K-Theory

In this section we first define the connective K-theory spectrum $\mathrm{k}(\mathcal{C})$ of a category with finite colimits, then extend it to the non-connective K-theory spectrum $\mathrm{K}(\mathcal{C})$. Connective K-theory Construction 1 (Cospan category). Let $\mathcal{C}$ be a category with pushouts. Applied to the saturated triple $(\mathcal{C}^{\mathrm{op}}, \mathcal{C}^{\mathrm{op}}, \mathcal{C}^{\mathrm{op}})$, the twisted arrow construction produces a complete Segal anima $\mathrm{N}\mathsf{Span}(\mathcal{C}^{\mathrm{op}})$; denote the corresponding category by $\mathsf{coSpan}(\mathcal{C})$. Concretely: Objects of $\mathsf{coSpan}(\mathcal{C})$ are the objects of $\mathcal{C}$. ...

Recall that for a category with finite colimits and idempotent-completeness, $\mathcal{C} \in \mathsf{Cat}^{\mathrm{rex,idem}}$, we defined the Calkin category $\mathsf{Calk}(\mathcal{C}) = (\mathsf{Ind}(\mathcal{C})^{\aleph_1}/\mathcal{C})^{\mathrm{idem}}$ and used the relation $\mathrm{k}(\mathsf{Calk}^n(\mathcal{C})) \simeq \tau_{\ge 0}\Omega\mathrm{k}(\mathsf{Calk}^{n+1}(\mathcal{C}))$ to construct non-connective algebraic K-theory $\mathrm{K}$. The aim of this note is to use the inclusion $\mathsf{Cat}^{\mathrm{rex}} \subset \mathsf{Cat}^{\mathrm{ca}} \simeq \mathsf{Pr}^L_{\mathrm{ca}}$ to extend algebraic K-theory to compactly assembled categories. The result is continuous (Efimov) K-theory. The continuous Calkin category First we extend the Calkin construction from small categories to compactly assembled categories, i.e. we want to produce a dashed arrow making commute. ...