Exponentiable Locales

• Exponentiable locales are pointfree spaces where exponentials exist for every locale if and only if they are locally compact.
• They are studied through explicit frame-level constructions using the way‐below relation and advanced categorical methods like Frobenius reciprocity and Beck–Chevalley.
• The theory unifies spatial topology and categorical logic by characterizing both object-level and slice exponentiability with concrete examples such as finite and discrete locales.

Searching arXiv for relevant papers on exponentiable locales and related locale-theoretic characterizations. I’m going to look up the cited arXiv papers to ground the article in the primary sources. Exponentiable locales are locales $X$ for which exponentials $Y^X$ exist for all locales $Y$, equivalently for which the functor $(-) \times X$ has a right adjoint $(-)^X$. In $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$, this is the pointfree analogue of exponentiable spaces, and the central theorem is that a locale is exponentiable if and only if it is locally compact. Modern treatments sharpen this in two directions: they give explicit frame-level constructions of $Y^X$ from the way-below relation, and they extend the analysis from objects of $\mathrm{Loc}$ to morphisms in slices $\mathrm{Loc}/B$, where exponentiability is controlled by dependent products, Frobenius reciprocity, Beck–Chevalley, and, over finite bases, a double-category description in terms of normal lax functors and doubly continuous vertical arrows (Huang, 21 Jul 2025, Niefield, 2011).

1. Categorical formulation in $\mathrm{Loc}$

A frame $Y^X$0 is a poset with finite meets and arbitrary joins, satisfying distributivity

$Y^X$1

with top $Y^X$2 and bottom $Y^X$3. A locale $Y^X$4 is given by its frame of opens $Y^X$5, and a continuous map $Y^X$6 is equivalently a frame homomorphism $Y^X$7. This identifies $Y^X$8 with $Y^X$9. The Sierpiński locale $Y$0 has frame $Y$1, and continuous maps $Y$2 correspond to opens of $Y$3:

$Y$4

For a locale $Y$5, exponentiability means that for every locale $Y$6 there exists $Y$7 with

$Y$8

naturally in $Y$9; the image of $(-) \times X$0 is the evaluation map $(-) \times X$1 (Huang, 21 Jul 2025).

For morphisms, the relevant notion is exponentiability in a slice. If $(-) \times X$2 is a locale map, then $(-) \times X$3 is exponentiable in $(-) \times X$4 when the pullback-product functor

$(-) \times X$5

admits a right adjoint $(-) \times X$6. Equivalently, there are natural bijections

$(-) \times X$7

for all $(-) \times X$8 over $(-) \times X$9. Thus object-level exponentiability is the special case $(-)^X$0, while slice exponentiability governs relative function locales and dependent products (Niefield, 2011).

2. Local compactness and the way-below relation

The object-level classification is the theorem usually attributed to Hyland: for a locale $(-)^X$1, the following are equivalent: $(-)^X$2 is exponentiable in $(-)^X$3, $(-)^X$4 exists, and $(-)^X$5 is locally compact. In this context, local compactness is expressed pointfreely by the way-below relation. For opens $(-)^X$6,

$(-)^X$7

Then $(-)^X$8 is locally compact precisely when

$(-)^X$9

The construction uses two structural properties of $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$0: stability under finite joins and interpolation, namely if $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$1, then there exists $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$2 with $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$3 (Huang, 21 Jul 2025).

The same equivalence can be expressed in the language of continuous lattices. For a frame $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$4, one writes $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$5 if every directed $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$6 with $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$7 contains some $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$8 with $\mathrm{Loc}\simeq \mathrm{Frm}^{\mathrm{op}}$9. Then $Y^X$0 is continuous if

$Y^X$1

A locale $Y^X$2 is locally compact iff $Y^X$3 is continuous, and this is equivalent to exponentiability in $Y^X$4. The classical reformulation is that $Y^X$5 exists if and only if $Y^X$6 is locally compact (Niefield, 2011).

A plausible implication is that the localic notion isolates exactly the compactness needed for right adjoints to product functors, without requiring $Y^X$7 itself to be cartesian closed. This is consistent with the standard observation that $Y^X$8 is not cartesian closed in general, while the full subcategory of locally compact locales is cartesian closed (Huang, 21 Jul 2025).

3. Construction of exponentials

A constructive route to exponentials reduces everything to $Y^X$9. The key result is that $\mathrm{Loc}$0 exists iff $\mathrm{Loc}$1 exists for all $\mathrm{Loc}$2. When $\mathrm{Loc}$3 is locally compact, $\mathrm{Loc}$4 can be presented by generators and relations. The generators are symbols

$\mathrm{Loc}$5 for $\mathrm{Loc}$6, interpreted as the open asking whether $\mathrm{Loc}$7 is well contained in the argument open $\mathrm{Loc}$8. They satisfy monotonicity in $\mathrm{Loc}$9, finite joins in $\mathrm{Loc}/B$0, and interpolation: $\mathrm{Loc}/B$1

$\mathrm{Loc}/B$2

$\mathrm{Loc}/B$3

The evaluation map $\mathrm{Loc}/B$4 is determined by

$\mathrm{Loc}/B$5

Given $\mathrm{Loc}/B$6, its transpose is defined on generators by

$\mathrm{Loc}/B$7

and the converse direction reconstructs $\mathrm{Loc}/B$8 from $\mathrm{Loc}/B$9 via

$\mathrm{Loc}$0

This yields the adjunction

$\mathrm{Loc}$1

The role of $\mathrm{Loc}$2, rather than a naive coefficient map based on $\mathrm{Loc}$3, is precisely to repair naturality (Huang, 21 Jul 2025).

For a general codomain $\mathrm{Loc}$4, the frame $\mathrm{Loc}$5 is generated by symbols $\mathrm{Loc}$6 with $\mathrm{Loc}$7 and $\mathrm{Loc}$8. The defining relations encode monotonicity, finite joins in $\mathrm{Loc}$9, finite meets in $Y^X$00, directed joins in $Y^X$01, and finite joins in $Y^X$02. In particular,

$Y^X$03

for directed $Y^X$04, and

$Y^X$05

These generators and relations provide an explicit pointfree presentation of $Y^X$06, and they show concretely how Scott continuity in the $Y^X$07-parameter is built into the exponential (Huang, 21 Jul 2025).

4. Exponentiable maps and slice locales

For a map $Y^X$08, the slice-theoretic formulation uses the adjunctions

$Y^X$09

where $Y^X$10 is dependent sum, $Y^X$11 is pullback, and $Y^X$12 is dependent product when it exists. In this setting, $Y^X$13 is exponentiable in $Y^X$14 iff $Y^X$15 exists and satisfies Frobenius reciprocity and Beck–Chevalley. Frobenius reciprocity is

$Y^X$16

and Beck–Chevalley says that for a pullback square

$Y^X$17

the canonical morphism $Y^X$18 is an isomorphism. These identities express that $Y^X$19 is the right adjoint $Y^X$20 to $Y^X$21 (Niefield, 2011).

Through the Joyal–Tierney equivalence $Y^X$22, exponentiability over a base is equivalent to local compactness internally in the topos of sheaves on $Y^X$23. Thus for $Y^X$24,

$Y^X$25

Over the terminal locale this recovers the classical criterion for objects. Over a general base, it identifies relative exponentials with internal local compactness of the locale of opens (Niefield, 2011).

The slice exponential law is therefore not an isolated formalism but the precise relative form of local compactness:

$Y^X$26

whenever $Y^X$27 is exponentiable. In the continuous and doubly continuous situations, the existence of $Y^X$28 with Frobenius and Beck–Chevalley supplies the right adjoint $Y^X$29 explicitly (Niefield, 2011).

5. Double-category and finite-base characterizations

The paper “Exponentiability via Double Categories” packages several exponentiability criteria into a single double-category theorem. For locales, the relevant double category $Y^X$30 has locales as objects, locale maps as horizontal morphisms, finite-meet-preserving maps $Y^X$31 as vertical morphisms, and 2-cells

$Y^X$32

in the pointwise frame order. For a finite poset $Y^X$33, normal lax functors $Y^X$34 encode families of locales and finite-meet-preserving transition maps; via glueing there is an equivalence

$Y^X$35

where $Y^X$36 is the down-set locale $Y^X$37 (Niefield, 2011).

The finite-base specialization states that for a map $Y^X$38 corresponding to a vertical normal lax functor $Y^X$39, the following are equivalent: $Y^X$40 is exponentiable in $Y^X$41 over $Y^X$42; $Y^X$43 is exponentiable in $Y^X$44; and each arrow $Y^X$45 is exponentiable in $Y^X$46, with $Y^X$47 preserving pseudo-functors for all $Y^X$48. The preservation condition is then shown to hold automatically under mild hypotheses when the arrows are exponentiable. This translates a global problem in a slice into local conditions on the vertical arrows and coherence of the associated pseudo-functor (Niefield, 2011).

The Sierpiński-base case is especially concrete. If $Y^X$49 is a vertical morphism in $Y^X$50 corresponding to $Y^X$51, then the following are equivalent: $Y^X$52 is exponentiable in $Y^X$53; $Y^X$54 is exponentiable in $Y^X$55; $Y^X$56 is doubly continuous; and $Y^X$57 is locally compact as an internal locale in $Y^X$58. Here doubly continuity is defined by introducing the Scott-topology functor

$Y^X$59

and a relative way-below relation

$Y^X$60

Then $Y^X$61 is doubly continuous when $Y^X$62 is continuous and every $Y^X$63 is the join of elements $Y^X$64 with $Y^X$65 for some $Y^X$66. This is the frame-theoretic form of relative local compactness over the Sierpiński locale (Niefield, 2011).

6. Double exponentiation and equivariant locale theory

Ordinary exponentiation and double exponentiation are distinct. In a category $Y^X$67 with finite products, an object $Y^X$68 is exponentiable when $Y^X$69 has a right adjoint $Y^X$70. By contrast, fixing an object $Y^X$71, one defines

$Y^X$72

and says that $Y^X$73 is double exponentiable when the presheaf exponential $Y^X$74 is representable by an object $Y^X$75, equivalently when

$Y^X$76

naturally in $Y^X$77 and $Y^X$78. For locales over an elementary topos, the Sierpiński object is the Sierpiński locale, $Y^X$79 is the double power locale functor, and its lower and upper submonads are the lower and upper power locale monads (Townsend, 2015).

The central stability theorem is that if $Y^X$80 is an order-enriched category with finite products, $Y^X$81 is an internal group in $Y^X$82, and $Y^X$83 is double exponentiable in $Y^X$84, then the trivial $Y^X$85-object $Y^X$86 is double exponentiable in $Y^X$87. This holds even if $Y^X$88 is not exponentiable. The proof uses the adjunction $Y^X$89 together with Frobenius reciprocity, and transports the double power monad structure to equivariant objects via its strength. In $Y^X$90, this means that the double power locale, and hence lower and upper power locales, persist equivariantly (Townsend, 2015).

This does not alter the criterion for ordinary exponentials in $Y^X$91: $Y^X$92 still exists precisely when $Y^X$93 is locally compact. What it does show is that the power-locale side of locale theory is robust under passage to categories of $Y^X$94-objects and, with slice-stability, to internal groupoids. The same framework retains open maps, triquotient surjections as regular epis, and connected-components adjunctions in settings that may lack coequalizers, and it applies even to categories of spaces not of the form $Y^X$95 (Townsend, 2015).

7. Examples, non-examples, and related settings

Typical exponentiable locales are exactly the locally compact ones. Finite locales are exponentiable because finite frames are continuous. Discrete locales are exponentiable because Boolean frames are locally compact. Spatial locales of locally compact Hausdorff spaces, compact Hausdorff spaces, Stone locales, spectral locales, intervals such as $Y^X$96, Euclidean spaces $Y^X$97, and Scott-open locales of continuous dcpos are all locally compact and hence exponentiable. At the morphism level, locally compact maps $Y^X$98 are exponentiable in $Y^X$99, and projections $Y$00 are exponentiable exactly when $Y$01 is locally compact (Huang, 21 Jul 2025, Niefield, 2011).

Standard non-examples are equally informative. The spatial locale of $Y$02 with the usual topology is not locally compact and is therefore not exponentiable; likewise, infinite-dimensional Banach spaces, including separable Hilbert spaces, are not locally compact, so their spatial locales are not exponentiable. More generally, maps that fail to have $Y$03, or fail Frobenius reciprocity or Beck–Chevalley, do not yield $Y$04 and are non-exponentiable (Huang, 21 Jul 2025, Niefield, 2011).

The relationship with topological spaces and posets is structural rather than merely analogical. In $Y$05, exponentiable spaces are core-compact; for sober spaces, this agrees pointfree with the localic condition that every open is a join of opens well below it. Over finite $Y$06 bases, the topological specialization of the double-category theorem says that $Y$07 is exponentiable iff the associated normal lax functor $Y$08 is a pseudo-functor and each fiberwise vertical arrow is exponentiable. For posets, exponentiable morphisms in $Y$09 correspond to pseudo-functors $Y$10, and for locales the slice $Y$11 is equivalent to $Y$12. These correspondences show that the localic theory simultaneously generalizes the classical space-level criterion and admits a purely order-theoretic reformulation (Niefield, 2011).

Taken together, these results yield a precise summary. Object-level exponentiability in $Y$13 is equivalent to local compactness of the frame of opens; morphism-level exponentiability in $Y$14 is equivalent to internal local compactness of $Y$15, or, over finite bases, to pseudo-functorial coherence plus exponentiable vertical arrows in the associated double category; and power-locale constructions based on double exponentiation remain available well beyond the cartesian closed fragment of locale theory (Huang, 21 Jul 2025, Niefield, 2011, Townsend, 2015).