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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.

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    $\begingroup$ 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. $\endgroup$ – Will Sawin Dec 12 '19 at 2:28
  • $\begingroup$ @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? $\endgroup$ – John Greenwood Dec 12 '19 at 2:47
  • $\begingroup$ My guess is there's probably a way to define the site to make it work. $\endgroup$ – Will Sawin Dec 12 '19 at 3:55
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    $\begingroup$ 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. $\endgroup$ – Denis-Charles Cisinski Dec 12 '19 at 8:05
  • $\begingroup$ @Denis-CharlesCisinski thanks! I guess "coarse topology" were the magic words I was looking for... $\endgroup$ – John Greenwood Dec 13 '19 at 22:55

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