The diagonal of a multicosimplicial object

TL;DR

This paper investigates the properties of the diagonal functor in multicosimplicial objects within model categories, establishing its status as a right Quillen functor and exploring implications for fibrancy, homotopy limits, and applications to functor calculus.

Contribution

It proves that the diagonal functor is a right Quillen functor, analyzes its limitations as a Quillen equivalence, and discusses homotopy limits and total objects in multicosimplicial contexts.

Findings

Diagonal functor is a right Quillen functor.

Diagonal of Reedy fibrant objects remains fibrant.

Homotopy limits of multicosimplicial objects are weakly equivalent under certain conditions.

Abstract

We show that the functor that takes a multicosimplicial object in a model category to its diagonal cosimplicial object is a right Quillen functor. This implies that the diagonal of a Reedy fibrant multicosimplicial object is a Reedy fibrant cosimplicial object, which has applications to the calculus of functors. We also show that, although the diagonal functor is a Quillen functor, it is not a Quillen equivalence for multicosimplicial spaces. We also discuss total objects and homotopy limits of multicosimplicial objects. We show that the total object of a multicosimplicial object is isomorphic to the total object of the diagonal, and that the diagonal embedding of the cosimplicial indexing category into the multicosimplicial indexing category is homotopy left cofinal, which implies that the homotopy limits are weakly equivalent if the multicosimplicial object is at least objectwise fibrant.