Groups

Classically, a group is a set ${\displaystyle G}$ with a group law ${\displaystyle m:G\times G\rightarrow G}$, with ${\displaystyle m(x,y)=x\cdot y}$, satisfying three axioms: the existence of an identity element, the existence of an inverse for each element of the group, and associativity.

To put this in a categorical framework, first define the identity and inverse as functions (morphisms of the set ${\displaystyle G}$) by ${\displaystyle e:1\rightarrow G}$ with ${\displaystyle e(*)=1}$, and ${\displaystyle i:G\rightarrow G}$ with ${\displaystyle i(x)=x^{-1}}$. Here ${\displaystyle 1}$ denotes the set with one element ${\displaystyle 1=\left\{*\right\}}$, which allows one to identify elements ${\displaystyle x\in G}$ with morphisms ${\displaystyle 1\rightarrow G}$.

It is then possible to write the axioms of a group in terms of functions (note how the existential quantifier is absent):

• ${\displaystyle \forall x\in G,\forall y\in G,\forall z\in G,m(m(x,y),z)=m(x,m(y,z))}$,
• ${\displaystyle \forall x\in G,m(e(*),x)=m(x,e(*))=x}$,
• ${\displaystyle \forall x\in G,m(i(x),x)=m(x,i(x))=e(*)}$.

Then this can be expressed with commutative diagrams:^[1][2]

Now use the coproduct (the disjoint union of sets) to glue the three morphisms in one: ${\displaystyle \alpha =e+i+m}$ according to

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

Thus a group is an ${\displaystyle F}$-algebra where ${\displaystyle F}$ is the functor ${\displaystyle F(G)=1+G+G\times G}$. However the reverse is not necessarily true. Some ${\displaystyle F}$-algebra where ${\displaystyle F}$ is the functor ${\displaystyle F(G)=1+G+G\times G}$ are not groups.

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

Algebraic structures

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.

Lattice

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, which are partial orders 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.

Recurrence

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.