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 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 : LC. A morphism of models h : MN 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

References

Further reading

This article is issued from Wikipedia - version of the 12/3/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.