Let $\mathbb{F}$ be a field. Let $K(\mathbb{F})$ be its algebraic (Quillen) $K$-theory spectrum. Let $X$ be a (nice, finite CW) topological space and let $\text{Rep}\Omega(X)$ be the DG category of (homologically) finite complexes of $\mathbb{F}$-valued representations of the DG algebra of chains on $\Omega(X)$ with values in $\mathbb{F}$ (equivalently, the $\infty$-category of $\mathbb{F}$-valued representations of the topological group $\Omega X$).
I'm hoping for an expression of the algebraic $K$ theory of $\text{Rep}\Omega(X)$ as some sort of cohomology of $X$ with values in the $K$ theory spectrum. If I understand correctly, this is too much to ask for, since Quillen $K$ theory does not commute with inverse limits in the category of (DG) categories. I would like to know if/when one could modify this picture to make such a statment true. In particular, I wonder whether negative or Bass $K$ theories (which I don't know much about) would give such a result.
The sort of theory I'm looking for should be a contravariant functor $F$ from the $(\infty, 1)$-category of (sufficiently nice) DG algebras over $\mathbb{F}$ to spectra with the following properties:
- $F$ takes finite direct limits in the category of algebras to inverse limits of spectra (i.e. can be computed from DG generators and relations).
- There is a map from the Grothendieck group of homologically finite-dimensional representations (i.e. representations with finitely many finite-dimensional $\mathbb{F}$-vector space homotopy groups) of $A$ to $F(A)$, which cannot be (naturally) extended to finitely-generated representations.
The reason I'm looking for such a theory is that an inverse limit of $K$ theory groups (of semisimple categories) is appearing in some algebraic $K$ theory calculations for group algebras (and also, of course, by analogy with classical $K$-theory, where one deals only with bundles with finite fibers).
