Kerodon

Warning 2.3.2.9. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Let us say that a $2$-simplex $\sigma $ of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is special if it corresponds to a diagram

where $h = g \circ f$ and $\gamma = \operatorname{id}_{g \circ f}$. Arguing as in the proof of Corollary 2.3.2.8, we see that a strictly unitary lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is strict if and only if it carries special $2$-simplices of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ to special $2$-simplices of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$. Beware, however, that the special $2$-simplices of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ and $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$ do not have an intrinsic description in terms of the simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ and $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{D}})$ themselves. In particular, it is possible to have an isomorphism of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ which does not preserve special $2$-simplices (corresponding to an isomorphism of $2$-categories which is strictly unitary but not strict).

In general, passage from a $2$-category $\operatorname{\mathcal{C}}$ to its Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ involves a slight loss of information. From the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, we can recover the objects of $\operatorname{\mathcal{C}}$ (these can be identified with vertices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$) and the collection of $1$-morphisms $f: X \rightarrow Y$ from an object $X$ to an object $Y$ (these can be identified with edges of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ having source $X$ and target $Y$). However, the composition $g \circ f$ of a pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ cannot be recovered from the structure of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ as an abstract simplicial set. The best we can do is to ask for a thin $2$-simplex $\sigma $ of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ satisfying $d^{2}_0(\sigma ) = g$ and $d^{2}_2(\sigma ) = f$. Such a simplex can be viewed as “witnessing” the presence of an isomorphism of the edge $h = d^{2}_1(\sigma )$ with the composition $g \circ f$. Put another way, the abstract simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ contains enough information to reconstruct the composition $g \circ f$ up to (unique) isomorphism, but not enough information to select a canonical representative of its isomorphism class. This can be viewed as a feature, rather than a bug: the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ often admits a more invariant description than the $2$-category $\operatorname{\mathcal{C}}$ itself (since the information lost by passing from $\operatorname{\mathcal{C}}$ to $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ depends on choices that one would prefer not make in the first place; see Remark 2.3.1.7).