nLab
Postnikov tower in an (infinity,1)-category

Idea

The notion of Postnikov tower in an $(\mathrm{\infty },1)$-category is the generalization of the notion of Postnikov tower from the archetypical (∞,1)-category Top$\simeq$ ∞Grpd to more general $(\mathrm{\infty },1)$-categories.

Definition

Proposition

For $C$ a presentable (∞,1)-category the subcategory $C_{\le n}$ of n-truncated objects is a reflective (∞,1)-subcategory

$$C_{\le n}\stackrel{\stackrel{{\tau }_{\le n}}{\leftarrow }}{\hookrightarrow }C.$$C_{\leq n} \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow} C \,.

This is (Lurie, prop. 5.5.6.18).

We write

$${\tau }_{\le n}:C\stackrel{{\tau }_{\le n}}{\to }{C}_{\le n}\hookrightarrow C$$\mathbf{\tau_{\leq n}} : C \stackrel{\tau_{\leq n}}{\to} C_{\leq n} \hookrightarrow C

for the corresponding localization. For $X\in C$, we say that ${\tau }_{\le n}C$ is the $n$-truncation of $C$.

The reflector of the reflective embedding provides morphisms

$$X\to {\tau }_{\le n}X$$X \to \mathbf{\tau}_{\leq n} X

from each object to its $n$-truncation.

Definition

A Postnikov tower for $X\in C$ is a diagram

$$X\to \cdots \to X_2\to X_1\to X_0$$X \to \cdots \to X_2 \to X_1 \to X_0

such that each $X\to X_n$ exhibits $X_n$ as the $n$-truncation of $X$.

This is HTT, def. 5.5.6.23.

Definition

A Postnikov pretower is a pre-tower

$$\cdots \to X_2\to X_1\to X_0$$\cdots \to X_2 \to X_1 \to X_0

(no initial $X$ on the left!) which exhibits each $X_n$ as the $n$-truncation of $X_{n+1}$.

We say Postnikov towers converge in the ambient (∞,1)-category if the forgetful (∞,1)-functor from Postnikov towers to Postnikov pretowers is an equivalence of (∞,1)-categories.

This is (Lurie, def. 5.5.6.23).

Examples

In $\mathrm{\infty }\mathrm{Grpd}$

When the archetypical (∞,1)-topos ∞Grpd is presented by the model structure on simplicial sets, truncation is given by the the coskeleton endofunctor ${\mathrm{cosk}}_{n+1}$ on sSet.

The unit of the adjunction $({\mathrm{tr}}_n⊣{\mathrm{cosk}}_n)$

$${\tau }_n:X\to {\mathrm{cosk}}_n(X)$$\mathbf{\tau}_n : \mathbf{}X \to \mathbf{cosk}_n(X)

sends an $\mathrm{\infty }$-groupoid modeled as a Kan complex simplicial set to its $n$-truncation.

Discussion of this can be found for instance in

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

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

In $\mathrm{\infty }$-Lie groupoids

…

Properties

Criteria for convergence

We discuss conditions that ensure that Postnikov towers converge.

Proposition

In an (∞,1)-topos which is locally of finite homotopy dimension, Postnikov towers converge.

This is (Lurie, prop. 7.2.1.10).

Relation to other concepts

• At least if the ambient $(\mathrm{\infty },1)$-category is a locally contractible (∞,1)-topos $H$, so that there is a notion of structured path ∞-groupoid-functor $\Pi :H\to H$, the homotopy fibers of the morphisms $X\to {\Pi }_n(X)$ into the Postnikov tower of $\Pi (X)$ form the

  • Whitehead tower in the (∞,1)-topos $H$ of the object $X$.

• In the context of nonabelian cohomology in (∞,1)-toposes the fact that we have Postnikov towers has been called the Whitehead principle of nonabelian cohomology.

References

Section 6.5…

• Jacob Lurie, Higher Topos Theory