# Simple characterization of Postnikov & Whitehead towers?

I'm asking this question in the most model-ambiguous way I can since this is the kind of answer i'm looking for.

There are various explicit constructions of the Whitehead and Postnikov towers. I'm trying to understand what exactly characterizes these construction.

Postnikov tower: A Postnikov tower of (a possibly non-pointed) space is a factorization of the terminal morphism $X \to *$ to a directed limit diagram:

$$X \cong\underset{\rightarrow_n}{\lim}X_n \to \dots \to X_2 \to X_1 \to X_0 \cong *$$ Such that $X_k$ is $(k-1)$-truncated and each morphism $X_{k} \to X_{k-1}$ is a $k$-equivalence (isomorphism on homotopy groups in degree smaller than $k$ and surjection on $k$).

Question 1: Does this property determine the Postnikov tower up to weak equivalence of diagrams?

Whitehead tower: A Whitehead tower of a space is a factorization of the initial morphism $* \to X$ to a directed limit diagram:

$$* \cong\underset{\rightarrow_n}{\lim}X_n \to \dots \to X_2 \to X_1 \to X_0 \cong X$$ Such that $X_k$ is $(k-1)$-connected and each morphism $X_{k} \to X_{k-1}$ is an isomorphism on homotopy groups in degree larger than $k-1$).

Question 2: Does this property determine the Whitehead tower up to weak equivalence of diagrams?

• The answer to both of your questions is yes, as you can see from the fact that the Postnikov truncation is a Bousfield localization. – Denis Nardin Jan 27 '17 at 12:58
• @DenisNardin Could you elaborate a bit? How does this follow from Bousfield localization? – Saal Hardali Jan 27 '17 at 13:06

Indeed, the properties you stated characterize Postnikov and Whitehead towers. A nice conceptual way of justifying this is by using $k$-connected / $k-$truncated factorization systems.
To fix terminology, a $k$-connected map is one whose all homotopy fibers are $k$-connected and a $k$-trunceted map is one whose all homotopy fibers are $k$-truncated. (Note that in the case of connectedness this disagrees with the traditional definition according to which a map is $k$-connected if and only if all its homotopy fibers are $(k-1)$-connected.)
Then every map factors as a composite of a $k$-connected map followed by a $k$-truncated map. Moreover, certain lifting property holds. On the point-set level you could say that $k$-connected cofibrations have the LLP with respect to $k$-truncated fibrations. However, more properly, a homotopy coherent / higher categorical lifting property is satisfied, i.e. for any $k$-connected map $A \to B$ and a $k$-truncated map $X \to Y$ the square of (derived) mapping spaces $\require{AMScd}$ \begin{CD} \mathrm{Map}(B, X) @>>> \mathrm{Map}(A, X) \\ @VVV @VVV \\ \mathrm{Map}(B, Y) @>>> \mathrm{Map}(A, Y) \\ \end{CD} is a homotopy pullback. (In other words, any lifting problem with $A \to B$ on the left and $X \to Y$ on the right has a contractible space of solutions.)
It follows directly that any map factors uniquely as a composite of a $k$-connected map followed by a $k$-truncated map, i.e. the space of such factorizations is contractible. Your statements are recovered as special cases of that. The $k$th Postnikov section of a space $X$ is obtained by factoring $X \to *$ as a composite of a $k$-connected map followed by a $k$-truncated map. In the based case, the $k$th connected cover of $X$ is obtained by factoring $* \to X$ as a composite of a $(k-1)$-connected map followed by a $(k-1)$-truncated map. (There is an offset here since $X \to *$ is $k$-connected if and only if $* \to X$ is $(k-1)$-connected.)
It should also be noted that a map $A \to B$ is $k$-connected if and only if it induces isomorphisms on homotopy groups up to dimension $k$ and an epimorphism in dimension $k + 1$. Similarly, a map $X \to Y$ is $k$-truncated if and only if it induces isomorphisms on homotopy groups above dimension $k + 1$ and a monomorphism in dimension $k + 1$. I believe your description of the Whitehead tower should be adjusted accordingly (although in this particular case, injectivity in dimension $k + 1$ may be vacuous).