Web Definition: Structure
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 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. that is, of structures on one type & 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 groups 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.
Read More »Dictionary Entry: The act of building - the practice of erecting buildings - construction.
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