# Lawvere 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.

## Definition

Let 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 preserving finite products.

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

## 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*′).

Lawvere theories together with maps between them form the category **Law**.

## Variations

Variations include **multisorted** (or **multityped**) **Lawvere theory**, **infinitary Lawvere theory**, **Fermat theory** (named Fermat's difference quotient), and **finite-product theory**.[1]