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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 13th 2018
   • (edited Sep 13th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexdiscovered this ancient entry (while searching for occurences of "permutation matrix" on the nLab). This was in very bad shape, with a ill-rendering floating toc and big query box right at the beginning, then a little bit of content, and then some speculation by a contributor who we had to persuade to leave, long ago. I did a minimum of cleaning up, in particular removed the query box, since it had been dealt with. This is what it had said: *** +-- {.query} Zoran: there are several things called "Birkhoff's theorem" in various field of mathematics and mathematical physics, and belong even to at least 2 different classical Birkhoff's. Even wikipedia has pages for more than one such theorem. To me the first which comes to mind is Birkhoff's factorization theorem, now also popular in Kreimer-Connes-Marcolli work and in connection to loop groups (cf. book by Segal nad Pressley). I would like that the $n$lab does not mislead by distinguishing one of the several famous Bikhoff labels without mentioning and directing to 2-3 others. [[Ian Durham]]: Good point. I think this probably ought to be renamed the "Birkhoff-von Neumann theorem." Is that a good enough label or should we get more specific with it? _Toby_: I have moved it. See also the new page [[Birkhoff's theorem]], which is basically just Zoran\'s comment above. =-- *** <a href="https://ncatlab.org/nlab/revision/diff/Birkhoff-von+Neumann+theorem/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/Birkhoff-von+Neumann+theorem/9">v9</a>, <a href="https://ncatlab.org/nlab/show/Birkhoff-von+Neumann+theorem">current</a>

   discovered this ancient entry (while searching for occurences of “permutation matrix” on the nLab). This was in very bad shape, with a ill-rendering floating toc and big query box right at the beginning, then a little bit of content, and then some speculation by a contributor who we had to persuade to leave, long ago.

   I did a minimum of cleaning up, in particular removed the query box, since it had been dealt with. This is what it had said:

   ---

   +– {.query}

   Zoran: there are several things called “Birkhoff’s theorem” in various field of mathematics and mathematical physics, and belong even to at least 2 different classical Birkhoff’s. Even wikipedia has pages for more than one such theorem. To me the first which comes to mind is Birkhoff’s factorization theorem, now also popular in Kreimer-Connes-Marcolli work and in connection to loop groups (cf. book by Segal nad Pressley). I would like that the $n$lab does not mislead by distinguishing one of the several famous Bikhoff labels without mentioning and directing to 2-3 others.

   Ian Durham: Good point. I think this probably ought to be renamed the “Birkhoff-von Neumann theorem.” Is that a good enough label or should we get more specific with it?

   Toby: I have moved it. See also the new page Birkhoff’s theorem, which is basically just Zoran's comment above. =–

   ---

   diff, v9, current

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 13th 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThis is one of those interesting results that I find myself drifting back to on occasion, part of a circle of ideas which includes Hall's marriage theorem, the latter of which I want to think about more with an eye to constructivity/categorification.

   This is one of those interesting results that I find myself drifting back to on occasion, part of a circle of ideas which includes Hall’s marriage theorem, the latter of which I want to think about more with an eye to constructivity/categorification.