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In the category of topological spaces, I would like to know that quotients of simplicial complexes (or $\Delta$-complexes) by equivalence relations which are "unramified" in a suitable sense still have the structure of simplicial complexes:

Say $X$ and $Y$ are finite disjoint unions of simplices (of varying dimension) and we have two maps $X \rightrightarrows Y$ such that restricted to each simplex $\Delta^k_i \subset X$, each map is a linear closed embedding into one of the simplices $\Delta^n_j \subset Y$. By linear closed embedding I mean that the map $\Delta^k_i \to \Delta^n_j$ is the inclusion of the convex hull of $k$ linearly independent points (which for my applications can be assumed to have rational coordinates) in $\Delta^n_j$.

It seems plausible that such a diagram $X \rightrightarrows Y$ can be identified with the geometric realization of a diagram of finite $\Delta$-sets, which would equip the coequalizer $coeq(X \rightrightarrows Y)$ with the structure of a finite $\Delta$-complex. Is there a counterexample to this claim that I'm missing? Does this appear in the literature?

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  • $\begingroup$ Doesn't the category of simplicial complexes have all small limits and colimits? Does this help? A reference is math.stackexchange.com/questions/492072/… $\endgroup$ – David White Aug 20 '15 at 13:01
  • $\begingroup$ I suspect the op wants to have the colimit agree with the one in top spaces. $\endgroup$ – Benjamin Steinberg Aug 20 '15 at 16:34
  • $\begingroup$ @DavidWhite, even though $X$ and $Y$ can be equipped with the structure of a simplical complex, by which I mean they admit homeomorphisms with the geometric realization of some abstract simplicial complex, the maps $X \rightrightarrows Y$ are not necessarily maps of simplicial complexes a priori (i.e. they need not come from maps of abstract simplicial complexes). If you could find triangulations of X and Y such that both maps are simplicial, then the coeq as topological spaces would agree with the geometric realization of the coeq in the category of abstract simplicial complexes. $\endgroup$ – user2700 Aug 21 '15 at 3:07

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