Fermat theory

In category theory, a Lawvere theory (named after American mathematician William Lawvere) is a category that can be considered a categorical counterpart of the notion of an equational theory. Intuitively, it is a categorical generalization of algebraic structures (e.g., a group or a ring), where there exists a "generic" object and all objects are isomorphic to an integer power of $x$ , representing the inputs for the $n$ -ary operations on $x$ (i.e., of the form $x^{n}\mapsto x$ , starting from the fact that $x:=x^{1}$ and so $x\circ x:=x^{1}\circ x^{1}:=x^{1}x^{1}:=x^{2}$ ; trivially generalizing inductively, we get the rest of the objects) where the operations come from the algebraic structure at hand (e.g., addition and/or multiplication).

The Lawvere theory of groups has as its generic object an underlying placeholder $x$ where the other objects are the inputs for $n$ -ary operations from those integer powers of $x$ of the form ( $x^{n}$ ) back to $x$ , where a model, a finite-product preserving functor, from this theory into a target category $C$ such as the category of sets or topological spaces, would map the abstract theory $x$ onto the category to create a concrete, "combined," group-based structure. This model would provide a set with group structure (a group) or a topological space with group structure (a topological group), supplying appropriate names to the generic object $x$ and its mappings ( $n$ -ary operations) according to whatever the theory and model at play are; in the model of sets for the Lawvere theory of groups, the generic object is a group and its mappings are group operations.

Definition

Let $\aleph _{0}$ be a skeleton of the category FinSet of finite sets and functions. Formally, a Lawvere theory consists of a small category $L$ with (strictly associative) finite products and a strict identity-on-objects functor $I:\aleph _{0}^{\text{op}}\rightarrow L$ preserving finite products.

A model of a Lawvere theory in a category $C$ with finite products is a finite-product preserving functor $M:L\rightarrow C$ . A morphism of models $h:M\rightarrow N$ where $M$ and $N$ are models of $L$ is a natural transformation of functors.

Model examples

Some examples of models of the Lawvere theory of groups (i.e., $L$ = $\mathbf {LawGrp}$ ):

$M:L\rightarrow \mathbf {Set}$ ( $M$ is a classical group)
$M:L\rightarrow \mathbf {Top}$ ( $M$ is a topological group)
$M:L\rightarrow \mathbf {Man}$ ( $M$ is a Lie group).

Some examples of models of the Lawvere theory of rings (i.e., $L$ = $\mathbf {LawRing}$ ):

$M:L\rightarrow \mathbf {Set}$ ( $M$ is a classical ring)
$M:L\rightarrow \mathbf {Top}$ ( $M$ is a topological ring)
$M:L\rightarrow \mathbf {Man}$ ( $M$ is a smooth ring).

Category of Lawvere theories

A map between Lawvere theories $(L,I)$ and $(L',I')$ is a finite-product preserving functor that commutes with $I$ and $I'$ . Such a map is commonly seen as an interpretation of $(L,I)$ in $(L',I')$ .