nLab regular hyperdoctrine

Idea

A regular hyperdoctrine, also called an elementary existential doctrine or a regular fibration, is a version of a first-order hyperdoctrine that is appropriate for regular logic.

Definition

Let $C$ be a category with finite limits. A regular hyperdoctrine, or elementary existential doctrine, over $C$ is a functor

$$P \;\colon\; C^{op} \to InfSemiLattice$$

from the opposite category of $C$ to the category of inf-semilattices, 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.

Remark

The original definition of Lawvere did not include the last two conditions and allowed the values of the functor to be arbitrary categories with finite products. Other references sometimes require the adjoints $\exists_f$ to be defined only along projections (corresponding to ordinary existential quantification in logic) and diagonals (corresponding to equality); Lawvere showed that in the presence of Frobenius and Beck-Chevalley more general quantifiers can be constructed in terms of these.

Related concepts

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

• William Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor

• Davide Trotta, The existential completion, 2021 (arxiv:2108.03416)

• Davide Trotta, An algebraic approach to the completions of elementary doctrines, 2021 (arxiv:2108.03415)

• Bart Jacobs, Categorical Logic and Type Theory, Studies in Logic and the Foundations of Mathematics 141, Elsevier (1998) [ISBN:978-0-444-50170-7, pdf, webpage]