Domain of left adjoint from condensed sets to anima

Question

$\DeclareMathOperator\Hom{Hom}$Let $X$ be a condensed set in the sense of Clausen-Scholze. If there is a universal anima $Y$ (or $\infty$ groupoid, or homotopy type) together with a map of condensed anima $X \to \pi^*Y$ that induces an equivalence $$\Hom(Y,Z) \to^\sim \Hom(X,\pi^* Z)$$ for every anima $Z$, we say that $Y = \pi_\#X$ is anima (or homotopy type) associated to $X$ and that the left adjoint $\pi_\#$ is defined on $X$

In Section 11 of "Lectures on Analytic Geometry" it is proved that $\pi_\#$ is defined on the subcategory of CW-complexes. Is it true more generally that $\pi_\#$ is defined (and equals the usual weak homotopy type) on the subcategory of locally contractible compactly generated Hausdorff spaces?

If not, what is the largest class of topological spaces on which this partial adjoint is defined?

Answer 1

Great question!

The answer is Yes. Let me elaborate a little. The question is more generally about the left adjoint to the inclusion $\mathrm{An}\to \mathrm{CondAn}$ from anima to condensed anima. This left adjoint is only partially defined; what exists in general is the functor $F: \mathrm{CondAn}\to \mathrm{Pro}(\mathrm{An})$ from condensed anima to pro-anima. Then $F(X)$ is an anima precisely when the left adjoint to $\mathrm{An}\to \mathrm{CondAn}$ exists on $X$, in which case it equals $F(X)$.

Lemma. The functor $F$ inverts the map $X\times [0,1]\to X$ for any condensed anima $X$.

In other words, for any condensed anima $X$ and any anima $Y$, the map $$ \mathrm{Hom}(X,Y)\to \mathrm{Hom}(X\times [0,1],Y) $$ is an isomorphism. Writing $X$ as a colimit of profinite sets $S$, this reduces to the case $X=S$. But then the condensed anima $\mathrm{Hom}(S,Y)$ is actually itself an anima (if $S=\mathrm{lim}_i S_i$, it is the colimit $\mathrm{colim}_i \mathrm{Hom}(S_i,Y)$), and by adjunction one reduces to the case that $X=\ast$. In this case, this is part of Lemma 11.9 in Analytic Geometry.

The lemma shows that $F$ factors over a functor $$\mathrm{CondAn}[W^{-1}]\to \mathrm{Pro}(\mathrm{An})$$ from the $\infty$-category obtained from $\mathrm{CondAn}$ by inverting homotopy equivalences.

In particular, any condensed anima $X$ which is homotopy equivalent to an anima $Y$ has the property that $F(X)=Y$. In particular, if $X$ comes from a contractible topological space, then $F(X)=\ast$.

Now if $X$ is just locally contractible, then one can find a cover $X=\bigcup_i U_i$ by contractible $U_i$, and then covers $U_i\cap U_j=\bigcup_k U_{ijk}$, etc., leading to a hypercover of $X$ by disjoint unions of contractible $U\subset X$. As $F$ commutes with colimits, this writes $F(X)$ as a colimit of disjoint unions of points, and hence $F(X)$ is an anima (which is the usual (weak) homotopy type).