nLab coherent functor

Disambiguation: there is a different notion of a coherent functor due Auslander as a finitely presented object in the category of additive functors from a fixed abelian category to the category of abelian groups, see abelianization of an additive category. This terminology is suggested by algebra where coherent algebras and modules have alike definition.

Definition

A functor $F : C \to D$ between coherent categories is called coherent if it is a regular functor and in addition preserves finite unions.

Properties

• For $\mathcal{T}$ a coherent theory and $\mathcal{C}_{\mathbb{T}}$ its syntactic category, coherent functors $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ into some topos $\mathcal{E}$ are precisely models of the theory in $\mathcal{E}$.

  For $\mathcal{C}_{\mathbb{T}}$ equipped with the structure of the syntactic site (the coherent coverage), this is in turn equivalent to geometric morphisms $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$ into the sheaf topos over $\mathcal{C}_{\mathbb{T}}$ (the classifying topos for the theory).

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

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• Peter Johnstone, Sketches of an Elephant