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.