Totalization of simplicial homotopy types

TL;DR

This paper investigates the obstructions to the functoriality and uniqueness of the totalization process for simplicial objects in homotopy categories, revealing that these obstructions can be non-zero, with implications from cyclic homology.

Contribution

It identifies and analyzes the obstructions to functoriality and uniqueness of totalization in simplicial homotopy types, connecting to cyclic homology results.

Findings

Obstructions to totalization can be non-zero.

Functoriality and uniqueness are not guaranteed.

Connections to cyclic homology of group algebras.

Abstract

We identify the obstructions for the functoriality and the uniqueness of the totalization functor, (partially) defined on the category of simplicial objects in the homotopy category of a stable model category, and we use a result from the cyclic homology of group algebras to show they can be non-zero.