CatDat

Structure

category of M-sets

notation: $M{-}\Set$
objects: sets with a left action of a monoid $M$
morphisms: maps that are compatible with the $M$ -action, meaning $f(m \cdot x)=m \cdot f(x)$ , also called $M$ -maps
Related categories: $\J_2$ , $R{-}\Mod$ , $\Set$
nLab Link

Here, $M$ can be any monoid. But the most important special case is that of a group. To settle (future) non-properties, we assume that $M$ is non-trivial, since otherwise we just get the category $\Set$ .

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

is not skeletal
is not trivial

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

—

Undecidable properties

There is 1 property for which it cannot be decided if it is satisfied or not.

is semi-strongly connected

Special objects

terminal object: singleton set with the unique action
initial object: empty set with the unique action
products: direct products with the evident $M$ -action
coproducts: disjoint union with obvious $M$ -action

Special morphisms

isomorphisms: bijective $M$ -maps
monomorphisms: injective $M$ -maps
epimorphisms: surjective $M$ -maps
regular monomorphisms: same as monomorphisms
regular epimorphisms: surjective homomorphisms