As far as I understand, whenever one has something (co)simplicial in spaces, one should take a sort of diagonal to reduce the study to a single space. I'm never sure though whether one preserves only the whole *homotopic* information or really the homeomorphic type.

My real question is the following: If $X^\bullet_\bullet$ is a simplicial object in cosimplicial spaces, does it make sense to speak about its **Borel-Moore homology**? Is it a limit/colimit of BM homologies? Does it matter in which order I take the diagonals/Tots/realisations?

I'm trying to get my multi(co)simplicial facts straight.

thatwould lead you to an appropriate definition. $\endgroup$ – Fernando Muro Nov 21 '13 at 18:46