nLab coherent hyperdoctrine

Definition

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

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

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

1. Frobenius reciprocity;

2. Beck-Chevalley condition.

Properties

The assignment (here) of a coherent hyperdoctrine $S(C)$ to a coherent category $C$ extends to a 2-adjunction

$$(A \dashv S) : CoherentCat \stackrel{\overset{A}{\leftarrow}}{\underset{S}{\hookrightarrow}} CoherentHyperdoctrine$$

with the right adjoint being a full and faithful 2-functor, hence exhibiting $CoherentCat$ as a reflective sub-2-category of $CoherentHyperdoctrine$.

(Here $CoherentCat$ has as 2-morphisms those natural transformations that preserve finite products.)

This appears as (Coumans, prop. 8).

Coherent hyperdoctrines are closed under canonical extension $(-)^\delta : DistLattice \to DistLattice$, in that for $P : C^{op} \to DistLattic$ a coherent hyperdoctrine, so is $(-)^\delta \circ P$.

This appears as (Coumans, prop. 9).

Examples

Powersets

The powerset functor

$$P := \{0,1\}^{(-)} : Set^{op} \to DistLattice$$

(sending a set to its power set and a function to the preimage-assignment) is a coherent hyperdoctrine.

Over a coherent category

Let $C$ be a coherent category. For every object $A \in C$ the poset of subobjects $Sub_C(A)$ is a distributive lattice.

The corresponding functor

$$C^{op} \to DistLattice$$

from the opposite category of $C$ to the category of distributive lattices is called the coherent hyperdoctrine of $C$.

For a coherent theory

Accordingly, there is a coherent hyperdoctrine associated with any coherent theory, where the objects of $C$ are lists of free variables in the theory, and the lattice assigned to them is that of propositions of the theory in this context.

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

• Dion Coumans, Generalizing canonical extensions to the categorical setting (Arxiv)

• Graham Manuell, Quantalic spectra of semirings, (arXiv:2201.06408)