equivalences in/of $(\infty,1)$-categories
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structures in a cohesive (∞,1)-topos
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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 DuskinSimplicial 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 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 $E_\infty$-ring spectra with an application to BP, preprint, 1993.
For simplicial commutative rings,
Last revised on March 15, 2015 at 21:05:18. See the history of this page for a list of all contributions to it.