Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 2.3.2.6. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category, so that the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category. Then:

  • Objects of the $\infty $-category $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with objects of the $2$-category $\operatorname{\mathcal{C}}$.

  • If $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}$, then morphisms from $X$ to $Y$ in the $\infty $-category $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with $1$-morphisms from $X$ to $Y$ in the $2$-category $\operatorname{\mathcal{C}}$.

  • If $f,g: X \rightarrow Y$ are $1$-morphisms in $\operatorname{\mathcal{C}}$ having the same domain and codomain, then $f$ and $g$ are homotopic when regarded as morphisms of the $\infty $-category $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ (Definition 1.4.3.1) if and only if they are isomorphic when viewed as objects of the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. More precisely, vertical composition with the left unit constraint $\lambda _{f}: \operatorname{id}_{Y} \circ f \xRightarrow {\sim } f$ induces a bijection

    \[ \xymatrix@R =50pt@C=50pt{ \{ \text{Isomorphisms from $f$ to $g$ in the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$} \} \ar [d]^{\sim } \\ \{ \text{Homotopies from $f$ to $g$ in the $\infty $-category $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$} \} . } \]