# Relationship between monoids and monads definition

### Monoid - Wikipedia

The definitions of monads and monoids on the one hand, a binary relation between X and Y. Thus, the Kleisli category SetP is isomorphic to. A monoidal action is a functor from a monoid to the category of endofunctors on a The monadic coherence relations come for free from your monoidal action . So, compared to Haskell, the monad is defined in terms of return, join and fmap. A functor is defined as a mapping of categories. . be pretty familiar with the definition of a monad as a monoid in the category of endofunctors.

Instead, each function carries a number to a list of numbers in a two-step process. Compute 0 or more results Combine those results unto a single answer somehow.

A simple way to combine function is composition: Let's take our examples above, and compose each function with itself: Without getting too much into type theory, the point is that you can combine two integers to get an integer, but you can't always compose two functions and get a function of the same type. So, let's define a different way of combining functions. When we combine two of these functions, we don't want to "double-wrap" the results.

Here is what we do. By treating a functor as an arrow, we forfeit the information about its action on the internals of a category its objects and morphismsjust like we forfeit the information about the action of a function on elements of a set when we treat it as an arrow in Set.

But functors between any two categories also form a category. This time you are asked to consider something that was an arrow in one category to be an object in another. In a functor category functors are objects and natural transformations are morphisms.

We have discovered that the same thing can be an arrow in one category and an object in another. Instead of switching between two views, we can try to merge them into one. This is how we get the concept of a 2-category, in which objects are called 0-cells, morphisms are 1-cells, and morphisms between morphisms are 2-cells.

## …, a monoid is a category, a category is a monad, a monad is a monoid, …

The category of categories Cat is an immediate example. We have categories as 0-cells, functors as 1-cells, and natural transformations as 2-cells.

The laws of a 2-category tell us that 1-cells between any two 0-cells form a category in other words, C a, b is a hom-category rather than a hom-set. This fits nicely with our earlier assertion that functors between any two categories form a functor category.

In particular, 1-cells from any 0-cell back to itself also form a category, the hom-category C a, a ; but that category has even more structure. Members of C a, a can be viewed as arrows in C or as objects in C a, a.

As arrows, they can be composed with each other. But when we look at them as objects, the composition becomes a mapping from a pair of objects to an object. In fact it looks very much like a product — a tensor product to be precise. This tensor product has a unit: It turns out that, in any 2-category, a hom-category C a, a is automatically a monoidal category with the tensor product defined as composition of 1-cells.

Associativity and unit laws simply fall out from the corresponding category laws.

The hom-category Cat a, a is the category of endofunctors on a. Endofunctor composition plays the role of a tensor product in it. The identity functor is the unit with respect to this product. You might recall that, in a general monoidal category, we did not insist on the monoid laws being satisfied on the nose. It was often enough for the unit laws and the associativity laws to be satisfied up to isomorphism.

In a 2-category, monoidal laws in C a, a follow from composition laws for 1-cells. These laws are strict, so we will always get a strict monoidal category.

It is, however, possible to relax these laws as well. We can say, for instance, that a composition of the identity 1-cell ida with another 1-cell, f:: The data defining a monad is thus one 1-cell and two 2-cells, subject to some commutative diagrams of 2-cells.

The fact that this data lives in Cat brings nothing important to the story: The 2-category Span Set is one of my favourites. The objects of Span Set are sets.

### category theory - Simple explanation of a monad - Mathematics Stack Exchange

Composition of spans involves taking pullbacks. The interesting question is what happens when we consider a monad in Span Set. Now we need to give two 2-cells, that is, morphisms of spans. So far so good. All this data has another name: X is the set of objects, Y is the set of arrows. So, a category is just a special case of the more general concept of monad, right?