Are higher etale homotopy groups topological groups in a natural way?

Question

Since etale fundamental group of a scheme $X$ is the group of natural automorphisms of the fibre functor of the category of finite etale covers of $X$, it comes with structure of a topological group. Is there some natural topology on higher etale homotopy groups (if we do not assume that $X$ is not geometrically unibranch)?

Answer 1

TL;DR The higher étale homotopy groups are the homotopy groups of the profinite completion of the shape of the étale topos. As such they are profinite groups. If you choose to see profinite groups as topological groups, group schemes or pro-systems is largely a matter of choice.

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How is the étale homotopy group defined? There are many ways, but possibly the most elegant understanding of it, can be obtained using the theory of the shape of an $∞$-topos. Under this theory, for any ($∞$-)topos $\mathscr{X}$ we can obtain a functor $$\mathrm{sh}(\mathscr{X}):\mathrm{Space}→\mathrm{Space}\,,$$ where $\mathrm{Space}$ is the $\infty$-category of spaces, by sending a Kan complex $K$ to $\Gamma(\mathscr{X},\underline{K})$, the (derived) global sections of the constant sheaf at $K$ [1]. Since this functor commutes with finite (homotopy) limits, it is of the form $$ \mathrm{sh}(\mathscr{X})(K)=\mathrm{colim}_i\mathrm{Map}(U_i,K)$$ for some (uniquely determined) pro-system of spaces $\{U_i\}_i$. [2]

Once we have the pro-space $\mathrm{sh}(\mathscr{X})$, we can consider its profinite completion, which in this language is simply the restriction of the functor to the ($∞$-)category of $\pi$-finite spaces (those spaces that have only finitely many nonzero homotopy groups, and such that all those homotopy groups are finite). This has also the nice technical advantage of ignoring the difference between a topos and its hypercompletion. This corresponds to replacing every $U_i$ in the above pro-system by "$\pi$-finite approximations" (i.e. by the pro-system of $\pi$-finite spaces with a map from $U_i$).

Long story short, we can associate to the étale topos of a scheme $X$ a (uniquely determined) pro-system $\{X_i\}_i$ of $\pi$-finite spaces. Moreover, since this construction is functorial in the scheme, to every geometric point $x:\mathrm{Spec}\,\Omega\to X$ we can associate a map of pro-systems $\ast→ \{X_i\}_i$, that is up to reindexing, a coherent choice of basepoint $x_i$ to each $X_i$. In particular, for any $n\ge 1$, we can consider the pro-system of groups $$ \{\pi_n(X_i,x_i)\}_i$$ Since all $X_i$ are $\pi$-finite, this is in fact a pro-system of finite groups. But pro-systems of finite groups are well known to be exactly the same thing as profinite groups.

In particular you could see $\pi_n(\mathscr{X},x)$ as some kind of topological group (since we know that profinite groups are the same thing as compact totally disconnected topological groups), although how useful that is in practice I couldn't say. You could also see it as a particular group scheme (and this can be useful to compare it to other constructions that naturally produce a group scheme).

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To see that this definition recovers the classical étale fundamental group when $n=1$, you need a theorem telling you that finite étale covers are equivalent to finite locally constant sheaves of sets over the étale site. Then it's just a matter of chasing universal properties along the various constructions.

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References

Shape theory for $\infty$-topoi is developed in section 7.1.6 of

Lurie, Jacob, Higher topos theory, Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14049-0/pbk; 978-0-691-14048-3/hbk). xv, 925 p. (2009). ZBL1175.18001.

(there you can find more references for the classical theory it generalizes)

Profinite homotopy theory, in particular the theory of profinite shape, is subsequently developed in Appendix E of

Lurie, Jacob Spectral Algebraic Geometry, preprint of the author's website.

In particular, to compare the profinite completion of the shape of $X_{ét}$ to the étale homotopy type constructed by Friedlander you need two things: first you need to compare the corresponding theories of pro-objects, and then see that Lurie's construction produces the same pro-object. The fact that, if $\mathscr{X}$ is a hypercomplete ∞-topos, the constant sheaf $\underline{K}$ is given by $$Y\mapsto \mathrm{colim}_{U_\bullet\to Y} \mathrm{Map}(\pi_0 U_\bullet,K)$$ where $\{U_i\}$ is the pro-system of all (basal) hypercovers, is treated in

Dugger, Daniel; Hollander, Sharon; Isaksen, Daniel C., Hypercovers and simplicial presheaves, Math. Proc. Camb. Philos. Soc. 136, No. 1, 9-51 (2004). ZBL1045.55007.

I'm not aware of a reference treating in detail the comparison of model categories of pro-objects.

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[1] Note that when $\mathscr{X}$ is the topos of a paracompact Hausdorff space $X$, $\Gamma(\mathscr{X},\underline{K})$ is just the space of maps from $X$ to $K$. So we can imagine this functor as a substitute for the (nonexistent) functor $\mathrm{Map}(\mathscr{X},-)$.

[2] You can see the classical Artin-Mazur description in terms of hypercovers as a formula for computing the sheafification of the constant presheaf. This is the same reason why hypercovers appear in Verdier formula for computing sheaf cohomology.