1 Locally Cartesian Closed Categories

Definition 1.0.1 exponentiable morphism

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Suppose $\mathbb {C}$ is a category with pullbacks. A morphism $f \colon A \to B$ in $\mathbb {C}$ is exponentiable if the pullback functor $f^* \colon \mathbb {C}/ B \to \mathbb {C}/ A$ has a right adjoint $f_*$. Since $f^*$ always has a left adjoint $f_!$, given by post-composition with $f$, an exponentiable morphism $f$ gives rise to an adjoint triple

$\begin{tikzcd} & {\catC / B} \\ \\ {} & {\catC / A} \arrow[""{name=0, anchor=center, inner sep=0}, "{f_*}", bend left, shift left=5, from=1-2, to=3-2] \arrow[""{name=1, anchor=center, inner sep=0}, "{f_!}"', bend right, shift right=5, from=1-2, to=3-2] \arrow[""{name=2, anchor=center, inner sep=0}, "{f^*}"{description}, from=3-2, to=1-2] \arrow["\dashv"{anchor=center}, draw=none, from=1, to=2] \arrow["\dashv"{anchor=center}, draw=none, from=2, to=0] \end{tikzcd}$

Definition 1.0.2 pushforward functor

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Let $f \colon A \to B$ be an exponentiable morphism in a category $\mathbb {C}$ with pullbacks. We call the right adjoint $f_*$ of the pullback functor $f^*$ the pushforward functor along $f$.

Theorem 1.0.3 exponentiable morphisms are exponentiable objects of the slices

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A morphism $f \colon A \to B$ in a category $\mathbb {C}$ with pullbacks is exponentiable if and only if it is an exponentiable object, regarded as an object of the slice $\mathbb {C}/ B$.

Definition 1.0.4 Locally cartesian closed categories

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A category with pullbacks is locally cartesian closed if is a category $\mathbb {C}$ with a terminal object $1$ and with all slices $\mathbb {C}/ A$ cartesian closed.