Extraordinary cohomology as a derived functor?

Question

The purpose of this question is to find out whether one can view the Atiyah-Hirzebruch spectral sequence as a particular case of the "composition of derived functors" spectral sequence.

The Leray spectral sequence of a continuous map $f:X\to Y$ of topological spaces can be constructed as follows. Let $a_X:X\to pt$ and $a_Y:Y\to pt$ be the maps from $X$ and $Y$ respectively to the one point space $pt$; we obviously have $a_X=a_Y\circ f$. So for any sheaf $L$ on $X$ we have $(a_X)_\ast L=(a_Y)_*f_\ast L$. But $(a_X)_\ast$ is just the functor of the global sections and so is $(a_Y)_*$. Recall that if $A,B$ and $C$ are abelian categories and $F:A\to B,G:B\to C$ are left exact functors, then (under mild hypotheses) $R_\ast(G\circ F)$ is isomorphic to $R_\ast\circ G_\ast$ and for any object $X$ of $A$ there exists a spectral sequence abutting $R^\ast (G\circ F) X$ with the $E_2$ sheet given by $E_2^{pq}=R^pG(R^q F(X))$. See e.g Gelfand-Manin, Methods of homological algebra, 3.7.

Applying this to the case when $A$, $B$ and $C$ are the categories of sheaves of $X$, $Y$ and the point respectively we get a spectral sequence $(E^{pq}_r,d_d)$ abutting to $H^*(X,F)$ with the $E_2$ term given by $$E_2^{pq}=H^p(Y,R^q f_\ast L).$$

If $X$ and $Y$ are sufficiently nice (say finite CW complexes), $F$ is constant and $f$ is a locally trivial fibration with fiber $F$, then we get (assuming for simplicity that $Y$ is simply-connected) $$E_2^{pq}=H^q(Y,H^q(F)).$$

Now, if we have an extraordinary cohomology theory $h^\ast$, we can construct (under the hypotheses of the previous paragraph) the Atiyah-Hirzebruch spectral sequence: the $E_2$ sheet is given by $$E_2^{pq}=H^q(Y,h^q(F))$$ and the spectral sequence abuts to $h^*(X)$. This looks pretty similar to the Leray spectral sequence, so it seems natural to ask whether it can be obtained in a way similar to the one described above.

Namely, given an extraordinary cohomology theory $h^\ast$ and a continuous map $f:X\to Y$ of topological spaces, is there an "extraordinary direct image" functor $f^{ex}_*$ from sheaves on $X$ to sheaves on $Y$ which would be "functorial in $f$" and which would give $h^{\ast}(X)$ after deriving when $Y$ is a point?

If not, is there still a way to view the Atiyah-Hirzebruch spectral sequence as (a version of) the spectral sequence of the composition of two derived functors? (It may happen that one has to consider something other than the categories of sheaves, but I have no idea what this could be.)

Answer 1

I'd like to propose that the answer is "No, but..". The viewpoint I'd like to suggest is that thinking of "derived functors" is probably insufficient here (because we're secretly interested in homotopical categories that are not derived categories of abelian categories), but that we are relying essentially on the observation that $R\Gamma(X, -) = R\Gamma(Y, Rf_* -)$. So what follows is a sketch-construction that gives a positive answer to a related question you could've been asking: "We can get the Serre spectral sequence from the Leray spectral sequence. Can we get the Atiyah-Hirzebruch spectral sequence in a similarly sheaf-theoretic way?"

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We get the Leray spectral sequence by studying sheaves of complexes of $\mathbb{Z}$-modules. The "derived" category (of sheaves of complexes of $\mathbb{Z}$-modules) is the derived category of its heart (sheaves of $\mathbb{Z}$-modules) w.r.t. the usual $t$-structure, so it's reasonable that we get lots of mileage from looking at derived functors, composites of derived functors, etc.

Analogously, we can think of more general Atiyah-Hirzebruch type spectral sequences as arising from studying sheaves of spectra. This is not the derived category of its heart w.r.t. the usual $t$-structure (this heart is again sheaves of $\mathbb{Z}$-modules). But, this does give a way of thinking about Atiyah-Hirzebruch type spectral sequences:

We start with (say) the "constant sheaf of spectra" $\mathbf{E}$ on $X$, and we're interested in computing the (homotopy groups of the spectrum) $$R\Gamma(X, \mathbf{E}) = R\Gamma(Y, Rf_* \mathbf{E})$$ The spectral sequence of interests arises by filtering $Rf_* \mathbf{E}$ using the $t$-structure (Postnikov sections "on values"), $$\cdots \to \tau_{\geq k} Rf_* \mathbf{E} \to \tau_{\geq (k-1)} Rf_* \mathbf{E}\to \cdots \to \mathbf{E}$$ The $k$-th "associated graded piece" of this filtration is $\pi_k Rf_* \mathbf{E}$ (which, recall, is an object in the heart --- i.e., a sheaf of $\mathbb{Z}$-modules on $Y$). This gives rise to a spectral sequence (excuse the funny indexing!, and no comment on convergence) $$ E^2_{p,q} = \pi_{-p} R\Gamma\left(Y, \pi_{-q} Rf_* \mathbf{E}\right) \Rightarrow \pi_{-p-q} R\Gamma\left(Y, Rf_*\mathbf{E}\right) = \pi_{-p-q} R\Gamma(X,\mathbf{E})$$ The funny indexing was picked so that I can rewrite it as $$ H^p(Y, \pi_{-q} Rf_* \mathbf{E}) \Rightarrow E^{p+q}(X) $$

To recover the usual form of AH-SS we need the following observation (analogous to what we'd need to get the Serre spectral sequence via sheaf theory): If $f$ is nice (i.e., a fibration between reasonable spaces), then $\pi_{-q} Rf_* E$ will be the locally constant sheaf associated to $E^{q}(F)$ with its monodromy action.