nLab Heyting category

Definition

A Heyting category (called a logos in Freyd-Scedrov) is a coherent category in which each base change functor $f^*:Sub(Y)\to Sub(X)$ has a right adjoint $\forall_f$ (the universal quantifier, in addition to the left adjoint existential quantifier that exists in any regular category). Thus, every Heyting category $\mathcal{C}$ is a first-order hyperdoctrine given by the subobject poset functor $\mathrm{Sub}:\mathcal{C}^{op} \to HeytAlg$.

It follows that each poset of subobjects $Sub(X)$ is a Heyting algebra and the base-change functors $f^*$ are Heyting algebra homomorphisms. The Heyting implication can be defined as follows: if $A$ and $B$ are subobjects of $X$, then we can define $(A\Rightarrow B) \in Sub(X)$ as $\forall_i(A\cap B) \hookrightarrow X$, where $i\colon A\hookrightarrow X$ is the subobject inclusion and we take $A \cap B\in Sub(A)$. Any Heyting category has an internal logic which is full (typed) first-order intuitionistic logic. The extra right adjoint $\forall_f$ provided by the above axiom gives the universal quantifier in this logic. On the other hand, any category defined in intuitionistic logic which satisfies bounded separation is a Heyting category.

A Heyting functor is a coherent functor between Heyting categories which additionally preserves the right adjoints $\forall_f$. Such functors also preserve the internal interpretation of first-order logic.

Examples

Any topos, and in fact any well-powered infinitary coherent category, is a Heyting category. Any Boolean category is also a Heyting category.

Every locally cartesian closed coherent category is a Heyting category, because the dependent product functors preserve monomorphisms. This implies that every Pi-pretopos is a Heyting pretopos.

In particular, the h-sets in Martin-Löf type theory with propositional truncations form a Heyting category.

Related concepts

• De Morgan Heyting category

• Heyting pretopos

• elementary topos

• Heyting algebra

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

• Peter Freyd, Andre Scedrov, Categories, Allegories