# Bringing cohomology recipes from algebra to topology?

In algebra, cohomology theories are often defined very roughly like this. Start with a scheme $$X$$ you want to study. Form a category-with-extra-structure (a site) $$\mathcal{C}$$ whose objects are, roughly, diagrams the form Spec$$R\rightarrow X$$, again possibly with extra structure and conditions. (eg. flat maps, etale maps, 'prismatic' maps...)

Authors like to say: we are "probing" $$X$$ with Spec$$R$$'s.

Then you define a functor from $$\mathcal{C}$$ to commutative rings (Spec$$R\rightarrow X)\mapsto R$$, and if the conditions are favorable this is a sheaf on $$\mathcal{C}$$. Finally you define your cohomology theory at $$X$$ to be the cohomology of that sheaf.

In algebraic topology, homology theories are often defined in a seemingly similar way: Start with a space $$X$$ you want to study. Form a category $$\mathcal{B}$$ whose objects look roughly like $$Y\rightarrow X$$ with extra structure and conditions. (e.g. $$Y$$ is a union of simplices, a stably almost complex manifold, a framed manifold...)

Again, authors like to say: we are "probing" $$X$$ with $$Y$$'s.

But here the path differs and you declare your homology theory evaluated at $$X$$ to be something roughly like: "equivalence" classes of objects in $$\mathcal{B}$$. (Consequently you get something covariant as opposed to contravariant above.)

My questions are:

1) is something like the first version plausible in topology? Has it been tried? For example, instead of $$X\mapsto MU_{*}X$$ = bordism group over $$X$$, could you try to define a (pre)sheaf over $$\mathcal{B}_{X} :=$$ (bordism category over $$X$$) by, say, sending an object $$M\rightarrow X$$ to $$K(M)$$ (or any contravariant functor applied to $$M$$, as a replacement for the algebraic version where that functor is Spec$$R\mapsto R$$).

2) Can at least singular cohomology be gotten this way? It seems like a good start if you make $$\mathcal{B}$$ some suitable category of unions of simplices $$Y$$ mapping to $$X$$ and the contravariant functor is $$Y\mapsto [Y,\mathbb{Z}]$$ , "locally constant $$\mathbb{Z}$$-valued functions." But then there's some homological algebra needed to get the chain complex to come out right.

• For nice topological spaces, singular cohomology is just the sheaf cohomology of the constant sheaf on the site of open subsets of $X$. This is what the whole idea of a site was supposed to be a generalization of - it didn't come naturally to anyone, or at least not anyone except Grothendieck. – Will Sawin Dec 12 '19 at 2:28
• @WillSawin Oh right! Do you know if the roundabout thing with the "site of simplices" will get it to work for general spaces, or is that already a subtle issue? – John Greenwood Dec 12 '19 at 2:47
• My guess is there's probably a way to define the site to make it work. – Will Sawin Dec 12 '19 at 3:55
• For a space X, we may take as a site the category of simplices over X with the coarse topology. Sheaf cocohomology on this site with constant coefficients literally is singular cohomology for all spaces. – Denis-Charles Cisinski Dec 12 '19 at 8:05
• @Denis-CharlesCisinski thanks! I guess "coarse topology" were the magic words I was looking for... – John Greenwood Dec 13 '19 at 22:55