Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 4.6.3.11 (Unitality). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X$ and $Y$. Then:

$(1)$

The composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \{ \operatorname{id}_{X} \} \hookrightarrow \operatorname{Hom})_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

is equal to the identity (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).

$(2)$

The composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \{ \operatorname{id}_{Y} \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \hookrightarrow \operatorname{Hom})_{\operatorname{\mathcal{C}}}(Y,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

is equal to the identity (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).

Proof. There is a diagram of Kan complexes

\[ \xymatrix { & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X,Y) \ar [dr] & \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^{ \{ \operatorname{id}_{X} \} } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \ar@ {-->}[r]^{\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), } \]

where the left diagonal arrow is induced by the map $\sigma ^0: [2] \rightarrow [1]$ of Notation 1.1.1.9 and the right diagonal arrow is induced by the map $\delta ^{1}: [1] \rightarrow [2]$ of Notation 1.1.1.8. Here the solid arrows are well-defined as morphisms of simplicial sets, while the dotted arrow is well-defined only as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We now observe that the triangle on the left is strictly commutative, the triangle on the right commutes up to homotopy (by the construction of the composition law $\circ $). Assertion $(1)$ follows from the observation that the composition of the diagonal arrows is the identity on the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (since $\sigma ^{0} \circ \delta ^{1}$ is the identity on the object $[1] \in \operatorname{{\bf \Delta }}$). Assertion $(2)$ follows by a similar argument. $\square$