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This is a question about mathematical writing.

I'm not super familiar with algebraic topology, but in something I'm writing I need to talk about abstract simplicial complexes, their geometric realizations as subsets of $\mathbb{R}^n$, and topological spaces that are homeomorphic to those geometric realizations.

So I have, if you like, abstract simplicial complexes, geometric simplicial complexes, and topological simplicial complexes. Just calling them all ``simplicial complexes'' seems unnecessarily confusing, but throwing in the qualifying adjectives everywhere makes the material hard to read, or at least unpleasant, in a different way. Is there a good solution to this problem?

Nik Weaver
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1 Answers1

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Here's a few suggestions.

Regarding geometric simplicial complexes: Sometimes people use terminology like polyhedra or polytope. That seems to work fine as long as you are staying away from a discussion of classical polyhedra.

Regarding topological simplicial complex: According to the definition you wrote, a topological space $X$ is a topological simplicial complex if there merely exists a homeomorphism from $X$ to a geometric simplicial complex. That usage feels off base to me. I would have expected that a specific choice of such a homeomorphism must be part of the structure of a topological simplicial complex. There is adjective for that concept as you wrote it, rather than a noun: $X$ is triangulable if such a homeomorphism exists. Also, sometime writers refer to a specific choice of such a homeomorphism as a triangulation of $X$, or a simplicial structure on $X$, or something like that (c.f. Hatcher's terminology of a $\Delta$ complex structure, which is a bit more general than a simplicial structure).

In general, another thing that happens is that a writer will make a choice, reserving the bare terminology simplicial complex for one of the more specialized cases. A combinatorist or a category theorist might reserve that bare terminology for an abstract simplicial complex. On the other hand someone writing a basic geometric topology textbook might reserve that bare terminology for a topological space $X$ equipped with a choice of triangulation (for example Moise's textbook Geometric Topology in Dimensions 2 and 3).

Lee Mosher
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