Lawvere theory

In category theory, a Lawvere theory (named after American mathematician William Lawvere) is a category which can be considered a categorical counterpart of the notion of an equational theory.

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 → 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 which 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.

See also

• Clone (algebra)

References

• (link) Hyland, Martin; Power, John (2007), The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
• Lawvere, William F. (1964), Functorial Semantics of Algebraic Theories (PhD Thesis)

Further reading

• Lawvere theory in nLab