Comment: I am asking several questions here. If I am expected to break them into separate posts, or mark this thread as community wiki, please let me know. Also, I am not entirely sure whether the at.algebraic-topology tag is appropriate.

I am attempting to find finitely cocomplete categories of compact Hausdorff topological spaces with good properties. The original motivation is to obtain categories of objects which behave like topological spaces, and which have convenient properties from the point of view of homotopy theory. $\newcommand{\Top}{\mathrm{Top}} \newcommand{\CHTop}{\mathrm{CHTop}} \newcommand{\into}{\hookrightarrow}$

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)? More stringently, is every finite colimit of compact Hausdorff spaces in $\Top$ a Hausdorff space?

I do not know if the first part of the question implies the second part. 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, and serves partly to deal with the possibility that its answer is no. This new question regards finding nice finitely cocomplete subcategories of compact Hausdorff spaces which still have "enough homotopies" to do homotopy theory conveniently. 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 $F$ (alternatively, 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?