# nLab Postnikov system

Contents

### Context

#### $(\infty,1)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## 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

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

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 $\leq k-2$. 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 $f \colon X \to Y$ the corresponding Postnikov system is a tower

$f \colon X \simeq im_\infty(f) \to \cdots \to im_2(f) \to im_1(f) \to im_0(f) \simeq Y$

of n-images of $f$ which interpolates between $X$ and $Y$ along $f$: the object $im_n(f)$ 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:

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 more discussion of the general abstract situation see at

### 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 $X^{(n)}$, $n \geq 1$, such that $\pi_r(X^{(n)})=0$ for $r \gt n$, 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

and then more generally for the

#### Absolute version

###### Definition

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

1. a sequence

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

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

$\array{ && X \\ & {}^{\mathllap{i_n}}\swarrow && \searrow^{\mathrlap{i_{n-1}}} \\ X_n && \stackrel{q_{n-1}}{\to} && X_{n-1} }$
2. such that for all vertices $v \in X_0$ we have for the homotopy groups

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

and

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

for $i \leq n$.

This appears for instance as (Goerss-Jardine, def VI 3.1ardine)).

#### Relative version

###### Definition

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

$\array{ X \\ \downarrow^{\mathrlap{\simeq}} \\ im_\infty(f) \\ \downarrow \\ \vdots \\ \downarrow \\ im_2(f) \\ \downarrow \\ im_1(f) \\ \downarrow \\ im_0(f) \\ \downarrow^{\mathrlap{\simeq}} \\ Y }$

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

1. $X \to im_n(f)$

1. induces an epimorphism on homotopy groups in degree $n-1$;

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

2. $im_n(f) \to Y$

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

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

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.

## Constructions

We discuss explicit constructions/presentations of Postnikov systems.

### For simplicial sets

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

#### Coskeleton tower

###### Proposition

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

$X \;\simeq\; \lim_n \big( cosk_n X \to \cdots \to cosk_{n+1} X \to cosk_{n} X \to \cdots \to * \big)$

exhibits a Postnikov tower for $X$.

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

#### Identification relative to skeleta

The following construction quotients 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

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

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

$sk_n(\alpha) = sk_n(\beta) : sk_n(\Delta^q) \hookrightarrow \Delta^q \to X \,.$

Write

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

for the quotient simplicial set.

There are evident morphisms

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

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

Moreover the canonical morphism

$X \to \lim_{\leftarrow_n} X(n)$

is an isomorphism, exhibiting $X$ as the limit (“inverse limit”) over this tower diagram.

This appears for instance as (Goerss-Jardine, theorem Vi 2.5).

##### Relative Postnikov tower

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

###### 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

$\alpha, \beta \colon \Delta^k \to X$

such that $\alpha \sim_n \beta$ if

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

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

Let

$im_{n+1}(f) \coloneqq X/\sim_n$

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

This canonically comes with morphisms $im_{n_1}(f) \to im_{n_2}(f)$ for $\infty \geq n_1 \gt n_2 \geq 0$.

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 $\tau_{\lt n}X$ be the simplicial set defined as the quotient

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

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 an object in sSet/∞Grpd that is in the image of $Str\omega Grpd$, hence given by a morphism $f \colon X \to Y$ of strict ∞-groupoids.

Then for $n \in \mathbb{N}$ the $(n-1)$-Postnikov stage of $f$ is given by the strict ∞-groupoid $im_n(f)$ with

$\left(im_n\left(f\right)\right)_k = \left\{ \array{ X_k & \forall k \lt n-1 \\ im(X_{n-1}) \subset Y_{n-1} & \forall \; k = n-1 \\ \ast & \forall k \geq n } \right.$

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

$X \to im_n(f) \to \ast$

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

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 $im_n(f)$ above that $X \to im_n(f)$ is surjective on $\pi_0$ and an isomorphism on $\pi_{k \lt n-1}$, and that $im_n(f)$ is a monomorphism on $\pi_{n-1}$ and an isomorphism on $\pi_{k \geq n}$.

### For chain complexes

The following gives a model for the $(n-1)$-stage of a relative Postnikov tower for the special case that the the morphism of Kan complexes is the image under the Dold-Kan correspondence of a chain map between chain complexes.

Before we give the explicit formule below as prop. , the following remark motivates the formula by regarding chain complexes as models for strict ∞-groupoids (by this prop.orrespondence#GlobularAndCubical))

###### Remark

Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$ be a chain map between chain complexes

For $n \in \mathbb{N}$, consider the abelian group

$(im_{n+1}(f))_n \;\coloneqq\; coker(\, ker(\partial_V) \cap ker(f_n) \to V_n \,)$

For the following it is helpful to think of this abelian group in the following equivalent ways.

Define an equivalence relation on $V_n$ by

$\left( v_n \sim v'_n \right) \;\Leftrightarrow\; \left( (\partial_V v_n = \partial_V v'_n) \;\text{and}\; (f_n(v_n) = f_n(v'_n)) \right) \,.$

Then

$(im_{n+1}(f))_n \simeq V_n/_\sim$

is equivalently the set of equivalence classes of this equivalence relation, which inherits abelian group structure since the eqivalence relation is linear.

This is because the equivalence relation says equivalently that

$\left( v_n \sim v'_n \right) \;\Leftrightarrow\; \left( v_n - v'_n \;\in\; ker(\partial_V) \cap ker(f_n) \right)$

and hence is generated under linearity by

$\left( v_n \sim 0 \right) \;\Leftrightarrow\; \left( v_n \in ker(\partial_V) \cap ker(f_n) \right) \,.$

Moreover, notice that the Dold-Kan correspondence

$DK \;\colon\; Ch_{\bullet \geq 0} \longrightarrow KanCplx$

factors through globular strict omega-groupoids (here). An n-morphism in the strict omega-groupoid $DK(V_\bullet)$ is of the form

$(v_{n-1}) \overset{\phantom{AA}v_n\phantom{AA}}{\longrightarrow} (v_{n-1} + \partial v_n) \,.$

In terms of these morphisms the equivalence relation above says that two of them are equivalent precisely if

1. they are “parallel morphisms” in that they have the same source and target;

2. they have the same image under $f$ in the n-morphisms of $DK(W_\bullet)$.

This suggests yet another equivalent way to think of $(im_{n+1}(f))_n$: it is the disjoint union over the target $(n-1)$-cells in $V_{n-1}$ of the images under $f$ of the sets of $n$-cells from zero to that target:

$(im_{n+1}(f))_n \simeq \underset{v_{n-1} \in V_{n-1}}{\sqcup} \left\{ f_n(v_n) \vert v_n \in V_n \,\text{and}\,\partial v_n = v_{n-1} \right\} \,.$
###### Proposition

Let $f_\bullet \colon V_\bullet \longrightarrow W_\bullet$ be a chain map between chain complexes and let $n \in \mathbb{N}$. Recall the abelian group $\underset{v_{n-1}}{\sqcup}\{f_n(v_n) \vert \partial v_n = v_{n-1}\}$ from remark .

The following diagram of abelian groups commutes:

$\array{ \vdots && \vdots && \vdots \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{W}}} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+3} &\overset{f_{n+3}}{\longrightarrow}& W_{n+3} &\overset{=}{\longrightarrow}& W_{n+3} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{W}}} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+2} &\overset{f_{n+2}}{\longrightarrow}& W_{n+2} &\overset{=}{\longrightarrow}& W_{n+2} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{ \partial_W } } && \downarrow^{\mathrlap{\partial_{W}}} \\ V_{n+1} &\overset{f_{n+1}}{\longrightarrow}& \left\{ w_{n+1} | \exists v_n : \partial_W w_{n+1} = f_n(v_n), \partial_V v_n = 0, \right\} &\overset{}{\longrightarrow}& W_{n+1} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\partial_W} && \downarrow^{\mathrlap{\partial_{W}}} \\ V_n &\overset{ (f_n, \partial_V) }{\longrightarrow}& \underset{v_{n-1}}{\sqcup} \left\{ f_n(v_n) \vert \partial_V v_n = v_{n-1} \right\} &\overset{ }{\longrightarrow}& W_n \\ \downarrow^{\mathrlap{\partial_V}} && \downarrow^{\mathrlap{(f_n(v_n),\partial_V v_n) \mapsto \partial_V v_n}} && \downarrow^{\mathrlap{\partial_W}} \\ V_{n-1} &\overset{=}{\longrightarrow}& V_{n-1} &\overset{f_{n-1}}{\longrightarrow}& W_{n-1} \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_W}} \\ V_{n-2} &\overset{=}{\longrightarrow}& V_{n-2} &\overset{f_{n-2}}{\longrightarrow}& W_{n-2} \\ \\ \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_{V}}} && \downarrow^{\mathrlap{\partial_W}} \\ \vdots && \vdots && \vdots }$

Moreover, the middle vertical sequence is a chain complex $im_{n+1}(f)_\bullet$, and hence the diagram gives a factorization of $f_\bullet$ into two chain maps

$f_\bullet \;\colon\; V_\bullet \longrightarrow im_{n+1}(f)_\bullet \longrightarrow W_\bullet \,.$

Finally, this is a model for the (n+1)-image factorization of $f$ in that on homology groups the following holds:

1. $H_{\bullet \lt n}(V) \overset{\simeq}{\to} H_{\bullet \lt n}(im_{n+1}(f))$ are isomorphisms;

2. $H_n(V) \to H_n(im_{n+1}(f)) \hookrightarrow H_n(W)$ is the image factorization of $H_n(f)$;

3. $H_{\bullet \gt n}(im_{n+1}(f)) \overset{\simeq}{\to} H_{\bullet \gt n}(W)$ are isomorphisms.

###### Proof

This follows by elementary and straightforward direct inspection.

### In simplicial (pre-)sheaves

The following gives a sufficient condition for modeling n-image factorizations in some (∞,1)-toposes with particularly convenient presentation.

###### Proposition

Let $C$ be a site with enough points, so that the weak equivalences in $sPSh(C)_{\mathrm{loc}}$ are detected on stalks (this prop.). Then given a morphism of Kan complex-valued simplicial presheaves

$f \colon X \longrightarrow Y$

such that both $X$ and $Y$ are homotopy k-types for some finite $k \in \mathbb{N}$, then its n-image factorization in the (∞,1)-topos $L_{lwhe} sPSh(C)_{loc}$ for any $n \in \mathbb{N}$ is presented by any factorization $X \longrightarrow im_{n}(f) \longrightarrow Y$ in $sPSh(C)$ through some Kan-complex valued simplicial presheaf $im_n(f)$ such that for each object $U \in C$ the simplicial homotopy groups satisfy the following conditions:

1. $\pi_{\bullet \lt n}\left(X(U) \to (im_{n}(f))(U)\right)$ are isomorphisms;

2. $\pi_n\left(X(U) \to (im_{n}(f))(U)\to Y(U)\right)$ is the (epi,mono) factorization of $\pi_n(f(U))$;

3. $\pi_{\bullet \gt n}\left((im_{n}(f))(U) \to Y(U)\right)$ are isomorphisms.

###### Proof

Evalutation on stalks is a filtered colimit which preserves the finite limits and finite colimits that go into the definition of simplicial homotopy groups. Therefore the global conditions assumed on the simplicial homotopy groups imply that the same kind of conditions holds for the stalkwise homotopy groups. These are the categorical homotopy groups in $L_{lwhe} sPSh(C)_{loc}$. By this prop. and this def. we may recognize $n$-truncation of morphisms on categorical homotopy groups (using the assumption that $X$ and $Y$ are $k$-truncated for some $k$). Therefore the claim now follows from the stalkwise long exact sequence of homotopy groups.

## 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 $1 \lt r \lt n$ 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

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

together with an isomorphism $Coker(C_2 \to C_1) \cong \pi_1$. 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. . For any vertex $v \in X_0$ write $K(n)$ for the pullback

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

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

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

This appears for instance as GoerssJardine, corollary VI 3.7.

###### Proof

Since $K(n) \to K(n-1)$ is a Kan fibration by prop. 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 tower in an $(\infty,1)$-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

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

Textbook accounts:

The original articles:

• 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)

• M. M. Postnikov, Issledovaniya po gomotopičeskoĭ teorii nepreryvnyh otobraženiĭ. I. Algebraičeskaya teoriya sistem. II. Naturalʹnaya sistema i gomotopičeskiĭ tip. (Russian) $[$Investigations in homotopy theory of continuous mappings. I. The algebraic theory of systems. II. The natural system and homotopy type.$]$ Trudy Mat. Inst. Steklov. no. 46. Izdat. Akad. Nauk SSSR, Moscow, 1955. (mathnet:tm1182)

Early development:

• 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

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

Analogous remarks are also in

• John DuskinSimplicial 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 spectra includes

Discussion in homotopy type theory is in

Discussion for homotopy types modeled by crossed complexes/strict ∞-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).

and for $n$-hypergroupoids in the thesis,

• M. A. García-Munoz, 2003, Un aceramiento algebraico a la theoría de Torres de Postnikov, thesis, Universidad de Granada.

This also contains a good discussion of the link with twisted cohomology and homotopy colimits.

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