# Tagged Questions

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**1**answer

151 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**

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**0**answers

98 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**

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**1**answer

282 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 ...

**9**

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**1**answer

381 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**

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**0**answers

204 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 ...

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**0**answers

181 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**

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**1**answer

786 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 ...

**14**

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**1**answer

950 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 ...

**32**

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**15**answers

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 ...

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**0**answers

310 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**

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**3**answers

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 ...

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**1**answer

274 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**

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**1**answer

558 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**

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**2**answers

464 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 ...

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**2**answers

274 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**

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**1**answer

490 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 ...

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**3**answers

386 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 ...

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**2**answers

350 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**

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**1**answer

389 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**

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**1**answer

282 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 ...

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**1**answer

355 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 ...

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**1**answer

196 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 ...

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**4**answers

866 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, ...

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**2**answers

996 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 ...

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**3**answers

604 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 ...

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**1**answer

633 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 ...