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I have been thinking about methods for constructing continuous paths locally in a space. These paths have domain the unit interval and map into "small" neighborhoods of points in a space. Moreover their images are constrained in ways which are unimportant for this question.

If the space is first countable (points have countable bases for the topology) and locally path connected (lpc) then the desired constrained paths are easily constructed.

Here is my question.

Are the spaces which arise naturally in stable homotopy theory (e.g. infinite loop spaces, spaces arising in "brave new algebra") locally nice in this sense (i.e. first countable and lpc)?

If not, what are some natural examples which fail either to be first countable or locally path connected (or both)?

Edit: My failure to offer more motivation for my question has led the discussion off in an unexpected (to me) direction. I now see that the question itself may be wrong-headed. Still, let me fill in the background which led to my question.

I am not motivated here by any homotopy theoretic considerations per se. Instead, I am studying the properties of functors from categories of point-set topological spaces to categories like CAT, the category of small categories and functors.

The simplest example is the Moore-path functor. This assigns to a space the small category whose objects are the points of the space and whose morphisms are the parametrized continuous paths in the space. I am wondering on just what subcategories of spaces is this functor full (it is trivially faithful).

I think one can show that the Moore-path functor is full on the subcategory of spaces whose objects are first countable and locally path connected.

My question then comes down to asking whether or not there is a class of "natural models" of stable homotopy types which lives in this category or whether instead weaker local hypotheses on the spaces are called for.

My actual interest involves related higher categorical notions but this seemed like a good place to start.

Thanks to everyone who has tried to illuminate the situation for me.

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    $\begingroup$ You misunderstood me. I don't work with geometric realizations of simplicial sets. I work with Kan complexes as my model of spaces. I never "geometrically realize" them and I consider that a very counterintuitive operation. For any topological space $X$ you are interested in you can form $Sing_\bullet X$, which is a Kan complex and which I call the (weak) homotopy type of $X$. Anyway this is only one possible approach, there are others that use actual topological spaces. But no one works with "all possible topological spaces" as their model, you always restrict to some "nice" category $\endgroup$ – Denis Nardin Apr 25 '16 at 18:52
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    $\begingroup$ I guess that what I meant is that the answer to this question depends on which topologists you're talking to, and that it might range easily from "of course yes!" to "of course not!" to "your question is meaningless". Your question sounds like asking which model of set theory should we prefer while doing algebraic topology. As long as it does the job, I'm quite happy with any of them. $\endgroup$ – Denis Nardin Apr 25 '16 at 19:15
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    $\begingroup$ I definitely don't know everyone in algebraic topology, but it seems like pretty much everyone working in stable homotopy theory only works with simplicial sets or CW complexes. $\endgroup$ – Jonathan Beardsley Apr 25 '16 at 19:34
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    $\begingroup$ Denis, what never? Consistency is the hobgoblin ... $\endgroup$ – Peter May Apr 25 '16 at 19:34
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    $\begingroup$ @PeterMay Well, I was maybe a little too absolute. I never had to so far, but of course it's not like I would stubbornly refuse if it became necessary. I just wanted to emphasize that I don't identify a Kan complex with its geometric realization (rather the opposite, I like to identify a topological space with its singular complex) $\endgroup$ – Denis Nardin Apr 25 '16 at 20:32
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When a certain kind of homotopy theorist says "space," they don't mean a topological space, or even an object which in any sense has an underlying topological space. The simplest translation of what "space" means in this sense is "weak homotopy type," and it's not a meaningful question to ask whether a weak homotopy type is locally nice: it may be that there are various topological spaces with the same weak homotopy type, some of which are locally nice and some of which are not. Since every topological space is weakly homotopy equivalent to a CW complex this should always happen.

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  • $\begingroup$ Yes, thanks, but other kinds of stable homotopy theorists say "space" and sometimes mean just that - see my reference to S-modules above. So I think it is reasonable to ask whether there are examples of S-modules or of infinite loops spaces arising "in nature" which at the point set level are locally pathological. $\endgroup$ – Carl Futia Apr 25 '16 at 21:06
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    $\begingroup$ @CarlFutia Even what is a loop space depends on the model... Of course we mean spaces of the form $Map_*(S^1,X)$, but the topology on the mapping space depends on the category you're working in. You'd have to ask "are the underlying spaces of S-modules in the sense of EKMM locally path connected", painfully specifying all the premises. $\endgroup$ – Denis Nardin Apr 25 '16 at 21:12
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    $\begingroup$ @Carl: okay, but once you decide you don't want to work model-independently, you need to ask much more specific questions. For example, an orthogonal spectrum, at the point-set level, consists of a bunch of pointed spaces and data relating them. You could ask whether, in natural examples, these spaces are locally nice, but it's unclear if this is the right question. You could pick a forgetful functor to topological spaces, but the most obvious one (taking one of those pointed spaces) is not the model-independently most natural one; for example, as an orthogonal spectrum, the sphere spectrum... $\endgroup$ – Qiaochu Yuan Apr 25 '16 at 21:20
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    $\begingroup$ ...is given by the spheres $S^n$ together with some extra data, but none of these spheres constitutes the model-independent notion of underlying or zeroth space of the sphere spectrum, which is an infinite loop space. Even here, whether this infinite loop space has local pathologies depends on what model of it you take. "The sphere spectrum," in isolation, just doesn't refer to a topological space. $\endgroup$ – Qiaochu Yuan Apr 25 '16 at 21:21
  • $\begingroup$ Thanks to all for these helpful responses. I am getting the impression that pretty much anything can happen. I have also edited my original question so that you have a better idea about my reasons for asking it. $\endgroup$ – Carl Futia Apr 25 '16 at 22:29

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