For instance an associative algebra may be obtained by imposing on the tensor algebra of the underlying vector space (which is the free algebra on that vector space) the product relations, which identify the formal juxtaposition of two elements with their intended product. This is for instance how Clifford algebras or universal enveloping algebras are traditionally defined.
in the category of -algebras. Here is the object of generators and is the object of relations.
The -algebra map corresponds to a unique morphism in , which we think of as the “inclusion” of the generators into (though in general it need not be any kind of monomorphism). Similarly, the -algebra maps correspond to unique morphisms in , which we think of as a binary relation on (though in general, the maps need not be jointly monic either).
The classical case is when Set. Then and are sets, and a presentation of a -algebra by generators and relations amounts to first constructing the free -algebra on the generators, then specifying (via the maps ) a set of pairs of elements of to be identified (the relations). The presented -algebra is the quotient set by these relations.
Probably the most classical case is when is the free group monad on .
Cat (to be precise, Str Cat) is the category of algebras for a monad on the category of quivers. Therefore, we have a notion of a presentation of a category by generators and relations.
Lawvere theories, operads, and even (accessible) monads can be realized as the algebras for monads on some categories. Thus, we can talk about presentations of these sorts of objects by generators and relations. In this context, each “generator” is an operation, and each “relation” is an axiom that the operations must satisfy. For instance, the usual definition of a group in terms of binary, nullary, and unary operations satisfying associativity, unitality, and inversion axioms can be regarded as a presentation of the Lawvere theory (or monad) of groups by generators and relations.
Every -algebra has a presentation by generators and relations.
For any -algebra , the following diagram is a coequalizer (also called the Beck coequalizer):
where is the structure map of the -algebra and is the multiplication of the monad .
This particular presentation is of importance in the monadicity theorem.
However, in practice one is usually interested in more “economical” presentations where and are smaller than . For example, for monads over Set, when and are finite sets, then we say that is finitely presented.
Note that knowing only the category of -algebras does not suffice to determine the notion of “presentation by generators and relations”; we also have to exhibit it as the category of algebras for a monad on some other category . Often the choice of is obvious, but there are cases in which the same category can be realized as the category of algebras for multiple monads on multiple different categories, yielding different notions of “generators and relations”. For example, the category of finitary monads on (= Lawvere theories) can be realized as monadic over the category of finitary endofunctors of Set, or over the category of -indexed sets.