Cofibrancy of operadic constructions in positive symmetric spectra

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."