A formal first order theory of mathematical structures

Question

This is in connection with a project that I am conducting as an undergraduate student of mathematics. The aim of my project is to give a concretized formal notion for mathematical structures. We have studied about various mathematical structures like groups, rings, topologies, etc. and also about structure preserving maps which we generally call homomorphisms. In my project I am trying to build a first order theory of mathematical structures. The theory will consist of a first order formal axiomatic system, whose interpretation in a particular model will give a particular mathematical structure. In other words, I am trying to provide a unified framework for different mathematical structures. But I am facing problem in formulating the non-logical axioms of my system, that is, some non-logical axioms which, in different models, will be interpreted as different axioms of the corresponding mathematical structure (like the topological axioms in a topological structure, the group axioms in a group structure). If anyone here can help or at least can give an idea, I will be obliged.

Answer 1

One approach is via category theory: classes of mathematical structures, together with the "structure-preserving" maps between them, form categories; and there is a natural first-order theory of categories. A model of that theory is exactly a category, and can be viewed as the class of some kind of mathematical structures "viewed from the outside".

However, this approach doesn't allow you to dig in to the structures themselves. I.e. there is no way to tell what an object "is" just by looking at the category it's in. Frequently this is a positive feature, since it lets us forget the "un-categorical" information; however, I suspect in this context you would find it a negative one.

If you want to "look inside" the objects, then maybe the right picture is closer to a category (the category of Things) together with a functor from that category to the category of sets; basically, the "external" information is in the category, and the functor tells you what an object "is". However, there are several drawbacks to this approach, which I'll say more about when I have more time (I have to run now).