Connective covers of based spaces

Question

I know that n-connective spectra form a coreflective subcategory of spectra. Namely, there is a n-connective cover functor $\tau_{>k}: Spectra\rightarrow Spectra^{n-connective}$ which has a fully faithful left adjoint. In particular, the counit gives a natural morphism $\tau_{>k}(E)\rightarrow E$ for each spectrum $E$.

I am wondering if the same holds unstably, for based spaces. Namely, I suspect that based n-connected spaces for a coreflective subcategory of based spaces $Space^*$, i.e. there is a n-connected cover functor $\tau_{>k}: Space^*\rightarrow Space^{*, n-connected}$ with a fully faithful left adjoint. I think the functor $\tau_{>n}$ can be obtained by taking the homotopy fiber of the unit $X \rightarrow \tau_{\le n} X$, where $\tau_{\le n}$ is the truncation functor $\tau_{\le n}: Space\rightarrow Space^{n-truncated}$, which is a localization. Is there a reference for this? (if it is true). I am specifically asking about the unstable case, as opposed to references for the case of spectra, which seems well-known. One potential issue I can foresee is that $\tau_{>n}$ may not be $n$-connected.

Note that to take the homotopy fiber functorially, we need to work with based spaces. Another reason that we need to work with based spaces is that unbased n-connected spaces do not have all colimits (for example, they do not have an initial object - see the discussion here Is there a homotopy theory of unbased simply connected spaces?). On the other hand, (an accessible) coreflective localization of a presentable category $Space^*$ would also be presentable and hence have all colimits.

Answer 1

Such a tower is usually called a Whitehead tower. More generally, there is the notion of a Moore-Postnikov factorization of a map $f: X \to Y$ of spaces: fixing an $n$, we can find a factorization $X \to M_n(f) \to Y$ such that $$ \begin{align*} \pi_k(X) \xrightarrow{\sim} \pi_k(M_n f)&\text{ for }k < n,\\ \pi_k(X) \twoheadrightarrow \pi_k(M_n f) \hookrightarrow \pi_k(Y)&\text{ for }k=n,\\ \pi_k(M_n f) \xrightarrow{\sim} \pi_k(Y)&\text{ for }k > n. \end{align*} $$ The Postnikov tower of $X$ is made up of Moore-Postnikov factorizations of $X \to \ast$, and the Whitehead tower of a pointed space $Y$ is made up of Moore-Postnikov factorizations of $\ast \to Y$. We can form $M_n f$ by attaching cells to $X$ of dimension at least $n+1$ to kill off the relative homotopy groups $\pi_k(f)$.

Moore gives a construction via simplicial sets in the paper "Semi-simplicial complexes and Postnikov systems". You can see this described in a way that is more explicitly functorial in May's "Simplicial objects in algebraic topology" (which I got from this answer).