Simplicial model category structures on the category of chain functors

Abstract

The model structure on the category of chain functors $\mathfrak{Ch}$, developed in [4], has the main features of a simplicial model category structure, taking into account the lack of arbitrary (co-)limits in $\mathfrak{Ch}$. After an appropriate tensor and cotensor structure in $\mathfrak{Ch}$ is established ($\S 1, \S 3$), Quillen’s axiom SM7 is verified in $\S 5$ and $\S 6$. Moreover, it turns out that in the definition of a simplicial model structure, the category of simplicial sets can be replaced by the category of simplicial spectra endowing $\mathfrak{Ch}$ with the structure of an approximate simplicial stable model structure (= approximate ss-model structure) ($\S 7$). In $\S 8$ the model structure on $\mathfrak{Ch}$ is shown to be proper.