nLab geometric hyperdoctrine

Idea

A geometric hyperdoctrine is a hyperdoctrine with respect to lattices that are frames.

 Definition

Let $C$ be a category with finite limits. A geometric hyperdoctrine over $C$ is a functor

$$P : C^{op} \to Frm$$

from the opposite category of $C$ to the category of frames, such that for every morphism $f : A \to B$ in $C$, the functor $P(A) \to P(B)$ has a left adjoint $\exists_f$ satisfying

1. Frobenius reciprocity;

2. Beck-Chevalley condition.

Examples

Let $OLoc$ be the category of overt locales, and let $Frm$ be the category of frames. The functor $\mathcal{O}:OLoc^\op \to Frm$ that takes an overt locale to its frame of opens is a geometric hyperdoctrine.

Let $Set$ be the category of sets, defined as a Grothendieck topos on the singleton, and let $Frm$ be the category of frames. The subobject poset functor $\mathrm{Sub}:Set^\op \to Frm$ that takes a set to its subobject poset is a geometric hyperdoctrine.

This could be generalized to any geometric category $C$: the subobject poset functor $\mathrm{Sub}:C^\op \to Frm$ that takes an object of $C$ to its subobject poset is a geometric hyperdoctrine.

See also

category	functor	internal logic	theory	hyperdoctrine	subobject poset	coverage	classifying topos
finitely complete category	cartesian functor	cartesian logic	essentially algebraic theory
lextensive category		disjunctive logic
regular category	regular functor	regular logic	regular theory	regular hyperdoctrine	infimum-semilattice	regular coverage	regular topos
coherent category	coherent functor	coherent logic	coherent theory	coherent hyperdoctrine	distributive lattice	coherent coverage	coherent topos
geometric category	geometric functor	geometric logic	geometric theory	geometric hyperdoctrine	frame	geometric coverage	Grothendieck topos
Heyting category	Heyting functor	intuitionistic first-order logic	intuitionistic first-order theory	first-order hyperdoctrine	Heyting algebra
De Morgan Heyting category		intuitionistic first-order logic with weak excluded middle			De Morgan Heyting algebra
Boolean category		classical first-order logic	classical first-order theory	Boolean hyperdoctrine	Boolean algebra
star-autonomous category		multiplicative classical linear logic
symmetric monoidal closed category		multiplicative intuitionistic linear logic
cartesian monoidal category		fragment $\{\&, \top\}$ of linear logic
cocartesian monoidal category		fragment $\{\oplus, 0\}$ of linear logic
cartesian closed category		simply typed lambda calculus

 References

• Graham Manuell, Uniform locales and their constructive aspects, (arXiv:2106.00678)