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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
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
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.
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
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 $k^{n+1} \in H^{n+1}(X^{(n)}, \pi_n)$.
We discuss the realization of Postnikov systems on simplicial sets/Kan complexes, first in the
and then more generally for the
Let $X$ be a simplicial set. A Postnikov tower for $X$ is
a sequence
with maps $i_n : X \to X_n$ such that all diagrams
such that for all vertices $v \in X_0$ we have for the homotopy groups
and
for $i \leq n$.
This appears for instance as (GoerssJardine, def VI 3.1).
Let $f : X \to Y$ be a homomorphism of simplicial sets. A (relative) Postnikov tower for $f$ is a tower
that factors $f : X \to Y$ such that for all $n \in \mathbb{N}$
$X \to im_n(f)$
induces an epimorphism on homotopy groups in degree $n-1$;
induces an isomorphism on homotopy groups in degree $\lt n-1$
$im_n(f) \to Y$
induces a monomorphism on homotopy groups in degree $n-1$;
induces an isomorphism on homotopy groups in degree $\geq n$.
This appears for instance as (Goerss-Jardine, def. VI 2.9).
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.
We discuss explicit constructions/presentations of Postnikov systems.
There are three main functorial models for the Postnikov tower of a simplicial set:
If $X$ is regarded as an ∞-groupoid modeled as a Kan complex, then the coskeleton sequence
exhibits a Postnikov tower for $X$.
This is observed for instance in (ArtinMazur)) or (DwyerKan). Also see coskeleton for more details.
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.
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
are equivalent if their restriction to the $n$-skeleton coincides
Write
for the quotient simplicial set.
There are evident morphisms
This is a Postnikov tower, def. 1, and all morphisms are Kan fibrations.
Moreover the canonical morphism
is an isomorphism, exhibiting $X$ as the limit (“inverse limit”) over this tower diagram.
This appears for instance as (GoerssJardine, theorem Vi 3.5).
We discuss a model for the relative Postnikov tower, def. 2.
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
such that $\alpha \sim_n \beta$ if
the $n$-skeleta $sk_n \Delta^k \to \Delta^k \stackrel{\alpha, \beta}{\to} X$ are equal;
the images $f(\alpha), f(\beta)\in Y$ are equal.
Let
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).
This construction gives indeed a relative Postnikov tower for $f$.
For instance (Goerss-Jardine, theorem VI 2.11).
For $X$ a Kan complex and $n \in \mathbb{N}$, let $\tau_{\lt n}X$ be the simplicial set defined as the quotient
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).
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 ω-groupoids.
Then for $n \in \mathbb{N}$ the $(n-1)$-Postnikov stage of $f$ is given by the strict ω-groupoid $im_n(f)$ with
equipped with the evident composition operations induced from those of $X$ and $Y$, and equipped with the canonical morphisms of strict $\omega$-groupoids
(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).
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 $\pi_{k \geq n}$.
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. 5, the following remark 2 motivates the formula by regarding chain complexes as models for strict ω-groupoids (by this prop.orrespondence#GlobularAndCubical))
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
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
Then
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
and hence is generated under linearity by
Moreover, notice that the Dold-Kan correspondence
factors through globular strict omega-groupoids (here). An n-morphism in the strict omega-groupoid $DK(V_\bullet)$ is of the form
In terms of these morphisms the equivalence relation above says that two of them are equivalent precisely if
they are “parallel morphisms” in that they have the same source and target;
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:
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 2.
The following diagram of abelian groups commutes:
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
Finally, this is a model for the (n+1)-image factorization of $f$ in that on homology groups the following holds:
$H_{\bullet \lt n}(V) \overset{\simeq}{\to} H_{\bullet \lt n}(im_{n+1}(f))$ are isomorphisms;
$H_n(V) \to H_n(im_{n+1}(f)) \hookrightarrow H_n(W)$ is the image factorization of $H_n(f)$;
$H_{\bullet \gt n}(im_{n+1}(f)) \overset{\simeq}{\to} H_{\bullet \gt n}(W)$ are isomorphisms.
This follows by elementary and straightforward direct inspection.
The following gives a sufficient condition for modeling n-image factorizations in some (∞,1)-toposes with particularly convenient presentation.
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
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:
$\pi_{\bullet \lt n}\left(X(U) \to (im_{n}(f))(U)\right)$ are isomorphisms;
$\pi_n\left(X(U) \to (im_{n}(f))(U)\to Y(U)\right)$ is the (epi,mono) factorization of $\pi_n(f(U))$;
$\pi_{\bullet \gt n}\left((im_{n}(f))(U) \to Y(U)\right)$ are isomorphisms.
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.
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
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.
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
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
This appears for instance as GoerssJardine, corollary VI 3.7.
Since $K(n) \to K(n-1)$ is a Kan fibration by prop. 2 the pullback $K(n)$ is the homotopy fiber of $X(n) \to X(n-1)$.
There are analogues in other setups, e.g.
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.
A standard textbook reference is section 8 of
and section VI of
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
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
reviewed around page 302 in
Discussion for spectra includes
Discussion in homotopy type theory is in
Discussion for homotopy types modeled by crossed complexes/strict ω-groupoids is in
and for $n$-hypergroupoids in the thesis,
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