Kerodon

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Definition 5.0.0.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories and let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{E}}$.

  • We say that $f$ is $U$-cartesian if, for every object $W \in \operatorname{\mathcal{E}}$, the diagram of sets

    \[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W,X) \ar [r]^-{f \circ } \ar [d]^-{U} & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(W,Y) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(W), U(X) ) \ar [r]^-{U(f) \circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(W), U(Y) ) } \]

    is a pullback square.

  • We say that $f$ is $U$-cocartesian if, for every object $Z \in \operatorname{\mathcal{E}}$, the diagram of sets

    \[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(Y,Z) \ar [r]^-{\circ f} \ar [d]^-{U} & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(Y), U(Z) ) \ar [r]^-{\circ U(f)} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Z) ) } \]

    is a pullback square.