nLab exponentiable morphism

Idea

An exponentiable morphism (sometimes called a powerful morphism) $f \colon A \to B$ in a category $\mathcal{C}$ is a morphism that is exponentiable as an object of the slice category $\mathcal{C}/B$. When pullbacks along $f$ exist, giving rise to a base change functor $f^* : \mathcal{C}/B \to \mathcal{C}/A$, this is equivalent to asking for $f^*$ to have a right adjoint $\Pi_f : \mathcal{C}/A \to \mathcal{C}/B$ (called the dependent product), whose universal property is explicitly described by that of a distributivity pullback. However, one can consider exponentiability even in the absence of pullbacks.

Relation to exponentiable objects

If $\mathcal{C}$ has a terminal object and $X$ is an object in $\mathcal{C}$, then if the unique morphism $!_X : X \to 1$ is exponentiable, then, for every object $Y$ for which the product $X \times Y$ exists, the exponentiable object $Y^X$ exists and is given by $\Pi_{!_X} (\pi_1 \colon X \times Y \to X)$.

Conversely, suppose $\mathcal{C}$ has a terminal object and $X$ is an exponentiable object in $\mathcal{C}$. Let $f \colon Z \to X$ be a morphism. If the following pullback exists, it exhibits the exponential morphism $\Pi_{!_X} f$.

Consequently, in a category with finite limits, an object $X$ is an exponentiable object if and only if $!_X : X \to 1$ is an exponentiable morphism.

Properties

• If $C$ has equalizers of coreflexive pairs, then any pullback of an exponentiable morphism is exponentiable. This follows from the adjoint triangle theorem, since the left adjoint $\Sigma_f$ of pullback is comonadic.

Examples

• Exponentiable functors are characterised by a factorisation property.

• The exponentiable morphisms in $Top$ were characterized by Niefield. In particular, a subspace inclusion $C \to D$ is exponentiable if and only if $C$ is locally closed?.

• The exponentiable morphisms in $Locale$ and $Topos$ which are embeddings were also characterized by Niefield. It seems that no complete characterization of exponentiable morphisms in $Locale$ or $Topos$ appears in the literature.

Related concepts

• A category is locally cartesian closed precisely when it has all pullbacks and in which every morphism is exponentiable.

• The universal property of $\Pi_f$ is described by a distributivity pullback, at least when pullbacks along $f$ exist.

• The type theoretic interpretation of exponential morphisms is the dependent product.

• exponentiable object

References

• Susan Niefield, Cartesianness, PhD thesis, Rutgers 1978 (proquest:302920643)

• Susan Niefield, Cartesian inclusion: locales and toposes., Communications in Algebra, 9.16, 1981, pp. 1639-1671 (doi:10.1080/00927878108822672)

• Susan Niefield, Cartesianness: topological spaces, uniform spaces, and affine schemes., Journal of Pure and Applied Algebra, 23.2, 1982, pp. 147-167.(doi:10.1016/0022-4049(82)90004-4, pdf)

• Mark Weber, Polynomials in categories with pullbacks, arXiv, TAC