1 Algebras over monads We recall from the last lecture the
definition of an Eilenberg-Moore algebra over a monad T =
(T,η,μ): Definition An algebra over a monad T in a category C (a
T-algebra) is a morphism , such that the diagrams below both
commute: While a monad corresponds to the imposition of some
structure on the objects in a category, an algebra over that
monad corresponds to some evaluation of that structure. 1.1
Example: monoids Let T be the Kleene star monad - the one we get
from the adjunction of free and forgetful functors between
Monoids and Sets. Then a T-algebra on a set A is equivalent to a
monoid structure on A. Indeed, if we have a monoid structure on
A, given by and , we can construct a T-algebra by - α() = u This
gives us, indeed, a T-algebra structure on A. Associativity and
unity follows from the corresponding properties in the monoid.
On the other hand, if we have a T-algebra structure on A, we can
construct a monoid structure by setting - u = α() - m(a,b) =
α([a,b]) It is clear that as