Cofibrancy of operadic constructions in positive symmetric spectra

TL;DR

This paper demonstrates that positive symmetric spectra allow for milder cofibrancy conditions in operadic constructions, especially for the relative composition product, leading to strengthened invariance results in spectral algebra.

Contribution

It provides new cofibrancy results for $n$-fold smash powers and analyzes the cofibrancy of the relative composition product in positive symmetric spectra, improving understanding of operadic invariance.

Findings

Cofibrancy conditions are milder in positive symmetric spectra.

Established new cofibrancy results for $n$-fold smash powers.

Strengthened Quillen invariance results for spectral operads.

Abstract

We show that when using the underlying positive model structure on symmetric spectra one obtains cofibrancy conditions for operadic constructions under much milder hypothesis than one would need for general categories. Our main result provides such an analysis for a key operation, the "relative composition product" $\circ_{\mathcal{O}}$ between right and left $\mathcal{O}$-modules over a spectral operad $\mathcal{O}$, and as a consequence we recover (and usually strengthen) previous results establishing the Quillen invariance of model structures on categories of algebras via weak equivalences of operads, compatibility of forgetful functors with cofibrations and Reedy cofibrancy of bar constructions. Key to the results above are novel cofibrancy results for $n$-fold smash powers of positive cofibrant spectra (and the relative statement for maps). Roughly speaking, we show that such $n$-fold powers satisfy a (new) type of $\Sigma_n$-cofibrancy which can be viewed as "lax $\Sigma_n$-free/projective cofibrancy" in that it determines a larger class of cofibrations still satisfying key technical properties of "true $\Sigma_n$-free/projective cofibrancy."