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.

[edit] 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.

[edit] 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.

[edit] See also

• Clone (algebra)

[edit] References

• 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) 

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