ScalingStacks

Remark 7.15. Recall that corollary 4.27 allow us to model the small $\infty$-category of exact functors $\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}}))$ as the pretriangulated spectral category $\mathrm{rep}({\mathcal{B}},{\mathcal{A}})$ of right-compact $\Upsilon({\mathcal{A}})^{\op}\wedge\Upsilon({\mathcal{B}})$-modules. Combined with proposition 2.10, this implies that the associated mapping space $(\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}})))_{\mathrm{iso}}$ can be calculated as $|w_{\bullet}\mathrm{rep}({\mathcal{B}},\Idem({\mathcal{A}}))|$. Moreover, $\mathrm{rep}({\mathcal{B}},\Idem({\mathcal{A}}))$ inherits a natural Waldhausen structure as a full subcategory of the cofibrant objects in the model structure on the category of ${\mathcal{B}}\text{-}\Idem({\mathcal{A}})$-bimodules. As such, we can also consider the algebraic $K$-theory space $|w_{\bullet}S_{\bullet}\mathrm{rep}({\mathcal{B}},\Idem({\mathcal{A}}))|$ and associated spectrum.