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I recently was wondering if there was a name for sheaves which were locally constant on the open simplexes in a simplicial complex. After some googling I stumbled across simplicial sheaves. I am alright with the definition as the presheaves to simplicial sets, but now I wonder, is this the answer to my question?

Are simplicial sheaves related to the locally-constant-on-simplex etale sheaves of a simplicial complex? If not, does this concept have a different name? Are there interesting places this sort of thing appears?

What are the other interesting places where simplicial sheaves appear?

I know that this question is fairly general, but I hope you understand what I am asking.

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up vote 4 down vote accepted

If I understand correctly, these are constructible sheaves with respect to the stratification of your simplicial complex by its skeleta. I think by a theorem of MacPherson the category of such sheaves is equivalent to the category of functors from the poset of faces and face inclusions to whatever category your sheaves take values in (maybe there are some conditions on the target category).

Simplicial sheaves are something else entirely—they're (pre)sheaves of simplicial sets on, say, a category equipped with a Grothendieck topology.

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There was a recent question here about the difference between simplicial sets and simplicial complexes that also might be instructive. –  Harry Gindi Nov 23 '09 at 6:40
    
The nLab software had a small hiccup after an update. Is running fine again. +1 for linking to the nLab (as one says around here) –  Urs Schreiber Nov 23 '09 at 22:34
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