Homology of the étale homotopy type

Question

$\DeclareMathOperator\Et{Et}$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.

Answer 1

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 pro-etale site on all schemes, which maps to the pro-etale 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 w-contractible $X_\bullet$, 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 finite-dimensional 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 pro-etale $\mathbb Q_\ell$-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 Artin--Mazur pro-(homotopy type).