CatDat

Unknown properties

There is 1 property for which the database doesn't have an answer if it is satisfied or not. Please help to contribute the data!

• is 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

Comments

• If this category is strongly connected depends on the choice of $M$. For $M = 1$ it is, for $M = \mathbb{Z}$ it is not. In general, if $G$ is a group, then $G{-}\mathbf{Set}$ is 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 $\mathbb{Z}/p^n$ or to $\mathbb{Z}/p^{\infty}$ for a prime $p$. See also MSE/5129804.