nLab HeytAlg

$Heyt Alg$ is the category whose objects are Heyting algebras and whose morphisms are Heyting algebra homomorphisms, that is lattice homomorphisms which also preserve the Heyting implication. $Heyt Alg$ is a subcategory of Pos.

$Hety Alg$ is given by a finitary variety of algebras, or equivalently by a Lawvere theory, so it has all the usual properties of such categories. By general abstract nonsense, the free Heyting algebra on a set $X$ exists, but it is not easy to describe in general.

• The free Heyting algebra on the empty set is the Boolean domain $\{\bot, \top\}$.
• The free Heyting algebra on a singleton $\{x\}$ is infinite, although it has a nice picture.

category: category