- 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 thecombinatorics classification of matters requiring special arguments.In

category theory "structure" is discussed "implicitly" - as opposed to the "explicit" discussion typical with the manyalgebraic structure s. Starting with a given class of algebraic structure, such as groups, one can build the category in which the objects are groups and themorphism s aregroup 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 standardabstract algebra notions. Structure in this sense is probably commensurate with the idea ofconcrete category that can be presented in a definite way - the topological case means that infinitary operations will be needed. "Presentation of a category" (analogously topresentation 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 byCayley's theorem . The analogue in category theory is theYoneda lemma . One concludes that knowledge on the 'structure' is bound up with what we can say about therepresentable functor s on "C". Characterisations of them, in interesting cases, were sought in the 1960s, for application in particular in themoduli problem s ofalgebraic geometry ; showing in fact that these are very subtle matters.

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