F-algebra

Classically, a group is a set G with a binary operation m : G × G → G, satisfying three axioms:

• associativity: ∀ x∈G, ∀ y∈G, ∀ z∈G, m(m(x, y), z) = m(x, m(y, z)),
• identity element: ∃ 1∈G, ∀ x∈G, m(1, x) = m(x, 1) = x,
• inverse element: ∃ 1∈G, ∀ x∈G, ∃ x⁻¹∈G, m(x⁻¹, x) = m(x, x⁻¹) = 1.

Note that the closure axiom is included in the symbolic definition of m.

To treat this in a categorical framework, we first define the identity and inverse as morphisms e and i respectively. Let C be an arbitrary category with finite products and a terminal object *. The group G is an object in C. The morphism e sends each element in * to 1, the identity element of the group G. The morphism i sends each element x in G to its inverse x⁻¹, satisfying m(x⁻¹, x) = m(x, x⁻¹) = 1. Then a group G can be defined as a 4-tuple (G, m, e, i), which describes a monoid category with only one object G. Each morphism f in this monoid category has an inverse f⁻¹ that satisfies f⁻¹ ∘ f = f ∘ f⁻¹ = Id.[1]

It is then possible to rewrite the axioms in terms of morphisms:

• ∀ x∈G, ∀ y∈G, ∀ z∈G, m(m(x, y), z) = m(x, m(y, z)),
• ∀ x∈G, m(e(*), x) = m(x, e(*)) = x,
• ∀ x∈G, m(i(x), x) = m(x, i(x)) = e(*).

Then remove references to the elements of G (which will also remove universal quantifiers):

• m∘(m, Id) = m∘(Id, m),
• m∘(e, Id) = m∘(Id, e) = Id,
• m∘(i, Id) = m∘(Id, i) = e.

Which is the same as claiming commutativity for the following diagrams:[2]

Now use the coproduct (the disjoint union of sets) to glue the three morphisms in one: α = e + i + m according to

${\displaystyle {\begin{matrix}\alpha$ :{1}+G+G\times G&\to &G,\\1&\mapsto &1,\\x&\mapsto &x^{-1},\\(x,y)&\mapsto &x\cdot y.\end{matrix}}}

This defines a group as a F-algebra where F is the functor F(G) = 1 + G + G × G.

Note 1: The above construction is used to define group objects over an arbitrary category with finite products and a terminal object *. When the category admits finite coproducts, the group objects are F-algebras. For example, finite groups are F-algebras in the category of finite sets and Lie groups are F-algebras in the category of smooth manifolds with smooth maps.

Going one step ahead of universal algebra, most algebraic structures are F-algebras. For example, abelian groups are F-algebras for the same functor F(G) = 1 + G + G×G as for groups, with an additional axiom for commutativity: m∘t = m, where t(x,y) = (y,x) is the transpose on GxG.

Monoids are F-algebras of signature F(M) = 1 + M×M. In the same vein, semigroups are F-algebras of signature F(S) = S×S

Rings, domains and fields are also F-algebras with a signature involving two laws +,•: R×R → R, an additive identity 0: 1 → R, a multiplicative identity 1: 1 → R, and an additive inverse for each element -: R → R. As all these functions share the same codomain R they can be glued into a single signature function 1 + 1 + R + R×R + R×R → R, with axioms to express associativity, distributivity, and so on. This makes rings F-algebras on the category of sets with signature 1 + 1 + R + R×R + R×R.

Alternatively, we can look at the functor F(R) = 1 + R×R in the category of abelian groups. In that context, the multiplication is a homomorphism, meaning m(x + y, z) = m(x,z) + m(y,z) and m(x,y + z) = m(x,y) + m(x,z), which are precisely the distributivity conditions. Therefore, a ring is an F-algebra of signature 1 + R×R over the category of abelian groups which satisfies two axioms (associativity and identity for the multiplication).

When we come to vector spaces and modules, the signature functor includes a scalar multiplication k×E → E, and the signature F(E) = 1 + E + k×E is parametrized by k over the category of fields, or rings.

Algebras over a field can be viewed as F-algebras of signature 1 + 1 + A + A×A + A×A + k×A over the category of sets, of signature 1 + A×A over the category of modules (a module with an internal multiplication), and of signature k×A over the category of rings (a ring with a scalar multiplication), when they are associative and unitary.

Not all mathematical structures are F-algebras. For example, a poset P may be defined in categorical terms with a morphism s:P × P → Ω, on a subobject classifier (Ω = {0,1} in the category of sets and s(x,y)=1 precisely when x≤y). The axioms restricting the morphism s to define a poset can be rewritten in terms of morphisms. However, as the codomain of s is Ω and not P, it is not an F-algebra.

However, lattices in which every two elements have a supremum and an infimum, and in particular total orders, are F-algebras. This is because they can equivalently be defined in terms of the algebraic operations: x∨y = inf(x,y) and x∧y = sup(x,y), subject to certain axioms (commutativity, associativity, absorption and idempotency). Thus they are F-algebras of signature P x P + P x P. It is often said that lattice theory draws on both order theory and universal algebra.

Consider the functor ${\displaystyle F:\mathrm {\bf {Set}} \to \mathrm {\bf {Set}} }$ that sends a set ${\displaystyle X}$ to ${\displaystyle 1+X}$. Here, ${\displaystyle \mathrm {\bf {Set}} }$ denotes the category of sets, ${\displaystyle +}$ denotes the usual coproduct given by the disjoint union, and ${\displaystyle 1}$ is a terminal object (i.e. any singleton set). Then, the set ${\displaystyle \mathbb {N} }$ of natural numbers together with the function ${\displaystyle [\mathrm {zero} ,\mathrm {succ} ]:1+\mathbb {N} \to \mathbb {N} }$—which is the coproduct of the functions ${\displaystyle \mathrm {zero} :1\mapsto 0}$ and ${\displaystyle \mathrm {succ} :n\mapsto n+1}$—is an F-algebra.