What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?

Question

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?

Answer 1

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

Answer 2

You can certainly form a category of sheaves of spectra on a space that is quite analogous to the derived category of sheaves of chain complexes. Roughly speaking, a presheaf of spectra is a called a (homotopy) sheaf (or stack) if it satisfies the sheaf condition, except where limits are replaced by homotopy limits. In this sense, algebraic $K$-theory is a sheaf of spectra in the Zariski topology. My definition is immediately fulfilled by Quillen's localization theorem. Brown and Gersten do more work to satisfy another definition, now sometimes called a hyper-sheaf (though sometimes a just a sheaf), and from this prove the Brown-Gersten-Quillen spectral sequence. Jardine, Joyal, and Thomason did further work on this definition. A basic theorem that illustrates the homotopy theory point of view without involving spectra is that the category of homotopy sheaves of chain complexes, up to quasi-isomorphism, is equivalent to the derived category of sheaves of abelian groups. Roughly speaking, the procedures of taking sheaves and taking derived categories commute.

But this category of sheaves of spectra is not that of May-Sigurdsson. Their category is a homotopy invariant of a space, while the category of sheaves knows much more about a space; consider skyscraper sheaves. Their category is that of locally constant sheaves. You can think of one of their functors as the usual functor followed by a functor that approximates a general sheaf by a locally constant one. Note that even in the setting of chain complexes of sheaves, the thick subcategory generated by the local systems is richer than the derived category of local systems; in the simply connected case, that's just $D(pt)$. But any element of $H^*(X)$ can be thought of as a nontrivial extension between (shifted) trivial local systems. Indeed, the thick subcategory generated by trivial local systems is the derived category of modules over the $A_\infty$-algebra $H^*(X)$, or the DGA $C^*(X)$.

A minor point: the category of chain complexes is not a subcategory of the category of spectra. It is best thought of as restriction of modules along a map of rings, thus unlikely to be fully faithful; and in the derived setting it is difficult to disentangle "full" from "faithful."

Answer 3

I feel like we gave up too quickly on parameterized spectra as a reasonable (if necessarily incomplete) answer to this question; we don't get the two formalisms in the question as special cases of an overarching formalism but we do manage to make the algebraic topology side of the story look at least deceptively more like the algebraic geometry side. So I'd like to campaign a bit for this point of view; the goal is just to remedy the impression that the two stories are "seemingly orthogonal."

I am going to try to say everything below in a model-independent way. Think of a space / homotopy type $X$ as an $\infty$-groupoid. Define an $\infty$-local system on $X$ to be an $\infty$-functor $L : X \to C$, where $C$ is an $\infty$-category. Then I claim that:

The $\infty$-colimit of $L$ should be thought of as the homology of $X$ with coefficients in $L$, while the $\infty$-limit of $L$ should be thought of as the cohomology of $X$ with coefficients in $L$.

This is the most general thing I know which deserves to be called the (twisted) (nonabelian) cohomology or homology of a homotopy type (as opposed to something like a scheme).

Example. Let $X$ be discrete and let $C$ be an ordinary category. Then a functor $L : X \to C$ is just a collection $c_x$ of objects in $C$. The colimit is the coproduct $\bigsqcup_x c_x$ and the limit is the product $\prod_x c_x$. In particular, if $L$ is a constant functor with constant value $c$, then the colimit is the tensor $X \otimes c$ and the limit is the cotensor $[X, c]$. Specializing to $C = \text{Ab}$ gives the usual homology resp. cohomology of a discrete space with coefficients in an abelian group.

Example. Let $X = BG$ be the classifying space of a discrete group $G$. Then a functor $L : BG \to C$ is essentially an object $c$ of $C$ together with an action of $G$. The colimit is the (homotopy) quotient of the action and the limit is the (homotopy) fixed points of the action. Specializing to $C = \text{Ch}$ (here I mean the $\infty$-category presented by chain complexes) gives usual group homology resp. cohomology.

Example. Any object $c \in C$ defines a constant local system on any space $X$. The colimit is the tensor $X \otimes c$ and the limit is the cotensor $[X, c]$ as in the discrete case. Specializing to $C = \text{Sp}$ gives usual homology resp. cohomology with coefficients in a spectrum.

Example. Let $C = \text{Sp}$. Then a functor $L : X \to \text{Sp}$ is a parameterized spectrum. If $X$ is pointed and connected, then we can think of such a thing as a spectrum $E$, namely the spectrum associated to the basepoint, together with an action of $\Omega X$. Here the colimit and limit reproduce twisted versions of homology and cohomology with coefficients in a spectrum. In particular, if $E$ is an Eilenberg-MacLane spectrum $HA$, then the only possible twists are given by actions of $\pi_1(X)$ on $A$, and we recover homology and cohomology with coefficients in local systems in the usual sense.

Example. Let $C = \text{Spaces}$ be the $\infty$-category of spaces. Then a functor $L : X \to C$ is, by a suitable version of the Grothendieck construction, the same thing as a bundle $\pi : Y \to X$. The total space $Y$ is in fact the homology / colimit of $L$, whereas the cohomology / limit is the space of sections of $\pi$. In particular, if $L$ is the constant local system with constant value $c$, then the homology / colimit is the tensor $Y = X \times c$, with $\pi : X \times c \to X$ the natural projection, and the cohomology / limit is the cotensor $[X, c]$ (here I mean the space of maps from $X$ to $c$).

To make this look a bit more like six functors, start by assigning to each space $X$ the $\infty$-category $\text{Loc}(X)$ of $C$-valued local systems on $X$. Every map $f : X \to Y$ induces a pullback map

$$f^{\ast} : \text{Loc}(Y) \to \text{Loc}(X)$$

and if $C$ is suitably nice this map will have left and a right adjoints (left and right Kan extension along $f$)

$$f_{!}, f_{\ast} : \text{Loc}(X) \to \text{Loc}(Y).$$

When $Y$ is a point these two pushforwards reproduce colimits resp. limits and hence reproduce homology resp. cohomology in the sense of the above definition. In particular when $C = \text{Sp}$ we are in fact assigning to a space $X$ the $X$-parameterized spectra, and pulling and then pushing from and then back to a point recovers the usual notion of homology resp. cohomology of $X$ with coefficients in a spectrum.

Answer 4

A very satisfactory answer to the question (and much more) recently appeared in a preprint of Marco Volpe, "Six functor formalism for sheaves with non-presentable coefficients". https://arxiv.org/abs/2110.10212

Answer 5

You can find an elementary categorical comparison of different contexts of the sort you ask about in a paper by Halvard Fausk, Po Hu, and myself, entitled ``Isomorphisms between left and right adjoints''. It is on my web page: http://www.math.uchicago.edu/~may/PAPERS/FormalFinalMarch.pdf

To quote from its abstract, ``One point is to differentiate between the classical framework that arises in algebraic geometry and a deceptively similar, but genuinely different, framework that arises in algebraic topology. Another is to make clear which parts of the proofs of such results are formal.''