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)?
1 Answer
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 prosystems is largely a matter of choice.
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) prosystem of spaces $\{U_i\}_i$. [2]
Once we have the prospace $\mathrm{sh}(\mathscr{X})$, 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 prosystem by "$\pi$finite approximations" (i.e. by the prosystem 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) prosystem $\{X_i\}_i$ 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 prosystems $\ast→ \{X_i\}_i$, 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 prosystem of groups $$ \{\pi_n(X_i,x_i)\}_i$$ Since all $X_i$ are $\pi$finite, this is in fact a prosystem of finite groups. But prosystems 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).
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.
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 9780691140490/pbk; 9780691140483/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 proobjects, and then see that Lurie's construction produces the same proobject. 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 prosystem 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, 951 (2004). ZBL1045.55007.
I'm not aware of a reference treating in detail the comparison of model categories of proobjects.
[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 ArtinMazur 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.

$\begingroup$ I've always been confused about how the finiteness conditions enter in. Is there a place where this modern perspective is written up carefully? $\endgroup$– Tim Campion ♦Dec 27, 2018 at 18:53

$\begingroup$ @TimCampion What finiteness conditions do you mean? The profinite completion is just for convenience (it's kind of hard to understand what information is actually contained in a prospace, a profinite space is more accessible) $\endgroup$ Dec 27, 2018 at 18:55

$\begingroup$ The theory in this language can be found in Higher Topos Theory, section 7.1.6. Another related paper where a generalization of this theory is considered is arxiv.org/abs/1807.03281 $\endgroup$ Dec 27, 2018 at 18:56

1$\begingroup$ I think I'm happy with the shape theory in HTT, it's really the part where one passes from a prospace to a pro($\pi$finite)space that confuses me (which I don't think is to be found in HTT). For instance, has anybody actually proved in the literature that this construction agrees with what Friedlander does? I see now from the citation in the exodromy paper that Appendix E of SAG seems to be about this, maybe it's there? $\endgroup$– Tim Campion ♦Dec 27, 2018 at 19:04

$\begingroup$ @TimCampion Yes, I forgot a lot of the theory was also there. The comparison with Friedlander should follow immediately from the description of the sheafification functor in the hypercomplete case (where the sheafification can be computed by a colimit over all the hypercovers). I'll try to hunt down a few more precise references and add them to this answer this evening or tomorrow $\endgroup$ Dec 27, 2018 at 19:44