Structure (category theory)

Structure (category theory)

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 structures. Starting with a given class of algebraic structure, such as groups, one can build the category in which the objects are groups and the morphisms are group homomorphisms: 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 functors 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.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Category theory — In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects and morphisms . Categories now appear in most branches of mathematics and in… …   Wikipedia

  • Monad (category theory) — For the uses of monads in computer software, see monads in functional programming. In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an (endo )functor, together with two natural transformations. Monads are used in …   Wikipedia

  • Higher category theory — is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Contents 1 Strict higher categories 2 Weak higher… …   Wikipedia

  • Allegory (category theory) — In mathematics, in the subject of category theory, an allegory is a category that has some of the structure of the category of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in… …   Wikipedia

  • Nerve (category theory) — In category theory, the nerve N(C) of a small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space, called the classifying space of the category C …   Wikipedia

  • Monoid (category theory) — In category theory, a monoid (or monoid object) (M,μ,η) in a monoidal category is an object M together with two morphisms called multiplication, and called unit, such that the diagrams and …   Wikipedia

  • Kernel (category theory) — In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel… …   Wikipedia

  • Outline of category theory — The following outline is provided as an overview of and guide to category theory: Category theory – area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as… …   Wikipedia

  • List of category theory topics — This is a list of category theory topics, by Wikipedia page. Specific categories *Category of sets **Concrete category *Category of vector spaces **Category of graded vector spaces *Category of finite dimensional Hilbert spaces *Category of sets… …   Wikipedia

  • Structure (mathematical logic) — In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it. Universal algebra studies structures that generalize the algebraic structures such as… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.