nLab
Postnikov system

Idea

The Postnikov system is the infinitary factorization system of (n-epi, n-mono)-factorizations through n-images in an (∞,1)-topos.

The basic (and historically first) example is the factorization in ∞Grpd/Top of morphisms to the point: here the Postnikov system assigns to a homotopy type $X$ a tower

$$X\to \cdots \to X_3\to X_2\to X_1\to X_0\simeq *$$X \to \cdots \to X_3 \to X_2 \to X_1 \to X_0 \simeq *

where each $X_k$ is a homotopy (k-2)-type/(n-2)-truncated object and such that each morphism $X_{k+1}\to X_k$ induces an isomorphism on homotopy groups in degree $\le k-2$. This may be thought of as decomposing $X$ into its “layers” as seen by the degree of homotopy groups. The characteristic classes which give the ∞-group extension in each layer are called the Postnikov invariants or k-invariants of $X$.

More generally, in any (∞,1)-topos and for any morphism $f:X\to Y$ the corresponding Postnikov system is a tower

$$f:X\simeq {\mathrm{im}}_{\mathrm{\infty }}(f)\to \cdots \to {\mathrm{im}}_2(f)\to {\mathrm{im}}_1(f)\to {\mathrm{im}}_0(f)\simeq Y$$f \colon X \simeq im_\infty(f) \to \cdots \to im_2(f) \to im_1(f) \to im_0(f) \simeq Y

of n-images of $f$ which interpolates between $X$ and $Y$ along $f$: the object ${\mathrm{im}}_n(f)$ is a homotopy type that looks like $X$ in low degrees (below $n-1$), looks like $Y$ in high degrees (above degree $n$) and looks like the ordinary (1-topos theoretic) image of $f$ on homotopy groups in degree $n-1$ itself.

Definition

In full generality, the Postnikov system is the infinitary factorization system of (n-epi, n-mono)-factorizations through n-images in an (∞,1)-topos.

The following spells this out explicitly for default homotopy theory, hence in ∞Grpd and in terms of the presentation by the model structure on topological spaces and the model structure on simplicial sets:

• For topological spaces

• For simplicial sets.

With a little bit of care this induces corresponding presentations of Postnikov systems in general (∞,1)-topos by prolonging to suitable model structures on simplicial presheaves:

• For simplicial presheaves.

For more discussion of the general abstract situation see at

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

For topological spaces

Historically, Postnikov systems were first described on the model of homotopy types constituted by topological spaces.

A Postnikov system or Postnikov tower or Moore-Postnikov tower/system of topological spaces, or rather of their homotopy types, 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

We discuss the realization of Postnikov systems on simplicial sets/Kan complexes, first in the

• Absolute version

and then more generally for the

• Relative version

Absolute version

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).

Relative version

Definition

Let $f:X\to Y$ be a homomorphism of simplicial sets. A (relative) Postnikov tower for $f$ is a tower

$$\begin{array}{c}X\\ {\downarrow }^{\simeq }\\ {\mathrm{im}}_{\mathrm{\infty }}(f)\\ \downarrow \\ ⋮\\ \downarrow \\ {\mathrm{im}}_2(f)\\ \downarrow \\ {\mathrm{im}}_1(f)\\ \downarrow \\ {\mathrm{im}}_0(f)\\ {\downarrow }^{\simeq }\\ Y\end{array}$$\array{ X \\ \downarrow^{\mathrlap{\simeq}} \\ im_\infty(f) \\ \downarrow \\ \vdots \\ \downarrow \\ im_2(f) \\ \downarrow \\ im_1(f) \\ \downarrow \\ im_0(f) \\ \downarrow^{\mathrlap{\simeq}} \\ Y }

that factors $f:X\to Y$ such that for all $n\in \mathbb{N}$

1. $X\to {\mathrm{im}}_n(f)$

   1. induces an epimorphism on connected components;

   2. induces an isomorphism on homotopy groups in degree $<n-1$

2. ${\mathrm{im}}_n(f)\to Y$

   1. induces a monomorphism on homotopy groups in degree $n-1$;

   2. induces an isomorphism on homotopy groups in degree $\ge n$.

This appears for instance as (Goerss-Jardine, def. VI 2.9).

Remark

By the long exact sequence of homotopy groups, the relative Postnikov tower is the tower of the (n-connected, n-truncated) factorization system of $f$ regarded as a morphism in the (∞,1)-category ∞Grpd: is is the n-epimorphism,n-monomorphism factorization through the n-image of a morphism.

For simplicial presheaves

Constructions

We discuss explicit constructions/presentations of Postnikov systems.

• For simplicial sets

• For strict ω-groupoids.

For simplicial sets

There are three main functorial models for the Postnikov tower of a simplicial set:

• Coskeleton tower;

• Identification relative to skeleta;

• Homotopy classes relative to skeleta

Coskeleton tower

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.

Identification relative to skeleta

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.

We first discuss the absolute tower and then the relative version.

Absolute Postnikov tower

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 (“inverse limit”) over this tower diagram.

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

Relative Postnikov tower

We discuss a model for the relative Postnikov tower, def. 2.

Definition

For $f:X\to Y$ a Kan fibration between Kan complexes, define for each $n$ and each $k$ an equivalence relation ${\sim }_n$ on $k$-simplices

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

such that $\alpha {\sim }_n\beta$ if

1. the $n$-skeleta ${\mathrm{sk}}_n{\Delta }^k\to {\Delta }^k\stackrel{\alpha ,\beta }{\to }X$ are equal;

2. the images $f(\alpha ),f(\beta )\in Y$ are equal.

Let

$${\mathrm{im}}_{n+1}(f)≔X/{\sim }_n$$im_{n+1}(f) \coloneqq X/\sim_n

be the simplicial set of equivalence classes under this equivalence relation.

This canonically comes with morphisms ${\mathrm{im}}_{n_1}(f)\to {\mathrm{im}}_{n_2}(f)$ for $\mathrm{\infty }\ge n_1>n_2\ge 0$.

For instance (Goerss-Jardine, def. VI 2.9).

Proposition

This construction gives indeed a relative Postnikov tower for $f$.

For instance (Goerss-Jardine, theorem VI 2.11).

Homotopy classes relative to skeleta

For $X$ a Kan complex and $n\in \mathbb{N}$, let ${\tau }_{<n}X$ be the simplicial set defined as the quotient

$${\tau }_{<m}X:k\mapsto X_k/\mathrm{homotopy}-\mathrm{rel}-(n-1)-\mathrm{skeleton},$$\tau_{\lt m}X : k \mapsto X_k / homotopy-rel-(n-1)-skeleton \,,

where two $k$-cells are identified if there is a simplicial homotopy between them that fixes their $(n-1)$-skeleton.

This is due to John Duskin. See for instance (Beke, pages 302-305).

For strict $\omega$-groupoids

Proposition

Consider a morphism in sSet/∞Grpd that is in the image of $\mathrm{Str}\omega \mathrm{Grpd}$, hence given by a morphism $f:X\to Y$ of strict ω-groupoids.

Then for $n\in \mathbb{N}$ the n-image factorization/$(n-1)$-Postnikov stage of $f$ is given by the strict ω-groupoid ${\mathrm{im}}_n(f)$ with

$${\left({\mathrm{im}}_n\left(f\right)\right)}_k=\{\begin{array}{cc}{X}_k& \forall k<n-1\\ \mathrm{im}(X_{n-1})\subset Y_{n-1}& \forall k=n-1\\ Y_k& \forall k\ge n\end{array}$$\left(im_n\left(f\right)\right)_k = \left\{ \array{ X_k & \forall k \lt n-1 \\ im(X_{n-1}) \subset Y_{n-1} & \forall \; k = n-1 \\ Y_k & \forall k \geq n } \right.

equipped with the evident composition operations induced from those of $X$ and $Y$, and equipped with the canonical morphisms of strict $\omega$-groupoids

$$X\to {\mathrm{im}}_n(f)\to Y$$X \to im_n(f) \to Y

(the left one being the identity in degree $k<n-1$, the quotent projection in degree $n-1$ and $f$ in degree $k\ge n$, and the right one being $f$ in degree $k<n-1$, the image inclusion in degree $n-1$ and the identity in degree $k\ge n$).

For $Y=*$ this is discussed in (BFGM).

Proof

The homotopy groups of a strict $\omega$-groupoid in any degree $k$ are simply given by the groups of $k$-automorphisms of the identity $(k-1)$-morphism on a given baspoint modulo $(k+1)$-morphisms (hence the homology of the corresponding crossed complex in that degree). Therefore it is clear from the construction of ${\mathrm{im}}_n(f)$ above that $X\to {\mathrm{im}}_n(f)$ is surjective on ${\pi }_0$ and an isomorphism on ${\pi }_{k<n-1}$, and that ${\mathrm{im}}_n(f)$ is a monomorphism on ${\pi }_{n-1}$ and an isomorphism on ${\pi }_{k\ge n}$.

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. 2. 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. 2 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.

Related concepts

• n-image, n-epimorphism, n-monomorphism

• Postnikov invariant/kinvariant?

• k-ary factorization system

References

A standard textbook reference is section 8 of

• Peter May, Simplicial methods in algebraic topology (djvu)

and section VI of

• Paul Goerss, Rick Jardine, Simplicial homotopy theory

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. J. Ellis and R. Mikhailov, A colimit of classifying spaces, arXiv:0804.3581.

Probably the earliest treatment of Postnikov systems for simplicial sets is in

• J. C. Moore, Semisimplicial complexes and Postnikov systems, Symposium Internacional de Topologia Algebraica, Mexico City, 1958, pp. 232-247,

and as a result, in that context, they are sometimes referred to as Moore-Postnikov systems

The coskeleton construction for the Postnikov tower of a Kan complex is already in

• Artin, Mazur, Étale homotopy, (Lecture Notes in Maths. 100).

Another classical article that amplifies the expression of Postnikov towers in terms of coskeleta is

• William Dwyer, Dan Kan, An obstruction theory for diagrams of simplicial sets (pdf), from 1984.

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)

reviewed around page 302 in

• Tibor Beke, Higher Čech theory, K-Theory, 32(4):293–322 (2004) (web)

Discussion for homotopy types modeled by crossed complexes/strict ω-groupoids is in

• M. Bullejos E. Faro, and M. A. García-Munoz, Postnikov Invariants of Crossed Complexes, Journal of Algebra Volume 285, Issue 1, 1 March 2005, Pages 238–291 (arXiv:math/0409339).

A pedagogical introduction to Postnikov systems with an eye towards their generalization from homotopy types to n-categories is in

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