Monoid - Relation To Category Theory

Relation To Category Theory

Group-like structures
Totality* Associativity Identity Inverses Commutativity
Magma Yes No No No No
Semigroup Yes Yes No No No
Monoid Yes Yes Yes No No
Group Yes Yes Yes Yes No
Abelian Group Yes Yes Yes Yes Yes
Loop Yes No Yes Yes No
Quasigroup Yes No No Yes No
Groupoid No Yes Yes Yes No
Category No Yes Yes No No
Semicategory No Yes No No No

Monoids can be viewed as a special class of categories. Indeed, the axioms required of a monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object. That is,

A monoid is, essentially, the same thing as a category with a single object.

More precisely, given a monoid (M,*), one can construct a small category with only one object and whose morphisms are the elements of M. The composition of morphisms is given by the monoid operation *.

Likewise, monoid homomorphisms are just functors between single object categories. So this construction gives an equivalence between the category of (small) monoids Mon and a full subcategory of the category of (small) categories Cat. Similarly, the category of groups is equivalent to another full subcategory of Cat.

In this sense, category theory can be thought of as an extension of the concept of a monoid. Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid.

Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms.

There is also a notion of monoid object which is an abstract definition of what is a monoid in a category. A monoid object in Set is just a monoid.

Read more about this topic:  Monoid

Famous quotes containing the words relation to, relation, category and/or theory:

    “Unaware of the absurdity of it, we introduce our own petty household rules into the economy of the universe for which the life of generations, peoples, of entire planets, has no importance in relation to the general development.”
    —Alexander Herzen (1812–1870)

    “Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.”
    —Henry David Thoreau (1817–1862)

    “I see no reason for calling my work poetry except that there is no other category in which to put it.”
    —Marianne Moore (1887–1972)

    “A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
    —Lionel Trilling (1905–1975)