What is the Branch of Mathematics that Maps Entire Sub-Disciplines?

[Lacy33]

A friend just emailed me the following guestion, would anyone know what this field is called please.

"A few years ago a friend and colleague of mine had told me about a branch of mathematics or metamathematics that dealt with mapping entire sub-disciplines of mathematics and, I think, physics.

This had seemed to me or perhaps he told me that this was an extension of topology, and I distinctly remember getting a text on the subject and starting, though only starting to read it.

The question is: what is the name of this subject. It's not category theory nor is it fuzzy set theory, though that's been ""'s primary focus for some time. I thought it was something relatively short beginning with a 'T'.

I've had no luck trying to find the term on the WEB."
Thank you,
Suzanne

 

[roger]

topos theory ?

 

[Lacy33]

topos theory ?

Thank you very much Roger.

Suzanne

 

[HallsofIvy]

Try "Category Theory". A "category" consists of a collection of objects together with functions, called "morphisms" from the collection of objects to itself. In the category of sets, the morphisms are functions from one set to another. In the category of groups the objects are groups, the morphism are homomorphisms. In the category of topologies, the objects are topological spaces, the morphisms are continuous functions. There are also "functors" from one category to another. One I remember was the "forgetful" functor. A group is a set with an operation defined. The "forgetful" functor from the category of groups to the category of sets mapped each group to its underlying set, each homomorphism to its underlying function, "forgetting" the group operation.

Warning! At least one text I saw referred to category theory, apparently quite seriously, as "abstract non-sense"!

 

[Lacy33]

OK.
Thank you HallsofIvy. I will pass this along to my friend.
He may be following this thread now.
Thank you all for your assistance.
So easy to get caught going down a dead end when time is so important and learning even more so.
S