Maths - Concrete Category - Monoids and Groups

monoid external

Monoids and Groups are categories with one object and an (endo-)functor going from that single functor back to itself. So this is the simplest type category that still has some structure.

This is useful to study because:

  • It gives a different approach to monoids and groups which are interesing in their own right.
  • Many definitions and theorems about monoids can be generalised to small categories with more than one object.
  • It allows monoids and groups to be defined over stuctures other than sets.

In Set theory Terms

First a reminder of the conventional set theory approach to these structures.

A monoid is a semigroup with an identity element or a group without inverse (these group-like structures are discussed in conventional terms on these pages).

monoid

The axioms required of a monoid operation are those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object.

A monoid is a category with a single object.

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 *.

Cayley's theorem states that every group is isomorphic to a subgroup of the symmetric group on G. That is to set of permutations or endo-mappings where the multipication operation is functional composition of these endo-mappings.

Endomap

When we looked at graphs we saw how an endomap is represented. If we overlay several graphs (shown here as red and blue) we get a Cayley graph.

  External diagram Internal mapping for a specific example
category diagram: monoid external monoid internal

So a single object with maps to itself can be a monoid but this does not really tell us much about the internal structure of the monoid/group.

Group in Category Theory Terms

In set theory terms we looked at a group as a set+structure, here we want to define this structure by its external properties (that is in terms of structure preserving mappings: functors) which is a more category theoretical approach and has the advantage that we can define it over structures other than sets.

How can we relate this to the structure in terms in the group identities:

We define:

Axioms:

  Set Theory Category Theory
Associatively (a b) c = a (b c) μ•(id×μ) = μ•(μ×id)
Identity element (unit) 'e' such that: e a = a = a e μ•(η×id) = id = μ•(id×η)
Inverse element 'a-1' such that: a a-1 = e = a-1 a μ•(id×δ)•Δ = η•ε = μ•(δ×id)•Δ

where:

These category theory axioms are derived from the set theory versions as follows,

Group
Associativity

μ(a,μ(b,c)) = μ(μ(a,b),c)

μ(id×μ)(a,b,c) = μ(μ×id)(a,b,c)

μ•(id×μ) = μ•(μ×id)

group associativity
Group
Identity (unit)

μ(e×a) = a = μ(a×e)

μ•(η×id) = id = μ•(id×η)

group identity
Group
Inverse


μ(a,a-1) = e
μ(id×δ)(a,a) = e
μ•(id×δ)•Δ = η•ε = μ•(δ×id)•Δ

group inverse

We discuss this idea of taking a group and extracting the structure from the underlying set on this page.

Lists Form a Free Monoid

list as a free monoid

list as free monoid Lists form a free monoid, that is, this diagram commutes.

This concept can be extended to other free structures see [R.M. Burstall and P.J. Landin, "Programs and their proofs, an algebraic approach", Machine Intelligence 4(1969)]

more about lists on this page.

Universal Properties of Monoid or Group

See this page for a discussion of universal properties.

Terminal Initial
terminal arrow category initial arrow category
trivial monoid (consisting of only the identity element) trivial monoid
Product
(pullback)

Sum
(Coproduct)
(pushout)

product arrow category sum arrow category
the product is given by the cartesian product with multiplication defined componentwise. free product
the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute.

 


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