(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
The notion of Postnikov tower in an -category is the generalization of the notion of Postnikov tower from the archetypical (∞,1)-category Top ∞Grpd to more general -categories.
This is (Lurie, prop. 220.127.116.11).
for the corresponding localization. For , we say that is the -truncation of .
The reflector of the reflective embedding provides morphisms
from each object to its -truncation.
A Postnikov tower for is a diagram
such that each exhibits as the -truncation of .
This is HTT, def. 18.104.22.168.
A Postnikov pretower is a pre-tower
(no initial on the left!) which exhibits each as the -truncation of .
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. 22.214.171.124).
When the archetypical (∞,1)-topos ∞Grpd is presented by the model structure on simplicial sets, truncation is given by the the coskeleton endofunctor on sSet.
The unit of the adjunction
sends an -groupoid modeled as a Kan complex simplicial set to its -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 -categories I: Nerves of bicategories , TAC 9 no. 2, (2002). (web)
In -Lie groupoids
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.
This is (Lurie, prop. 126.96.36.199).
Relation to other concepts
For -rings: section 7.4 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 -ring spectra with an application to BP, preprint, 1993.
For simplicial commutative rings,