A Quillen model structure of local homotopy equivalences

Abstract

In this note, we construct a closed model structure on the category of $\mathbb{Z}/2\mathbb{Z}$-graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field $F$ of a complete discrete valuation ring $V$, the homotopy category of this model structure is the derived category of the quasi-abelian category $\overleftarrow{\mathsf{Ind}(\mathsf{Ban}_F)}$. This homotopy category is the appropriate target of the local and analytic cyclic homology theories for complete, torsionfree $V$-algebras and $\mathbb{F}$-algebras. When the base field is $\mathbb{C}$, the homotopy category is the target of local and analytic cyclic homology for pro-bornological $\mathbb{C}$-algebras, which includes the subcategory of pro-$C^*$-algebras.