Comments on: The why and the what of homological algebra http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/ Because my LiveJournal is too silly Mon, 15 Mar 2010 18:50:59 +0100 http://wordpress.org/?v=2.8.6 hourly 1 By: alpheccar http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-20278 alpheccar Tue, 17 Jul 2007 15:27:54 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-20278 Thanks to all. It is helpful. Thanks to all. It is helpful.

]]> By: Michi http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-20197 Michi Tue, 17 Jul 2007 07:02:37 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-20197 John: Of course I can figure out module categories over a poset, and of course I can view poset cohomology over the category cohomology construction as well! You're missing my point right now, viz. that this particular kind of poset cohomology does not seem to deal with a module structure over poset directly. John: Of course I can figure out module categories over a poset, and of course I can view poset cohomology over the category cohomology construction as well! You’re missing my point right now, viz. that this particular kind of poset cohomology does not seem to deal with a module structure over poset directly.

]]> By: John Armstrong http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-20082 John Armstrong Mon, 16 Jul 2007 23:16:43 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-20082 Ahh, but a poset <em>is</em> a category, which is a generalization of a monoid, and a ring is a monoid object in the category of abelian groups :D Ahh, but a poset is a category, which is a generalization of a monoid, and a ring is a monoid object in the category of abelian groups :D

]]> By: Michi http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-20074 Michi Mon, 16 Jul 2007 22:00:34 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-20074 So that leaves with poset homology, among the things I can think of, and that isn't really that much of a good example either. We take a poset, and we equate this with a cellular complex that has this poset as a face poset. And then, boom, we're back in topology and can take the usual route. The main hook is that the poset isn't even remotely close to a ring, the modules of which we want to study, which might make this just barely a decent example. So that leaves with poset homology, among the things I can think of, and that isn’t really that much of a good example either. We take a poset, and we equate this with a cellular complex that has this poset as a face poset. And then, boom, we’re back in topology and can take the usual route.

The main hook is that the poset isn’t even remotely close to a ring, the modules of which we want to study, which might make this just barely a decent example.

]]> By: John Armstrong http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-20072 John Armstrong Mon, 16 Jul 2007 21:57:44 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-20072 <i> This characterization breaks down ... for Khovanov homology</i> Well, not really (in my view). Basically, he replaces a tangle with a cobordism in the cobordism <a href="http://unapologetic.wordpress.com/2007/07/12/the-temperley-lieb-category/" rel="nofollow">Temperley-Lieb</a> bicategory. Then he replaces the cobordism with a complex of graded modules to get a bicomplex whose graded Euler characteristic is the Jones polynomial. And it's really not so surprising. Remember that (for a post-Jones knot theorist like me) a knot, a link, or a tangle <em>is</em> an algebraic gadget: a morphism in the <a href="http://unapologetic.wordpress.com/2007/07/13/braided-monoidal-categories-with-duals/" rel="nofollow">free braided monoidal category with duals on one unframed, self-dual object</a>. This characterization breaks down … for Khovanov homology

Well, not really (in my view). Basically, he replaces a tangle with a cobordism in the cobordism Temperley-Lieb bicategory. Then he replaces the cobordism with a complex of graded modules to get a bicomplex whose graded Euler characteristic is the Jones polynomial.

And it’s really not so surprising. Remember that (for a post-Jones knot theorist like me) a knot, a link, or a tangle is an algebraic gadget: a morphism in the free braided monoidal category with duals on one unframed, self-dual object.

]]> By: Michi http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-20067 Michi Mon, 16 Jul 2007 21:13:55 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-20067 So, the different theories deal with various contexts. For one, there is the case that you can read different amounts out of your Exts depending on what kind of algebra you started with. And things developed historically slightly differently - so that it need not have been immediately obvious that group- and Lie cohomology are the same thing, for instance. On the topological side, we have similar things going on: singular and simplicial and cellular (co)homologies coincide whenever they all exist, but they cover different classes of spaces. And then there is Tate - where you have a double ended resolution. This is a bit like approximating a function with power series in a variable <i>and its inverse</i>, instead of just power series as such: you go in both directions, and get a slightly different presentation that way. All theories (well ... kinda) tend to be used to study algebraic gadgets by studying properties of a corresponding module category. This characterization breaks down, among other places, for Khovanov homology and for certain kinds of poset homology - unless I missed something about these. But for most the algebraic kind of theories, we do intend all along to study the module categories anyway. So, the different theories deal with various contexts. For one, there is the case that you can read different amounts out of your Exts depending on what kind of algebra you started with. And things developed historically slightly differently – so that it need not have been immediately obvious that group- and Lie cohomology are the same thing, for instance.

On the topological side, we have similar things going on: singular and simplicial and cellular (co)homologies coincide whenever they all exist, but they cover different classes of spaces.

And then there is Tate – where you have a double ended resolution. This is a bit like approximating a function with power series in a variable and its inverse, instead of just power series as such: you go in both directions, and get a slightly different presentation that way.

All theories (well … kinda) tend to be used to study algebraic gadgets by studying properties of a corresponding module category. This characterization breaks down, among other places, for Khovanov homology and for certain kinds of poset homology – unless I missed something about these. But for most the algebraic kind of theories, we do intend all along to study the module categories anyway.

]]> By: alpheccar http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-20066 alpheccar Mon, 16 Jul 2007 21:06:09 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-20066 So, if I have understood, in the context of the category of modules, homological algebra is used as a kind of Xray machine. Instead of studying modules and the morphisms between modules we replace them by injective/projective/free resolutions and study the morphisms in this new category of chain complexes. And, thanks to that we get a lot of information about the original problem. The last point which is remaining fuzzy for me is why we have so many different homological and cohomological theories. Are they all used with the category of modules or are they needed to study different algebraic gadgets ? So, if I have understood, in the context of the category of modules, homological algebra is used as a kind of Xray machine. Instead of studying modules and the morphisms between modules we replace them by injective/projective/free resolutions and study the morphisms in this new category of chain complexes. And, thanks to that we get a lot of information about the original problem.

The last point which is remaining fuzzy for me is why we have so many different homological and cohomological theories. Are they all used with the category of modules or are they needed to study different algebraic gadgets ?

]]> By: alpheccar http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-19424 alpheccar Thu, 12 Jul 2007 21:12:49 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-19424 Thank you so much Mikael ! I was not expecting such a quick and such a long answer to my email. I am not understanding everything and I'll have to read this post again. But, it helps me to connect the dots. It is much better than anything I have read so far. I have the Weibel book but I never found the energy to decrypt it. I needed some explanation of why homological algebra is useful, and how before reading that book. Your post is a great help. I'll post more comments when I have digested this first post :-) Thank you so much Mikael ! I was not expecting such a quick and such a long answer to my email. I am not understanding everything and I’ll have to read this post again. But, it helps me to connect the dots. It is much better than anything I have read so far.

I have the Weibel book but I never found the energy to decrypt it. I needed some explanation of why homological algebra is useful, and how before reading that book. Your post is a great help.

I’ll post more comments when I have digested this first post :-)

]]> By: Michi’s blog » Blog Archive » Today seems to be a day for posting… http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-19412 Michi’s blog » Blog Archive » Today seems to be a day for posting… Thu, 12 Jul 2007 19:20:24 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-19412 [...] asked me about the origins of my intuitions for homological algebra in my recent post. The answer got a bit lengthy, so I’ll put it in a [...] [...] asked me about the origins of my intuitions for homological algebra in my recent post. The answer got a bit lengthy, so I’ll put it in a [...]

]]> By: Michi http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-19410 Michi Thu, 12 Jul 2007 19:00:57 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-19410 John: I can, at times, sympathize with that. However, I did the other way around: I said "screw it" and wrote so that some of my posts are, really, only ever accessible to me and me alone. John: I can, at times, sympathize with that. However, I did the other way around: I said “screw it” and wrote so that some of my posts are, really, only ever accessible to me and me alone.

]]> By: John Armstrong http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-19409 John Armstrong Thu, 12 Jul 2007 18:57:42 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-19409 Well, it's sort of become that. Really I just wanted to talk about my stuff. But I don't know what everybody else knows, and I don't want it to just be accessible to a narrow cluster of experts, so I have to back up a bit. And after you back up a bit more and a bit more, you just say "screw it" and start at the beginning. Well, it’s sort of become that. Really I just wanted to talk about my stuff. But I don’t know what everybody else knows, and I don’t want it to just be accessible to a narrow cluster of experts, so I have to back up a bit.

And after you back up a bit more and a bit more, you just say “screw it” and start at the beginning.

]]> By: ComplexZeta http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-19408 ComplexZeta Thu, 12 Jul 2007 18:57:28 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-19408 This is amazing stuff! What books/articles did you read to learn this stuff (or is much of it your own work)? Somehow Weibel doesn't make it seem this interesting, but maybe it's all there if I can work through it. This is amazing stuff! What books/articles did you read to learn this stuff (or is much of it your own work)? Somehow Weibel doesn’t make it seem this interesting, but maybe it’s all there if I can work through it.

]]> By: Michi http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-19406 Michi Thu, 12 Jul 2007 18:46:47 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-19406 John: Are you seriously trying to write a blog that you can typeset, staple and hand in as a textbook in your field? Sometimes I kinda get that feeling. John: Are you seriously trying to write a blog that you can typeset, staple and hand in as a textbook in your field? Sometimes I kinda get that feeling.

]]> By: John Armstrong http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/comment-page-1/#comment-19403 John Armstrong Thu, 12 Jul 2007 18:42:23 +0000 http://blog.mikael.johanssons.org/archive/2007/07/the-why-and-the-what-of-homological-algebra/#comment-19403 Don't worry, I'll back you up a bit once I can say what an Abelian category is, which will require enriched categories, which is one of my goals in going through monoidal categories. And one of these days I might even start in on topology... Don’t worry, I’ll back you up a bit once I can say what an Abelian category is, which will require enriched categories, which is one of my goals in going through monoidal categories.

And one of these days I might even start in on topology…

]]>