Definition. Normalized T-monoid [mde-HGO7]

February 20, 2025
Matt Earnshaw
[crutwell-shulman]

A \(T\)-monoid is normalized if its unit cell is cartesian.

Related

Definition. \(T\)-monoid [mde-NO9S]

November 4, 2024
Matt Earnshaw
[crutwell-shulman-2010]

A \(T\)-monoid in a virtual double category, for a monad \((T, \eta , \mu )\) on a virtual double category, is a monoid in the loose (horizontal) Kleisli virtual double category of \(T\).

It comprises in other words,

An object \(X_0\),
a loose morphism \(X : X_0 \nrightarrow TX_0\),
a unit cell,
a multiplication cell,
satisfying unitality and associativity equations.