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The following quote is found in the (~1969) book of Saunders MacLane, "Categories for the working mathematician" "All told, this suggests that in Top we have been studying the wrong mathematical objects.The right ones are the spaces in CGHaus." CGHaus is the category of compactly generated Hausdorff spaces. It is advocated that it is a better category than Top because it is cartesian closed.

Almost 50 years later, what is the general feeling on that point? Is CGHaus the "right" topology to study algebraic topology (which is what MacLane is interested in)? Is there a better choice? Or no consensus on the question? Is CGHaus used in other fields of mathematics?

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    $\begingroup$ I think that for the purposes of homotopy theory every convenient category of topological spaces (in the sense of Steenrod) is acceptable (see ncatlab.org/nlab/show/convenient+category+of+topological+spaces for a thorough discussion of the issue). These technicalities are often swept under the rug and considered unimportant (for example many topics in homotopy theory can be done as easily using Kan complexes instead of topological spaces, obtaining equivalent theorems) $\endgroup$
    – Denis Nardin
    Commented Aug 30, 2016 at 14:16
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    $\begingroup$ As I understand it, in homotopy type theory space (synonymous with $\infty$-groupoid) is a primitive notion. Perhaps if these foundations are widely accepted then one will be able to work "invariantly" or "model-independently" in a perfectly rigorous manner. As Denis Nardin points out, this is already common practice. $\endgroup$
    – Justin Campbell
    Commented Aug 30, 2016 at 16:02
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    $\begingroup$ I feel like 'spaces' arise for two completely distinct reasons. There are manifolds, and there are $\infty$-groupoids. The latter are combinatorial and can be modeled by simplicial sets, and the former are very intuitive and behave how we want them to behave (no pathological point-set nonsense). Sometimes we want to do space-y things with objects that aren't quite manifolds, but they're close enough... I think most of the examples are captured if we copy motivic folk and look at sheaves (of $\infty$-groupoids) on the site of manifolds. $\endgroup$
    – Dylan Wilson
    Commented Aug 30, 2016 at 17:49
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    $\begingroup$ (As a sanity check, it is indeed true that if you kill the real line in the above homotopy theory, you recover the homotopy theory of spaces as we know it.) In any case, topological spaces work just fine, and are probably not going anywhere any time soon. Though people are getting good at avoiding them when they're unnecessary. $\endgroup$
    – Dylan Wilson
    Commented Aug 30, 2016 at 17:51
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    $\begingroup$ Concerning the interest of weak Hausdorff spaces, see my answer http://mathoverflow.net/a/204627/24563. $\endgroup$
    – Philippe Gaucher
    Commented Sep 1, 2016 at 12:26

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The convenient category CGH of compactly generated Hausdorff spaces has some poor colimits, since Hausdorffification may change the underlying point sets. The category CGWH of compactly generated weak Hausdorff spaces is even better behaved. The advantages are discussed in Chris McCord's paper "Classifying Spaces and Infinite Symmetric Products", Trans. A.M.S. (1969), which credits John Moore for these ideas. Lewis-May-Steinberger and Elmendorf-Kriz-Mandell-May do their serious work on spectra and S-modules in CGWH. This is probably the category of topological spaces (as opposed to simplicial sets) in which most algebraic topologists really work. A search for "CGWH" leads to several related questions on this site, including a link to Neil Strickland's note http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf . (I think his reference to Jim McClure's thesis might really be to Gaunce Lewis' thesis.)

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  • $\begingroup$ You meant "even worse behaved"? $\endgroup$
    – Bombyx mori
    Commented Oct 9, 2016 at 3:04
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    $\begingroup$ @Bombyxmori No, I believe he means that although the individual spaces in CGWH are worse, the category as a whole is better. $\endgroup$
    – Noah Schweber
    Commented Oct 9, 2016 at 17:44
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    $\begingroup$ @Bombyx mori: By using Steenrod's term "convenient" about the category CGH, I meant to say that it is already a rather well-behaved category. The need to change the underlying point sets of some colimits (compared to those in CG or Top) is a drawback, but a minor one. By "even better behaved" I meant to say that CGWH has the good properties of CGH, plus better point set behavior for colimits. I should perhaps have started by emphasizing the good properties of CGH more explicitly than I did by saying "convenient", but given the topic of the question I probably thought this was clear. $\endgroup$
    – John Rognes
    Commented Oct 13, 2016 at 17:00
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I'm a little late to the party. I agree with Dennis Nardin's comment that really any convenient category of spaces should be good for homotopy theory. By this, I mean any complete and cocomplete cartesian closed full subcategory of $\mathsf{Top}$ which includes the CW complexes and whose limits and colimits are not too far from those in $\mathsf{Top}$; in particular, homotopy groups of CW complexes should not change from those in $\mathsf{Top}$. $k$-spaces and its variants are rather large categories as these convenient categories go.[1]

But sometimes you actually want to work with a smaller category of topological spaces -- usually asking for there to be a universal way to turn any topological space into one in your category without changing the weak homotopy type. For example, sequential spaces or delta-generated spaces, or variants with separation conditions, are good options for point-set level categories. These smaller categories have the advantage, unlike $k$-spaces and variants, of being locally presentable, which gives even better categorical control, and in particular allows all the techniques of combinatorial model categories to be brought to bear. An additional advantage of delta-generated spaces is that connected components are the same as path components, and there is a connected component functor, which is left-adjoint to the discrete space functor.

There is a general recipe which allows one to cook up convenient categories of topological spaces. Start with a subcategory $\mathcal{C} \subseteq \mathsf{Top}$, and consider the category $\mathsf{Top}_\mathcal{C}$ which is the closure of $\mathcal{C}$ under colimits in $\mathsf{Top}$ [2]. If $\mathcal{C}$ consists of locally compact Hausdorff spaces [3] and is closed under finite products in $\mathsf{Top}$ [4], then $\mathsf{Top}_\mathcal{C}$ will be cartesian closed, and coreflective in $\mathsf{Top}$, so its colimits are computed as in $\mathsf{Top}$, and limits are computed by taking the ordinary limit and then making the topology finer to land back in $\mathsf{Top}_\mathcal{C}$. So as long as $\mathsf{Top}_\mathcal{C}$ includes the unit interval, it will also contain all CW complexes and have the right homotopy groups thereof, and hence be a convenient category of topological spaces. If in addition $\mathcal{C}$ is small, then $\mathsf{Top}_\mathcal{C}$ will have the added benefit of being locally presentable. If desired, additional separation conditions can also be added without much fuss.

Delta-generated spaces are $\mathsf{Top}_\mathcal{C}$ when $\mathcal{C}$ is the category of topological simplices [5]. Sequential spaces are $\mathsf{Top}_\mathcal{C}$ when $\mathcal{C}$ consists of all metric spaces [6]. And $k$-spaces are $\mathsf{Top}_\mathcal{C}$ when $\mathcal{C}$ is all compact Hausdorff spaces.

[1] To be fair, it's maybe an unnatural restriction to require the category to be a subcategory of $\mathsf{Top}$ - e.g. a supercategory like quasitopological spaces might do as well. But I'll let others discuss these options.

[2] Another way to describe $\mathsf{Top}_\mathcal{C}$, familiar in the case of $k$-spaces, is the following. If $X$ is a space, say that $U\subseteq X$ is $\mathcal{C}$-open iff for every $C \in \mathcal{C}$ and map $f: C \to X$, the set $f^{-1}(U)$ is open. Then $X \in \mathsf{Top}_\mathcal{C}$ iff every $\mathcal{C}$-open subset of $X$ is open. An equivalent description (again familiar in the case of $k$-spaces) says that $X \in \mathsf{Top}_\mathcal{C}$ iff for every $Y \in \mathsf{Top}$ and every function $f: X \to Y$, if the induced map $\mathsf{Top}(C,X) \to \mathsf{Top}(C,Y)$ is continuous for every $C \in \mathcal{C}$, then $f$ is continuous. Finally, $\mathsf{Top}_\mathcal{C}$ is also equivalently described as the closure of $\mathcal C$ in $\mathsf{Top}$ under coproducts and quotients.

[3] Or more generally, exponentiable spaces.

[4] This condition can be weakened to say that finite products in $\mathsf{Top}$ of objects of $\mathcal{C}$ are at least in $\mathsf{Top}_\mathcal{C}$.

[5] Or equivalently, you could take $\mathcal{C}$ to be the category of CW complexes, or the category of manifolds, or you even just the one-object category consisting of the unit interval, or the real line; in all these cases $\mathsf{Top}_\mathcal{C}$ will still be exactly the delta-generated spaces.

[6] Or equivalently, you could take $\mathcal{C}$ to be the category of first-countable spaces, or the category of second-countable spaces, or just the one-object category consisting of the one-point compactification of the natural numbers.

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  • $\begingroup$ Thanks for including all the variants in comments. These are useful if one wants to cook up a small(er) site out of these spaces. $\endgroup$
    – David Roberts ♦
    Commented Oct 9, 2016 at 8:31
  • $\begingroup$ An example of this is in Johnstone's "On a Topological Topos", where he looks at the relationship between sequential spaces and the topos of sheaves on the one-object site of consisting of the one-point compactification of the natural numbers. He mentions that Lawvere looked at sheaves on the one-object site consisting of the unit interval (related to delta-generated spaces), but Isbell pointed out there are some bad colimits here -- the unit interval mod its endpoints doesn't yield the circle. But now I wonder -- maybe you could fix this by working with the site of all finite CW complexes! $\endgroup$
    – Tim Campion
    Commented Oct 9, 2016 at 17:00
  • $\begingroup$ it makes one wonder whether if one takes simplicial sheaves on the site consisting of the unit interval then one gets the correct homotopy theory, even if the cohesive theory is incorrect. A bit like how one can try to consider diffeological spaces as particular sheaves on the category consisting of the real line and its endomorphisms, or using all cartesian spaces. Odd things happen with other, non-concrete sheaves, like n-forms for n > 1. $\endgroup$
    – David Roberts ♦
    Commented Oct 10, 2016 at 0:12
  • $\begingroup$ You claim that the category of $k$-spaces is not locally presentable without further argument; how do you prove it? $\endgroup$
    – fosco
    Commented Jan 15, 2018 at 23:10
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    $\begingroup$ @Fosco Let $\kappa$ be a cardinal. Say that a space $X$ is $\kappa$-tight if $U \subseteq X$ is open in $X$ iff $U \cap A$ is open in $A$ for every $A \subseteq X$ with $|A| < \kappa$. The $\kappa$-tight spaces are closed under colimits. If $\mathcal D \subseteq \mathsf{Top}$ is coreflective and locally presentable, then it is the colimit closure of a small set of spaces and so the tightness of spaces in $\mathcal D$ is bounded by some cardinal $\kappa$. But there are compact Hausdorff spaces of arbitrarily large tightness, e.g. the Stone-Cech compactifications of discrete spaces. $\endgroup$
    – Tim Campion
    Commented Jan 16, 2018 at 1:34
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You should look at this ncatlab exposition on "convenient categories of topological spaces", and references. One problem is still the difficulty of getting a locally cartesian closed convenient category; Spanier's "quasitopologies" seemed a possibility but have not been taken up, partly because the quasicategories on the 2 point set formed a class (comment from E. Dyer).

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    $\begingroup$ Presumably this notion of 'quasicategory' is different from what we now call quasicategories, but were called by Boardman and Vogt 'weak Kan complexes'? $\endgroup$
    – David Roberts ♦
    Commented Sep 1, 2016 at 8:24
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    $\begingroup$ My mistake: it should have been "quasitopologies"! $\endgroup$
    – Ronnie Brown
    Commented Sep 1, 2016 at 10:18
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    $\begingroup$ Yeah, Spanier's quasi-topological spaces are not popular largely because of hassles to do with size (set/class) conditions. For example, there is a proper class of quasi-topological structures on a two-element set, which is a bit weird. But there are some interesting quasitoposes (which by definition are nice locally cartesian closed categories) that contain the category of topological spaces as a full subcategory, e.g., the category of pseudotopological spaces, the category of subsequential spaces, and the category of equilogical spaces. All are described in the nLab. $\endgroup$
    – Todd Trimble
    Commented Sep 1, 2016 at 15:16
  • $\begingroup$ One can also just use compactly generated spaces (without Hausdorffness). This is Cartesian closed. It's not locally Cartesian closed, but, if $X$ is Hausdorff, then $CG/X$ is Cartesian closed. Not so bad. $\endgroup$
    – David Carchedi
    Commented Sep 5, 2016 at 3:17
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    $\begingroup$ @DavidCarchedi I would have looked, but it's behind a paywall. Thanks for your additional explanation. $\endgroup$
    – Todd Trimble
    Commented Oct 6, 2016 at 12:25
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You might look at On a topological topos, n-category café.

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    $\begingroup$ While that's an interesting construction, I don't think anyone actually uses that topos to do homotopy theory and if I understand correctly there is no connected component functor, which sounds like a problem to me. $\endgroup$
    – Denis Nardin
    Commented Aug 30, 2016 at 16:58
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This might be useful along with some of the comments above.

Chapter 8 of Gray's book does provide some explanation into this, which hopefully will convince the reader that $\mathbf{K}$, the category of compactly generated spaces, is a good category to work within for the purpose of doing homotopy theory. There is retraction, say, $\mathbf{Top}\to \mathbf{K}$. We may also find similar material in introductory chapters of G. W. Whitehead's book. The homotopy theory in these books, is built within this category. When falling out of this category, then one tends to filter a given object by objects of $\mathbf{K}$ and use limits, etc. And, most of what at least I know (with a very limited scope of the subject) as homotopy theory is built on the homotopy theory of such classic books. So, I think, implicitly, for the purpose of homotopy theory, the answer is that $\mathbf{K}$ is of interest, although we don't mention it!

In particular, the case of topology of function spaces, and the adjointness relation between to functors, is treated nicely in Gray's books, which explains why equipping a space with a compactly generated topology is useful.

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