category of M-sets

  • notation: M−SetM{-}\mathbf{Set}
  • objects: sets with a left action of a monoid MM
  • morphisms: maps that are compatible with the MM-action, meaning f(m⋅x)=m⋅f(x)f(m \cdot x)=m \cdot f(x), also called MM-maps
  • nLab Link
  • Related categories: R−ModR{-}\mathbf{Mod}

Here, MM can be any monoid. But the most important special case is that of a group.

Properties

Properties from the database

Deduced properties

Non-Properties

Non-Properties from the database

Deduced Non-Properties*

*This also uses the deduced properties.

Unknown properties

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Special morphisms

  • Isomorphisms: bijective MM-maps
  • Monomorphisms: injective MM-maps
  • Epimorphisms: surjective MM-maps