Kerodon

Remark 2.3.5.8. It is not difficult to show that the category $\operatorname{2Cat}_{\operatorname{Str}}$ of strict $2$-categories admits small colimits (beware that this is not true for the larger category $\operatorname{2Cat}$). Combining Corollary 2.3.5.7 with Proposition 1.2.3.15, we deduce that the Duskin nerve functor $\operatorname{N}^{\operatorname{D}}_{\bullet }: \operatorname{2Cat}_{\operatorname{Str}} \rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint $\operatorname{Set_{\Delta }}\rightarrow \operatorname{2Cat}_{\operatorname{Str}}$, which carries a simplicial set $S_{\bullet }$ to the generalized geometric realization $| S_{\bullet } |^{\operatorname{Path}_{(2)}[\bullet ] }$. Composing this left adjoint with the fully faithful embedding $\operatorname{N}^{\operatorname{D}}_{\bullet }: \operatorname{2Cat}_{\operatorname{ULax}} \rightarrow \operatorname{Set_{\Delta }}$ (Theorem 2.3.4.1), we deduce that the inclusion functor $\operatorname{2Cat}_{\operatorname{Str}} \hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}$ has a left adjoint, given by the construction $\operatorname{\mathcal{C}}\mapsto | \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) |^{ \operatorname{Path}_{(2)}[\bullet ] }$. We can regard Theorem 2.3.5.6 as providing an explicit description of this left adjoint in a special case: it carries each partially ordered set $Q$ to the strict $2$-category $\operatorname{Path}_{(2)}[Q]$ given by Construction 2.3.5.1.