A Quillen model structure of local homotopy equivalences

TL;DR

This paper constructs a model structure on complexes of projective systems of ind-Banach spaces, linking it to derived categories relevant for local and analytic cyclic homology theories across various algebraic contexts.

Contribution

It introduces a Quillen model structure on graded complexes of ind-Banach spaces, connecting homotopy categories to cyclic homology for different base fields and algebra types.

Findings

Homotopy category matches derived category of quasi-abelian categories.

Applicable to cyclic homology theories for complete, torsionfree V-algebras.

Includes pro-$C^*$-algebras as a special case.

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.