Is this an algebraic theory? I suspect that this is true. The "operations" will be weighted sums, ...
12
votes
2answers
1k views

What is a projective space?

Is there a "recognition principle" for projective spaces? What categories are there with projective spaces for objects? Background: Although the title is a nod to What is a metric space?, this ...
12
votes
3answers
637 views

What are the right categories of finite-dimensional Banach spaces?

This is inspired partly by this question, especially Tom Leinster's answer. Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...
11
votes
1answer
642 views

Gelfand-Naimark from the category-theoretic point of view

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative C* algebra (with unit) A and the C* algebra of continuous complex-valued functions on its ...