Warning 4.8.1.5. We have now given several a priori different definitions for the notion of $(2,1)$-category:
- $(1)$
According to Definition 2.2.8.5, a $(2,1)$-category is a $2$-category $\operatorname{\mathcal{C}}$ in which every $2$-morphism is an isomorphism.
- $(2)$
According to Definition 4.8.1.8 (or Definition 4.8.1.1), a $(2,1)$-category is a simplicial set which satisfies some additional conditions.
However, these definitions are compatible with one another. If $\operatorname{\mathcal{C}}$ is a $(2,1)$-category in the sense of $(1)$, then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is a $(2,1)$-category in the sense of $(2)$ (Example 4.8.1.4). We will later see that the converse is also true: every $(2,1)$-category in the sense $(2)$ is isomorphic to the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is a $2$-category in which every $2$-morphism is an isomorphism (in this case, Theorem 2.3.4.1 guarantees that $\operatorname{\mathcal{C}}$ is unique up to non-strict isomorphism). See Proposition 
.