There are at least two approaches to algebra in category theory:
- Monads and algebras
- Lawvere algebraic theories
The relationship between algebras and monads is discussed on the page here. On the remainder of this page we discuss Lawvere algebraic theories.
Lawvere Algebraic Theories
In this approach, instead of using signatures as we would in universal algebra, we define arrows from powers of our structure 'T'.
This means that we can only define algebras on a structure with associative products.
|We can then use this to define the axioms like this:|
Models of Algebraic Theories
See page here for a general discussion of algebraic theories and models of those theories.
A model of the theory 'T' is a functor F: T->Set which preserves finite products.
Initial and Final Algebras
Category Theory and W-types
W-types are initial algebras for polynomial endofunctors.