Most striking applications of category theory? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T18:00:37Z http://mathoverflow.net/feeds/question/19325 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory Most striking applications of category theory? muad 2010-03-25T16:15:51Z 2012-07-28T09:05:19Z <p>What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:</p> <ul> <li>Joyals Combinatorial Species</li> <li>Grothendieck's Galois Theory</li> <li>Programming (unification as computing a coequalizer, Tatsuya Haginos categorical construction of functional programming)</li> </ul> <p>I am sure that these only touch on the surface so I would be most grateful to hear of more examples, thank you!</p> <p>edit: To try and be more precise. "Application" in the context of this question means that it makes use of slightly deeper results form category theory in a natural way. So we are not just trying to make a list of 'maths that uses category theory' but some of the results which exemplify it best, and might not have been possible without it.</p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/19334#19334 Answer by Kristal Cantwell for Most striking applications of category theory? Kristal Cantwell 2010-03-25T17:43:09Z 2011-03-28T03:54:12Z <p>The finite vector space analog to Ramsey's theorem was proved using categories the paper is available <a href="http://www.math.ucsd.edu/~ronspubs/72_03_ramsey_categories.pdf" rel="nofollow">here</a></p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/19343#19343 Answer by Bruce Westbury for Most striking applications of category theory? Bruce Westbury 2010-03-25T19:14:50Z 2010-03-25T19:14:50Z <p>This question is too vague. Off the top of my head: algebraic topology, homological algebra, etale cohomology (Weil conjectures), homotopical algebra, topological field theory, Mackey functors, Kazhdan-Lusztig theory, ...</p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/19401#19401 Answer by Shizhuo Zhang for Most striking applications of category theory? Shizhuo Zhang 2010-03-26T10:25:37Z 2011-11-10T23:12:43Z <p>Noncommutative algebraic geometry(in the sense of Gabriel-Rosenberg, Artin-Zhang,Van den Berg) are based on category(abelian or Grothendieck category). They consider category as category of quasi coherent sheaves on some noncommutative space. This idea was proposed by Grothendieck and then re-quoted by Manin. Without category theory, this subject can not be built. More information is in <a href="http://mathoverflow.net/questions/7917/non-commutative-algebraic-geometry" rel="nofollow">Noncommutative Algebraic Geometry</a> and <a href="http://mathoverflow.net/questions/10512/theories-of-noncommutative-geometry" rel="nofollow">Theories of Noncommutative geometry</a></p> <p>Another kind of Noncommutative algebraic geometry is based on Functorial POV. It was proposed by Gabriel in the theory of algebraic group and then developed by Grothendieck in commutative algebraic geometry and then Kontsevich-Rosenberg developed noncommutative stack theory via this POV.</p> <p>Noncommutative derived algebraic geometry is also based on category(triangulated category)theory. </p> <p>The relevant names(maybe I will miss some of them)are Manin-Beilinson-Drinfeld, Kapranov, Deligne, Bernstein, Bondal-Orlov-Lunts,Kontsevich-Soibelman,Toen,Van den berg, Lurie, Keller,Neeman and others </p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/80628#80628 Answer by Jan Weidner for Most striking applications of category theory? Jan Weidner 2011-11-10T20:26:55Z 2011-11-10T20:26:55Z <p>One can find a couple of "concrete" striking applications of category theory in algebraic geometry. For example:</p> <ul> <li><p><a href="http://en.wikipedia.org/wiki/Descent_%28category_theory%29" rel="nofollow">Faithfully flat descent</a> follows from <a href="http://en.wikipedia.org/wiki/Beck%27s_monadicity_theorem" rel="nofollow">Beck's monadicity theorem</a>. </p></li> <li><p><a href="http://ncatlab.org/nlab/show/Grothendieck+duality" rel="nofollow">Grothendieck duality</a>, follows from Brown representability, by work of Neeman.</p></li> </ul> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/80662#80662 Answer by Ryan Reich for Most striking applications of category theory? Ryan Reich 2011-11-11T06:34:39Z 2011-11-11T06:34:39Z <p>For a while, my answer to this question was algebraic K-theory; what little I know of it, I learned from Quillen's paper, and it was a relief to finally see an example of category theory being used in an essential way to do something that was <em>not</em> just linguistic. Quillen defines the higher K-groups of an exact category by forming a quite different category in some combinatorial manner that seems to strip away any vestige of a connection to something non-categorical, and then taking its geometric realization and homotopy groups. The whole process: ring to module category to Q-construction to geometric realization, was the first argument I'd seen that category theory could do more than just rephrase perfectly good theorems confusingly.</p> <p>(Now my answer would be "perverse sheaves", though.)</p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/83646#83646 Answer by Ronnie Brown for Most striking applications of category theory? Ronnie Brown 2011-12-16T18:00:23Z 2011-12-16T18:00:23Z <p>First, a comment on studying category theory for its own sake': this slur was very much setting up a straw man. Those accessing the category theory discussion list will know that the discussion there ranges very widely, and actually discusses issues in mathematics, in contrast to other email discussion lists I access. </p> <p>Second, I have found some elementary facts from category theory very useful; examples are left adjoints preserve colmits, right adjoints preserve limits'. Many years ago, listening to Albrecht Dold on half exact functors made me realise how I could cut down considerably a proof from my thesis by using the basic idea of representable functor: this automatically led to the existence of a homotopy equivalence making a diagram commutative. Again, the theory of ends and coends does make life simpler in discussing geometric realisations. </p> <p>Third, I have fairly recently realised that the general framework of fibred and cofibred categories is specially useful for discussing pullbacks and pushouts for certain hierarchical structures with which I have dealt. A basic example here is the bifibration (Groupoids) $\to$ (Sets) given by the object functor. </p> <p>I wish I had a good application in my work of some of the deeper theorems! </p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/103291#103291 Answer by Timo Keller for Most striking applications of category theory? Timo Keller 2012-07-27T10:10:27Z 2012-07-27T10:10:27Z <p>Schlessinger's criteria and deformation of Galois representations, see e.g. Mazur's article in Cornell-Silverman-Stevens.</p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/103334#103334 Answer by none for Most striking applications of category theory? none 2012-07-27T18:12:32Z 2012-07-27T18:12:32Z <p>Toposes and categorical logic. ZOMG.</p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/103336#103336 Answer by Denis Chaperon de LauziÃ¨res for Most striking applications of category theory? Denis Chaperon de LauziÃ¨res 2012-07-27T18:38:30Z 2012-07-27T18:38:30Z <p>The most recent book of Nick Katz [see <a href="https://web.math.princeton.edu/~nmk/mellin398.pdf" rel="nofollow">https://web.math.princeton.edu/~nmk/mellin398.pdf</a> ] proves extremely concrete equidistribution theorems for certain families of exponential sums. Categories enter in three essential ways (at least): (1) all the work going to Deligne's Weil II version of the Riemann Hypothesis over finite fields; (2) the theory of perverse sheaves; (3) the Tannakian formalism to recover a group from a category. In this, the new contribution of Katz in this book is (3): essentially, the equidistribution is proved using the Weyl equidistribution criterion, and all analytic estimates follow from (1). But if one doesn't know that there is a group underlying the families of sums (or rather the unitarized Frobenius automorphisms which give rise to these sums), one doesn't know what these estimates are really proving. </p> <p>For more traditional families of sums, one uses instead Deligne's Equidistribution Theorem, where the group is given concretely as monodromy group of a lisse sheaf, but Katz's family are not parameterized by an algebraic variety, and the Tannakian category arises by looking at a category of perverse sheaves equiped with a suitable form of multiplicative convolution.</p> <p>This is, I think, completely amazing...</p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/103370#103370 Answer by Philippe Gaucher for Most striking applications of category theory? Philippe Gaucher 2012-07-28T09:05:19Z 2012-07-28T09:05:19Z <p>The recent developments in homotopical algebra (after 1990) would not be possible without the use of category theory, and more precisely the theory of locally presentable and accessible categories. I am talking about the theory of combinatorial model categories (model categories such that the underlying category is locally presentable). </p>