Maps of simplicial spectra whose realizations are cofibrations

TL;DR

This paper establishes practical conditions under which the geometric realization of a map of simplicial symmetric spectra is a cofibration, extending classical results to the stable homotopy category.

Contribution

It provides new, easily checkable criteria ensuring cofibrations in the geometric realization of simplicial symmetric spectra, not derived from model category formalism.

Findings

Conditions for cofibrations are simple and space-dependent.

Results apply to symmetric spectra in the stable homotopy category.

No need for Reedy theory or latching object computations.

Abstract

Given a map of simplicial topological spaces, mild conditions on degeneracies and the levelwise maps imply that the geometric realization of the simplicial map is a cofibration. These conditions are not formal consequences of model category theory, but depend on properties of spaces, and similar results have not been available for any model for the stable homotopy category. In this paper we prove such results for symmetric spectra. Consequently, we get a set of conditions which ensure that the geometric realization of a map of simplicial symmetric spectra is a cofibration. These conditions are very user-friendly in that they are simple, often easily checked, and do not require computation of a latching object or any other knowledge of Reedy theory.