# Does compactly supported cohomology make sense for cosimplicial spaces?

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.

• I find this question somewhat unclear, the BM-homology is defined for locally finite simplicial complexes, right? Not even for simplicial sets in general. – Fernando Muro Nov 21 '13 at 18:11
• @FernandoMuro: OK, let's say I have a cosimiplicial space, which at each level is homeomorphic to a locally finite simplicial complex. Can I define BM homology for that? – John Salvatierrez Nov 21 '13 at 18:43
• John, probably you can, but maybe you want your definition to satisfy some properties, and hopefully that would lead you to an appropriate definition. – Fernando Muro Nov 21 '13 at 18:46
• Do you have any suggestions? I was hoping people smarter than me had already thought about it, that's why I asked. In general, is there a good place to read about cosimplicial spaces? What about simplicial-cosimplicial ones? – John Salvatierrez Nov 21 '13 at 18:55
• I don't really have any nice suggestion. I'd like to be able to say something like: take levelwise the complex of locally finite chains, in an ideal world this would give rise to a cosimplicial chain complex, then totalize, and consider the homology of that complex. The problem is that I think that assuming each space to be a locally compact polyhedron doesn't help that much because the chain complex of locally finite chains I think it's only functorial with respect to simplicial maps, so you would have to try to take many compatible simplicial approximations. Who knows... – Fernando Muro Nov 22 '13 at 18:40