Abstract Math: Beyond Category Theory

[tgt]

Category theory is considered extremely abstract. What are some other branches of mathematics which are considered as abstract or even more abstract then category theory?

 

[olliemath]

I don't think I'm in a position to answer this fully. Things like sheaf theory, cohomology theories, algebraic geometry can be pretty abstract.. they use category theory to some extent, but I don't know if they're 'more abstract'. Logic seems to me to be on an equal footing with category theory in terms of how abstract it can get, but a logician might disagree. Universal algebra is another example, though I've never learned any.

 

[HallsofIvy]

I rather expect that, in order to determine which of "category theory", "set theory", "logic", or "universal algebra" are more or less abstract, you will need an abstract definition of "abstract"!

 

[tgt]

For ease of comparison, let's just have category theory as a standard for what is abstract.

How is logic up there with category theory?

 

[olliemath]

If you take category theory as the standard then since nothing is more like category theory than category theory everything else falls a bit short.

I suppose I'd put logic up there because while you use e.g. set theory and category theory to study mathematical structures, logic can be used to study both of these theories (set theory is a branch of logic). Model theory is another branch which studies structures in a similar way to universal algebra. To me the study of different types of logic purely for their own sake is as abstruse as studying categories. Just my opinion though.