nLab cartesian monoidal preordered object




Category theory

Limits and colimits

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



In a finitely complete category CC, a cartesian 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(-)\wedge(-):X \times X \to X and a global unit :*X\top:* \to X, where ** is the terminal object in CC, with

  • a function τ:(*X)(*R)\tau:(* \to X) \to (* \to R) such that for all global elements a:*Xa:* \to X, sτ(a)=as \circ \tau(a) = a and tτ(a)=t \circ \tau(a) = \top.

  • functions

    λ l:((*X)×(*X))(*R)\lambda_l:((* \to X) \times (* \to X)) \to (* \to R)
    λ r:((*X)×(*X))(*R)\lambda_r:((* \to X) \times (* \to X)) \to (* \to R)

    such that for all global elements a:*Xa:* \to X and b:*Xb:* \to X, sλ l(a,b)=abs \circ \lambda_l(a,b) = a \wedge b, tλ l(a,b)=at \circ \lambda_l(a,b) = a, sλ r(a,b)=abs \circ \lambda_r(a,b) = a \wedge b, and tλ r(a,b)=bt \circ \lambda_r(a,b) = b.

See also

Last revised on May 14, 2022 at 14:32:59. See the history of this page for a list of all contributions to it.