M-Set.yaml

id: M-Set
name: 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
description: 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 <a href="/category/Set">$\Set$</a>.
nlab_link: https://ncatlab.org/nlab/show/MSet

tags:
  - algebra

related:
  - J2
  - R-Mod
  - Set

satisfied_properties:
  - property: locally small
    proof: There is a forgetful functor $M{-}\Set \to \Set$ and $\Set$ is locally small.

  - property: Grothendieck topos
    proof: It is the category of sheaves on the opposite of the one-object category associated to $M$.

  - property: finitary algebraic
    proof: Take the algebraic theory of an $M$-sets (having a unary operation for each $m \in M$).

unsatisfied_properties:
  - property: skeletal
    proof: This is trivial.

  - property: trivial
    proof: This is trivial.

undecidable_properties:
  - property: semi-strongly connected
    proof: If this category is semi-strongly connected depends on the choice of $M$. For $M = 1$ it is, for $M = \IZ$ it is not. In general, if $G$ is a group, then $G{-}\Set$ is semi-strongly connected if and only if for all subgroups $H,K \subseteq G$, $H$ is subconjugated to $K$ or $K$ is subconjugated to $H$. If $G$ is abelian, this means that the poset of subgroups is linear, in which case $G$ is either isomorphic to $\IZ/p^n$ or to $\IZ/p^{\infty}$ for a prime $p$. See also <a href="https://math.stackexchange.com/questions/5129804" target="_blank">MSE/5129804</a>.

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

special_morphisms:
  isomorphisms:
    description: bijective $M$-maps
    proof: This characterization holds in every algebraic category.
  monomorphisms:
    description: injective $M$-maps
    proof: 'This holds in every finitary algebraic category: the forgetful functor to $\Set$ is faithful, hence reflects monomorphisms, and it is continuous (even representable), hence preserves monomorphisms.'
  epimorphisms:
    description: surjective $M$-maps
    proof: This holds in every functor category $[\C,\Set]$, here applied to the case that $\C$ has just one object.
  regular monomorphisms:
    description: same as monomorphisms
    proof: This is because the category is mono-regular.
  regular epimorphisms:
    description: surjective homomorphisms
    proof: This holds in every finitary algebraic category.