Groups form a full subcategory of monoids

From Groupprops
Jump to: navigation, search

Statement

The category of groups forms a full subcategory of the category of monoids, via the functor that sends a group to its underlying monoids forgetting the inverse map.

In other words, the forgetful functor from the category of groups to the category of monoids is full, faithful and injective.

Related facts

Proof

Injectivity

This follows from the fact that the inverse map on a group is completely determined by its binary operation and identity element. Further information: Equivalence of definitions of group

Faithfulness

This follows from the fact that a group homomorphism is completely described by what it does at the set level, hence, it is completely described by the corresponding monoid homomorphism.

Fullness

This follows from the fact that, to check that a set-theoretic map of groups is a group homomorphism, it suffices to check that it preserves the binary operation.