In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.

The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory.

Wiki's definitions seem to be inconsistent. What's a structure which does not have an underlying set?

Does that mean a structure can arise on its own, or have a underlying class instead, like a category in which both objects and morphisms are actually proper classes and not sets?

  • $\begingroup$ I'm not asking about the difference between the two concepts. That's beside the point here. I'm asking what's meant by "a structure which does not have an underlying set" in that particular context. @DietrichBurde $\endgroup$ – Kinzle B Aug 23 '17 at 12:50
  • $\begingroup$ I suspect the Wikipedia article is referring to the fact that categories don't need to look like sets with functions between them. For instance, the fundamental groupoid of a topological space is a category with points as its objects and homotopy classes of paths as its morphisms. (But I wouldn't say that an object in the fundamental groupoid is meant to represent a structure by itself - it's the whole category we're interested in) $\endgroup$ – Milo Brandt Aug 23 '17 at 13:47

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