Abstract Nonsense

Crushing one theorem at a time

Tensor Algebra and Exterior Product (Pt. VI)


Point of Post: This is a continuation of this post.

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May 10, 2012 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , | 1 Comment

Exact Sequences and Homology (Pt. IV)


Point of Post: This is a continuation of this post.

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April 10, 2012 Posted by | Algebra, Homological Algebra | , , , , , | Leave a comment

Exact Sequences and Homology (Pt. III)


Point of Post: This is a continuation of this post.

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April 10, 2012 Posted by | Algebra, Homological Algebra | , , , , , , | 2 Comments

Exact Sequences and Homology (Pt. II)


Point of Post: This is a continuation of this post.

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April 10, 2012 Posted by | Algebra, Homological Algebra | , , , , , | 1 Comment

Exact Sequences and Homology (Pt. I)


Point of Post: In this post we discuss how to define exactness for chains in a general abelian category and then discuss the homology objects associated to a chain.

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Motivation

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Last time we discussed the notions of chain complexes in abelian categories (amongst other things). This time we are going to discuss the notion of exact sequences which, in essence, are the “best kind” of exact sequences one can reasonably expect from a general chain complex. We should have a pretty good idea about what exactness means in our favorite categories like \mathbf{Ab}–it’s just the old image equals kernel routine. Of course, going from our favorite abelian category to general ones is a task which, by now, should be obvious isn’t always quite easy or obvious. Indeed, how exactly do we define “image equals kernel” when a) our objects aren’t necessarily sets, b) kernels are objects defined only up to isomorphism, and so even if they were sets there is no reason that kernel has to be literally contained inside image, c) . As has been a theme in our development of abelian categories we can replace the notion of “literal equality” in our more standard, tame categories with the notion of “canonical isomorphism”. Though, we shall see that while \text{im }f=\ker g shall be meaningless in a general abelian category, that there will be a canonical maps \text{im }f\to\ker g whose invertibility shall be equivalent to being exact. Here is where it shall be extremely important that we are dealing with abelian categories. Namely, we shall see that in general we shall only get a canonical map \text{coim }f\to\ker g and it’s the fact that there is a natural isomorphism \text{coim }f\xrightarrow{\approx}\text{im }f that allows us to construct our canonical map \text{im }f\to \ker g.

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Once we have defined exact sequences we shall define the homology objects of a given chain. Roughly, these will be measures of how far away a given point in a chain is from being exact. What this shall mean is that for each chain \mathbf{C} in \mathbf{Ch}(\mathscr{A}) we shall associated objects H_n(\mathbf{C}) for n\in\mathbb{Z} such that \mathbf{C} will be exact at C_n if and only if H_n(\mathbf{C})=0.

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Ok, so I think it’s about time that I tried (no doubt, to no avail) to explain the reasons that I have come to understand homology is important and why we care about it.

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April 10, 2012 Posted by | Algebra, Homological Algebra | , , , , | 2 Comments

Abelian Categories (Pt. II)


Point of Post: This is a continuation of this post.

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April 2, 2012 Posted by | Algebra, Category Theory | , , , , , , , | 3 Comments

Abelian Categories (Pt. I)


Point of Post:  In this post we define the notion of abelian category, motivating it completely.

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Motivation

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We have finally come to defining the notion of abelian categories. Cool–so what are they? Well, we have slowly been building up to this point, defining \mathbf{Ab}-categories, and then preadditive categories, and then finally additive categories. So, what’s the next step? Depending on how finely we want to parse the steps we’d actually now be taking about “pre-abelian categories” (this was actually the level of fineess to which Peter Freyd cut up the definition”, but for the sake of time we’re going to concatenate the two to go straight from additive to abelian. Ok, so what exactly do we need to add to additive categories to make them abelian?

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Well, our goal the entire time we’ve been creating this progression was to create classes of categories that come closer and closer to modeling \mathbf{Ab}. So, the first thing we did was to demand that it was enriched over \mathbf{Ab} (i.e. have abelian Hom sets, with addition distributing over addition)–next we decided that we wanted zero objects, a reasonable choice–we then decided that we wanted products and coproducts to always exists, be the same, and have a characterization in terms of the additive. These are all good, they all definitely start to give us the feeling of \mathbf{Ab}–in fact, there is only one major ingredient missing. Now, I want you to take a second (don’t peek ahead!) and think what construction is ubuiquitous with our work in \mathbf{Ab} that we have yet to define? If you’re having difficulty let me give you a hint. You are on an exam, and you are given two groups (abelian if you’d like, but they don’t have to be) A,B and a group map f:A\to B. Regardless of what you are going to do with this map, or where it came from, there is always two pretty natural things to check. Ring any bells? Well, hopefully you answered kernel, and dually, cokernel. It seems that hardly one group theoretic/module theoretic argument goes by where one doesn’t use the notion of a kernel. Thus, the first thing we’d like to throw into our ever-growing list of axioms of an abelian category is that every morphism has a kernel and a cokernel.

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Ok, that surely must be it, right? What more could we possibly want out of a category? Well, unfortunately for our line of categories hopeful to get into Club Abelian, we have one more round of clip-board check-offs. Namely, we want to require something extra of our kernel and cokernel baring categories. Think about it, we definitely do use kernels and cokernels constantly in our work with abelian groups/modules–but we often use something more. Namely, if I said to you “Hey ____, here’s an epimorphism f:A\to B of abelian groups” Your first reaction is not just to examine \ker f and \text{coker} f=0. No, our well-trained (inculcated?) mathematical minds immediately jump to the theorem which has been so used, so seared into our minds that it’s a near automatic response–first isomorphism theorem. Aha! We know that X/\ker f\cong Y by a “natural” isomorphism sending x+\ker f to f(x).  It is doubtless to say that we use this first ismomorphism theorem–a lot. Thus it seems natural to require that not only do our categories have kernels and cokernels, but that their favorite associated theorem (whatever that will end up meaning in a general abelian category) should hold true.

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So, we finally get right down to brass tacks–an abelian category should be an additive one that has both kernels and cokernels and satisfies the “first isomorphism theorem”.

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Ok, besides the obvious shoring up that this definition needs in the rigor department it begs one question. We have, up until this point, convinced (hopefully) the readership that all of these restrictions on categories (requiring they be enriched over \mathbf{Ab}, have kernels and cokernels, etc.) were necessary by a pure “well, obviously” argument–if we want something to look like \mathbf{Ab} we obviously must require that it satisfies so-and-so. While this line of argument is difficult to deal with, it opens up floodgate for the eventuality that is “and that’s all the requirements we need!” I mean, if at every step along the formation of our abelian category axiom list we said “not enough!” why can we just stop now? What impartial verification do we have that this is just the right amount of requirements to say it’s “pretty close to \mathbf{Ab}.

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Well, in fact, we do have a somewhat convincing impartial argument as to why this list of axioms is enough. Namely, we have the Freyd Embedding Theorem (not going to prove that baby here) which says that every small abelian category can fully and faithfully embedded into R\text{-}\mathbf{Mod} for some ring R. While this doesn’t give us exactly what we want (it doesn’t work for the full category in general) it allows us to state that given an abelian category, the category “locally” looks like \mathbf{Ab}. So, for example, all statements involving a diagram with finitely many objects in an abelian category \mathcal{A}  can be proven just by proving it in \mathbf{Ab} since the diagram in question sits inside a small abelian subcategory of \mathcal{A}. This is precisely the kind of validation that let’s us know–yes, we have the correct definition.

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April 2, 2012 Posted by | Algebra, Category Theory | , , , , , , | 4 Comments

Differential Geometry of Curves and Surfaces


Point of Post: In this post we motivate the upcoming discussion of the differential geometry of curves and surfaces.

Motivation

So, we begin discussing another topic, “Differential Geometry of Curves and Surfaces”. The first questions is (as it always should be)–what and why? Namely, what is the differential geometry of curves and surfaces and why is it interesting/important. The first of these questions is easily enough answered. Differential geometry of curves and surfaces (just called differential geometry when there is no confusion, or even just geometry) is the study of surfaces, mostly living in \mathbb{R}^3, as well as their lower dimensional counterparts, curves. In particular, we’d like to study the aspects of curves and surfaces which are preserved under the kinds of transformations that don’t ‘stretch’. But, let’s try not to just relegate the motivation to ‘studying maps that don’t stretch’, let’s really talk about what we are trying to preserve. Ostensibly topology and differential geometry seem quite similar–they are both studying ‘geometric objects’ and the properties of these objects that are invariant under certain ‘admissible’ transformations. So, what precisely is the difference–what makes geometry..geometry? The key distinction to make between geometry and topology is the local vs. global phenomenon. Namely, one could say that a topologist is concerned with ‘global’ properties of geometric objects and geometers are concerned with ‘local’ properties. Take for example the classic example of the sphere and doughnut. Everyone and their kid sister  knows that the sphere and the doughnut are not ‘homeomorphic’, but that they are ‘locally homeomorphic’. A reasonable way to explain what this means is that locally the two objects look alike in the sense that if you pick your two favorite points, one on the coffee cup and one on the doughnut, and cut out an extremely tiny piece of the object around each point, the cut-outs can be made to look exactly alike with the right amount of topologically admissible massaging (i.e. bending and stretching but no tearing of holes). That said, the entire objects themselves aren’t homeomorphic because one has ‘a hole’ and the other doesn’t. Geometers are much more discerning. You give two small little neighborhoods on each of the objects to the geometer, and they’d be able to tell you immediately that the surface of a doughnut and a sphere are not ‘isomorphic’ (whatever that means).  To beat a dead horse, geometers could tell you that two spaces are different by putting them on each and locally measuring things (what they measure, we shall soon see). Hopefully this gives one a very, very, very vague idea of what geometry is about.

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So, why is it interesting? As is mathematicians plight, to convince ninety-nine percent of the population as to why a subject is interesting one must appeal to either physics or shakey real-world parlor tricks. So, let’s get that out of the way. For the first of these, one cannot do modern physics without knowing relativity (or at least I am told). Notions of curving spacetime and the like are entirely within the realm of differential geometry. Trying to build turns in roads such that a car going ___ miles and hour will not skid off of it, differential geometry. The list goes on and on. So, what about the second? What ‘parlor tricks’ (i.e. applications of mathematical theorems to the real world, that assume a lot about the situation) can we product with differential geometry? Well, have you ever tried gift wrapping a basketball? If so, or if you have watched someone else do so, you know that it’s not an easy task. Go ahead and try if you haven’t. If you can’t, don’t feel bad…you can’t do it. Literally, differential geometry will show that one cannot wrap a piece of paper around a ball without ripping it at some point–cool huh?

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So, why would a mathematician find geometry (at this level) interesting? Besides the obvious fact that all math is useful (you’d be shocked how often I hear professional mathematicians say “Oh, I was doing ___ in subject X and couldn’t possibly solve it until I remembered this random theorem from subject Y!” there is one other very particularly useful reason for studying geometry. Namely, geometry of curves and surfaces provides excellent motivation and intuitive backing for more advanced differential geometry and differential topology. And, to be more down to earth, it’s just plain cool.

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September 11, 2011 Posted by | Differential Geometry, Uncategorized | , , , | Leave a comment