Let $\mathcal{S}$ be the ($\infty$-)category of spaces and $\mathcal{S}_{p-\text{finite}}$ the full subcategory spanned by the $p$-finite spaces (that is, the spaces with finitely many connected components, each having finitely many non-vanishing homotopy groups all of which are finite $p$-groups). One can consider the pro-completion (in the $\infty$-cateogrical sense, or any model category representation of it) of $\mathcal{S}_{p-\text{finite}}$ which we denote by $\mathcal{S}_p^{\vee}$ and call the ($\infty$-)category of $p$-profinite spaces. In this setup, there is a canonical adjunction: $$ (-)_p^{\vee}: \mathcal{S} \leftrightarrows \mathcal{S}_p^{\vee}:M $$

Another fact is that although $\mathcal{S}_p^{\vee}$ is not closed, for a $p$-finite space $K$, viewed as a $p$-profinite space, and any $p$-profinite space $X$, there is an exponential object $\underline{Map}(K,X) \in \mathcal{S}_p^{\vee}$ (satisfying the usual properties).

Now, given a $p$-finite space and a space $X$, we have a canonical map $$ Map(K,X)_p^{\vee} \to \underline{Map}(K,X_p^{\vee}) $$

Question: Under which conditions on $K$ and $X$, this map is an equivalence?

I am mainley intereseted in the case that $K = B(\mathbb{Z}/p^k)$ and $X$ is $\pi$-finite (i.e. like $p$-finite, but the homotopy groups are only required to be finite and not $p$-finite), but I am also interested in more general cases.

Some observations:

1) For $X$ a $p$-finite space, this is a tautology.

2) For $X$ a nilpotent $\pi$-finite space this is is also easy since $X$ is a product of finitely many $q$-finite spaces for some primes $q$ and this reduces to (1).

3) Since the functor $M$ is conservative (e.g. Dag XIII 3.2.3) it is equivalent to ask when the map of spaces $$ Map(K,X)_p^{\wedge} \to Map(K,X_p^{\wedge}) $$

is an equivalence, where $(-)^{\wedge}_p$ is the $p$-adic completion (i.e. the composition of $M$ with $(-)^{\vee}_p$).

4) For good $X$ (I think it includes $\pi$-finite), the $p$-adic completion is a Bousfield localization with respect to mod $p$-homology.

  • $\begingroup$ This could be related: "Mapping Spaces of Compact Lie Groups and p-adic completion", Proc. AMS. 117 (1993), pp. 251-258. $\endgroup$ – Ofra Nov 15 '17 at 19:20
  • $\begingroup$ @Ofra, thanks for the reference. It is related, though not exactly what I am looking for as it covers a range of examples with small overlap with what I am interested in (In particular, they study maps from a space which is not p-finite). It is somewhat encouraging though, as they do prove something similar in spirit to what I am looking for, although in a different context. $\endgroup$ – KotelKanim Nov 15 '17 at 19:49

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