Concrete category of algebras

Summary

A concrete category of algebras is a category whose objects are algebras of a fixed signature and whose morphisms are homomorphisms. It comes with the forgetful functor to sets. It is the main example of a concrete category: objects are sets with structure, and morphisms are functions that respect the structure. Varieties, quasivarieties and other classes of algebras are studied in this setting. There, notions such as monomorphism, epimorphism, free object and reflection can be compared with their set-based meanings.