Simplicial Sheaves?

Question

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.

Answer 1

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.

Answer 2

I think you make reference to what is called cellular sheaves (restricte to cw-complex given by the simplicial complex structure). These are presheaves on the poset of faces.