Hi guys, I am back! Sorry for the naive question:

We have that the derived category (homotopy category if you prefer) of Top and of sSet (wrt the Quillen model structure) are quillen equivalent. On the other side, there is a quillen adjunction between Cat and sSet (wrt the Joyal model structure). I'd like to compare functors (up to equivalence) between groupoids and maps up to homotopy between the geometric realization of their nerve.

In the other direction, I'd like to compare maps between 1-truncated spaces up to homotopy and functors between their fundamental groupoid

At the beginning I thought this was a derived equivalence, meaning that functors up to equivalence correspond to map up to homotopy on the other side (say our spaces are always CW complexes).

But then I realized that it would imply $ \pi_1(X) = $(? not pointed ?) $= [S^1, X] = Fun( \hat{\pi}_1(S^1), \hat{\pi}_1(X) ) / \sim $. Now groupoids are equivalent to their automorphism groups as categories, so $= Hom_{Grp}(\pi_1(S^1), \pi_1(X) ) /conj $ which is in bijection with the conjugacy classes of $\pi_1(X)$ rather than with $\pi_1(X)$ itself.

I guess the error is that I am using two different model structures in the middle, so a priori they yield different derived morphisms. But is there a way to recover $[X,Y]$ from $Fun(\hat{pi}_1(X), \hat{\pi}_1(Y) )$ without modding out by equivalence? Or maybe more wisely: is there a model structure on cat that makes $(N,ho)$ a quilen adjunction between the quillen model structure on sSets?

Or maybe the problem is in $\pi_1(X) = [S^1,X]$?

@AndreaMarino Well, that statement is false (the rhs is the set of conjugacy classes of the lhs)

$π_1$ is the set of pointed maps up to pointed homotopy, that's not the same as the set of maps up to homotopy, as you've just discovered ;)

If I'm not mistaken, there are no nontrivial left-exact localizations of the $\infty$-topos $\mathsf S$ of spaces (i.e. there are exactly two left-exact localizations of $\mathsf S$, namely the terminal category and $\mathsf S$ itself). This stands in stark contrast to the stable case: the Bousfield lattice (i.e. the lattice of localizations of the $\infty$-category $\mathsf{Sp}$ of spectra) is quite rich. I find this strange.

There's a whole theory of unstable localization, but the fact that these localizations can't be left-exact seems to take us outside the realm of $\infty$-topos theory.

So what's the next fallback position of not-quite-left-exact-localizations? What types of localizations of $\mathsf S$ should we be especially interested in?

Should we be interested in semi-left-exact localizations, which interact nicely with the locally cartesian closed structure?

Or should we stop thinking in terms of localizations? Maybe particularly nice factorization systems should be the object of study. Factorization systems where the left class is closed under base change have been called modalities and seem to have some kind of homotopy-type-theory interpretation.

I guess I'm wondering what the "correct" "unstable analog" of the Bousfield lattice is.

@TimCampion I was wondering some time ago whether or not modalities which contain all the acyclic maps can be classified in terms of the bousfield lattice of spectra. I mean any bousfield localization of spectra gives you a class of morphisms of spaces consisting of maps that induce equivalence on $\Sigma^{\infty}_+$ post-composed with the localization. We can then ask whether or not this defines a modality. I wonder if this property is equivalent to something more familiar...

Of course many bousfield localizations of spectra would correspond to the same class of maps in the above construction. For example $H\mathbb{F}_p$-local and $p$-complete. We may ask for the smallest possible bousfield class generating the same equivalences of spaces to make things neater.