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 The why and the what of homological algebra

  • July 12th, 2007
  • 7:35 pm

I seem to have become the Goto-guy in this corner of the blogosphere for homological algebra.

Our beloved Dr. Mathochist just gave me the task of taking care of any readers prematurely interested in it while telling us all just a tad too little for satisfaction about Khovanov homology.

And I received a letter from the Haskellite crowd – more specifically from alpheccar, who keeps on reading me writing about homological algebra, but doesn’t know where to begin with it, or why.

I have already a few times written about homological algebra, algebraic topology and what it is I do, on various levels of difficulty, but I guess – especially with the carnival dry-out I’ve been having – that it never hurts writing more about it, and even trying to get it so that the non-converts understand what’s so great about it.

So here goes.

 Reading Merkulov: Differential geometry for an algebraist (4 in a series)

  • April 24th, 2006
  • 2:52 pm

Suppose we have a presheaf \mathcal F of abelian groups over M and pick a point x. On the collection of all abelian groups defined over some neighbourhood of x (disjoint union) we put an equivalence relation which identifies f\in\mathcal F(U) and g\in\mathcal F(V) precisely if there is some open W in the intersection where f and g coincide. (or more precisely, their restrictions coincide). The set of equivalence classes turns out to be an Abelian group \mathcal F_x called the stalk of the presheaf \mathcal F at x.

So, with more fluff introduced, the stalk is all the elements in the presheaf that are defined above any neighbourhood of the point, and counted as the same if they seem to be.

For an open set U and a point x\in U there is a canonical group morphism \rho_x:\mathcal F(U)\to\mathcal F_x which sends an element f\in\mathcal F(U) to its equivalence class. This image is the germ of f at x.

 Reading Merkulov: Differential geometry for an algebraist (3 in a series)

  • February 13th, 2006
  • 1:55 pm

Since I cannot concentrate anyway, here comes the third installment of my reading Merkulov. We now get to the funky stuff – introducing sheaves and getting to what should be at the very start of basically any modern geometry course. At least if I am to believe the geometers I know.

So, let’s launch straight to it. A presheaf \mathcal F on the topological space \mathcal M is just a contravariant functor from Top(\mathcal M) to Ab, where Top(\mathcal M) is the category of open subsets of \mathcal M with morphisms being inclusion maps.

So that’s the one-line definition. But what does it mean?
Well, a functor is a map between categories that takes objects to objects and morphisms to morphisms. So we have that \mathcal F(U) is an abelian group for any open set U\subset\mathcal M. For such a map to really be a functor, it has to be sane in a rather precisely defined sense: namely morphism composition should still be associative and the identity endomorphism on a group shouldn’t actually, ya’know, change the morphisms before or after it.
For the functor to be contravariant means precisely that for f:U\to V we get \mathcal F(f):\mathcal F(V)\to\mathcal F(U) – all arrows reverse by application of the functor.

 Reading John M. Lee – Introduction to Smooth Manifolds (1 of 1)

  • February 12th, 2006
  • 11:14 pm

If I’m going to take this renewed interest in trying to understand differential geometry more seriously, I might as well read more than one source on it. So, I’ll start a sequence of posts on this book as well.

Just as Merkulov, Lee starts with a short definition of what a topological manifold is, including definitions for the terms needed for the treatment.

Definition

An n-dimensional topological manifold is a second countable Haussdorff space of local Euclidean dimension n.

Next, Lee goes on to define coordinate charts. I won’t repeat the treatment, since he doesn’t really bring anything Merkulov hasn’t talked about in some manner or other. Atlases, smooth structures and equivalence of atlases also merits some treatment.

 Reading Merkulov: Differential geometry for an algebraist (2 in a series)

  • February 11th, 2006
  • 11:03 pm

So, in the last installment, we got to know smooth manifolds and charts, atlases and some nice topological tricks and tweaks. For this round, we follow Merkulov onward, and pretty soon stumble across category theory and sheaves. The notes I’m following here are from the link on Merkulov’s website. It starts, however, with a nice discussion of temperatures in archipelagos. Go read it – I imagine I’m almost comprehensible at that part of the text.

A map from a subset of a smooth manifold to \mathbb R is called a smooth function on the subset if for every x in the subset and a coordinate chart at x, the n-to-1 variable function f\circ\phi^{-1} is smooth at the point \phi(x).

 Reading Merkulov: Differential geometry for an algebraist (1 in a series)

  • February 11th, 2006
  • 2:23 pm

I’ll do this in posts and not pages on further thought…

Sergei Merkulov at Stockholm University gives during the spring 2006 a course in differential geometry, geared towards the algebra graduate students at the department. The course was planned while I was still there, and so I follow it from afar, reading the lecture notes he produces.

At this page, which will be updated as I progress, I will establish my own set of notes, sketching at the definitions and examples Merkulov brings, and working out the steps he omits.

Familiar parts in unfamiliar language

Merkulov begins the paper by introducing in swift terms the familiar definitions from topology of topology, continuity, homeomorphisms, homotopy, and then goes on to discuss homotopy groups, and thereby introducing new names for things I already knew. Thus, I give you, for a pointed topological space M