nLab
cocartesian monoidal preordered object

Contents

Context

Relations

Category theory

Limits and colimits

(0,1)(0,1)-Category theory

Contents

Definition

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

  • a function β:(*X)(*R)\beta:(* \to X) \to (* \to R) such that for all global elements a:*Xa:* \to X, sβ(a)=s \circ \beta(a) = \bot and tβ(a)=at \circ \beta(a) = a.

  • functions

    κ l:((*X)×(*X))(*R)\kappa_l:((* \to X) \times (* \to X)) \to (* \to R)
    κ r:((*X)×(*X))(*R)\kappa_r:((* \to X) \times (* \to X)) \to (* \to R)

    such that for all global elements a:*Xa:* \to X and b:*Xb:* \to X, sκ l(a,b)=as \circ \kappa_l(a,b) = a, tκ l(a,b)=abt \circ \kappa_l(a,b) = a \vee b, sκ r(a,b)=bs \circ \kappa_r(a,b) = b, and tκ r(a,b)=abt \circ \kappa_r(a,b) = a \vee b.

See also

Last revised on May 14, 2022 at 10:31:50. See the history of this page for a list of all contributions to it.