category of M-sets

 algebra
Notation M−SetM{-}\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 Children Z−Set\IZ{-}\Set Related Set\Set, R−ModR{-}\Mod, J2\J_2 External nLab link
Here, MM can be any monoid. But the most important special case is that of a group. To settle (future) non-properties, we assume that MM is non-trivial, since otherwise we just get the category Set\Set.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Undecidable properties

There is 1 property for which it cannot be decided whether it is satisfied.

Special objects

  • terminal object: singleton set with the unique action
  • initial object: empty set with the unique action
  • products: direct product with the obvious action
  • coproducts: disjoint union with the obvious action

Special morphisms