On the profinite homotopy type of log schemes

We complete the program, initiated in [8], to compare the many different possible definitions of the underlying homotopy type of a log scheme. We show that, up to profinite completion, they all yield the same result, and thus arrive at an unambiguous definition of the profinite homotopy type of a log scheme. Specifically, in [8], we define this to be the profinite étale homotopy type of the infinite root stack, and show that, over $C$, this agrees up to profinite completion with the Kato-Nakayama space. Other possible candidates are the profinite shape of the Kummer étale site $X_{k\acute{e}t}$, or of the representable étale site of

. Our main result is that all of these notions agree, and moreover the profinite étale homotopy type of the infinite root stack is not sensitive to whether or not it is viewed as a pro-system in stacks, or as an actual stack (by taking the limit of the pro-system). We furthermore show that in the log regular setting, all these notions also agree with the étale homotopy type of the classical locus $X^{\mathrm{triv}}$ (up to an appropriate completion). We deduce that, over an arbitrary locally Noetherian base, the étale homotopy type of $G_m^N$ agrees with that of $B{\mu }_{\infty }^N$ up to completion.