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 ${\displaystyle \aleph _{0}}$ 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 ${\displaystyle I:\aleph _{0}^{\text{op}}\rightarrow L}$ 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 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, and finite-product theory.^[1]

See also[]

• Algebraic theory
• Clone (algebra)
• Monad (category theory)

Notes[]

References[]

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