Model theory, category theory and universal algebra

[Fredrik]

If someone is willing to briefly explain what these three branches of mathematics are about, I'd appreciate it. I don't even understand if they are three completely different things or if they're overlapping a lot. I understand the definition of a category, and Landau just made me aware of the definition of a structure yesterday. Other than that, I know close to nothing about these things.

 

[Newtime]

I originally thought you meant "module theory" instead of "model theory"; I didn't even know there was such a discipline. I too would like to know the difference between all these since their definitions all sound remarkably similar.

 

[TMM]

Universal algebra is very much an aspect of model theory, so I'll just define the other two.

Model theory studies the abstract notion of a mathematical theory. With model theory you can study things like the consistency of certain statements, how to construct models of more complicated languages from simpler models, and how proofs are conducted in whatever theory you'd like to study.

Category theory is a little more particular. In category theory you have classes of objects and morphisms among those objects. Algebraic theories often share this common structure of studying objects (eg. groups) and special maps between them (eg. homomorphisms). This makes category theory useful for quantifying what aspects certain types of objects will share. If, say, we discover something interesting about semigroups and we want to see if it applies elsewhere, a good bet is to come up with a categorical definition of the phenomenon and see if it is interesting in other categories. Functors in this sense also become a nice way of studying relationships between categories.

 

[MathematicalPhysicist]

Isn't universal algebra and algebraic logic rather similar if not the same?

 

[Hurkyl]

Category theory is somewhat broad.

One of the recurring themes is that algebra isn't just for things like numbers and vectors -- it's also for things like sets, groups, vector spaces, and topological spaces.

e.g. if you're studying an interesting topological space, you might want to know its homology groups. One approach to finding them is:

• find a way to build your interesting space out of simpler spaces,
• convert that into a construction involving the homology groups of the simpler spaces,
• try to perform the calculation.

Many interesting structures themselves are categories -- matrix algebra, for example. It can be awkward to try and study it as a "normal" algebraic structure -- e.g. if you want to use ring theory, you have to restrict yourself to square matrices of a particular size. However, Abelian category theory is up to the task!

Another interesting example of categories-as-structures is the fundamental groupoid of a topological space. There are a few similar examples of things -- e.g. the path groupoid of a manifold, or the category of arrows in an affine space.