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• CommentRowNumber1.
• CommentAuthorTim_Porter
• CommentTimeFeb 13th 2012
• (edited Feb 13th 2012)

I mentioned (in another thread) the school this summer on mathematics and philosophy and, for my sins, I have been asked to give some talks. I suggested the title ’Where do spaces come from?’ or similar and now have the task of trying to organise my thoughts on the matter beyond the point that I got to in the answer I gave in MO. At the risk of you lot ’disorganising’ my thoughts (such as they are) I thought ’Why not ask the others what they think?’ I think we can be fairly free as to the meaning of space and can interpret the question in several different ways, but that is the fun of it. (I hope David C. is surveying the nForum from time to time and can give some views form a better informed viewpoint than mine.)

The point of the question is serious. We have various models for homotopy types, and I wonder if the notion of space is not partially a means of organising such data into a nice neat object. (One extreme case may be the use of topological techniques in topological data analysis., for extracting sense from a data cloud.) However we also have non compact spaces, (and proper homotopy theory) and the spaces studied by shape theory, and the $\infty$-cat models are less good for them.

So let me ’declare the session open’ and thank people in advance.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 13th 2012

Just for the record: a pretty good answer to the question of that MO thread

Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?

is given in Lurie’s Structured Spaces.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeFeb 13th 2012
• (edited Feb 13th 2012)

I wish the discussion would extend to noncommutative algebras, where the aspects of spectra which as usual topological spaces usually give too meager picture.

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeFeb 13th 2012

One of the points that I would hope to make is that ‘space’ interpreted as CW-complex, or compact Hausdorff space, etc. is probably inadequate. For instance, a dyadic Chu space gives a simplicial complex or a T_0-space but why dyadic (i’e’ two-valued) I am not advocating fuzzy or something like that but querying if our attitude to space is ’fit for purpose’… which means that the purpose of spaces may have to be looked at. So yes non-commutative algebras etc should be around.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 13th 2012

One of the points that I would hope to make is that ‘space’ interpreted as CW-complex, or compact Hausdorff space, etc. is probably inadequate.

In general there is an underlying locale, or more generally an underlying topos. This is really very nicely discussed in Structured Spaces.

• CommentRowNumber6.
• CommentAuthorTim_Porter
• CommentTimeFeb 13th 2012
• (edited Feb 13th 2012)

I agree that Lurie gives a nice account of structured spaces, but that does not really answer my question. Where do spaces come from?

In many article that treat ’spaces’ there is some variant of ’Let $X$ be a topological space, locale, …. or what ever’. I was once asked (many years ago when I was doing shape theory), why was I studying compact metric spaces, since the only interesting spaces had the homotopy type of a CW-complex. In the MO answer, I asked ’ when someone starts a theorem with ’Given a space X…’, how is the space ’given’? ’ There is a mismatch (partially handled by Lurie’s approach but not completely if my feelings are correct) between the methods available for studying a ’space’ which from my viewpoint are those of algebraic topology and are optimised for use with CW-complexes (or dually CW-complexes are optimised for use with those methods) and the spaces that come up elsewhere, (general compact spaces, ‘non-commutative’ spaces of various types, quantales, directed spaces, spaces given as subsets of a high dimensional ${ℂ}^{n}$, spectra of operator algebras, spaces specified by a dynamical system (attractors), etc.)

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 13th 2012
• (edited Feb 13th 2012)

Where do spaces come from?

We had a lengthy discussion of this just recently.

Especially here, where we need a notion of space more general than locale/topological space I feel that there is a neat answer provided by topos theory.

But opinions differ.

• CommentRowNumber8.
• CommentAuthorTim_Porter
• CommentTimeFeb 13th 2012
• (edited Feb 13th 2012)

Ah! but that assumes that spaces are to be topological and I am not really assuming that. The question of observations was raised in that answer and then one has the reply that spaces help one to organise observations. I quite like that as an answer, but then I ask myself about spaces specified by a dynamical system or similar.Does the observational aspect ’explain’ them?

TO put it another way, if the $\infty$-topos theoretic approach is somehow optimal, then more or less all of these other situations should have a (useful?) encoding into that language. I am not conversant enough with that theory to see if that is the case.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeFeb 13th 2012

Ah! but that assumes that spaces are to be topological and I am not really assuming that.

No, not for the approach that I have been advocating there.

then I ask myself about spaces specified by a dynamical system or similar.

Which concept exactly do you have in mind here?

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeFeb 13th 2012

It might be worth stressing that (as you know) “space” has a very different meaning when it is used “up to homeomorphism” versus “up to homotopy”. When algebraic topologists say “space” they usually mean “homotopy type” or $\infty$-groupoid, and it’s for that purpose that CW-complexes are well-adapted. The relationship between these two meanings of “space” is one place that I think higher topos theory (building on shape theory) does really help to clarify; I wrote about it here. (Which is not, of course, to assert that every notion of “space up to homeomorphism” can be modeled by an $\left(\infty ,1\right)$-topos.)

From that point of view, it’s almost an accident that every $\infty$-groupoid is ${\Pi }_{\infty }$ of some topological space (or 1-topos), and that by restricting the spaces to CW-complexes we can compute effectively with $\infty$-groupoids presented in this way. In fact, our ability to present $\infty$-groupoids by topological spaces is a “classicality” property that will fail in other contexts, such as homotopy type theory or when internalized in an ambient higher topos that is nonclassical.

• CommentRowNumber11.
• CommentAuthorTim_Porter
• CommentTimeFeb 13th 2012

Thanks I think this is clarifying my thoughts a bit.

My thought on dynamical systems, or for that matter with regards to the end spaces of non-compact manifolds is that there should be some, I will say $\infty$-topos model for the dynamical system (say a flow on a manifold) which would encode the limiting behaviour or for the proper homotopy theory the homotopy type at the end. This might be related to some generalisation of the Edwards-Hastings theory that DEFINED strong shape by a sort of duality. If we now have a $\infty$-topos theory based strong shape theory then is there some duality giving proper homotopy theory. (Zoran knows this stuff well and hopefully has some thoughts on it???)

There are results using shape theoretic methods in dynamical system theory but also the fact that strange attractors are a long way away from the type of behaviour that traditional alg. top can handle. TO be more precise I would need to do some ferreting in the literature, so that would be for later. The idea is to have a nice space with a process on it and to use a model of the process to (eventually) get information on the limit set of the process.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeFeb 13th 2012

Hi Tim,

so you are after notions of space that have a notion of eventual image, is that the idea?

I haven’t been thinking about that. Does Topos have eventual images?

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeFeb 13th 2012

I have forwarded that question to the $n$-Café here.

• CommentRowNumber14.
• CommentAuthorTim_Porter
• CommentTimeFeb 13th 2012

I am not sure if the eventual image idea works, but that is an interesting idea to throw to the mob! One set of examples come from the solenoids. these occur in may contexts, Dynamical systems as such, Fractal geometry, and of course, shape theory. They are the limit/ intersection of some neighbourhoods in the solid torus, and are the limits of a flow (I think). We should be able to analyse their structure from some idea of their definition. (Shape theory gives a complete classification up to homeomorphism.)

• CommentRowNumber15.
• CommentAuthorzskoda
• CommentTimeFeb 14th 2012

There is an active thread at MathOverflow on sheaf characterizations of a manifold. One of the quite sensitive and knowledgeable answers however ends with words

However, I think that sheaf theory and locally ringed spaces are the wrong software for differential geometry and differential topology.

I have the feeling that such attitudes dwell just on the beginnings of the story, while the true power is much later, in allowing for advanced points of view and techniques, e.g. access to the tools of derived geometry. Any thoughts ?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeFeb 14th 2012

Hi Zoran,

thanks for pointing this out. I would have missed this otherwise. Yes, I agree with your sentiment.

We have discussion of this question at smooth manifold in this section.