• Interaction With Classifying Spaces. We have an isomorphism of groupoids
    \[ \mathrm{K}_{0}\webleft (\mathcal{C}\webright ) \cong \Pi _{\leq 1}\webleft (\left\lvert \mathrm{N}_{\bullet }\webleft (\mathcal{C}\webright )\right\rvert \webright ), \]

    natural in $\mathcal{C}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$; i.e. the diagram

    commutes up to natural isomorphism.


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