Algebraic 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 $\mathcal{C}$. ...