Fibers of partial totalizations of a pointed cosimplicial space

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.

14 pages. Final version, to appear in Proc. Amer. Math. Soc

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