Structure (category theory)
Encyclopedia
In mathematics
, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications. In fact this is a normal way of proceeding; in the absence of recognisable structure (which might be hidden) problems tend to fall into the combinatorics
classification of matters requiring special arguments.
In category theory
structure is discussed implicitly - as opposed to the explicit discussion typical with the many algebraic structure
s. Starting with a given class of algebraic structure, such as group
s, one can build the category in which the objects are groups and the morphism
s are group homomorphism
s: that is, of structures on one type, and mappings respecting that structure. Starting with a category C given abstractly, the challenge is to infer what structure it is on the objects that the morphisms 'preserve'.
The term structure was much used in connection with the Bourbaki
group's approach. There is even a definition. Structure must definitely include topological space
as well as the standard abstract algebra
notions. Structure in this sense is probably commensurate with the idea of concrete category
that can be presented in a definite way - the topological case means that infinitary operations will be needed. Presentation of a category (analogously to presentation of a group
) can in fact be approached in a number of ways, the category structure not being (quite) an algebraic structure in its own right.
The term transport of structure is the 'French' way of expressing covariance or equivariance as a constraint: transfer structure by a surjection and then (if there is an existing structure) compare.
Since any group is a one-object category, a special case of the question about what the morphisms preserve is this: how to consider any group G as a symmetry group
? That is answered, as best we can by Cayley's theorem
. The analogue in category theory is the Yoneda lemma
. One concludes that knowledge on the 'structure' is bound up with what we can say about the representable functor
s on C. Characterisations of them, in interesting cases, were sought in the 1960s, for application in particular in the moduli problems of algebraic geometry
; showing in fact that these are very subtle matters.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications. In fact this is a normal way of proceeding; in the absence of recognisable structure (which might be hidden) problems tend to fall into the combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
classification of matters requiring special arguments.
In category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
structure is discussed implicitly - as opposed to the explicit discussion typical with the many algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s. Starting with a given class of algebraic structure, such as group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s, one can build the category in which the objects are groups and the morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s are group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s: that is, of structures on one type, and mappings respecting that structure. Starting with a category C given abstractly, the challenge is to infer what structure it is on the objects that the morphisms 'preserve'.
The term structure was much used in connection with the Bourbaki
Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
group's approach. There is even a definition. Structure must definitely include topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
as well as the standard abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
notions. Structure in this sense is probably commensurate with the idea of concrete category
Concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...
that can be presented in a definite way - the topological case means that infinitary operations will be needed. Presentation of a category (analogously to presentation of a group
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
) can in fact be approached in a number of ways, the category structure not being (quite) an algebraic structure in its own right.
The term transport of structure is the 'French' way of expressing covariance or equivariance as a constraint: transfer structure by a surjection and then (if there is an existing structure) compare.
Since any group is a one-object category, a special case of the question about what the morphisms preserve is this: how to consider any group G as a symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
? That is answered, as best we can by Cayley's theorem
Cayley's theorem
In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
. The analogue in category theory is the Yoneda lemma
Yoneda lemma
In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
. One concludes that knowledge on the 'structure' is bound up with what we can say about the representable functor
Representable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
s on C. Characterisations of them, in interesting cases, were sought in the 1960s, for application in particular in the moduli problems of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
; showing in fact that these are very subtle matters.