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 $X$?



to what extent does the category Cov(X) determine a topological space X?
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](http://math.uoregon.edu/~ddugger/spre.html) 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](http://pages.uoregon.edu/ddugger/hyper.html) 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](http://math.uoregon.edu/~ddugger/spre.html) (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](http://pages.uoregon.edu/ddugger/univ.html) D. Dugger, *Universal homotopy theories*, Adv. Math. 164 (2001), no. 1, 144-176.


