Let $\Delta$ be a simplicial complex. I believe the standard definition of $\Delta$ being contarctible is if the topological realization $|\Delta|$ is contractible. However, I am seeking a definition of a contractible simplicial complex that is in terms of the simplicial complex qua simplicial complex, possibly having something to do with its simplicial homology. In other words, something in terms of the faces or its simplicial homology or something that can be checked while staying in the category of simplicial complexes. Call such a definition $S$-contractible. I would then hope that a simplicial complex is $S$-contractible if and only if $|\Delta|$ is contractible. Is anyone aware of such a definition and if so, has it been shown to agree with the standard definition of contractible? Many thanks in advance.
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You want the simplicial approximation theorem. One part, or perhaps I should say version, says that any continuous map between the geometric realizations of two simplicial complexes is homotopic to a simplicial map (possibly after subdivision). Another part/version says that there is a simplicial definition of what it means for two simplicial maps to be homotopic and that it agrees with the usual definition (possibly after subdivision).
Given that, you can write down a simplicial definition of what it means for two simplicial complexes to be homotopy equivalent, and you can write down a simplicial definition of what it means for there to exist a map from a point to your simplicial complex which is a homotopy equivalence. That's contractibility by simplicial approximation. But maybe you want something that doesn't require subdivision?