Postnikov tower in an (infinity,1)-category


(,1)(\infty,1)-Category theory

(,1)(\infty,1)-Topos theory

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos

Factorization systems



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



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

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

This is (Lurie, prop.

We write

τ n:Cτ nC nC \mathbf{\tau}_{\leq n} : C \stackrel{\tau_{\leq n}}{\to} C_{\leq n} \hookrightarrow C

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

The reflector of the reflective embedding provides morphisms

Xτ nX X \to \mathbf{\tau}_{\leq n} X

from each object to its nn-truncation.


A Postnikov tower for XCX \in C is a diagram

XX 2X 1X 0 X \to \cdots \to X_2 \to X_1 \to X_0

such that each XX nX \to X_n exhibits X nX_n as the nn-truncation of XX.

This is HTT, def.


A Postnikov pretower is a pre-tower

X 2X 1X 0 \cdots \to X_2 \to X_1 \to X_0

(no initial XX on the left!) which exhibits each X nX_n as the nn-truncation of X n+1X_{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.


In Grpd\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 cosk n+1\mathbf{cosk}_{n+1} on sSet.

The unit of the adjunction (tr ncosk n)(tr_n \dashv cosk_n)

τ n:Xcosk n(X) \mathbf{\tau}_n : \mathbf{}X \to \mathbf{cosk}_n(X)

sends an \infty-groupoid modeled as a Kan complex simplicial set to its nn-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 nn-categories I: Nerves of bicategories , TAC 9 no. 2, (2002). (web)

In \infty-Lie groupoids

In E 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.


Criteria for convergence

We discuss conditions that ensure that Postnikov towers converge.


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

This is (Lurie, prop.

Relation to other concepts


Section 6.5…

For E E_\infty-rings: section 8.4 of

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 E_\infty-ring spectra with an application to BP, preprint, 1993.

For simplicial commutative rings,

Revised on March 3, 2015 17:25:23 by Adeel Khan (