Definition
Category of groups
Groups and identity-preserving multiplicative maps.
The category of groups has groups as objects and group homomorphisms as morphisms. Thus a map obeys . It automatically sends identity to identity and inverses to inverses. Identities and composition are the corresponding functions.
Convention and comparison
This category imposes no topology or differentiability. The constant map with value the target identity is always a morphism; a constant map with any other value is not.