# nLab bicartesian closed preordered object

Contents

category theory

## Applications

#### Limits and colimits

limits and colimits

(0,1)-category

(0,1)-topos

# Contents

## Definition

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

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

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 $b:* \to X$,

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