nLab cocartesian monoidal preordered object

Definition

In a finitely complete category $C$, a cocartesian monoidal preordered object $X$ is a preordered object with internal preorder $R\stackrel{(s,t)}\hookrightarrow X \times X$ and a monoid object in $C$ with multiplication $(-)\vee(-):X \times X \to X$ and a global unit $\bot:* \to X$, where $*$ is the terminal object in $C$, with

• a function $\beta:(* \to X) \to (* \to R)$ such that for all global elements $a:* \to X$, $s \circ \beta(a) = \bot$ and $t \circ \beta(a) = a$.

• functions

  $$\kappa_l:((* \to X) \times (* \to X)) \to (* \to R)$$
  $$\kappa_r:((* \to X) \times (* \to X)) \to (* \to R)$$

  such that for all global elements $a:* \to X$ and $b:* \to X$, $s \circ \kappa_l(a,b) = a$, $t \circ \kappa_l(a,b) = a \vee b$, $s \circ \kappa_r(a,b) = b$, and $t \circ \kappa_r(a,b) = a \vee b$.

See also

• preordered object

• monoid object

• cartesian monoidal preordered object

• bicartesian preordered object

• join-semilattice object

• cocartesian monoidal category