nLab
Postnikov system

Definition

For topological spaces

A Postnikov system or Postnikov tower of topological spaces (M M Postnikov, 1951) is a sequence of path connected, pointed topological spaces $X^{(n)}$, $n\ge 1$, such that ${\pi }_r(X^{(n)})=0$ for $r>n$, together with a sequence ${\pi }_n$ of modules of the fundamental group ${\pi }_1(X^{(1)})$ of $X^{(1)}$ and fibrations $p_n:X^{(n+1)}\to X^{(n)}$ classified up to homotopy type by a specified cohomology class $k^{n+1}\in H^{n+1}(X^{(n)},{\pi }_n)$.

For simplicial sets

Definition

Let $X$ be a simplicial set. A Postnikov tower for $X$ is

1. a sequence

   $$\cdots \to X_2\stackrel{q_1}{\to }{X}_1\stackrel{q_0}{\to }{X}_0$$\cdots \to X_2 \stackrel{q_1}{\to} X_1 \stackrel{q_0}{\to} X_0

   with maps $i_n:X\to X_n$ such that all diagrams

   $$\begin{array}{ccc}& & X\\ & {}^{i_n}\swarrow & & {\searrow }^{i_{n-1}}\\ X_n& & \stackrel{q_{n-1}}{\to }& & X_{n-1}\end{array}$$\array{ && X \\ & {}^{\mathllap{i_n}}\swarrow && \searrow^{\mathrlap{i_{n-1}}} \\ X_n && \stackrel{q_{n-1}}{\to} && X_{n-1} }

   commute;

2. such that for all vertices $v\in X_0$ we have for the homotopy groups

   $${\pi }_{>n}(X_n,v)=0$$\pi_{\gt n}( X_n, v) = 0

   and

   $$(i_n{)}_{*}:{\pi }_i(X,v)\stackrel{\simeq }{\to }{\pi }_i{X}_n$$(i_n)_* : \pi_i (X,v) \stackrel{\simeq}{\to} \pi_i X_n

   for $i\le n$.

This appears for instance as (GoerssJardine, def VI 3.1).

Constructions

For simplicial sets

Proposition

If $X$ is regarded as an ∞-groupoid modeled as a Kan complex, then the coskeleton sequence

$$X=\underset{n}{\lim }{\mathrm{cosk}}_{n}X\to \cdots \to {\mathrm{cosk}}_{n+1}X\to {\mathrm{cosk}}_{n}X\to \cdots \to *$$X = \lim_n cosk_n X \to \cdots \to cosk_{n+1} X \to cosk_{n} X \to \cdots \to *

exhibits a Postnikov tower for $X$.

This is observed for instance in (DwyerKan). Also see coskeleton for more details.

The following construction qotients out the relations encoded by the cells that are thrown in in the above construction, such as to make the maps in the Postnikov tower into Kan fibrations.

Definition

Let $X$ be a Kan complex. Define for each $n\in \mathbb{N}$ an equivalence relation ${\sim }_n$ on the simplices of $X$ as follows: two $q$-simplices

$$\alpha ,\beta :{\Delta }^q\to X$$\alpha, \beta : \Delta^q \to X

are equivalent if their restriction to the $n$-skeleton coincides

$${\mathrm{sk}}_n(\alpha )={\mathrm{sk}}_n(\beta ):{\mathrm{sk}}_n({\Delta }^q)\hookrightarrow {\Delta }^q\to X.$$sk_n(\alpha) = sk_n(\beta) : sk_n(\Delta^q) \hookrightarrow \Delta^q \to X \,.

Write

$$X(n):=X{/}_{{\sim }_n}$$X(n) := X/_{\sim_n}

for the quotient simplicial set.

There are evident morphisms

$$X(n)\to X(n-1).$$X(n) \to X(n-1).

Proposition

This is a Postnikov tower, def. 1 and all morphisms are Kan fibrations.

Moreover the canonical morphism

$$X\to \underset{{\leftarrow }_n}{\lim }X(n)$$X \to \lim_{\leftarrow_n} X(n)

is an isomorphism, exhibiting $X$ as the limit over this tower diagram.

This appears for instance as (GoerssJardine, theorem Vi 3.5).

Properties

General

It is known that Postnikov systems classify all weak, pointed connected homotopy types. In particular, if $X$ satisfies ${\pi }_r(X)=0$ for $1<r<n$ then the first non trivial Postnikov invariant is an element $k^{n+1}$ of group cohomology (with twisted coefficients of course). Such elements are also determined by $n$-fold crossed extensions of ${\pi }_n$ by ${\pi }_1$, which are exact crossed complexes of the form

$$0\to {\pi }_n\to C_n\to C_{n-1}\to \cdots \to C_2\to C_1$$0 \to \pi_n \to C_n \to C_{n-1} \to \cdots \to C_2 \to C_1

together with an isomorphism $\mathrm{Coker}(C_2\to C_1)\cong {\pi }_1$. This gives an algebraic model of such an $n$-type. Advantages of algebraic models are that algebraic constructions can be made on them, such as forming limits or colimits. The various higher homotopy van Kampen theorems are useful in the latter case. For example, it may be difficult or well nigh impossible to write down a determination of the Postnikov invariant of a pushout of crossed modules, even if the pushout consists of finite groups.

A Postnikov system is easiest to understand in the 2-stage case, i.e. two non vanishing homotopy groups, and focuses attention on the cohomology of Eilenberg-Mac Lane spaces, which also determine all cohomology operations. Basic work on this area was done by Eilenberg and Mac Lane, and by H. Cartan, while the theory of cohomology operations, including Steenrod operations, is itself a large area.

The reference below shows the problems in the 3-stage systems.

For homotopy 3-types, the algebraic model of crossed squares is more explicit than the corresponding Postnikov system, and more calculable. However, not much work has been done on, say, cohomology operations using the algebraic model of $n$-fold groupoids, and it is not clear if that would help.

For simplicial sets

Proposition

Let $X$ be a Kan complex and $\{X(n)\}$ the model for its Postnikov tower from prop. 1. For any vertex $v\in X_0$ write $K(n)$ for the pullback

$$\begin{array}{ccc}K(n)& \to & *\\ \downarrow & & {\downarrow }^b\\ X(n)& \to & X(n-1)\end{array}.$$\array{ K(n) &\to& * \\ \downarrow && \downarrow^{\mathrlap{b}} \\ X(n) &\to& X(n-1) } \,.

Let $K({\pi }_n(X,v),n)$ be the Eilenberg-MacLane object on the $n$-homotopy group of $X$. Then there is a weak homotopy equivalence

$$K(n)\stackrel{\simeq }{\to }K({\pi }_n(X,v),n).$$K(n) \stackrel{\simeq}{\to} K(\pi_n(X,v),n) \,.

This appears for instance as GoerssJardine, corollary VI 3.7.

Proof

Since $K(n)\to K(n-1)$ is a Kan fibration by prop. 1 the pullback $K(n)$ is the homotopy fiber of $X(n)\to X(n-1)$.

Generalizations

There are analogues in other setups, e.g.

• Postnikov systems in triangulated categories

• motivic homotopy theory (M. Levine, The Postnikov tower in motivic stable homotopy theory).

Postnikov tower in an $(\mathrm{\infty },1)$-category

We may think of Top as being the archetypical (∞,1)-category.

In every (∞,1)-category there is a notion of n-truncated object and accordingly a notion of

• Postnikov tower in an (∞,1)-category.

The traditional case of Postnikov towers in Top is a special case of this more general concept.

References

Orginal references include

• M.M. Postnikov, Determination of the homology groups of a space by means of the homotopy invariants, Doklady Akad. Nauk SSSR (N.S.) 76: 359–362 (1951)
• George Whitehead, Elements of homotopy theory, chapter 9
• Donald W. Kahn, The spectral sequence of a Postnikov system, Comm. Math. Helv. 40, n.1, 169–198, 1965 doi
• P. I. Booth, An explicit classification of three-stage Postnikov towers, Homology, homotopy and applications 8 (2006), No. 2, 133–155
• G. Ellis and R. Mikhailov, A colimit of classifying spaces, arXiv:0804.3581.

A standard textbook reference is section VI of

• Paul Goerss, Rick Jardine, Simplicial homotopy theory

A classical article that amplifies the expression of Postnikov towers in terms of coskeletons is

• William Dwyer, Dan Kan, An obstruction theory for diagrams of simplicial sets (pdf)

Analogous remarks are also in

• John Duskin Simplicial matrices and the nerves of weak $n$-categories I: Nerves of bicategories , TAC 9 no. 2, (2002). (web)

A pedagogical introduction to Postnikov systems with an eye towards their $\mathrm{\infty }$-groupoid incarnation under the correspondence given by the homotopy hypothesis is in

• John Baez, Mike Shulman, Lectures on n-Categories and Cohomology