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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?

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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