category of monoids

 algebra
Notation Mon\Mon Objects monoids Morphisms monoid homomorphisms Related CMon\CMon, Ring\Ring, Cat\Cat, Grp\Grp, SemiGrp\SemiGrp External nLab link
This category is a typical example of a one-sorted finitary algebraic category.

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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Special objects

  • terminal object: trivial monoid
  • initial object: trivial monoid
  • products: direct products with pointwise operations
  • coproducts: free products

Special morphisms

  • isomorphisms: bijective morphisms
  • monomorphisms: injective morphisms
  • epimorphisms: A monoid map f:T→Sf : T \to S is an epimorphism iff SS equals the dominion of U≔f(T)⊆SU \coloneqq f(T) \subseteq S, meaning that for every s∈Ss \in S there are u1,…,um+1∈Uu_1,\dotsc,u_{m+1} \in U, v1,…,vm∈Uv_1,\dotsc,v_m \in U, x1,…,xm∈Sx_1,\dotsc,x_m \in S and y1,…,ym∈Sy_1,\dotsc,y_m \in S such that s=x1u1s = x_1 u_1, u1=v1y1u_1 = v_1 y_1, xi−1vi−1=xiuix_{i-1} v_{i-1} = x_i u_i, uiyi−1=viyiu_i y_{i-1} = v_i y_i, xmvm=um+1x_m v_m = u_{m+1} and um+1ym=su_{m+1} y_m = s.
  • regular monomorphisms:
  • regular epimorphisms: surjective morphisms

Functors

The database stores 6 functors based on the category of monoids.