The goal of this post is to compute the cohomology of the -torus in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is by default. At the end we will interpret this computation in terms of cohomology operations.
Archive for the ‘topology’ Category
Often in mathematics we define constructions outputting objects which a priori have a certain amount of structure but which end up having more structure than is immediately obvious. For example:
- Given a Lie group , its tangent space at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.
- Given a space , its cohomology is a priori a graded abelian group, but it ends up having the structure of a graded ring.
- Given a space , its cohomology over is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod- Steenrod algebra.
The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?
In this post we will answer this question for the homotopy groups of a space: that is, we will show that, in a suitable sense, each individual homotopy group is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)
Posted in category theory, differential geometry, functional analysis, topology, universal algebra, tagged abstract nonsense, adjoint functors, duality, Lawvere theories, pointless topology, reconstruction theorems on June 9, 2013 | 10 Comments »
Groups are in particular sets equipped with two operations: a binary operation (the group operation) and a unary operation (inverse) . Using these two operations, we can build up many other operations, such as the ternary operation , and the axioms governing groups become rules for deciding when two expressions describe the same operation (see, for example, this previous post).
When we think of groups as objects of the category , where do these operations go? They’re certainly not morphisms in the corresponding categories: instead, the morphisms are supposed to preserve these operations. But can we recover the operations themselves?
It turns out that the answer is yes. The rest of this post will describe a general categorical definition of -ary operation and meander through some interesting examples. After discussing the general notion of a Lawvere theory, we will then prove a reconstruction theorem and then make a few additional comments.
Suitably nice groupoids have a numerical invariant attached to them called groupoid cardinality. Groupoid cardinality is closely related to Euler characteristic and can be thought of as providing a notion of integration on groupoids.
There are various situations in mathematics where computing the size of a set is difficult but where that set has a natural groupoid structure and computing its groupoid cardinality turns out to be easier and give a nicer answer. In such situations the groupoid cardinality is also known as “mass,” e.g. in the Smith-Minkowski-Siegel mass formula for lattices. There are related situations in mathematics where one needs to describe a reasonable probability distribution on some class of objects and groupoid cardinality turns out to give the correct such distribution, e.g. the Cohen-Lenstra heuristics for class groups. We will not discuss these situations, but they should be strong evidence that groupoid cardinality is a natural invariant to consider.
Previously we introduced string diagrams and saw that they were a convenient way to talk about tensor products, partial compositions of multilinear maps, and symmetries. But string diagrams really prove their use when augmented to talk about duality, which will be described topologically by bending input and output wires. In particular, we will be able to see topologically the sense in which the following four pieces of information are equivalent:
- A linear map ,
- A linear map ,
- A linear map ,
- A linear map .
Using string diagrams we will also give a diagrammatic definition of the trace of an endomorphism of a finite-dimensional vector space, as well as a diagrammatic proof of some of its basic properties.
Below all vector spaces are finite-dimensional and the composition convention from the previous post is still in effect.
Today I would like to introduce a diagrammatic notation for dealing with tensor products and multilinear map. The basic idea for this notation appears to be due to Penrose. It has the advantage of both being widely applicable and easier and more intuitive to work with; roughly speaking, computations are performed by topological manipulations on diagrams, revealing the natural notation to use here is 2-dimensional (living in a plane) rather than 1-dimensional (living on a line).
For the sake of accessibility we will restrict our attention to vector spaces. There are category-theoretic things happening in this post but we will not point them out explicitly. We assume familiarity with the notion of tensor product of vector spaces but not much else.
Below the composition of a map with a map will be denoted (rather than the more typical ). This will make it easier to translate between diagrams and non-diagrams. All diagrams were drawn in Paper.
My current top candidate for a mathematical concept that should be and is not (as far as I can tell) consistently taught at the advanced undergraduate / beginning graduate level is the notion of a groupoid. Today’s post is a very brief introduction to groupoids together with some suggestions for further reading.
There are various natural questions one can ask about monomorphisms and epimorphisms all of which lead to the same answer:
- What is the “easiest way” a morphism can be a monomorphism (resp. epimorphism)?
- What are the absolute monomorphisms (resp. epimorphisms) – that is, the ones which are preserved by every functor?
- A morphism which is both a monomorphism and an epimorphism is not necessarily an isomorphism. Can we replace either “monomorphism” or “epimorphism” by some other notion to repair this?
- If we wanted to generalize surjective functions, why didn’t we define an epimorphism to be a map which is surjective on generalized points?
For the last few weeks I’ve been working as a counselor at the PROMYS program. The program runs, among other things, a course in abstract algebra, which was a good opportunity for me to get annoyed at the way people normally introduce normal subgroups, which is via the following unmotivated
Definition: A subgroup of a group is normal if for all .
It is then proven that normal subgroups are precisely the kernels of surjective group homomorphisms . In other words, they are precisely the subgroups you can quotient by and get another group. This strikes me as backwards. The motivation to construct quotient groups should come first.
Today I’d like to present an alternate conceptual route to this definition starting from equivalence relations and quotients. This route also leads to ideals in rings and, among other things, highlights the special role of the existence of inverses in the theory of groups and rings (in the latter I mean additive inverses). The categorical setting for this discussion is the notion of a kernel pair and of an internal equivalence relation in a category, but for the sake of accessibility we will not use this language explicitly.
Posted in algebraic topology, category theory, higher category theory, module theory, ring theory, tagged 2-categories, abstract nonsense, Eckmann-Hilton, enriched categories, homotopy groups on February 6, 2012 | 6 Comments »
The center of a group is an interesting construction: it associates to every group an abelian group in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor (unlike the abelianization ). We might wonder, then, exactly what kind of construction the center is.
Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category we may associate to an object its automorphism group or endomorphism monoid ), and this is a canonical construction which again doesn’t extend in an obvious way to a functor or . (It merely reflects some special part of the bifunctor .)
It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group , but of the identity functor , where is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.