"The Z/2-cohomology functor from Top to GrVecSpaces factors through Unstable-A-Mod, and this is the largest algebraic category through which it factors."

Question

(Here A is the Steenrod algebra.)

I have it on good authority (p. 23) that this is true, but I can't quite make sense of it. The crux of the matter is something I've been wondering for a while:

Why exactly is it that cohomology operations are "the best we can do" (i.e./more generally, why is it that the finest structure we can impose on $E$-cohomology is that it be a module over $E^*E$)?

There might be something tied up in the qualification that we're talking about algebraic categories, which are defined here. I've never heard of these before. Presumably their properties show up in GrVecSpaces as manifestations of actual things on the topological side, which makes it seem like we're only restricting ourselves to categories that are going to preserve some information that we obviously (?) want to preserve, but probably there's more to it than that. And in any case I don't understand why those should be exactly the properties of GrVecSpaces that we care about.

(I'd love to hear about category theory stuff of course, but my main goal is to understand the boxed question.)

Answer 1

Here's a thought. Let $\newcommand{\hT}{h\mathrm{Top}}\hT$ be the homotopy category of spaces. I'll write $K_p=K(Z/2,p)$ for an Eilenberg MacLane space; it represents the functor $H^p=H^p({-},Z/2)$, so $\hT(X,K_p)=H^pX$.

Let $\newcommand{\K}{\mathcal{K}}\K$ denote the full subcategory of $\hT$ consisting of finite products of the $K_p$s.

Let $\newcommand{\A}{\mathcal{A}}\A$ denote the category of product preserving functors $\newcommand{\Set}{\mathrm{Set}}\K\to \Set$. Let $\newcommand{\grSet}{\mathrm{grSet}}U:\A\to \grSet$ be the functor defined by evaluating a functor at the objects $K_p$ of $\K$, so $U(F)_p=F(K_p)$.

The category $\A$ is precisely the category of unstable algebras over the steenrod algebra! You can think of an object $F$ in $\A$ as the collection of sets $U(F)_p=F(K_p)$, together with, for each "cohomology operation" $K_{p_1}\times\cdots \times K_{p_r}\to K_q$, a function $U(F)_{p_1}\times\cdots \times U(F)_{p_r}\to U(F)_q$, which satisfy all the identities they need to satisfy.

This shows the precise sense in which unstable algebras are an algebraic category.

The functor $\newcommand{\op}{\mathrm{op}}H^\*\colon \hT^\op\to \grSet$ factors tautologically through $U:\A\to \grSet$, via the functor $A\colon \hT^\op\to \A$ defined by $$A(X)(\prod K_{p_i}) = \hT(X, \prod K_{p_i}).$$

We'd like to show the functor $A$ is universal in some sense. So lets suppose we have some other small category $\newcommand{\L}{\mathcal{L}}\L$, that we let $\newcommand{\B}{\mathcal{B}}\B$ denote the category of product preserving functors $\L\to \Set$, and that we have a collection of objects $L_p\in \L$, at which we can evaluate to get a forgetful functor $U'\colon \B\to \grSet$. And that furthermore there is a functor $B\colon \hT^\op\to \B$ and a natural isomorphism $U'\circ B\approx H^\*$. In other words, $B$ is another factorization of $H^\*$ through an algebraic category.

Can we construct a comparison functor $C\colon \A\to\B$, ideally so that $C\circ A\approx B$ and $U'\circ C\approx U$? I don't know ... I suspect you need some additional conditions in order to do this. However, there is an obvious candidate, given by the coend construction. Thus, if $F\in \A$, then perhaps we ought to have $C(F)\in \B$ be defined by the coend in $\B$ $$ F\otimes_{\K} (B\circ i^\op)=\mathrm{Cok}(\coprod_K F(K)\times B(i(K)) \leftleftarrows \coprod_{K\to K'} F(K)\times B(i(K')))$$ where $i\colon \K\to \hT$ is the inclusion functor. But now I'm stuck ...

Answer 2

A little more information is at the page on infinitary Lawvere theories on the nLab. Basically, Charles is right: as cohomology is defined by the Eilenberg-Mac Lane spaces, everything about the resulting algebraic category is contained in the Eilenberg-Mac Lane spaces and their morphisms. In particular, if that category had more structure then that would be visible in the cohomology of the Eilenberg-Mac Lane spaces and would then be already in the algebraic category we were already in. In short it's because (as I proved on that page, in generality) the cohomology of the Eilenberg-Mac Lane spaces are the free algebras in this algebraic category.

Furthermore, to extend a point that Sam Isaacson makes, it's not true to say that $E^\star(X)$ is a module over $E^\star(E)$ with the usual interpretation of module as an abelian group with a (suitable) action of a ring. (Charles' answer is correct in his language, but you may not have noticed his choice of language.) What is correct is to say that $E^\star(X)$ is a module for the Tall-Wraith monoid $E^\star(\underline{E}_\star)$. This use of module generalises the idea of a module over an algebra. In particular, the fact that $E^\star(X)$ is an $E^\star$-algebra is included in this description. More on the nLab page Tall-Wraith monoids and the papers linked therein.

(All of this holds for any generalised cohomology theory.)

Answer 3

My impression is that Haynes Miller uses the term "algebraic category" in the informal sense in his notes (but I may be wrong). So let me answer this informal version first: one could argue that there often is an Adams spectral sequence ${\rm Ext}_{E^\ast E}\ (E^\ast Y, E^\ast Y) \Rightarrow [L_E X, L_E Y\ ]$ that allows you to (almost) reconstruct the $E$-localized homotopy picture from the $E^\ast E$-module information. In that (admittedly vague) sense the $E^\ast E$-module $E^\ast X$ already contains all of the information you might hope for and an algebraic refinement is neither necessary nor plausible.

Note that for many $E$ (in particular for $E=H{\mathbb Z}_p$), you can compute $[E^{\land n}, E]$ in terms of $E^\ast E$ and the multiplication $E\land E \rightarrow E$. In those cases $E^\ast E$ actually describes all possible $n$-ary natural transformations $E^\ast X \otimes \cdots \otimes E^\ast X\rightarrow E^\ast X$, not just the unitary ones.

Now back to the formal question, whether $H:{\rm Top} \rightarrow A-{\rm Mod}$ admits a factorization through a different algebraic category in the precise sense of The Joy of Cats: I'm pretty ignorant about these issues, but Mike Mandell has studied an enrichment of the singular cochain functor using $E_\infty$-operads, see for example Cochains and Homotopy Type. I wonder if his target category is algebraic in the formal sense?