Oh actually... I guess this sort of follows from Aaron's paper, because Quillen equivalences induce equivalences after (co)fibrant replacement.

@JonathanBeardsley A presentable quasicategory C is automatically tensored over spaces, but not over pointed spaces. For example, the quasicategory of spaces itself. The symmetric monoidal structure on C is not relevant.

@Gasterbiter Ah nice, great thanks!

@JonathanBeardsley An adjunction of quasi-categories is an equivalence if and only if the unit and counit transformations are both equivalences. So what you want should indeed follow from Mazel-Gee’s paper

@Gasterbiter thanks! do you know of a reference for that fact?

Looking in Lurie, there's one place where he says an equivalence of categories is one which is fully faithful and essentially surjective, but then he doesn't prove that this coincides with the obvious notion (i.e. being a weak equivalence of simplicial sets, or marked simplicial sets).

I would have to look up the precise reference but it must be in HTT somewhere, or even joyal’s notes. Sorry!

@Gasterbiter Ah sorry you're right, it's required to be a fully faithful on "enriched" homotopy categories, it seems.

Heh, "it must be in HTT somewhere."

Just joking though, thanks for the help! :-)

The theory of “categorical equivalences” of simplicial sets is one of the more nontrivial parts of quasi-category theory, a great part of HTT is dedicated to it. Sorry again I can’t be more help

Haha I got excited about that post, now it's gone!

Is every simply connected CW complex with finite dimensional homology of finite type? That is, is such a thing always homotopy equivalent to a finite CW complex? Is the simply connectedness necessary (can we replace this hypothesis with the assumption that the fundamental group is finitely presented for example)

For the simply-connected case, this follows from the construction in Proposition 4C.1 of Hatcher's book

@ThomasRot If the fundamental group is non-trivial you get a K-theoretic obstruction called "Wall's finiteness obstruction". Its vanishing is equivalent to being finite type. His original papers are not haed to read.

(...Under the ground assumption you can check that you are a retract up to homotopy of a finite space)

great thanks!

But in the non-simply-connected case, is every CW complex with finite dimensional homology a retract of a finite CW complex?

Does any retract of a finite CW complex have finite dimensional homology with arbitrary local coefficients?

I think retract of finite CW is equivalent to 'all local system homology is finitely generated' but I think this condition is usually hard to check in practice