to what extent does the category Cov(X) determine a topological space X?

Question

Given a topological space $X$, form the category $\mathrm{Cov}(X)$ consisting of open subsets $U \subset X$ as objects and inclusions as morphisms. To what extend can we recover (properties of) $X$ from $\mathrm{Cov}(X)$?

Example: $\mathrm{Cov}(X)$ determines the Cech cohomology groups of $X$, and so for nice $X$ its homology groups, and thus for $\pi_1(X)$ abelian by the Hurewicz theorem the homotopy type of $X$.

Can we also read of $\pi_1(X)$ from $\mathrm{Cov}(X)$? This should be possible if there is a nice covering of $X$ by open contractible subsets with intersections connected (by the Seifert-van-Kampen theorem).

How is $B\mathrm{Cov}(X) = |N\mathrm{Cov}(X)|$ related to $X$?

Answer 1

In order to have enough freedom, I would prefer to consider a basis $\mathcal{U}$ of open subsets of $X$. In this case, considering $\mathcal{U}$ as a partially ordered set (for the inclusion), there is a canonical inclusion functor into topological spaces

$$\mathcal{U}\to \mathit{Top} \ , \ U\mapsto U$$

whose colimit is precisely (and obviously) the space $X$. But, in fact, we have a better property. the natural map

$$hocolim_{U\in\mathcal{U}} \ U\to colim_{U\in\mathcal{U}} \ U=X$$

is a weak homotopy equivalence (i.e. induces an isomorphism of homotopy groups); for a sketch of proof, see below.

Of course, we might consider the case where $\mathcal{U}$ consists of all the open subsets of $X$, but, as noticed in Todd's answer, we then get a partially ordered set with initial and terminal object, which is not very interesting, homotopy theoretically. However, if $X$ is locally contractible (e.g. if $X$ is a CW-complex), then it might be interesting to consider for $\mathcal{U}$ the contractible open subsets of $X$. In that case, as the maps $U\to pt$ are weak homotopy equivalences, then the natural map

$$hocolim_{U\in\mathcal{U}} \ U \to hocolim_{U\in\mathcal{U}} \ pt=: B\mathcal{U}$$

is a weak homotopy equivalence as well. In other words, in this case, $X$ and $B\mathcal{U}$ are canonically isomorphic in $Ho(Top)$, which is a nice way of seeing that partially ordered sets are models for homotopy types.

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Edit:

Here is a sketch of proof of the fact that the map $hocolim_{U\in\mathcal{U}} \ U\to colim_{U\in\mathcal{U}} \ U$ is a weak homotopy equivalence for any $X$. The point is that we may consider the model category $P(X)$ obtained as the left Bousfield localization of the projective model category of simplicial presheaves on $X$ by the class of hypercovers; see

1 D. Dugger, S. Hollander and D. Isaksen, Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9-51.

Then, there is an obvious left Quillen functor from $P(X)$ to the model category of topological spaces which sends a representable $U$ to the corresponding subspace of $X$: indeed, we have a left Quillen functor from the projective model structure on $P(X)$, and we conclude using Theorem 1.3 of

2 D. Dugger and D. Isaksen, Topological hypercovers and $\mathbf{A}^1$-realizations, Math. Z. 246 (2004), no. 4, 667-689.

To finish the proof, as left Quillen functors preserve homotopy colimits (up to weak equivalences), it is thus sufficient to prove that $hocolim_{U\in\mathcal{U}} \ U$ (seen as a simplicial preasheaf) is weakly equivalent to the terminal object in $P(X)$ (for the local model structure). This follows from Theorem 6.2 of 1 (which allows to replace $P(X)$ by the model category of simplicial presheaves on $\mathcal{U}$) and from from the fact that the homotopy colimit of all representable presheaves is always weakly equivalent to the terminal presheaf; see for instance Lemma 3.4.27 and Theorem 3.4.34 in my book Les préfaisceaux comme modèles des types d'homotopie, Astérisque 308, 2006; this can also be obtained easily from Proposition 2.9 in

3 D. Dugger, Universal homotopy theories, Adv. Math. 164 (2001), no. 1, 144-176.

Answer 2

If $X$ is a sober space, you can retrieve $X$ up to homeomorphism from $Cov(X)$. (Nitpick: this is not very good notation; it is very easy to misread it as the category of covering spaces over $X$. I prefer to denote this as $Open(X)$ or $\mathcal{O}(X)$, and will do so below.) A basic reference for this is Johnstone's Stone Spaces.

The lattice $\mathcal{O}(X)$ is a frame, i.e., a complete lattice for which finite meets distribute over arbitrary joins. Let $\mathbf{2}$ be the topology of a one-point space $\{1\}$. For any frame $F$, there is a topological space $Spec(F)$ whose points are frame homomorphisms $F \to \mathbf{2}$; for each $f \in F$, there is a corresponding set

$$U_f = \{x \in Spec(F): x(f) = \{1\}\}$$

and these $U_f$ generate the topology of $Spec(F)$. If $X$ is any space, there is a canonical continuous map

$$X \to Spec(\mathcal{O}(X))$$

which sends a point $x \in X$ to the frame homomorphism $\phi_x: \mathcal{O}(X) \to \mathbf{2}$ defined by $\phi_x(U) = \{1\}$ if and only if $x \in U$. This map is a homeomorphism precisely when $X$ is a sober space, i.e., when every closed irreducible set is the closure of a unique point of $X$.

Examples of sober spaces include all Hausdorff spaces and all underlying topological spaces of spectra of rings as defined in algebraic geometry (so in other words, just about all spaces that arise in practice).

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Edit: There is also the question about what relations exist between $X$ and $B \mathcal{O}(X)$. This I am less sure of; notice that $B\mathcal{O}(X)$ is contractible because $\mathcal{O}(X)$ has a top element (and a bottom element). It is sometimes possible to get interesting information about the geometry of a lattice from the homology of the (classifying space of the) poset obtained by removing the top and bottom element; for example, if the lattice is a finite geometric lattice, then a well-known result of Folkman is that the nerve of that poset is a bouquet of spheres; the number of spheres is the Möbius number $\mu(0, 1)$ of the lattice.

However, we can get something interesting in the case of finite topological spaces if we change the constructions slightly. First, the category of finite topological spaces is equivalent to the category of finite preorders (if $X$ is a topological space, we can order the points by $x \leq y$ if $x$ is contained in the closure of $y$, and this contains all the topological information in the finite case). If $P(X)$ is this associated preorder, then one can define a canonical map $BP(X) \to X$ which is a weak homotopy equivalence in the case where $X$ is finite. See also this (brief) discussion at the Secret Blogging Seminar, including the comments.

Answer 3

$\mathrm{Cov}(X)$ determines $X$ if $X$ is sober.