Edit: Zhen Lin incisively observes in a comment below that the category of compact Hausdorff spaces is monadic over the category of sets, hence is cocomplete. That answers the first part of question 1 as it was stated previously. I have changed that question accordingly.

Final edit: Chris Schommer-Pries answered the rest of the question in the negative with a very simple idea which I overlooked when thinking about this matter. Many thanks to Zhen Lin and Chris. To everyone else, I humbly apologize for asking such a simple question.

I am attempting to find finitely cocomplete categories of compact Hausdorff topological spaces with some convenient homotopical properties. In particular, I want to keep the nice feature of topological spaces that one can often perform homotopical constructions without using fibrant or cofibrant replacements (partly because all objects are fibrant). $\newcommand{\Top}{\mathrm{Top}} \newcommand{\CHTop}{\mathrm{CHTop}} \newcommand{\into}{\hookrightarrow}$

Let $\Top$ denote the category of topological spaces, and $\CHTop$ its full subcategory generated by the compact Hausdorff spaces. The first question is:

Question 1: Is the category $\CHTop$ finitely cocomplete (i.e. has all finite colimits)? Is every finite colimit of compact Hausdorff spaces in $\Top$ a Hausdorff space? In other words, does the inclusion $\CHTop\into\Top$ preserve finite colimits?

Regarding the plausibility of the question, it is certainly easy to construct quotients of compact Hausdorff spaces which are not Hausdorff. However, only certain types of quotients appear when computing finite colimits of spaces. I would still be inclined to believe the answer to the above question is no, yet I found no counterexample in my limited search. Also, I would be most interested to learn of simple conditions one can impose on spaces such that the corresponding full subcategory of $\CHTop$ is finitely cocomplete.

My second question is related to question 1. It concerns nice finitely cocomplete subcategories of compact Hausdorff spaces which still have "enough homotopies" to be able to avoid using fibrant/cofibrant replacements. As an example, simplicial sets do not have enough homotopies in the sense that one cannot in general concatenate homotopies since $\Delta^1\coprod_{\Delta^0} \Delta^1$ is not isomorphic to $\Delta^1$.

Question 2: Give examples of categories $C$ with a faithful functor $F:C\to\CHTop$ such that:

  1. there exists an object $1_C$ in $C$ such that $F(1_C)$ is a singleton space;
  2. there exists an object $I_C$ in $C$ such that $F(I_C)$ is homeomorphic to $I$;
  3. $C$ admits all products with $I_C$, and the functor $F$ preserves those products;
  4. $C$ is finitely cocomplete, and the composition $C\overset{F}{\to}\CHTop\into\Top$ preserves all finite colimits.

Bonus points if the category $C$ verifies the following strengthening of condition 3: $C$ has all finite products, and the functor $F$ preserves them.

Independently of the answer to question 1, I would be very interested to hear about any examples you know that would fit, or at least approximate, the requirements in question 2. I would be particularly interested in examples consisting of small, manageable spaces.

My own ill-determined idea to give an example as in question 2 is the subject of the following vague question.

Question 3: Does there exist a category as in question 2 which consists of finite polyhedron-like spaces, and where the morphisms are some sort of piecewise linear maps?

  • $\begingroup$ I don't really understand your motivations here. In particular, I don't understand what you dislike about simplicial sets. The category of finite simplicial sets has the structure of a cofibration category and this gives you tools to work with homotopy theory of finite simplicial sets including finite homotopy colimits. The category of finite simplicial sets and the geometric realization functor satisfy all the requirements of your Question 2. $\endgroup$ Mar 19 '13 at 7:59
  • $\begingroup$ (Except that I am not sure whether $\mathrm{CHTop} \into \mathrm{Top}$ really preserves arbitrary finite colimits but I think it preserves pushouts along cofibrations which is good enough from the perspective of homotopy theory.) $\endgroup$ Mar 19 '13 at 7:59
  • 6
    $\begingroup$ The category of compact Hausdorff spaces in fact has all small colimits, because $\textbf{CHTop} \to \textbf{Set}$ is monadic. (Any category monadic over $\textbf{Set}$ has all small colimits, under the axiom of choice.) $\endgroup$
    – Zhen Lin
    Mar 19 '13 at 8:23
  • $\begingroup$ @Zhen Lin: That is a very good point. I will change the question accordingly. Thank you very much. $\endgroup$ Mar 19 '13 at 8:47
  • $\begingroup$ @Karol: Rereading my question, I see what you meant by your comment. I rephrased it to better reflect my intentions. Sorry. $\endgroup$ Mar 19 '13 at 9:11

As Zhen Lin points out in the comments to your question, the category of compact Hausdorff spaces has all small colimits. However the inclusion functor from CHTop to Top does not preserve these colimits in general. It seems from your other questions that you are interested in a category, either CHTop or something similar, so that finite colimits exist and the inclusion functor preserves finite colimits.

This will not be possible. In particular the answer to question 2 (hence also question 3) is no, as the following example illustrates.

Let $X = Y = I = [0,1]$ and consider a coequalizer $Y \rightrightarrows X$ where the first map is the identity and the second map is: $$ t \mapsto 2(t - t^2) $$ Even though Y is compact, this generates an equivalence relation with non-closed equivalence classes. The equivalences classes are essentially the solution sets to the above non-linear recurrence relation. In particular nearly every equivalence class cannot be separated from the middle fixed point $t = 0.5$, and hence the quotient is non-Hausdorff.

  • $\begingroup$ @Chris: It seems you meant $t\mapsto t^2$ or something like that. In any case, the idea stands, and I see that my conditions are way too strict. That certainly settles the question as I stated it. Thank you very much. $\endgroup$ Mar 19 '13 at 9:16
  • $\begingroup$ @Ricardo: Yes I miss typed the formula the first time. It is supposed to be an upside down parabola which goes through (0,0), (0.5, 0.5) and (1, 0). I think it is fixed now. As you increase the factor 2 the behavior changes and becomes chaotic. see physics.udel.edu/~jim/PHYS460_660_13S/oscillations&chaos/… This must mean quotients become really really bad non-Hausdorff spaces. $\endgroup$ Mar 19 '13 at 9:21
  • $\begingroup$ @Chris: No problem. Thanks again. Also, sorry for the simple question. $\endgroup$ Mar 19 '13 at 9:26
  • $\begingroup$ A reference for the fact that every monadic category over $\mathsf{Set}$ is cocomplete can be found on the nlab, Corollary 1. Alternatively, use the observation of Linton (discussed on the same page, Theorem 1) that if $C$ is cocomplete and $C^T$ has reflexive coequalizers then $C^T$ is cocomplete. After all, compact spaces are closed under quotients, and Hausorffification is another quotient, so compact Hausdorff spaces have all coequalizers, formed by taking the coequalizer and Hausdorffifying. $\endgroup$
    – Tim Campion
    Jul 10 '16 at 14:00

The examples given are part of the case for homotopy colimits. For example the pushout of the two maps $$S^1 \leftarrow S^1 \to S^!$$ given by $z\mapsto z^2, z \mapsto z^3$ is not Hausdorff. But the double mapping cylinder $M$ is a nice CW-complex. Amusingly, this is rel;ated to the case for groupoids, where the pushout in groups of the two maps $$ \mathbb Z \leftarrow \mathbb Z \to \mathbb Z$$ is the trefoil group $T$ with generators $x,y$ and relation $x^2=y^3$, but the homotopy pushout in groupoids is the "trefoil groupoid", say $T'$, with two objects $0,1$, generators $x,y$ at $0,1$ respectively and one arrow $\iota :0 \to 1$ conjugating $x^2$ to $y^3$. The advantage of this is that it "separates" the two group generators $x,y$, and of course $T'$ is the fundamental groupoid of $M$ on two base points, one at each end of the cylinder.


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