In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a Lawvere theory (named after
American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the "United States" or "America"
** Americans, citizens and nationals of the United States of America
** American ancestry, pe ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
William Lawvere
Francis William Lawvere (; born February 9, 1937) is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.
Biography
Lawvere studied continuum mechanics as an undergraduate with Clifford Truesdell ...
) is a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
that can be considered a categorical counterpart of the notion of an
equational theory
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, ...
.
Definition
Let
be a
skeleton
A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
of the category
FinSet In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are a ...
of
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. Th ...
s and
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s. Formally, a Lawvere theory consists of a
small category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
''L'' with (strictly
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
) finite
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
s and a strict identity-on-objects
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
preserving finite products.
A model of a Lawvere theory in a category ''C'' with finite products is a finite-product preserving functor . A morphism of models where ''M'' and ''N'' are models of ''L'' is a
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
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.
See also
*
Algebraic theory Informally in mathematical logic, an algebraic theory is a theory that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset o ...
*
Clone (algebra) In universal algebra, a clone is a set ''C'' of finitary operations on a set ''A'' such that
*''C'' contains all the projections , defined by ,
*''C'' is closed under (finitary multiple) composition (or "superposition"): if ''f'', ''g''1, …, ''gm ...
*
Monad (category theory)
In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor mapping a category to itself, and a monad is an ...
Notes
References
*
* {{Citation , last1=Lawvere , first1=William F. , authorlink=William Lawvere , date=1963 , title=Functorial Semantics of Algebraic Theories , publisher=Columbia University , work=PhD Thesis , volume=50 , issue=5 , pages=869–872 , doi=10.1073/pnas.50.5.869 , pmid=16591125 , pmc=221940 , bibcode=1963PNAS...50..869L , url=http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html, doi-access=free
Categorical logic