Fibers of partial totalizations of a pointed cosimplicial space

TL;DR

This paper investigates the fibers of partial totalizations of pointed cosimplicial spaces, revealing they depend only on certain looped objects and are characterized as k-fold loop objects, using explicit obstruction theory in quasicategories.

Contribution

It provides a new explicit obstruction-theoretic description of fibers of partial totalizations in terms of loop objects within pointed cosimplicial spaces.

Findings

Fibers depend only on the looped cosimplicial object $\

$ ext{Fiber} o ext{Total}_n$ is a $k$-fold loop object.

Approach uses explicit obstruction theory with quasicategories.

Abstract

Let $X^\bullet$ be a cosimplicial object in a pointed $\infty$-category. We show that the fiber of $\mathrm{Tot}_m(X^\bullet) \to \mathrm{Tot}_n(X^\bullet)$ depends only on the pointed cosimplicial object $\Omega^k X^\bullet$ and is in particular a $k$-fold loop object, where $k = 2n - m+2$. The approach is explicit obstruction theory with quasicategories. We also discuss generalizations to other types of homotopy limits and colimits.