Simplicial Sheaves? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:06:57Z http://mathoverflow.net/feeds/question/6535 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/6535/simplicial-sheaves Simplicial Sheaves? B. Bischof 2009-11-23T05:33:49Z 2009-11-23T06:01:47Z <p>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?</p> <p>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?</p> <p>What are the other interesting places where simplicial sheaves appear?</p> <p>I know that this question is fairly general, but I hope you understand what I am asking.</p> http://mathoverflow.net/questions/6535/simplicial-sheaves/6537#6537 Answer by Reid Barton for Simplicial Sheaves? Reid Barton 2009-11-23T06:01:47Z 2009-11-23T06:01:47Z <p>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).</p> <p><a href="http://ncatlab.org/nlab/show/simplicial+presheaf" rel="nofollow">Simplicial sheaves</a> are something else entirely&mdash;they're <i>(pre)sheaves of simplicial sets</i> on, say, a category equipped with a Grothendieck topology.</p>