$\DeclareMathOperator\Et{Et}$Let $X$ be a scheme and denote by $\Et(X)$ the associated étale homotopy type. Then by the work of Artin–Mazur, we know that for an abelian group $A$, we have
$$H^n(\Et(X),A)=H^n_{\text{ét}}(X,A)$$
and
$$\pi^1(\Et(X))\cong \pi^1_{\text{alg}}(X).$$
Therefore I wonder what $H_n(\Et(X),A)$ is? Naïvely I would hope that it is $H_{n,\text{ét}}(X,A)$, however, it seems that no survey or reference mentions this.

1$\begingroup$ What's your definition of étale homology? $\endgroup$– Denis NardinAug 3, 2020 at 8:23

$\begingroup$ In the case that $A=\mathbb{Q}_p$ (which is the case I'm most interested in), I know it as the dual of the étale cohomology. $\endgroup$– curious math guyAug 3, 2020 at 17:33

$\begingroup$ In that case, doesn't the theorem basically follow from the theorem for cohomology by taking duals? $\endgroup$– Denis NardinAug 4, 2020 at 5:10

$\begingroup$ It does absolutely, but I'm wondering if this agrees with more general coefficients. The way I've heard about constucting the étale homology is to consider étale cosheaves and their derived functor. $\endgroup$– curious math guyAug 4, 2020 at 13:59
1 Answer
I'm sure there are easier and better ways to think about this, but here's how I like to think about it.
Work on the big proetale site on all schemes, which maps to the proetale site of a point, $\pi: \mathrm{Sch}_{\mathrm{proet}}\to \ast_{\mathrm{proet}}$. Also, let's consider (hypercomplete) sheaves of anima. Then $\pi^\ast$ has a left adjoint $\pi_\natural$, and (for our given scheme $X$) $\pi_\natural(X)$ is a condensed anima that "is" the etale homotopy type of $X$. Concretely, if $X_\bullet\to X$ is a proetale hypercover by wcontractible $X_\bullet$, then $\pi_\natural(X)$ is represented by the simplicial extremally disconnected profinite set $\pi_0(X_\bullet)$. By adjunction, there is a natural map $X\to \pi^\ast \pi_\natural (X)$.
Now giving a $\mathbb Q_\ell$local system $\mathbb L$ (same for other coefficient rings) is the same as giving a map $X\to \pi^\ast B\mathrm{GL}_n(\mathbb Q_\ell)$, i.e. equivalently a map $\pi_\natural(X)\to B\mathrm{GL}_n(\mathbb Q_\ell)$, i.e. a $\mathbb Q_\ell$local system on the condensed anima $\pi_\natural(X)$.
Now what is homology of $X$ with coefficients in $\mathbb L$? One definition uses the formalism of solid $\mathbb Q_\ell$sheaves $D_\blacksquare(X,\mathbb Q_\ell)$ (no reference for them in the case of schemes yet, sorry!). In that setting, pullback $D_\blacksquare(\ast,\mathbb Q_\ell)\to D_\blacksquare(X,\mathbb Q_\ell)$ has a left adjoint, which takes $\mathbb L$ to the homology of $X$ with coefficients in $\mathbb L$. Here $D_\blacksquare(\ast,\mathbb Q_\ell)$ is the "usual" derived category of solid $\mathbb Q_\ell$modules.
Working with the condensed anima $\pi_\natural(X)$ instead, one can apply a similar procedure, and the two answers will agree, so indeed the homology of $X$ with coefficients in $\mathbb L$ agrees with the homology of $\pi_\natural(X)$ with coefficients in $\mathbb L$.
In practice, if $X$ is sufficiently nice, then these homology groups will be finitedimensional over $\mathbb Q_\ell$ and dual to cohomology, so this would indeed follow from the usual statements about cohomology. For general $X$, this statement about homology is however slightly finer (as cohomology is always the dual of homology, but not conversely).
Edit: Here is a way to phrase the answer so that it does not involve a reference to $D_\blacksquare(X,\mathbb Q_\ell)$. One can instead use the full derived category of proetale $\mathbb Q_\ell$sheaves $D(X_{\mathrm{proet}},\mathbb Q_\ell)$; then, just like for sheaves of anima, the pullback along $f: X_{\mathrm{proet}}\to \ast_{\mathrm{proet}}$ has a left adjoint $f_\natural$, and $f_\natural \mathbb L\in D(\ast_{\mathrm{proet}},\mathbb Q_\ell)$ is a complex of condensed $\mathbb Q_\ell$vector spaces that can be considered as the homology of $\mathbb L$. Again, a similar construction can be done for $\pi_\natural(X)$, and these two notions of homology agree.
On the other hand, I expect that it is extremely difficult to compute this notion of homology even for $X=\mathbb P^1_k$ for $k$ an algebraically closed field. However, passing to the solidification, one can compute it in practice, and I guess it usually agrees with the homology of the ArtinMazur pro(homotopy type).

$\begingroup$ I do like very much everything you do on analytic rings and condensed mathematics, but I have trouble to see why solid modules are necessary here. I mean that the derived category of proétale ladic sheaves, as documented in your joint paper with Bhatt, is sufficient to make sense of the left adjoint of the pullback functor: we can simply apply the $l$adic version of $\pi_\sharp$ to $\mathbb{L}$ to get homology with coefficients. Could you explain what to expect from the consideration of solid modules everywhere? $\endgroup$ Mar 17, 2021 at 12:30

$\begingroup$ As long as $X$ is nice, you are right, the left adjoint exists on the usual level, but not for general $X$. If $X=\mathrm{Spec} \mathbb Q$ for example, $H^1(X,\mathbb F_2)$ is infinite, and dually $H_1(X,\mathbb F_2)$ ought to be an infinite product of $\mathbb F_2$'s. So you need some solid formalism, I think. $\endgroup$ Mar 17, 2021 at 12:35

$\begingroup$ On the other hand, I wouldn't be completely sure how to define homology of the proetale homotopy type with coefficients. I guess usually one would use coefficients that come from stage in the prolimit, and then take the projective limit of the homologies? I think this implicit passage to the projective limit is what's mirrored here by the use of solid modules, specifically $\mathbb Z[S]^\blacksquare = \varprojlim_i \mathbb Z[S_i]$ for profinite $S=\varprojlim_i S_i$. $\endgroup$ Mar 17, 2021 at 12:37

$\begingroup$ I should maybe mention that one advantage of using the condensed anima $\pi_\natural(X)$ in place of the ArtinMazur prohomotopy type is that $\pi_\natural(X)$ can see general $\mathbb Q_\ell$local systems (as in the argument above), which I think is not the case for the latter (whose $\pi_1$ is given by the SGA3 $\pi_1$ which is too small for say nodal curves of positive genus). But then to define the homology of a condensed anima, I think I really need the solid formalism. $\endgroup$ Mar 17, 2021 at 12:47

1$\begingroup$ Right  this is the left adjoint I was referring to in the beginning of my comments. But if you take the one on general proetale $\mathbb Q_\ell$ (or even $\mathbb F_\ell$)sheaves, it gets more complicated. So you have three options: Restrict to nice $X$ and nice coefficients, and get a good left adjoint; take general $X$ and solid sheaves, and get a good left adjoint; take general $X$ and all proetale sheaves, and get a weird left adjoint. $\endgroup$ Mar 17, 2021 at 13:36