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Sophie El Agami (Sep 20 2021 at 18:01):Hello there,
I'm looking for paper/reading recommendations on the application of sheaves in algebraic topology, and of course, real-world applications too.
I would be delighted if you posted some of your favourites! :penguin:
Cheers!
Tim Hosgood (Sep 20 2021 at 18:28):https://arxiv.org/abs/1303.3255 is a very nice thesis that talks a lot about applications :-)
Patrick Nicodemus (Sep 20 2021 at 18:30):Hi Sophie; A recommendation I've heard is the book by Dimca, Sheaves in Topology. (I haven't read it.) I think they expect the reader knows some basic sheaf theory but it is meant more as an introduction to the applications. It is short, not a massive technical treatise, they say in the foreword that the book can be considered as preparation for more technical and in depth treatments such as Kashiwara and Schapira's book Sheaves on Manifolds.
The book by Iversen, "Cohomology of Sheaves" describes using sheaves in topology to study Poincare duality and its generalizations due to Poincare. I have started it recently and I like it.
There is a beautiful book by Godement if you have knowledge of French, "Topologie algebrique et theorie des faisceaux". It is one of my favorite books. I cannot really say that it deals too much with "applications". But in it he shows how to use (co)simplicial sheaves to give constructions like the cup product in a way that very closely resembles how they are defined in singular homology. This influenced me a lot, I find it intriguing that we can give such strong correspondences between the theories using simplicial methods.
As for real world applications I have heard recently of sheaves being used to model signal processing.
Sophie El Agami (Sep 21 2021 at 19:41):Tim Hosgood said:
https://arxiv.org/abs/1303.3255 is a very nice thesis that talks a lot about applications :-)
Thanks, I saw this, that's a great one! It's mostly focused cellular sheaves though, have you happened to come across something aside of this?
I guess cellular sheaves have those applications because they are combinatorical and sort of contained, but how about genreal sheaves? Are there applications that are particularly successful (apart from Ghrists work and his students)?
Patrick Nicodemus said:
Hi Sophie; A recommendation I've heard is the book by Dimca, Sheaves in Topology. (I haven't read it.) I think they expect the reader knows some basic sheaf theory but it is meant more as an introduction to the applications. It is short, not a massive technical treatise, they say in the foreword that the book can be considered as preparation for more technical and in depth treatments such as Kashiwara and Schapira's book Sheaves on Manifolds.
The book by Iversen, "Cohomology of Sheaves" describes using sheaves in topology to study Poincare duality and its generalizations due to Poincare. I have started it recently and I like it.
There is a beautiful book by Godement if you have knowledge of French, "Topologie algebrique et theorie des faisceaux". It is one of my favorite books. I cannot really say that it deals too much with "applications". But in it he shows how to use (co)simplicial sheaves to give constructions like the cup product in a way that very closely resembles how they are defined in singular homology. This influenced me a lot, I find it intriguing that we can give such strong correspondences between the theories using simplicial methods.
As for real world applications I have heard recently of sheaves being used to model signal processing.
Thank you Patrick! Godements book is really interesting, and looking at the content table I see what you mean - I will take a closer look!
As you say, Sheaves in Topology looks more like a preparation and seemed to have a lot of perverse sheaves, that I'm not especially interested in at moment. I stumbled upon the signal processing in Robinsons and Ghrists work and it at least partly lead me to ask (-:
Fawzi Hreiki (Sep 21 2021 at 22:57):Sheaf cohomology has been mentioned above which is the most obvious use of sheaves in topology.
There’s also the equivalence between locally constant sheaves and covering spaces which connects with homotopy theory that way.
John Baez (Sep 22 2021 at 02:56):What's a good easy introduction to sheaf cohomology that actually gets around to doing some concrete and interesting calculations with it? The best answer I have is Griffiths and Harris' Principles of Algebraic Geometry. This starts with a lot of examples involving complex manifolds, which are in some ways closer in flavor to geometric topology.
Bott and Tu's Differential Forms in Algebraic Topology doesn't actually talk about sheaves, but they use a lot of ideas that could be expressed in the language of sheaves.
I have the feeling I'm forgetting the name of a very nice ntroductory book that uses sheaves to prove things about de Rham and Dolbeault cohomology.
John Baez (Sep 22 2021 at 02:58):I haven't looked at it recently, but I get the feeling that Bredon's Sheaf Theory is rather dry, introducing a lot of useful machinery but not doing enough with it to make it fun.
John Baez (Sep 22 2021 at 03:00):Anyway, if one is near a library it's good to check out all these books and read them sort of simultaneously for a year or so.
Fawzi Hreiki (Sep 22 2021 at 09:45):If you're more comfortable with differential rather than algebraic geometry, there's a nice book 'Manifolds, Sheaves, and Cohomology' by Torsten Wedhorn (an algebraic geometer)
Fawzi Hreiki (Sep 22 2021 at 09:45):Its basically an introduction to manifolds and their cohomology by an algebraic geometer
John Baez (Sep 22 2021 at 19:12):That sounds nice! I only brought up algebraic geometry because most of what I've read about sheaves and cohomology works with algebraic varieties or at least complex-analytic manifolds. (I find complex-analytic manifolds pleasant because I get to use some of my smooth manifold intuitions while still getting into some of the issues typical of algebraic geometry. This is why I like Griffiths and Harris.)
Sophie El Agami (Sep 23 2021 at 12:17):Yes, exactly, apart from sheaf cohomology most things I found are rather algebraic geometry, which is why I was wondering if there is anything aside from this. I guess it makes sense to look at the different resources while learning and pick the most tangible. The 'Manifold, Sheaves, and Cohomology' looks indeed more comfortable, and using Bredons book as a reference point is a good suggestion as it appearss quite comprehensive in terms of machinery.
Thank you for the suggestions! :smile:
Peiyuan Zhu (Oct 22 2021 at 18:20):Hi, does anyone has recommendation to understand cohomology structure of information? I think I need to start from the most basic definitions and go from there.
John Baez (Oct 22 2021 at 18:38):I guess I have to ask: does anyone here know what the work trying to apply cohomology theory to information actually accomplishes?
John Baez (Oct 22 2021 at 18:39):I didn't understand the explanations I've seen.
Peiyuan Zhu (Oct 23 2021 at 01:34):I've seen some applications here: https://www.youtube.com/watch?v=-KrPrjBUPyo&t=3s
Patrick Nicodemus (Oct 27 2021 at 03:13):John Baez said:
I have the feeling I'm forgetting the name of a very nice ntroductory book that uses sheaves to prove things about de Rham and Dolbeault cohomology.
this was a while ago but were you thinking of Lectures on Riemann Surfaces by Otto Forster? That's a great introduction to sheaf cohomology on Riemann surfaces.
John Baez (Oct 27 2021 at 21:11):No, I've never seen that book.
John Baez (Oct 27 2021 at 21:13):It sounds nice!