Exploring the Link between Math Structures with Category Theory

[Shaun Culver]

Does 'category theory' link different mathematical structures? Can you give a superficial example of how this has been done (eg. Name two mathematical structures and, with waving of hands, explain how they have been linked)? Could you direct me to a link where this has been done elegantly in 'category theory' notation? Could it, for example, give insight into how group theory & differential geometry are intertwined?

 

[waht]

I know little about category theory, but it sort of links relations between different abstract categories. A functor is a map between categories that satisfies some conditions.

One functor I'm familiar with is the fundamental group functor $$\pi_1$$ that assigns to a topological space its fundamental group.$$\pi_1 : TOP_* \longrightarrow GROUP$$So if a the topological space is a surface of sphere, then you can assign to it a fundamental group, which in this case is trivial (which means you can shrink any loop on the surface to a single point).
There is dozen of lectures on category theory on youtube actually

 

[mathwonk]

on my webpage: http://www.math.uga.edu/~roy/

there are free algebra notes. in the last section (10) of the 843.1 notes is a few remarks on what categories and functors are with a few examples.

 

[gel]

There's a correspondence between commutative C*-algebras and locally compact topological spaces (http://en.wikipedia.org/wiki/Gelfand_representation" ). The idea is that instead of studying a topological space directly, you can instead look at the algebra of continuous real (or complex) valued functions on that space. Conversely, instead of studying an algebra, you can look at the the set of characters on that algebra.

Then, many concepts in geometry have their counterparts in algebra, and vice-versa. For example, vector bundles in differential geometry and projective modules in commutative algebra (http://en.wikipedia.org/wiki/Swan%27s_theorem" ).

 

[mathwonk]

the previous example recalls that basically there is only one functor in the world, and it is called HOM.

i.e. to study any object X, we look at maps of that object into other objects, or maps of other objects into it.the fundamental group of X, is the study of maps of closed intervals into X, mod an equivalence relation called homotopy.

homology is the study of maps of simplicial cells into X, plus the algebraic trick of taking formal linear combinations of them, modulo the equivalence relation imposed by taking boundaries.

the dual functor of a vector space is the study of maps of the vector space into the scalar field.

characters are a fancy name for certain maps into a, let's see, some group of units?

we just can't think of very many other functors than maps in and out of spaces.

when we do think of them, then we try to sklve the problem of "representability", i.e., of finding a HOM functor that is equivalent to our functor.

e.g. we can define a functor of curves over spaces, whicha ssigns to each space X, th set of all families of curves over X, i.e. of all maps Y-->X whose fibers are all curves.

representing this functor means finding a uniuversal "moduli" space M with a universal family of curves over it, such that every other family of curves arises from pull back from this one. some creativity is needed to find sucha thing since strictly speaking it does not exist.

then when such a representing object does not exist we change the definition of th words, and introduce terms like "stacks" instead of spaces so that representing objects wille exist.

point: to ubnderstand fucntors study HOM.

 

[HallsofIvy]

Does 'category theory' link different mathematical structures?

That is, indeed, the whole point of category theory!

Can you give a superficial example of how this has been done (eg. Name two mathematical structures and, with waving of hands, explain how they have been linked)? Could you direct me to a link where this has been done elegantly in 'category theory' notation? Could it, for example, give insight into how group theory & differential geometry are intertwined?

Functions between categories are call "functors" as mentioned above. One inportant functor is the "forgetful functor". A functor from the category of "topological spaces" to the category of "sets" just maps each underlying set of a topological space into the corresponding set while "forgetting" the topology.

As far as "group theory" and "differential geometry" are concerned, those are so far apart I doubt one could get more useful results than a chain of functors that reduces both to the set constituent.

 

[masnevets]

As far as "group theory" and "differential geometry" are concerned, those are so far apart I doubt one could get more useful results than a chain of functors that reduces both to the set constituent.

Well you could study homology groups of smooth manifolds, for instance.

 

[Shaun Culver]

As far as "group theory" and "differential geometry" are concerned, those are so far apart I doubt one could get more useful results than a chain of functors that reduces both to the set constituent.

Don't 'Lie Groups' connect 'group theory' and 'd. geometry'?

 

[HallsofIvy]

Yes, to both of those responses. But I was thinking of the general question as phrased: insight into connections between group theory and topology in general. What you are referring to are particular applications of both.