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Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both categories are complete, cocomplete, and have a closed symmetric monoidal structure, given by the projective tensor product (see here). The forgetful functor $\mathsf{Ban}_1 \to \mathsf{NormVect}_1$ is continuous (but not cocontinuous), in fact has a left adjoint (which is symmetric monoidal), the Cauchy completion.

Question. Can you name a categorical property of $\mathsf{Ban}_1$ which is useful in practice, but which is not satisfied by $\mathsf{NormVect}_1$?

Background: There is a branch called categorical Banach space theory, and I really wonder why there one does not consider the larger category of all normed vector spaces somehow as a first approximation. In functional analysis it is well-known that (and why) Banach spaces are more useful than normed vector spaces. I would like to know if or why this is also true for the corresponding categories.

$^{\dagger}$ Notice that the subscript $1$ indicates that we restrict ourselves to short linear maps, which is quite important for having the mentioned categorical properties. For me, the moral of this choice is that if you use continuous linear maps, you don't take the whole structure of the objects into account, which tends to be bad.

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    $\begingroup$ Could you please give an introductory reference to " categorical Banach space theory"? $\endgroup$ – magma Apr 17 '15 at 18:06
  • $\begingroup$ Well, the functional analysis arguments which make Banach spaces preferable over NormedVectorSpaces can't be described categorically? If yes, then these are (some of) the special properties that differentiate the 2 categories. Perhaps you could exemplify some of these arguments and we can see if there is a way to express them categorically. $\endgroup$ – magma Apr 17 '15 at 18:13
  • $\begingroup$ A bijective continuous linear map between Banach spaces is an isomorphism. But I'm not sure what "bijective" means categorically, and if this is an interesting statement in $\mathsf{Ban}_1$. Notice that monomorphisms are the "obvious" ones, but epimorphisms are those continuous (resp. short) linear maps with dense image. A good reference for categorical Banach space theory is "Banach modules and functors on categories of Banach spaces" by by Cigler, Losert, Michor. $\endgroup$ – Martin Brandenburg Apr 17 '15 at 18:36
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    $\begingroup$ I can't resist mentioning Geoff Cruttwell's thesis. Just as Lawvere showed observed that metric spaces are categories enriched in the monoidal category $([0,\infty],\geq,+)$, Crutwell observes that a normed abelian group is a (discrete) compact closed category equipped with a monoidal functor to $([0,\infty],\geq,+)$. Being Banach means that the induced enriched category structure is Cauchy-complete, of course. In the conclusion, he has some musings about doing analysis with normed spaces using a double-categorical framework. $\endgroup$ – tcamps Apr 17 '15 at 19:29
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    $\begingroup$ I didn't know this thesis, thank you very much. $\endgroup$ – Martin Brandenburg Apr 17 '15 at 20:49
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The category of Banach spaces is locally $\aleph_1$-presentable.

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    $\begingroup$ Hm, isn't the category of normed vector spaces even locally finitely presentable? $\endgroup$ – Martin Brandenburg Apr 17 '15 at 11:41
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    $\begingroup$ I don't see why the category of normed vector spaces (or rather their unit balls) should be locally finitely presentable. There's no obvious essentially algebraic axiomatisation. $\endgroup$ – Zhen Lin Apr 17 '15 at 23:17
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    $\begingroup$ Well, I thought that $\mathbb{C}^{\oplus n}$ (caution: this differs from $\mathbb{C}^{\times n}$) is finitely presentable for $n \in \mathbb{N}$ and that the set of these objects is dense, but I will have to check this. I was quite sure because this seems to be used in the proof of the Eilenberg-Watts-Theorem for enriched functors between Banach modules. $\endgroup$ – Martin Brandenburg Apr 18 '15 at 10:28
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    $\begingroup$ Zhen Lin is right. And $\mathbb{C}$ is not finitely presentable. This is actually a nice exercise in going through the construction of directed colimits of normed vector spaces. But I am pretty sure that the category of normed vector spaces is locally $\aleph_1$-presentable, which means that the answer does not fit to the question. $\endgroup$ – Martin Brandenburg Apr 18 '15 at 17:36
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    $\begingroup$ @tcamps: Yes, I've also realized this. Usually these objects are called seminormed spaces, by the way. $\endgroup$ – Martin Brandenburg Apr 18 '15 at 23:15

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