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Notation 2.1.4.19 (Oriented Fiber Products). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be categories, and suppose we are given a pair of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. We let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the iterated pullback $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{E}}) } \operatorname{Fun}( [1], \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{E}}) } \operatorname{\mathcal{D}}$. We will refer to $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ as the oriented fiber product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ over $\operatorname{\mathcal{E}}$. More concretely:

  • An object of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is a triple $(C, D, \eta )$ where $C$ is an object of the category $\operatorname{\mathcal{C}}$, $D$ is an object of the category $\operatorname{\mathcal{D}}$, and $\eta : F(C) \rightarrow G(D)$ is a morphism in the category $\operatorname{\mathcal{E}}$.

  • If $(C,D,\eta )$ and $(C', D', \eta ')$ are objects of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, then a morphism from $(C,D,\eta )$ to $(C', D', \eta ')$ is a pair $(u,v)$, where $u: C \rightarrow C'$ is a morphism in the category $\operatorname{\mathcal{C}}$, $v: D \rightarrow D'$ is a morphism in the category $\operatorname{\mathcal{D}}$, and the diagram

    \[ \xymatrix { F(C) \ar [r]^-{\eta } \ar [d]^{ F(u) } & G(D) \ar [d]^{G(v)} \\ F(C') \ar [r]^-{\eta '} & G(D') } \]

    commutes in the category $\operatorname{\mathcal{E}}$.