Universal cohomology theories

TL;DR

This paper constructs universal (co)homology theories within categories, using representability of functors in R-linear abelian categories of motivic type, unifying various homological concepts.

Contribution

It introduces a universal framework for (co)homology theories applicable to any category, generalizing existing concepts through representability and motivic categories.

Findings

Universal (co)homologies are obtained via functor representability.

Universal homology on a point yields hieratic R-modules.

Grothendieck $oundary$-functors are recovered as additive relative homologies.

Abstract

We furnish any category of a universal (co)homology theory. Universal (co)homologies and universal relative (co)homologies are obtained by showing representability of certain functors and take values in $R$-linear abelian categories of motivic nature, where $R$ is any commutative unitary ring. Universal homology theory on the one point category yields "hieratic" $R$-modules, i.e. the indization of Freyd's free abelian category on $R$. Grothendieck $\partial$-functors and satellite functors are recovered as certain additive relative homologies on an abelian category for which we also show the existence of universal ones.