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

Idea

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

Definition

Proposition

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

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

This is (Lurie, prop. 5.5.6.18).

We write

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

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

The reflector of the reflective embedding provides morphisms

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

from each object to its $n$-truncation.

Definition

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

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

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

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

(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 $\infty Grpd$

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

The unit of the adjunction $(tr_n \dashv cosk_n)$

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

sends an $\infty$-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 DuskinSimplicial matrices and the nerves of weak $n$-categories I: Nerves of bicategories , TAC 9 no. 2, (2002). (web)

In $\infty$-Lie groupoids

…

In $E_\infty$-rings

The Postnikov tower of a connective E-∞ ring is a sequence of square-zero extensions. See Basterra 99 and Lurie “Higher Algebra”, section 8.4 (the result is due to Kriz).

In simplicial commutative rings

(A special case of the above:) The Postnikov tower of a simplicial commutative ring is a sequence of square-zero extensions. See Toen-Vezzosi.

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

  • Whitehead tower in the (∞,1)-topos$\mathbf{H}$ 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

For $E_\infty$-rings: section 7.4 of

• Jacob Lurie, Higher Algebra

and in more classical language, section 8 of

• M. Basterra?, Andre-Quillen cohomology of commutative S-algebras, J. Pure Appl. Algebra 144 (1999), no. 2, 111–143.

• Igor Kriz, Towers of $E_\infty$-ring spectra with an application to BP, preprint, 1993.

For simplicial commutative rings,

• Bertrand Toen, Gabriele Vezzosi, Homotopical Algebraic Geometry II: geometric stacks and applications, arXiv:math/0404373.