Proposition 2.12 (0KEI) · ScalingStacks

[0KEI]

Proposition 2.12. The inclusion $\mathbf{Cat}_{d}\hookrightarrow\mathbf{Cat}_{\infty}$ admits a left adjoint $h_{d}\colon\mathbf{Cat}_{\infty}\to\mathbf{Cat}_{d}$ with unit map given by $\theta_{d}\colon\mathcal{C}\to h_{d}\mathcal{C}$ .

[0KEJ]

Proof. By T.2.3.4.18, every essentially $d$ -category is equivalent to a $d$ -category and for every $d$ -category $\mathcal{D}$ , the map

\operatorname{Fun}\left(h_{d}\mathcal{C},\mathcal{D}\right)\to\operatorname{Fun}\left(\mathcal{C},\mathcal{D}\right)

is an isomorphism by 2.11. Restricting to the maximal Kan sub-complexes, the map of simplicial sets

\theta_{d}^{*}\colon\operatorname{Map}_{\mathbf{Cat}_{d}}\left(h_{d}\mathcal{C},\mathcal{D}\right)\to\operatorname{Map}_{\mathbf{Cat}_{\infty}}\left(\mathcal{C},\mathcal{D}\right)

is a homotopy equivalence. It now follows that $\theta_{d}$ exhibits $h_{d}\mathcal{C}$ as the $\mathbf{Cat}_{d}$ -localization of $\mathcal{C}$ in the sense of T.5.2.7.6. Thus, the claim about the existence of a left adjoint follows from T.5.2.7.8 and the claim about the unit follows from the proof of T.5.2.7.8. ∎