I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to every "space" $X$ a triangulated category $\newcommand{\D}{\mathsf D} \D (X)$, its derived category, and to each morphism $f \colon X \to Y$ derived pushforward/pullback maps $f_\ast, f_!, f^\ast, f^!$ between the derived categories (as well as $\mathcal Hom$ and $\otimes$), which are required to satisfy a list of formal properties: adjunctions, base change theorems, projection formula, etc. The usual cohomology of a space is given by the functor $(a_X)_\ast (a_X)^\ast$ applied to our choice of coefficients in $\D(\mathrm{pt})$, where $a_X$ is the map from $X$ to a point. But the formalism also incorporates sheaf cohomology, and allows us to talk in a uniform language also about other cohomology theories (like Borel-Moore, intersection homology) by other combinations of the six functors, or to work freely in a relative setting (e.g. there are relative versions of things like Künneth theorem) or to talk about enhanced versions of cohomology (like mixed Hodge theory) by an appropriate other choice of functor $\D(-)$.
However my impression is that this approach is more popular among for instance algebraic geometers than honest topologists. Topologists who talk about generalized cohomology theories of course talk about K-theory, cobordism, elliptic cohomology… I understand much less of this. In any case, here a generalized cohomology theory is considered to be an object of the stable homotopy category $\mathrm{SH}$.
I have been wondering for a while what (if anything) it means that there are these two seemingly orthogonal ways of thinking about what cohomology is, which seem to allow for generalizations in different directions. Is there any way to unify the two approaches?
To ask a more precise question, is there a functor $\D$ which assigns to a nice enough space $X$ a triangulated category $\D(X)$, together with a "six functors" formalism satisfying the usual properties, such that $\D(\mathrm{pt}) \cong \mathrm{SH}$? Even better, can one in that case also find a subfunctor $\D' \subset \D$, stable under six functors, which assigns to a space $X$ the (for instance unbounded) derived category of abelian sheaves on $X$?
Some more speculative comments: If $\D(\mathrm{pt}) \cong \mathrm{SH}$ then possibly $\D(X)$ should be the category of spectra parametrized by the base space $X$ (as in parametrized homotopy theory), but I'm very ignorant about such things. I've understood that a large part of May-Sigurdsson's book is devoted to constructing functors $f^\ast$, $f_\ast$ and $f_!$ in parametrized homotopy theory – can these be considered as some kind of lifts of those in the usual derived category of abelian sheaves, or do they just have the same names? Is there a reason that $f^!$ does not appear; does Verdier duality fail in this context?
Best Answer
The answer is motivic stable homotopy theory! This assigns to any reasonable scheme $S$ a triangulated category $SH(S)$ whose objects represent generalized cohomology theories for $S$-schemes, and there is also a similar category $DM(S)$ whose objects represent ordinary cohomology theories.
These theories do almost everything you ask for, in particular they give a nice framework for cohomology and homology including Borel-Moore and compact support versions.
I could say a lot about this, but I have to go to sleep. Let me just point what is probably the best reference, namely the introduction to this preprint of Déglise and Cisinski:
http://www.math.univ-toulouse.fr/~dcisinsk/DM.pdf