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Tag 0040

Chapter 4: Categories > Section 4.31: Categories over categories

Lemma 4.31.3. Let $\mathcal{C}$ be a category. The $(2, 1)$-category of categories over $\mathcal{C}$ has 2-fibre products. Suppose that $F : \mathcal{X} \to \mathcal{S}$ and $G : \mathcal{Y} \to \mathcal{S}$ are morphisms of categories over $\mathcal{C}$. An explicit 2-fibre product $\mathcal{X} \times_\mathcal{S}\mathcal{Y}$ is given by the following description

  1. an object of $\mathcal{X} \times_\mathcal{S} \mathcal{Y}$ is a quadruple $(U, x, y, f)$, where $U \in \mathop{\rm Ob}\nolimits(\mathcal{C})$, $x\in \mathop{\rm Ob}\nolimits(\mathcal{X}_U)$, $y\in \mathop{\rm Ob}\nolimits(\mathcal{Y}_U)$, and $f : F(x) \to G(y)$ is an isomorphism in $\mathcal{S}_U$,
  2. a morphism $(U, x, y, f) \to (U', x', y', f')$ is given by a pair $(a, b)$, where $a : x \to x'$ is a morphism in $\mathcal{X}$, and $b : y \to y'$ is a morphism in $\mathcal{Y}$ such that
    1. $a$ and $b$ induce the same morphism $U \to U'$, and
    2. the diagram $$ \xymatrix{ F(x) \ar[r]^f \ar[d]^{F(a)} & G(y) \ar[d]^{G(b)} \\ F(x') \ar[r]^{f'} & G(y') } $$ is commutative.

The functors $p : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{X}$ and $q : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{Y}$ are the forgetful functors in this case. The transformation $\psi : F \circ p \to G \circ q$ is given on the object $\xi = (U, x, y, f)$ by $\psi_\xi = f : F(p(\xi)) = F(x) \to G(y) = G(q(\xi))$.

Proof. Let us check the universal property: let $p_\mathcal{W} : \mathcal{W}\to \mathcal{C}$ be a category over $\mathcal{C}$, let $X : \mathcal{W} \to \mathcal{X}$ and $Y : \mathcal{W} \to \mathcal{Y}$ be functors over $\mathcal{C}$, and let $t : F \circ X \to G \circ Y$ be an isomorphism of functors over $\mathcal{C}$. The desired functor $\gamma : \mathcal{W} \to \mathcal{X} \times_\mathcal{S} \mathcal{Y}$ is given by $W \mapsto (p_\mathcal{W}(W), X(W), Y(W), t_W)$. Details omitted; compare with Lemma 4.30.4. $\square$

    The code snippet corresponding to this tag is a part of the file categories.tex and is located in lines 5555–5593 (see updates for more information).

    \begin{lemma}
    \label{lemma-2-product-categories-over-C}
    Let $\mathcal{C}$ be a category.
    The $(2, 1)$-category of categories
    over $\mathcal{C}$ has 2-fibre products.
    Suppose that
    $F : \mathcal{X} \to \mathcal{S}$ and
    $G : \mathcal{Y} \to \mathcal{S}$ are morphisms of categories over
    $\mathcal{C}$.
    An explicit 2-fibre product
    $\mathcal{X} \times_\mathcal{S}\mathcal{Y}$ is given by the following
    description
    \begin{enumerate}
    \item an object of $\mathcal{X} \times_\mathcal{S} \mathcal{Y}$ is a quadruple
    $(U, x, y, f)$, where $U \in \Ob(\mathcal{C})$,
    $x\in \Ob(\mathcal{X}_U)$, $y\in \Ob(\mathcal{Y}_U)$,
    and $f : F(x) \to G(y)$ is an isomorphism in $\mathcal{S}_U$,
    \item a morphism $(U, x, y, f) \to (U', x', y', f')$ is given by a pair
    $(a, b)$, where $a : x \to x'$ is a morphism in $\mathcal{X}$, and
    $b : y \to y'$ is a
    morphism in $\mathcal{Y}$ such that
    \begin{enumerate}
    \item $a$ and $b$ induce the same morphism $U \to U'$, and
    \item the diagram
    $$
    \xymatrix{
    F(x) \ar[r]^f \ar[d]^{F(a)} & G(y) \ar[d]^{G(b)} \\
    F(x') \ar[r]^{f'} & G(y')
    }
    $$
    is commutative.
    \end{enumerate}
    \end{enumerate}
    The functors $p : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{X}$
    and $q : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{Y}$ are the
    forgetful functors in this case. The transformation $\psi : F \circ p \to
    G \circ q$ is given on the object $\xi = (U, x, y, f)$ by
    $\psi_\xi = f : F(p(\xi)) = F(x) \to G(y) = G(q(\xi))$.
    \end{lemma}
    
    \begin{proof}
    Let us check the universal property: let
    $p_\mathcal{W} : \mathcal{W}\to \mathcal{C}$
    be a category over $\mathcal{C}$, let $X : \mathcal{W} \to \mathcal{X}$ and
    $Y : \mathcal{W} \to \mathcal{Y}$ be functors over $\mathcal{C}$, and let
    $t : F \circ X \to G \circ Y$ be an isomorphism of functors over $\mathcal{C}$.
    The desired functor
    $\gamma : \mathcal{W} \to \mathcal{X} \times_\mathcal{S} \mathcal{Y}$
    is given by $W \mapsto (p_\mathcal{W}(W), X(W), Y(W), t_W)$.
    Details omitted; compare with Lemma \ref{lemma-2-fibre-product-categories}.
    \end{proof}

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