In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?

Question

In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra.

He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie de mon énergie a été consacrée à un travail de réflexion sur les fondements de l'algèbre (co)homologique non commutative, ou ce qui revient au même, finalement, de l'algèbre homotopique." (Beginning of section 7. English version here: "Since the month of March last year, so nearly a year ago, the greater part of my energy has been devoted to a work of reflection on the foundations of non-commutative (co)homological algebra, or what is the same, after all, of homotopic[al] algebra.)

In a letter to Thomason written in 1991, he states: "[P]our moi le “paradis originel” pour l’algèbre topologique n’est nullement la sempiternelle catégorie ∆∧ semi-simpliciale, si utile soit-elle, et encore moins celle des espaces topologiques (qui l’une et l’autre s’envoient dans la 2-catégorie des topos, qui en est comme une enveloppe commune), mais bien la catégorie Cat des petites caégories, vue avec un œil de géomètre par l’ensemble d’intuitions, étonnamment riche, provenant des topos." [EDIT 1: Terrible attempt of translation, otherwise some people might miss the reason why I have asked this question: "To me, the "original paradise" for topological algebra is by no means the never-ending semi-simplicial category ∆∧ [he means the simplex category], for all its usefulness, and even less is it the category of topological spaces (both of them imbedded in the 2-category of toposes, which is a kind of common enveloppe for them). It is the category of small categories Cat indeed, seen through the eyes of a geometer with the set of intuitions, surprisingly rich, arising from toposes."]

If $Hot$ stands for the classical homotopy category, then we can see $Hot$ as the localization of $Cat$ with respect to functors of which the topological realization of the nerve is a homotopy equivalence (or equivalently a topological weak equivalence). This definition of $Hot$ still makes use of topological spaces. However, topological spaces are in fact not necessary to define $Hot$. Grothendieck defines a basic localizer as a $W \subseteq Fl(Cat)$ satisfying the following properties: $W$ is weakly saturated; if a small category $A$ has a terminal object, then $A \to e$ is in $W$ (where $e$ stands for the trivial category); and the relative version of Quillen Theorem A holds. This notion is clearly stable by intersection, and Grothendieck conjectured that classical weak equivalences of $Cat$ form the smallest basic localizer. This was proved by Cisinski in his thesis, so that we end up with a categorical definition of the homotopy category $Hot$ without having mentionned topological spaces. (Neither have we made use of simplicial sets.)

I personnally found what Grothendieck wrote on the subject quite convincing, but of course it is a rather radical change of viewpoint regarding the foundations of homotopical algebra.

A related fact is that Grothendieck writes in "Esquisse d'un programme" that "la "topologie générale" a été développée (dans les années trente et quarante) par des analystes et pour les besoins de l'analyse, non pour les besoins de la topologie proprement dite, c'est à dire l'étude des propriétés topologiques de formes géométriques diverses". ("[G]eneral topology” was developed (during the thirties and forties) by analysts and in order to meet the needs of analysis, not for topology per se, i.e. the study of the topological properties of the various geometrical shapes." See the link above.) This sentence has already been alluded to on MO, for instance in Allen Knutson's answer there or Kevin Lin's comment there.

So much for the personal background of this question.

It is not new that $Top$, the category of all topological spaces and continuous functions, does not possess all the desirable properties from the geometric and homotopical viewpoint. For instance, there are many situations in which it is necessary to restrict oneself to some subcategory of $Top$. I expect there are many more instances of "failures" of $Top$ from the homotopical viewpoint than the few I know of, and I would like to have a list of such "failures", from elementary ones to deeper or less-known ones. I do not give any example myself on purpose, but I hope the question as stated below is clear enough. Here it is, then:

In which situations is it noticeable that $Top$ (the category of general topological spaces and continuous maps) is not adapted to geometric or homotopical needs? Which facts "should be true" but are not? And what do people usually do when encountering such situations?

As usual, please post only one answer per post so as to allow people to upvote or downvote single answers.

P.S. I would like to make sure that nobody interpret this question as "why should we get rid of topological spaces". This, of course, is not what I have in mind!

Answer 1

The smash product of pointed topological spaces is not associative, i.e., $(X \wedge Y)\wedge Z$ need not be homeomorphic to $X \wedge (Y \wedge Z)$. (It fails, for example, for $X = Y = \mathbb{Q}$ and $Z = \mathbb{N}$.)

Answer 2

The geometric realisation functor (read: homotopy colimit for nice situations) from simplicial spaces to $Top$ preserves pullbacks only when you take the $k$-ification of the product in $Top$, or work with compactly generated spaces (Edit: or a convenient category of spaces). This is false in the category of all spaces with the ordinary product.

See e.g. here in the nLab.

Answer 3

My answer is in agreement with Grothendieck that topological spaces may be seen as inadequate for many geometric, and in particular, homotopical purposes. Round about 1970, I spent 9 years trying to generalise the fundamental groupoid of a topological space to dimension 2, using a notion of double groupoid to reflect the idea of ``algebraic inverse to subdivision'' and in the hope of proving a 2-dimensional van Kampen type theorem. In discussion with Philip Higgins in 1974 we agreed that:

1) Whitehead's theorem on free crossed modules, that $\pi_2(X \cup \{e^2_\lambda\},X,x)$ was a free crossed $\pi_1(X,x)$-module, was an instance of a 2-dimensional universal property in homotopy theory.

2) If our proposed theories were to be any good, then Whitehead's theorem should be a corollary.

However we observed that Whitehead's theorem was about relative homotopy groups. So we tried to define a homotopy double groupoid of a pair of pointed spaces, mapping a square into $X$ in which the edges go to $A$ and the vertices to the base point, and taking homotopy classes of such maps. This worked like a dream, and we were able to formulate and prove our theorem, published after some delays (and in the teeth of opposition!) in 1978.

We could then see how to generalise this to filtered spaces, but the proofs needed new ideas, and were published in 1981; this and subsequent work has evolved into the book ``Nonabelian algebraic topology'' published last August.

Contact with Loday who had defined a special kind of $(n+1)$-fold groupoid for an $n$-cube of spaces led to a more powerful van Kampen Theorem, with a totally different type of proof, published jointly in 1987. This allows for calculations of some homotopy $n$-types, and has as a Corollary an $n$-ad connectivity theorem, with a calculation of the critical (nonabelian!) $n$-ad homotopy group, as has been made more explicit by Ellis and Steiner, using the notion of a crossed $n$-cube of groups.

Thus we could get useful strict homotopy multiple groupoids for kinds of structured spaces, allowing calculations not previously possible.

In this way, Grothendieck's view is verified that as spaces with some kind of structure arise naturally in geometric situations, there should be advantages if the algebraic methods take proper cognisance of this structure from the start. That is, one should consider the data which define the space of interest.