Category of topological spaces with open or closed maps

Question

Consider the category whose objects are topological spaces and whose morphisms are the open maps (or closed maps, open continuous maps, closed continuous maps … that is, one whose isomorphisms are precisely the homeomorphisms). How does such a category compare with the usual one whose objects are topological spaces and whose morphisms are continuous maps? For example, what limits and colimits exist?

I'm probably missing something obvious, but why don't products typically exist in the category with open maps? The projections from the usual product (in the category with continuous maps) are open, yielding a canonical open map from the usual product to the putative unusual product.

Todd's observation is true enough: the product in the usual topology (continuous maps) typically fails to realize the corresponding universal property in the unusual topology (open maps). Nevertheless, some other object might realize that universal property. Is it even clear that if such a space exists its underlying set should be naturally identifiable with the underlying set of the factors? After all, while one point spaces are still terminal, maps out of such objects tend not to be open: it seems one would thereby only extract the subset of isolated points. In any event, http://christianmarks.wordpress.com/category/bagatelle
treats the special case of squares. The appropriate space is $X\times X$ with the weakest topology (stronger than the usual) which makes the diagonal embedding open. This construction is clearly not available for products of distinct spaces. My question concerns whether there isn't (as that post suggests there isn't) some devious workaround.

Answer 1

This seems difficult to answer in any kind of uniform way, i.e., it seems we have to work on a case-by-case basis. Let's start with $\mathrm{Top}_{\mathrm{open}}$, the category of topological spaces and open continuous maps.

A useful heuristic is to let oneself be guided by locales instead of topological spaces. (For general information on locales, see for instance Mac Lane-Moerdijk, Sheaves in Geometry and Logic, chapter IX.) Recall that the category of locales is (by definition) the category opposite to the category of frames (sup-lattices in which finite meets distribute over arbitrary joins); the frame associated with a locale (by viewing the same object in the opposite category) should be thought of as its topology. Now a frame automatically carries Heyting algebra structure, and it turns out that open continuous maps of locales correspond to frame maps that are also Heyting algebra maps (i.e., Heyting algebra maps which are sup-preserving). Thus, the category of locales and open continuous maps is opposite to the category of cocomplete Heyting algebras, which is a category of algebras of an infinitary algebraic theory. It follows that the category of locales and open continuous maps is complete. Moreover, limits of cocomplete Heyting algebras are computed just as they are in $Set$, and so the calculation of colimits in locales and open continuous maps will be similarly straightforward.

Guided by this heuristic, one might at the very least expect that colimits in $\mathrm{Top}_{\mathrm{open}}$ are calculated just as they are in $\mathrm{Top}$. Indeed, one can easily check this directly for arbitrary coproducts (where the coproduct topology on a disjoint union $X = \sum_\alpha X_\alpha$ is the largest one such that all the canonical inclusions $X_\alpha \to X$ are continuous; notice they are open). As for coequalizers, if two maps

$$f, g: R \stackrel{\to}{\to} X$$

are open and continuous, and if we consider the coequalizer $q: X \to Y$ in $\mathrm{Top}$ (where the topology on $Y$ is the usual quotient topology induced by $q$), then we claim that $q$ is also open and this is the coequalizer in the category of open continuous maps. For, if $V$ is open in $X$, then $q(V)$ is open in $Y$ iff $q^{-1}(q(V))$ is open in $X$, and clearly this holds since

$$q^{-1}(q(V)) = V \cup \bigcup_n (g f^{-1})^n(V) \cup (f g^{-1})^n(V)$$

where the right side is open because $f$ and $g$ are open and continuous.

Pausing briefly to consider the category of topological spaces and closed continuous maps, it is certainly true that finite coproducts exist and are computed just as they are in $\mathrm{Top}$. Also, it is true that quotients of equivalence relations are computed as in $\mathrm{Top}$. That is to say: if $E \subseteq X \times X$ is an equivalence relation and the two projection maps $\pi_1: E \to X$ and $\pi_2: E \to X$ are closed, then the coequalizer $q = \mathrm{coeq}(\pi_1, \pi_2)$ in $Top$ is the coequalizer in the category of closed continuous maps. Indeed, if $C$ is closed in $X$, then $q(C)$ is closed because

$$q^{-1}(q(C)) = \pi_1(\pi_2^{-1}(C))$$

is closed.

Limits are trickier, and I believe do not always exist in $\mathrm{Top}_{\mathrm{open}}$ (see below). Even when they do exist, one should not expect to calculate limits as one would in $\mathrm{Top}$ (for example, Cantor space as a countable product of copies of the discrete two-element space could not possibly be the correct product in $\mathrm{Top}_{\mathrm{open}}$ since Cantor space has no isolated points), nor even as in $\mathrm{Set}$, so this makes limits a little weird.

Here's a partial result: if we have a diagram $F: D \to \mathrm{Top}_{\mathrm{open}}$ and all the spaces $F(d)$ are sober (i.e., of the form $\mathrm{Pt}(\mathcal{O}(X))$, where $\mathrm{Pt}: \mathrm{Frame}^{op} \to \mathrm{Top}$ is right adjoint to $\mathcal{O}: Top \to \mathrm{Frame}^{op}$), then the limit of $F$ in $\mathrm{Top}_{\mathrm{open}}$ can be constructed as $\mathrm{Pt}(\mathrm{colim} \; \mathcal{O}(F d))$ where the colimit is computed in the category of cocomplete Heyting algebras, provided that colimit exists. This uses the fact that for any topological space $X$, the set of open continuous maps $X \to \mathrm{Pt}(H)$ ($H$ a cocomplete Heyting algebra) is in natural one-one correspondence with cocomplete Heyting algebra maps $H \to \mathcal{O}(X)$, together with elementary categorical manipulations.

I conjecture that finite colimits of cocomplete Heyting algebras exist (so that finite limits of sober spaces in $\mathrm{Top}_{\mathrm{open}}$ exist). But inasmuch as the countable coproduct of copies of the free cocomplete Heyting algebra on one generator does not exist (it would be the free cocomplete Heyting algebra on countably many generators, which is known not to exist), it seems to me that $\mathrm{Top}_{\mathrm{open}}$ does not have countable products.

Answer 2

A nice recent paper of Bezhanishvili and Kornell (which actually references this MathOverflow question specifically) has shown that the existence of products fails very badly in $\mathrm{Top}_{\mathrm{open}}$. In particular, the product of Sierpiński space $\mathbb{S}$ (i.e., the set $2 = \{0,1\}$ with the topology $3 = \{0,1,2\} = \{\varnothing, \{0\},\{0,1\}\}$) with itself does not exist in $\mathrm{Top}_{\mathrm{open}}$ (or in the full subcategories of $T_0$ and of sober spaces with open maps). Interestingly enough, they do this by mimicking the construction of the von Neumann hierarchy inside the product $\mathbb{S} \times \mathbb{S}$, showing that no small space satisfies the required universal property. As a consequence they show that the coproduct of the three-element Heyting algebra (i.e., the frame of opens of $\mathbb{S}$) with itself does not exist in the category of complete Heyting algebras. I think this resolves Todd's conjecture negatively, although I might be getting confused by the completeness/cocompleteness terminology.

This would also seem to contradict what you mention about the WordPress post, since the counterexample is a square, but I'm not entirely sure since the post has vanished from the internet.

Answer 3

$\newcommand\Top{\mathit{Top}}\newcommand\Mfd{\mathit{Mfd}}$The category can be drastically different. For example, suppose your morphisms are precisely local homeomorphisms. Call this category $\Top^\text{ét}$. This category is locally a topos, which is something certainly not the case for the category $\Top$ of all continuous maps. Moreover, $\Top^\text{ét}$ lacks a terminal object (which is something which will happen for many variants). In fact, $\Top^\text{ét}$ behaves much more like a category of sheaves on a single space, than a category of spaces; to see this, if we let $\mathfrak{Top}^\text{ét}$ denote the bicategory of (étale) topological stacks and local homeomorphisms, this is equivalent to the bicategory of stacks on some (filtered colimit of) étale topological stack(s). The bicategory $\mathfrak{Top}^\text{ét}$ contains $\Top^{ét}$ as a full subcategory, and (if we restrict to a set of topological spaces) is a $2$-topos, so has all the limits, colimits etc. you can imagine. Using this, it can be shown that $\Top^{ét}$ does have at least binary products, but they behave very differently than in $\Top$. For example, instead of $\Top$, consider the category $\Mfd$ of smooth manifolds. Given an $n$-manifold $N$ and an $m$-manifold $M$, their product $N \times^\text{ét} M$ in $\Mfd^\text{ét}$ is empty if $n \ne m$, and if $n=m$, their product is a highly non-Hausdorff smooth $n$-manifold. This is discussed in arXiv:1212.2282.