Modules over the integral dual Steenrod algebra as linear functors

Question

Let $\text{Latt}$ denote the category of lattices, i.e., finitely generated free abelian groups. In the appendix to Lecture 4 of Condensed.pdf, Scholze considers functors $F \colon \text{Latt} \to \mathcal D(\mathbb Z)$ that are additive: $F(A \oplus B) \cong F(A) \oplus F(B)$.

It seems to be a folklore result that the category of such functors is equivalent to the category of modules over the integral dual Steenrod algebra $\mathbb Z \otimes_{\mathbb S} \mathbb Z$. (Here $\mathbb S$ denotes the sphere spectrum.) As I'm not very well-versed in homotopy theory/higher algebra, I have no idea how this equivalence works.

Q: How does this equivalence work? Is there a reference that explains it?

Answer 1

First I should say that Clausen-Scholze are not considering functors which preserve direct sums, but rather all functors. (This is likely why, in the end, one needs to know something about the homology of Eilenberg-MacLane spaces rather than the homology of the Eilenberg-MacLane spectrum. That guess/observation I learned from Piotr Pstragowski.)

But if you still want to know about functors that preserve direct sums, then it is indeed true that there's an equivalence $\mathsf{Fun}^{\oplus}(\mathsf{Latt}, \mathsf{D}(\mathbb{Z})) \simeq \mathsf{Mod}_{\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}}$. The 'intuitive' explanation is the following: on the one hand, if we have a functor $F$ on the left hand side, then its value on $\mathbb{Z}$ is some $\mathbb{Z}$-module $M$ and it's equipped with a bunch of extra structure, as a $\mathbb{Z}$-module, coming from the functoriality. This extra structure turns out to be a second $\mathbb{Z}$-module structure which `commutes' with the first, hence $M$ gets the structure of a bimodule, or a $\mathbb{Z}\otimes_{\mathbb{S}} \mathbb{Z}$-module. In the other direction, given a bimodule $M$, we can define a functor by $\Lambda \mapsto M \otimes_{\mathbb{Z}} \Lambda$, using the leftover $\mathbb{Z}$-module structure on $M$ to make this a functor from $\mathbb{Z}$-modules to $\mathbb{Z}$-modules.

To be more precise, the equivalence proceeds in three steps.

1. There is an inclusion $\mathsf{Latt} \to \mathsf{Mod}_{\mathbb{Z}}^{\mathrm{cn}}$ into connective $\mathbb{Z}$-modules (i.e. chain complexes concentrated in nonnegative degrees). Left Kan extending along this gives an equivalence $\mathsf{Fun}^{\oplus}(\mathsf{Latt}, \mathsf{Mod}_{\mathbb{Z}})\stackrel{\simeq}{\to} \mathsf{Fun}^{\mathrm{colim}}(\mathsf{Mod}^{\mathrm{cn}}_{\mathbb{Z}}, \mathsf{Mod}_{\mathbb{Z}})$
2. There is a canonical equivalence $\mathsf{Fun}^{\mathrm{colim}}(\mathsf{Mod}^{\mathrm{cn}}_{\mathbb{Z}}, \mathsf{Mod}_{\mathbb{Z}}) \simeq \mathsf{Fun}^{\mathrm{colim}}(\mathsf{Mod}_{\mathbb{Z}}, \mathsf{Mod}_{\mathbb{Z}})$.
3. Every $\mathbb{Z}$-bimodule, i.e. $\mathbb{Z} \otimes_{\mathbb{S}} \mathbb{Z}$-module, $M$, defines a colimit preserving functor $N \mapsto M\otimes_{\mathbb{Z}} N$. This procedure gives a functor $\mathsf{Mod}_{\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}} \to \mathsf{Fun}^{\mathrm{colim}}(\mathsf{Mod}_{\mathbb{Z}}, \mathsf{Mod}_{\mathbb{Z}})$ which turns out to be an equivalence.

Here (1) is a formal consequence fo the fact that Latt gives compact projective generators of $\mathsf{Mod}^{\mathrm{cn}}_{\mathbb{Z}}$ (see, e.g., Higher Topos Theory 5.5.8.22 for the equivalence once you know these are compact projective generators), (2) is a formal consequence fo the fact that $\mathsf{Mod}_{\mathbb{Z}}$ is stable (so there is a unique way to extend a colimit-preserving functor on connective modules to all modules), and (3) can be proven using the Barr-Beck theorem, but a reference (for a stronger statement) would be Higher Algebra 4.8.5.16 (though you'll have to do some translation of the notation...).

Answer 2

Theorem A from Jibladze and Pirashvili's 1991 paper "Cohomology of Algebraic Theories," http://www.rmi.ge/~jib/pubs/jipicat.pdf , identifies Mac Lane cohomology with the cohomology of additive functors from finitely generated projective $R$-modules to $R$-modules. In the case $R=\mathbb{Z}$, the finitely generated projective $R$-modules are simply the lattices that you mentioned.

As far as I know, that 1991 paper is the correct one to cite for the result you mentioned. But the result also needs the connection between Mac Lane cohomology and the Steenrod algebra, and that came earlier. Sometimes people define Mac Lane cohomology in a way which does not make any obvious reference to Eilenberg-Mac Lane spaces or the Steenrod algebra at all, so it is nontrivial to see the relationship. Mac Lane cohomology is isomorphic to the stable cohomology of Eilenberg-Mac Lane spaces with appropriate coefficients; when those coefficients are suitably trivial, then of course the stable cohomology of Eilenberg-Mac Lane spaces is simply the Steenrod algebra. The right citation here is probably Eilenberg and Mac Lane "Homology theories for multiplicative systems" from 1951, or Mac Lane "Homologie des anneaux et des modules" from 1956, but the result is described a nice concise way on pg. 257 of the Jibladze-Pirashvili paper. Another nice modern exposition of the relationship between Mac Lane cohomology and Eilenberg-Mac Lane spaces is (the dual of) Definition 1.1 in Pirashvili-Waldhausen's 1992 "Mac Lane homology and topological Hochschild homology", here: https://core.ac.uk/download/pdf/82212669.pdf .