In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a Lawvere theory (named after
American 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, mathematical structure, structure, space, Mathematica ...
William Lawvere
Francis William Lawvere (; February 9, 1937 – January 23, 2023) was an American mathematician known for his work in category theory, topos theory and the philosophy of mathematics.
Biography
Born in Muncie, Indiana, and raised on a farm outsi ...
) is a
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
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 in general, not specific types of algebraic structures.
For instance, rather than considering groups or rings as the object of study ...
.
Definition
Let
be a
skeleton
A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...
of the category
FinSet 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
functions. 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 that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
) finite
products 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 subse ...
*
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, … ...
*
Monad (category theory)
In category theory, a branch of mathematics, a monad is a triple (T, \eta, \mu) consisting of a functor ''T'' from a category to itself and two natural transformations \eta, \mu that satisfy the conditions like associativity. For example, if F, ...
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