Coefficients of cohomology theories can come from a variety of categories: fields, rings, sheaves ...

I wonder: what are the properties an object must satisfy in order to be a legitimate candidate for coefficients of a cohomology theory? What is the most general such object or structure in some suitable sense?


closed as unclear what you're asking by Fernando Muro, Stefan Kohl, abx, Francois Ziegler, Sean Lawton Apr 15 at 1:44

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    $\begingroup$ This is a very vague question: cohomology theories on what? If the answer is "topological spaces", or possibly "topoi", the most general thing you can take the cohomology of is a sheaf of spectra, but of course you can have (co)homology of many other weirder objects $\endgroup$ – Denis Nardin Apr 14 at 6:58
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    $\begingroup$ According to Grothendieck your categories of coefficients should always satisfy a six functors formalism. But I agree that it's not really a well posed question. $\endgroup$ – Dan Petersen Apr 14 at 7:12
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    $\begingroup$ @DanPetersen Well, then I want a precise definition of a six functors formalism :) (I know one for the category of schemes, of course, but if we are being ridiculously general...). $\endgroup$ – Denis Nardin Apr 14 at 11:43