nLab bicartesian closed preordered object

Idea

The notion of a bicartesian closed preordered object or Heyting prealgebra object is the generalization of that of bicartesian closed preordered set or Heyting prealgebra as one passes from the ambient category of sets into more general ambient categories with suitable properties.

Definition

In a finitely complete category $C$, a bicartesian closed preordered object or Heyting prealgebra object is a bicartesian preordered object

$$(X, R, s, t, \rho, \tau_p \wedge, \top, \tau, \lambda_l, \lambda_r, \vee, \bot, \beta, \kappa_l, \kappa_r)$$

with a morphism $(-)\Rightarrow(-):X \times X \to X$ and functions

$$\epsilon_l:((* \to X) \times (* \to X)) \to (* \to R)$$

$$\epsilon_r:((* \to X) \times (* \to X)) \to (* \to R)$$

such that for all global elements $a:* \to X$ and $b:* \to X$,

• $s \circ \epsilon_l(a, b) = (a \Rightarrow b) \wedge a$
• $t \circ \epsilon_l(a, b) = b$
• $s \circ \epsilon_r(a, b) = a$
• $t \circ \epsilon_r(a, b) = b \Rightarrow (a \wedge b)$

See also

• Heyting prealgebra

• bicartesian preordered object

• bicartesian closed category

• Boolean prealgebra object

• Heyting algebra object