It’s been a long time in coming, but here it is. Can anyone fggive me a basic rundown on catergory theory and topos. To give you the kind of I’ve done some differential geometry, basic group theory, reasonably advance linear algebra and some analysis (i.e basically mathematics for physics), so can you gear any answers to about that level.
That is to say don’t assume any topology or homolgoy other than that which I’ve picked up unknowingly.
Categories are actually pretty easy. The exact definitions vary depending on the author. Since Jacobson is the closest reference at hand, I’ll quote him.
A category C consists of
[ol]
[li]A class ob C of objects (usually denoted as A, B, c, etc.)[/li][li]For each ordered pair of objects (A,B), a set hom[sub]C/sub (or simply hom(A,B) if C is clear) whose elements are called morphisms with domain A and codomain B (or from A to B).[/li][li]For each ordered triple of objects (A,B,C), a map (f,g) -> fg of the product set hom(A,B) x hom(B,C) into hom(A,C).[/li][/ol]
It is assumed that the objects and morphisms satisfy the following conditions:
[list=1]
[li]If (A,B) != (C,D), then hom(A,B) and hom(C,D) are disjoint.[/li]li. If f in hom(A,B), g in hom(B,C) and h in hom(C,D), then f(gh) = (fg)h. (As usual we simplify this to fg*h)[/li]li. for every object A we have an element 1[sub]A[/sub] in hom(A,A) such that 1[sub]A[/sub]f = f for all f in hom(A,B) and g1[sub]A[/sub] = g for all g in hom(B,A). (1[sub]A[/sub] is of course unique.)[/li]
Note that if the class of objects has only one element, this reduces to the notion of a monoid. I choose to omit the assumption of units, but this is a very new addition to the field.
The classical archetype of a category is Set, the category of sets. This is why the objects of C are assumed to form a class. The morphisms hom[sub]Set/sub are all functions from A to B and composition is simple composition of functions. There are similarly defined categories Grp (groups with homomorphisms), Ab (abelian groups with homomorphisms), Ring (unital rings with homomorphisms), Rng (rings in general with homomorphisms), Top (topological spaces with continuous maps), C[sup]k[/sup]mfld (C[sup]k[/sup] manifolds with C[sup]k[/sup] maps), and so on.
The classic use of category theory is metamathematical. There’s a lot of classical results that really don’t depends much on the structure of the things involved; just on the maps between them. For instance: the first, second, and third homomorphisms theorems of both group and ring theory can be proven in one swoop by noting that both Grp and Rng have certain features (namely kernels and a zero object), and that in any category with these features the appropriate theorems hold. This is essentially a further level of abstraction, since group theory (for instance) shows that any system which has the structure of a group will behave in certain ways.
This use essentially was kicked off in the development of homology theory, but had heavy influences in the attempt to pin down what was meant by a “natural” construction. For example, any vector space V over a division ring D has a natural map into its double-dual (V[sup][/sup])[sup][/sup]. Given an x in V, t(x) will be a linear functional on the space of linear functionals on V. In particular, t(x) = f(x). The “naturality” is summed up in the idea of a “natural transformation” of “functors”. Functors are to categories what homomorphisms are to semigroups: ways of mapping one category to another (objects to objects, morphisms to morphisms) so that the notion of composition is preserved. The technical definition is a little more involved, but this gives the gist. A natural transformation between two functors F and G (each of which send C to D) is a collection of morphisms in D indexed by the objects of C so that
t[sub]C[/sub]:F©->G©
and that given any c:A->B in C
t[sub]A[/sub]*G© = F©*t[sub]B[/sub]
In particular, the identity functor from the category of vector spaces over D to itself and the double-dual functor from the category to itself have a natural transformation t[sub]V[/sub]:V->(V[sup][/sup])[sup][/sup] as defined above. That this commutes with the apropriate linear maps is left to the reader.
The theory of functors and natural transformations on one hand highlights the parts of mathematics which are “functorial” and “natural”, while throwing those which are not into stark relief. As an example, the assignment sending a ring R to its center Z® is not functorial, and as such is a very interesting concept indeed. On the other hand, functors can show when various fields are “really the same thing”. If two fields are working with two different categories and one can show a pair of functors going back and forth between them making them equivalent (a little weaker than isomorphism, but it’s a technicality), then problems in one field can be translated into the other and back.
The more modern use (though somewhat ignored since many mathematicians think of categories as “abstract nonsense”) plays heavily on the analogy with semigroups and monoids. A category is basically a notion of composition that simply isn’t defined for all pairs of elements. The canonical example I like to give here is the category of matrices over R. The objects of Mat® are the positive integers. The morphisms hom[sub]Mat®/sub are the m-by-n matrices of real numbers with matrix multiplication as the composition.
This is closer to anything with any application. In particular, the category n-Bord whose objects are all smooth n-manifolds and whose morphisms between two manifolds M and N are all bordisms from M to N (basically, an n+1-manifold whose boundary consists of a copy of M and a copy of N with its orientation reversed) is considered interesting in physics. A functor from this category to that of complex vector spaces is called a “topological quantum field theory”. One might attempt to understand the structure of the category of bordisms and in so doing get a description of all possible such functors. This would tell you a lot in physics if you knew what sort of manifolds you were dealing with and that some element of your theory constituted a topological quantum field theory.
Topoi (some authors have lost the faith in Latinate forms and write “toposes”) are actually much heavier a subject. Basically, for two different reasons Alexandre Grothendieck and William Lawvere identified certain important properties of Set: that it “has all finite limits and colimits”, that it “has exponentials”, and that it “has a subobject classifier”. Limits and colimits are rather difficult to describe in general, but the other bits I can wave my hands at in terms of sets.
Given sets A and B, the functions from A to B again form a set. In a general category, these are just collected in hom(A,B), but in Set we sometimes identify this as another object B[sup]A[/sup]. In a general topos, B[sup]A[/sup] will be another object of the topos, but will not in general be interpreted as “maps” from A to B. This is the exponential in sets.
Given an object C in a category, a subobject is a “monomorphism” f:B->C. In group theory a monomorphism is an injective function, but in general morphisms are not to be interpreted as functions. A morphism f:B->C is “mono” if it is “right-cancellable”. That is, given g and h:A->B, gf = hf implies that g = h.
Since a topos has finite limits and colimits it has a “terminal object” 1, to which every object has a unique morphism. For Set this is the set with a single element. Now, given any injective map f:B->C, the unique map u:B->{!}, and the map t:{!}->{0,1} with t(!)=1, there exists a unique function c[sub]B[/sub]:C->{0,1} sending the objects in the image of f to 1 and the rest to 0, the “characteristic function” of B in C. If a category has a certain way of assigning a characteristic morphism to every subobject, it is said to have a subobject classifier.
A topos is a category with all finite limits and colimits, an exponential, and a subobject classifier. It turns out this is all one needs to do a surprising amount of what has classically been called “set theory”. In fact, logicians run with this, finding an interpretation that different interpretations of fundamental mathematics (intuitionism and classicism, for example) are “just” the different consequences of working in two different topoi.
Anyhow, I’ve rambled on long enough and probably burned you out completely.
Thanks! I can’t claim to understand all of it straight off the bat, but I can understand at leats some of it.
So the concept of a morphism is just a genarlization of homomorphisms, isomorphisms, etc?
Originally, yes. As I said, though, the modern view is more that a category is essentially a notion of composition that is only defined for certain pairs of arrows – those where one’s domain is the other’s codomain like the case for functions between sets or like the way matrix multiplication is only defined for certain pairs of matrices.