Exponentiable morphisms for a clan

Joseph Hua

Joint work with: Reid Barton

Many categories, such as $\mathrm{Top}$ and $\mathrm{Cat}$ , are not locally Cartesian closed, but still have certain exponentiable morphisms [1, 2]. We propose a more fine-grained condition of being exponentiable relative to a clan [5].

A clan $(\mathcal{C},\mathcal{R})$ consists of a category $\mathcal{C}$ with a class of maps $\mathcal{R}$ whose members are called $\mathcal{R}$ -maps, such that pullbacks of $\mathcal{R}$ -maps along all morphisms exist and are $\mathcal{R}$ -maps, all isomorphisms are in $\mathcal{R}$ , and $\mathcal{R}$ is closed under composition. Modifying the original definition in [5], we do not require that $\mathcal{C}$ has a terminal object, nor that maps to the terminal object are $\mathcal{R}$ -maps. Following [5], we use $\mathcal{R}(X)$ to denote the full subcategory of the slice $\mathcal{C}/X$ consisting of objects whose underlying map is in $\mathcal{R}$ .

Fix a clan $(\mathcal{C},\mathcal{R})$ and $f:Y\to X$ a morphism in $\mathcal{C}$ with all pullbacks, and such that for any pullback $f^{\prime}:Y^{\prime}\to X^{\prime}$ of $f$ , the restricted pullback functor $(f^{\prime})^{*}:\mathcal{R}(X^{\prime})\to\mathcal{R}(Y^{\prime})$ has a right adjoint $f^{\prime}_{*}:\mathcal{R}(Y^{\prime})\to\mathcal{R}(X^{\prime})$ . We say the Beck–Chevalley condition holds at $f$ if every pullback of $f$ induces a canonical isomorphism:

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We say the partial right adjoint condition holds at $f:Y\to X$ when there is a bijection of hom-sets

\operatorname{\mathrm{Hom}}_{\mathcal{C}/Y}(f^{*}X^{\prime},A)\cong% \operatorname{\mathrm{Hom}}_{\mathcal{C}/X}(X^{\prime},f_{*}A)

natural in $X^{\prime}\in(\mathcal{C}/X)^{\mathsf{op}}$ and $A\in\mathcal{R}(Y)$ . Note that this is stronger than $f_{*}$ merely being right adjoint to the restricted functor $f^{*}:\mathcal{R}(X)\to\mathcal{R}(Y)$ .

Theorem 1.

The following are equivalent. ( $i$ ) The Beck–Chevalley condition holds at all pullbacks of $f$ . ( $i i$ ) The partial right adjoint condition holds at all pullbacks of $f$ . ( $i i i$ ) The partial right adjoint condition holds at $f$ and the pushforward functor (in presheaves) $\Pi_{f}:\widehat{\mathcal{C}}/Y\to\widehat{\mathcal{C}}/X$ preserves (Yoneda images of) $\mathcal{R}$ -maps.

The map $f:Y\to X$ is called $\mathcal{R}$ -exponentiable if it satisfies the equivalent conditions in Theorem 1. Note that $f$ itself need not belong to $\mathcal{R}$ . When $\mathcal{C}$ has finite limits, this notion is a special case of properness with respect to a fibration over $\mathcal{C}$ [6], namely the fibration corresponding to the indexed category $\mathcal{R}(-)$ . A prototypical example is $\mathcal{C}=\mathrm{Top}$ and $\mathcal{R}$ the class of local homeomorphisms, so that $\mathcal{R}(X)\simeq\mathrm{Sh}(X)$ . Then the proper base change theorem [3] says that proper maps are $\mathcal{R}$ -exponentiable. An example of similar flavor: in $\mathcal{C}=\mathrm{Cat}$ , discrete fibrations are exponentiable with respect to the clan structure of discrete opfibrations (and vice versa). A different flavor of example is given by the category of simplicial sets with $\mathcal{R}$ the class of Kan fibrations: the $\mathcal{R}$ -exponentiable maps are exactly the sharp maps of Rezk [4].

[1] Niefield, S. Cartesianness: topological spaces, uniform spaces, and affine schemes. Journal Of Pure And Applied Algebra. 23, 147-167 (1982).
[2] Giraud, J. Méthode de la descente. (Société mathématique de France, 1964).
[3] Stacks project authors, T. The Stacks project. (2025), stacks.math.columbia.edu.
[4] Rezk, C. Fibrations and homotopy colimits of simplicial sheaves. ArXiv Preprint Math/9811038. (1998)
[5] Joyal, A. Notes on Clans and Tribes. (2017), arXiv:1710.10238
[6] Anel, M. & Weinberger, J. Smooth and proper maps with respect to a fibration. MSCS special issue “Advances in Homotopy Type Theory”. (2025)