Structure (category theory)

In mathematics, in the absence of recognizable structure (which may, however, be hidden) problems tend to fall into that combinatorial classification of subjects that require special arguments.

In category theory structure is implicitly discussed - as opposed to the explicit discussion typical with many algebraic structures. Starting with a given class of algebraic structures, say groups, one can construct the category in which the objects are groups and the morphisms are the group homomorphisms: that is, of structures of a type, and of functions that respect that type. structure. Starting with an abstractly given C category, the challenge is to deduce what structure "is there" on objects that maps 'preserve'.

The term structure was used a lot in connection with the Bourbaki group's approach. There is even a definition. The structure must clearly include both the topological space as well as the standard notions of abstract algebra. The structure in this sense is similar to the idea of a concrete category that can be presented in a definite way - the topological case means that infinitary operations will be necessary. The presentation of a category (analogous to the presentation of a group) can in fact be approached in several ways, the category structure is not strictly an algebraic structure.

The term frame transport is the 'French' to express covariance or equivariance as a constraint: transfer the structure by an overlay and then (if there is an already existing structure) compare.

Since any group is a category of a single object, a special case of the question about what morphisms preserve is this: how to consider a group G as a symmetry group? The best answer we can give is Cayley's theorem. The analogue in category theory is Yoneda's lemma. One concludes that knowledge of the 'structure' it is bounded by what we can say about the representable functors in C. His characterizations, in interesting cases, were sought after in the 1960s, for use in particular in the moduli problems of algebraic geometry; proving in fact that these are very subtle matters.