Category of algebras in a variety

Definition

Let ${\mathcal {V}}$ be a variety of algebras. In other words, ${\mathcal {V}}$ is the collection of all algebras with a particular operator domain (each algebra has the same collection of operation arities) satisfying a set of universal identities. In particular, ${\mathcal {V}}$ is closed under taking subalgebras, quotient algebras, and arbitrary direct products.

The category of algebras in ${\mathcal {V}}$ is defined as follows:

1. The objects of the category are the algebras in ${\mathcal {V}}$.
2. The set of morphisms between any two objects of the category is precisely the set of algebra homomorphisms between them.
3. The identity map is the identity morphism.
4. Composition of morphisms is done by function composition.

The category of algebras in a variety has the natural structure of a concrete category via the functor sending each algebra to its underlying set and sending each morphism to its underlying set function. In particular, it is a locally small category.

Examples

• The category of sets is the category corresponding to the variety of sets: here, a set is a set with no operations and no universal identities.
• The category of sets is the category corresponding to the variety of pointed sets: here, a pointed set is a set with a single 0-ary operation that returns the distinguished point of the set. There are no universal identities.
• The category of groups is the category corresponding to the variety of groups.
• The category of Abelian groups is the category corresponding to the variety of Abelian groups.
• The category of unital rings is the category corresponding to the variety of unital rings.