equivalences in/of $(\infty,1)$-categories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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.
For $C$ a presentable (∞,1)-category the subcategory $C_{\leq n}$ of n-truncated objects is a reflective (∞,1)-subcategory
This is (Lurie, prop. 5.5.6.18).
We write
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
from each object to its $n$-truncation.
A Postnikov tower for $X \in C$ is a diagram
such that each $X \to X_n$ exhibits $X_n$ as the $n$-truncation of $X$.
This is HTT, def. 5.5.6.23.
A Postnikov pretower is a pre-tower
(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).
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 $(tr_n \dashv cosk_n)$
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 Duskin Simplicial matrices and the nerves of weak $n$-categories I: Nerves of bicategories , TAC 9 no. 2, (2002). (web)
…
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).
(A special case of the above:) The Postnikov tower of a simplicial commutative ring is a sequence of square-zero extensions. See Toen-Vezzosi.
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. 7.2.1.10).
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
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.
Section 6.5…
For $E_\infty$-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 $E_\infty$-ring spectra with an application to BP, preprint, 1993.
For simplicial commutative rings,