Notation 7.6.2.15 (04K5): Relative Diagonals

Notation 7.6.2.15 (Relative Diagonals). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: Y \rightarrow X$ be a morphism in $\operatorname{\mathcal{C}}$. Suppose that there exists a pullback square

7.64

\begin{equation} \begin{gathered}\label{equation:relative-diagonal} \xymatrix { Y \times _{X} Y \ar [r]^{ \pi } \ar [d]^{\pi '} & Y \ar [d]^{f} \\ Y \ar [r]^{f} & X } \end{gathered} \end{equation}

in the $\infty $-category $\operatorname{\mathcal{C}}$. Let us abuse notation by identifying $Y$ with an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/X}$, so that $Y \times _{X} Y$ can be viewed as a product of $Y$ with itself in $\operatorname{\mathcal{C}}_{/X}$ (Proposition 7.6.2.14). Applying the construction of Notation 7.6.1.13, we obtain a morphism $\delta _{Y/X}: Y \rightarrow Y \times _{X} Y$, which we will refer to as the relative diagonal of $f$. It is characterized (up to homotopy) by the requirement that ( 7.64) can be extended to a commutative diagram

\[ \xymatrix { Y \ar [dr]^{ \delta _{Y/X} } \ar [drr]^{\operatorname{id}_ Y} \ar [ddr]^{ \operatorname{id}_{Y} } & & \\ & Y \times _{X} Y \ar [r]^{\pi } \ar [d]^{\pi '} & Y \ar [d]^{f} \\ & Y \ar [r]^{f} & X, } \]

where the outer square is the commutative diagram given by the composition

\[ \Delta ^1 \times \Delta ^1 \xrightarrow {(i,j) \mapsto ij} \Delta ^1 \xrightarrow {f} \operatorname{\mathcal{C}}. \]