  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.
