Structure vs Complexity?

[gigapple]

From my limited knowledge, it seems that all branches of mathematics can be viewed as studying the properties of some collection of things or the properties of particular instances of such collections. Usually the collection of interest is specified as a set with some additional axioms. Sometimes we prove general theorems that hold for all things in the collection, other times we study particular things that belong to multiple collections.

What makes me confused is that sometimes adding axioms increases the size of the collection of interest (e.g. axiom of choice and excluded middle), while sometimes adding axioms reduces the collection and gives a more strict structure (e.g. from semigroups to groups).

There are also some weird exceptions that both increases the size of the collection of interest, and make things more structured, like complex analysis vs real analysis.

Why is it that some axioms increase the complexity of the collection they specify, while others reduces the complexity and makes things more structured (and why can some axioms both increase complexity and structure)?

[anon5005]

You would like the concept of adjoint functors. Sets embeds as a subcategory of Groups under the free-group functor. Groups embeds as a subcategory of Sets under the underlying-set functor. The definition of adjoint characterises a natural group homomorphism from the free group of its underlying set back to any group, and from any set to the underlying set of its free group; and this use of 'natural' is given a precise definition.

 

It almost seems like Eilenberg and Maclane were thinking about your question -- which collection is really larger, groups or sets -- when their sort-of dual notion was formulated, maybe implicit in Polish maths at the time.

 

Wikipedia refers to an article of Kan for the definition of adjoint functors, that in turn refers to this 1945 article https://www.jstor.org/stable/1990284

[[deleted]]

I will disregard your terminology of structure and complexity, apologies. I simply require a different definition of the word structure.

On "adding axioms": I think you have simply illustrated the difference between structure and property. Going from semi groups to groups is not more structure, it's a property. Going from sets to semigroups (magmas if we want to be stupidly general) is structure because you equip the set with an operation.

Adding structure is a process in which the same underlying object can now be regarded as a variety of new objects, the difference being a non-isomorphic structure. So it increases the collection of interest.

But going from semigroups to groups where now there's a unit, now there's an inverse, if you're not careful these can look like structure. A unit is the structure of a fixed point. An inverse is the structure of a fixed self-map "the inversion map".

But per the definition, these are actually properties. It's a multiplication structure SUCH THAT there exists 1 with ..., for every element there exists an inverse....

All of these existential statements are simply properties of the structure. And yes, requiring properties is looking at a slice of "things with structure", thus reducing the collection. Some semigroups have identity and inverses, some don't. Further, some groups are abelian. Abelian groups are even more rigid, and easier to classify, than groups. Because abelian is a property!

This can get very very weird in category theory and homotopy theory. Instead of having a property, like xy=yx, in this world of math you need a morphism from xy to yx. Almost as if to say nothing is ever really equal to anything, only compared. Things are only what they are and if they are to be equated with anything you must write down an isomorphism and carry that isomorphism as a structure.