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Started bornological set. Some people call it a bornological space, but that conflicts with the terminology in functional analysis which refers to a locally convex TVS with a suitable “bounded = continuous” property. I quickly wrote that uniformly continuous maps between metric spaces induce bounded maps, but I’ll recheck when I have a free moment.
I have added pointers to
and
and created a stub for bornological group. Added the statements proven in in these articles that complex bornological vector spaces and also bornological abelian groups form a quasi-abelian category (and added this as examples there).
Lawvere has spoken often about bornology, though I didn’t really get his point. I see I mentioned something of this here.
In Volterra’s functionals and covariant cohesion of space, he writes
Grothendieck remarked to me in 1981 that already around 1950 he had had great difficulty in persuading analysts that there can be more than one natural topology for the same family of linear spaces. This remark in effect presupposes that there is some more basic cohesive structure which can be the ‘same’ for various notions of neighborhood. Perhaps this situation could be schematically represented as [diagram], where the family F of basically-cohesive linear spaces is functorially parameterized by a category C of problems, and we consider liftings across the forgetful functor from basically-cohesive spaces which are moreover compatibly equipped with a given topology.
But what could such a more basic notion of cohesiveness amount to? A very strong candidate is bornology.
Yakov Kremnitzer advocates the point of view that the good category to look at for all purposes of analysis is $Ind(Ban)$ (ind-objects in Banach spaces). Everything else should nicely sit in there, in particular bornological spaces.
Then for analysis generally, the corresponding category would seem to be $Ind(Met)$ (or perhaps $Ind(Q Met)$ for full generality). This surprises me, since I know that a lot of things can be done with (quasi)-gauge spaces, which generalize to prometric spaces, but these are pro-objects rather than ind-objects.
I shouldn’t maybe be talking about this at this point. But since we got this far now I need to add:
the suggestion is that $Comm(Ind(Ban))$ is the good context for analytic rings and then $(Comm(Ind(Ban)))^{op}$ the good site for analytic spaces. So this does match your pro-space feelings.
Is terminology standardised in this area? Bornological set has the category of bornological sets as a quasitopos, where Lawvere speaks in
of
…the topos of bornological sets (in which linear algebra becomes functional analysis).
and in
of
my bornological topos (sheaves for ﬁnite coverings of countable sets)
and in
of
bornology in its own sequential version has a very simple topos incarnation which could be considered parallel to Johnstone’s sequential-convergence topos.
I don’t know how standard the terminology is. What is in the nLab as bornological set is in some contexts better known as a bornological space, although that term could easily be confused with bornological topological vector space. I have a vague memory that this naming issue was discussed here in the nForum.
I should probably take another look to see how this quasitopos compares with Lawvere’s topos. Maybe the two are closely connected.
Does Lawvere ever explain his construction? At CT2010, there was a talk by Montañez Puentes:
Starting from the space of the natural numbers $\mathbb{N}$ with the discrete bornology and imposing the extensive topology on its endomorphism monoid, we determined the Bornological topos that coincides with the proposed by F. W. Lawvere during an on published conference, in Bogota in 1983 and studied subsequently by L. Español, C. Minguez and L. Lamban professors of the Universidad de la Rioja (Spain ).
Espanol had given a talk three years earlier defining the bornological topos as sheaves over a site.
@Urs #6: What does $Comm$ mean here?
@Toby,
“$Comm$” is meant to stand for the category of commutative monoids.
My apologies, I realize that I made comments above without making the context from which i am speaking clear.
So I have been chatting with people about what the right perspective on global analytic geometry would be, which is in parts about the question which analytic constructions one wants to take as fundamental, which as derived.
There is the feeling that the existing definitions of rigid analytic space and Berkovich space and Huber space etc. are good in themselves, but should eventually be subsumed in something more general and maybe more natural. More general abstract.
In this spirit for instance Bambozzi 14 “On a generalization of affinoid varieties” proposes to base the theory on bornological algebras.
On the other hand, Ben-Bassat & Kremnitzer 13 had observed that traditional rigid/Berkovich/Huber analytic geometry is all neatly realized inside a topos over $(Comm(Ban))^{op}$ in the style of generalized algebraic geometry based on symmetric monoidal categories, here that of Banach spaces.
An evident question then was: to get a more general picture, should one consider $Comm(Born)^{op}$ instead? And Yakov Kremnitzer’s suggestion – which I had tried to refer to, but clearly very insufficiently so – is that: no, the right perspective should be to consider $Comm(Ind(Ban))^{op}$.
@David#9: the richest discussion of bornology in Lawvere is probably in his 1989 treasure box ’Qualitative Distinctions between some toposes of generalized graphs’ -paper. I think he mentions his and the Johnstone topos as well in the last section of the 2008 Como lectures albeit briefly.
Ah, I see, so $Comm(Ind(Ban))^op$ is a category of spaces much as $Comm(Ab)^op$ is (being the category of affine schemes). I'm going to have to think about this to see how this all works out.
@Thomas#12: Is that paper only available through the book ’Categories in computer science and logic’? My institution does give me access to it.
Re #11, it seems there’s a relationship between $Ban$ and bornological spaces
Every bornological space $E$ is the inductive limit of a family of normed spaces (and Banach spaces if $E$ is quasi-complete).
And from here
Proposition 2.3. An inductive limit of Banach spaces (and in particular a strong algebra) is bornological and barrelled.
Yes, that’s the statement that one should consider $Ind(Ban)$.
Of course this suggestion to use $Ind(Ban)$ is somewhat reminiscent of the category of LF spaces.
Does LF stand for limit of Fréchet?
I see
An LF space is barrelled and bornological (and thus ultrabornological).
generalizing #16, since Fréchet generalizes Banach.
What a complicated set of concepts!
And from the Lawvere paper (#14) that Thomas kindly sent me
the category ab(bor) of bornological abelian groups contains the usual categories of Fréchet nuclear spaces, Banach spaces, etc., with continous maps as full subcategories.
What a complicated set of concepts!
Yes!
Would be nice to have a cleanup. The claim seems to be that everything is nicely cleaned up by organizing it inside $Ind(Ban)$ and more generally $Comm(Ind(Ban))^{op}$.
So what’s this ’nuclear’ property? From Wikipedia, it seems ’orthogonal’ to Banach:
There are no Banach spaces that are nuclear, except for the finite-dimensional ones. In practice a sort of converse to this is often true: if a “naturally occurring” topological vector space is not a Banach space, then there is a good chance that it is nuclear.
That was Grothendieck’s main area before algebraic geometry:
Produits tensoriels topologiques et espaces nucléaires.
Yes, LF means an inductive limit of Frechet spaces as computed in locally convex TVS. The inductive limits might be over countable diagrams; I’m not sure what is absolutely standard in this area. A somewhat typical example is smooth functions with compact support defined over an open set of Euclidean space, and there are many others.
TVS theory is, I believe, notoriously complicated (a mathematical menagerie if you will). I’m all for a clean-up, as long as there are good solid supporting theorems in place.
Where’s Andrew Stacey in this discussion? :-)
Minor additions:
’Anonymous’ has posted a query box after
X is any topological space, there is a bornology consisting of all precompact subsets of X (subsets whose closure is compact). Any continuous map is bounded with respect to this choice of bornology.
there is now
+– {: .query} This can’t be right. Consider the left order topology on the reals. This has no nonempty compact closed sets. =–
It would have been less fuss just to add the obvious missing condition: that the space should be $T_1$. I have added that and removed the query box (bornological set).
Re #6, I see that the $Comm(Ind(Ban))$ approach appeared in
and in a talk
Is there a suitable page to add these?
And how does this perspective relate to the perfectoid geometry of Scholze et al?
Same question as David, and unfortunately I’m too much of a noob in both fields to ask anything more specific: how does this relate to Perfectoid geometry?
(I see in the notes for this talk that there is a version of the fargues-fontaine curve in global analytic geometry http://www-personal.umich.edu/~snkitche/Conference/notes/Kremnitzer-2.pdf + audio… so I’m going to try to understand that and the perfectoid geometry fargues fontaine curve to get an idea of how these two fields are comparable/different)
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