internalization and categorical algebra
algebra object (associative, Lie, …)
internal category ($\to$ more)
There are different, related ways in which one could view the notion of algebra over a monad.
Algebras over a monad are usually objects equipped with extra structure, not just properties. (They can also be seen as algebras over the underlying endofunctor, satisfying extra compatibility properties.)
Let $(T,\mu,\eta)$ be a monad on a category $C$. An algebra over $T$, or $T$-algebra, consists of an object $A$ of $C$ together with a morphism $a:TA\to A$ of $C$, such that the following diagrams commute.
The diagram on the left is sometimes called the unit triangle, and the diagram on the right the multiplication square or algebra square.
The corresponding dual notion is that of a coalgebra over a comonad.
In the case of a commutative monad, one can define a tensor product of algebras.
Let $(A,a)$ and $(B,b)$ be $T$-algebra. A morphism of $T$-algebras is a morphism $f:A\to B$ of $C$ which makes the following diagram commute.
The category of $T$-algebras and their morphisms is called the Eilenberg-Moore category and denoted by $C^T$.
Given a monad $(T,\mu,\eta)$ on a category $C$, for every object $X$ of $C$, the object $T X$ is canonically equipped with a $T$-algebra structure, given by the multiplication map $\mu$. The relevant diagrams commute by the monad axioms.
Algebras of this sort are called free algebras.
Given any morphism $f:X\to Y$ of $C$, the map $T f:T X\to T Y$ is a morphism of algebras, by naturality of $\mu$. In general, not every morphism of algebras between $T X$ and $T Y$ arises this way.
The subcategory of free algebras and their morphisms is (equivalent to) the Kleisli category.
Many monads are named after their (free) algebras.
In these cases, the notion of free group, free monoid, et cetera coincide with the notion of free algebra given above.
Given a monoid or group $M$, the algebras of the $M$-action monad on Set are the $M$-sets, i.e. sets equipped with an action of $M$. The morphisms are the equivariant maps.
The example above generalizes to action monads given by monoid objects in a general monoidal category. Famous examples of this construction in mathematics are smooth actions of Lie groups on manifolds and actions of rings on their modules.
The algebras of the maybe monad $(-)_*\colon Set \to Set$, which adds a disjoint point, are the pointed sets.
The algebras of the power set monad are the sup-semilattices.
The algebras of the distribution monad are convex spaces, and more generally algebras of probability monads correspond to generalized convex spaces or conical spaces (see probability monad - algebras).
An algebra over a monad is a special case of a module over a monad in a bicategory. See there for more information.
The Eilenberg-Moore and Kleisli categories are also special cases of more general 2-dimensional universal constructions, namely the Eilenberg-Moore object? and the Kleisli object. See those pages for more information.
algebra over a monad, module over a monad, algebra over an endofunctor, coalgebra over an endofunctor, algebra over a profunctor
Eilenberg-Moore category, Kleisli category, Eilenberg-Moore object?, Kleisli object
An introduction to the basic ideas, which gives some intuition for newcomers, can be found in
See also the references of the article on monads.
Last revised on June 30, 2021 at 20:06:06. See the history of this page for a list of all contributions to it.