User david corfield - MathOverflow

Answer by David Corfield for A terminal coalgebra of a certain functor on Mes

2013-02-14T13:59:35Z

Final coalgebras for functors on measurable spaces, Lawrence S. Moss and Ignacio D. Viglizzo:

"We prove that every functor on the category Meas of measurable spaces built from the identity and constant functors using products, coproducts, and the probability measure functor Δ has a final coalgebra."

The set-theoretic multiverse as a (bi)category

2011-09-19T11:53:46Z

Joel Hamkin's The set-theoretic multiverse has featured in MO questions before, e.g., here and here. But I was wondering about the best category theoretic angle to take on it.

In the paper Joel writes, rather poetically,

Set theory appears to have discovered an entire cosmos of set-theoretic universes, revealing a category-theoretic nature for the subject, in which the universes are connected by the forcing relation or by large cardinal embeddings in complex commutative diagrams, like constellations filling a dark night sky. (p. 3)

He has given us a couple of kinds of morphism here, but what is the best way to capture this multiverse category theoretically? Which morphisms should we allow?

Is it right to stay at the level of ordinary categories? Since each universe, a model of ZFC, is a category, one might expect the multiverse to be at least a bicategory, as suggested here. Do set theorists consider, say, arrows between two forcing relations between two models?

Answer by David Corfield for Incidences of rigorous proofs used in legal proceedings

2012-07-11T07:46:36Z

For a case which very nearly came to court disputing a claim made by a company to have built a computer chip mathematically proven to meet its specification, take a look at this section of an article by Donald MacKenzie: http://books.google.co.uk/books?id=rx3oUTzjh8sC&pg=PA134. It would have been wonderful to have had barristers contest the nature of mathematical proof. Mackenzie writes about the case at greater length in 'Mechanizing Proof: Computing, Risk, and Trust' MIT Press 2001.

Answer by David Corfield for A "simple" explanation of the concept of D-separation in a Bayesian Network?

2012-06-19T21:29:28Z

Try this tutorial http://www.andrew.cmu.edu/user/scheines/tutor/d-sep.html

Answer by David Corfield for Essential reads in the philosophy of mathematics and set theory

2012-04-18T09:30:04Z

The Stanford Encyclopedia of Philosophy is a good resource, especially for 'analytic' approaches. See the Philosophy of Mathematics entry, and links at the bottom of the page. You might also want to browse through the dedicated journal Philosophia Mathematica. If you're interested in approaches which look to broaden the range of questions asked by philosophers about mathematics, you could try Mancosu (ed.) The Philosophy of Mathematical Practice.

Answer by David Corfield for Formalizing "no junk, no confusion"

2012-03-13T11:52:51Z

Belatedly, an answer in set-based situations to

What would be the corresponding slogan to "no junk, no confusion" for final coalgebras?

Given an initial algebra, any algebra will have a special subobject which is the image of the initial structure. The subobject may have confused elements (terms) of the initial algebra, and the object may have extra junk. A map between objects will map the first special subobject to the second, possibly confusing more. The initial algebra has no confusion and no junk.

Given a final/terminal coalgebra, the elements of any coalgebra will have a special colouring in terms of images in the elements of the terminal structure. The object may have more than one element with the same colour, and the object may not use all the colours. A map between objects will preserve the colouring, the domain possibly using fewer colours. The terminal coalgebra colours without ambiguity and without redundancy. If two things behave the same way, they are the same; all behaviours are covered.

No junk, no confusion; No redundancy, no ambiguity.

Answer by David Corfield for Failures that lead eventually to new mathematics

2011-10-07T07:30:02Z

An important moment for chaos theory and dynamical systems was the discovery by Phragmén that there was a problem with the convergence of a series in Poincaré's original submission to a competition organised as part of the 60th-anniversary celebration of the birth of Oscar II, King of Sweden and Norway. The rewritten paper is seminal. The story is well told by June Barrow-Green in Poincaré and the three body problem (1997).

Answer by David Corfield for Applications of PDE in mathematical subjects other than geometry & topology

2011-09-01T10:16:32Z

Klainerman writes in PDE as a Unified Subject

...the range of applications of specific PDE's is phenomenal, many of our basic equations being in fact at the heart of fully fledged fields of Mathematics or Physics such as Complex Analysis, Several Complex Variables, Minimal Surfaces, Harmonic Maps, Connections on Principal Bundles, Kahlerian and Einstein Geometry, Geometric Flows, Hydrodynamics, Elasticity, General Relativity, Electrodynamics, Nonrelativistic Quantum Mechanics, etc. Other important subjects of Mathematics, such as Harmonic Analysis, Probability Theory and various areas of Mathematical Physics are intimately tied to elliptic, parabolic, hyperbolic or Schrodinger type equations. Specific geometric equations such as Laplace-Beltrami and Dirac operators on manifolds, Hodge systems, Pseudoholomorphic curves, Yang-Mills and recently Seiberg-Witten, have proved to be extraordinarily useful in Topology and Symplectic Geometry. The theory of Integrable systems has turned out to have deep applications in Algebraic Geometry; the spectral theory Laplace-Beltrami operators as well as the scattering theory for wave equations are intimately tied to the study of automorphic forms in Number Theory. (p. 2)

Answer by David Corfield for The concept of Duality

2011-09-01T08:23:03Z

What would be useful here is a list of mechanisms lying behind these appearances of duality. So we have (at least)

Duality pairing
Dualizing object
Maximal fixed subcategories of an adjunction
Arrow reversal

Then we could look at any relations between these mechanisms, such as between 2 and 3, maps into a dualizing object form the functors for an adjunction.

Atiyah in his talk Duality in Mathematics and Physics says

"Fundamentally, duality gives two different points of view of looking at the same object. There are many things that have two different points of view and in principle they are all dualities."

So perhaps we need

5 . Something is seen in two different ways

The Dynkin diagram for $SL_n$ is a string of $n-1$ dots, we can view it from either end as point, line, plane, etc. Put another way, the symmetry of the diagram corresponds to an outer automorphism which account for the duality of projective geometry.

I wonder if 'deeper' dualities come from more intricate processes of seeing something from two points of view. Frenkel gives a very accessible talk What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common? where he explains that the duality of Geometric Langlands arises from compactifying a 6d quantum field theory in two different ways onto 2d surfaces.

Is there a common framework for Tannaka and Gabriel-Ulmer reconstruction theorems?

2011-07-14T16:31:43Z

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory from the category of models of that theory, for example, the theory of groups from its category of models. There are other related dualities corresponding to other 'doctrines', see A Duality Relative to a Limit Doctrine.

This idea of recovering something from its category of models seems to resemble closely the Tannakian reconstruction of an algebraic entity from its category of modules or comodules or whatever.

So I was wondering whether there is a way to see doctrinal, Gabriel-Ulmer style reconstruction and Tannaka style reconstruction as instances of something more general.

The only explicit reference I have found is Brian Day in Enriched Tannaka reconstruction writing "Our approach provides a synthesis of Tannaka reconstruction and Gabriel-Ulmer duality," but I haven't found the paper very helpful.

I've posed this question at greater length here.

Answer by David Corfield for Logic in mathematics and philosophy

2011-04-21T16:01:17Z

The inclusion by Neel of a computer scientist is indicative of the ways in which that discipline mediates between mathematical and philosophical logic. Temporal and modal logics are good cases of a situation where initial philosophical motivation gets swept up by more practical concerns, e.g., model checking in the case of temporal logic. Mathematical sophistication then usually increases. Now to work on (first-order) modal logic you had better have your sheaf theory up to scratch.

When and why do universal objects have extra properties?

2009-11-24T17:43:06Z

I'm interested in situations where universal objects come with more structure than their definitions suggest. A classic case of this is where the free abelian group on one element has a ring structure. Proving this is a straightforward exercise using the free-underlying adjunction. So I'd like to know of other cases of this phenomenon, and if possible an explanation as to why the extra structure comes about.

Plenty of examples are given in Hazewinkel's paper Niceness Theorems, but how about very familiar examples such as the rationals? Does, say, the characterisation of $\langle \mathbb{Q}, \gt \rangle$ as the Fraïssé limit of the category of finite linearly ordered sets and order preserving injections tell us why it should support a compatible group, ring and even field structure? Do characterisations of the reals relate to each other?

My question is not completely unrelated to Theorems for nothing (and the proofs for free), as shown by the example given there of subgroups of free groups being free, which also occurs in Hazewinkel's paper.

Answer by David Corfield for Problems where we can't make a canonical choice, solved by looking at all choices at once

2010-12-21T13:37:18Z

You could say that treating all models of a first-order theory is a way of avoiding the arbitrary selection of a particular completion of that theory. There are other situations where it may be best to treat all completions - http://ncatlab.org/nlab/show/completion#nonunique.

Answer by David Corfield for Reference request: 2-Grothendieck Construction

2010-10-01T09:39:31Z

I. Bakovic, Grothendieck construction for bicategories.

Answer by David Corfield for How do you decide whether a question in abstract algebra is worth studying?

2010-09-24T09:17:26Z

There are surely no hard and fast rules as to assessing the importance of a generalization of a concept. I once took a look (chap. 9) at debates surrounding the move from groups to groupoids. One important step up for a concept is being deemed essential rather than merely useful. To achieve this it must find its place in an array of good storylines.

Which languages could appear on Weil's Rosetta Stone?

2010-09-08T15:05:23Z

André Weil's likening his research to the quest to decipher the Rosetta Stone (see this letter to his sister) continues to inspire contemporary mathematicians, such as Edward Frenkel in Gauge Theory and Langlands Duality.

Remember that Weil's three 'languages' were: the 'Riemannian' theory of algebraic functions; the 'Galoisian' theory of algebraic functions over a Galois field; the 'arithmetic' theory of algebraic numbers. His rationale was the desire to bridge the gap between the arithmetic and the Riemannian, using the 'Galoisian' curve-over-finite-field column as the best intermediary, so as to transfer constructions from one side to the other. (See also 'De la métaphysique aux mathématiques' 1960, in volume II of his Collected Works.)

That fitted rather neatly with demotic Egyptian mediating between priestly Egyptian (hieroglyphs) and ordinary Greek on the real Rosetta Stone. But just as one might have expressed that text in a range of other contemporary languages - Sanskrit, Aramaic, Old Latin, why should there not be other columns in Weil's story? Frankel himself adds a fourth column (p. 11) 'Quantum Physics'.

So now the questions:

Are there other candidate languages for Weil's stone? Might there be a further language for which we would need intermediaries back to the arithmetic? Could there be a meta-viewpoint which determines all possible such languages.

Presumably the possession of a zeta function is too weak a condition as that would allow the language of dynamical systems.

Answer by David Corfield for Direct proof of irrationality?

2010-07-15T15:00:42Z

Wikipedia has a constructive proof. You can bound $\sqrt 2$ away from $p/q$.

Answer by David Corfield for Arithmetic fixed point theorem

2010-07-13T14:46:51Z

For a unified account which subsumes the First Incompleteness theorem, Russell's paradox and Cantor's theorem, try Yanofsky's paper A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points.

What are hypergroups and hyperrings good for?

2010-07-02T08:29:19Z

I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a weakening of the ring concept, but where the addition is allowed to be multivalued. Indeed the additive part of a hyperring forms a 'canonical hypergroup'.

A canonical hypergroup is a set, $H$, equipped with a commutative binary operation,

$$ + : H \times H \to P^*(H) $$

taking values in non-empty subsets of $H$, and a zero element $0 \in H$, such that

$+$ is associative (extended to allow addition of subsets of $H$);
$0 + x = {{x}} = x + 0, \forall x \in H$;
$\forall x \in H, \exists ! y \in H$ such that $0 \in x + y$ (we denote this $y$ as $-x$);
$\forall x, y, z \in H, x \in y + z$ implies $z \in x - y$ (where $x - y$ means $x + (-y)$ as usual).

(NB: $x$ may be written for the singleton {$x$}.)

I know that hyperrings occur whenever a ring is quotiented by a subgroup of its multiplicative group, but I'd like to know more about where and how hyperrings and hypergroups have cropped up in different branches of mathematics. How is a canonical hypergroup to be thought of as canonical? Are noncanonical hypergroups important? Is there a category theoretic way to see these hyperstructures as natural?

Answer by David Corfield for The ring of integers looks like the 3-dimensional sphere viewed as the Hopf fibration

2010-07-01T14:07:58Z

Various pieces of exposition and references are to be found - here, here, here, and here.

Answer by David Corfield for What are other theories of causality besides graphical models and Bayesian networks?

2010-06-15T09:16:56Z

With several variables connected by asymmetric causal relations, it's not so likely that a mathematical theory of causality will escape graphical representation. Neel mentioned above Lewis's counterfactual analysis, and this has a close affinity with aspects of Judea Pearl's work on casual Bayesian networks, see p. 239 of http://books.google.co.uk/books?id=wnGU_TsW3BQC.

Among quantitative approaches, a variant on the usual statistical approach is Janzing and Schoelkopf's use of algorithmic dependence to determine causal relations: http://www.kyb.mpg.de/publications/attachments/paper_IEEE_version3_webseite_6526%5B1%5D.pdf.

Answer by David Corfield for A question about Chapter 12 (Vapnik-Chervonenkis Theory) of 'A Probabilistic Theory of Pattern Recognition'

2010-05-25T09:19:14Z

Yes, a hyperrectangle is a generalisation of rectangle to higher dimensions. Here the data is given by points in $\mathbb{R}^d$, so the hyperrectangles are all of that dimension. As with all such algorithms you need to find a way to get a handle on the set of classifiers, so rather than the infinite class of all hyperrectangles of dimension $d$, the choice will be from the $n \choose 2 d$ hyperrectangles of dimension $d$, each of which is the smallest for some choice of $2 d$ of the data points.

For example, if the data consisted of 1000 points in $\mathbb{R}^2$, rather then the infinite class of all rectangles, we confine ourselves to the $1000 \choose 4$ rectangles which minimally contain a subset of 4 of the data points.

One task then is to show that the best of this finite set is almost as good as the best of all the hyperrectangles -- good in the sense that were the data points each labelled $+$ or $-$, the $+$s would be best separated from the $-$s. The argument claims that for each hyperrectangle there will be one from the finite set agreeing with it except for a small number of points on the boundary equal to the number of faces of the hyperrectangles. In the example above, it says that for any rectangle, there is one in the set of $1000 \choose 4$ which encloses exactly the same points, except possibly for 4 on the boundary. Not completely obvious, I agree.

Edit: If hyperrectangles are restricted to have lines parallel to the axes, then it is obvious. Judging from problem 11.6 on page 183, this may well be the case.

Answer by David Corfield for What are some examples of interesting uses of the theory of combinatorial species?

2010-04-26T08:19:10Z

One further line of response would again invoke Rota:

"What can you prove with exterior algebra that you cannot prove without it?" Whenever you hear this question raised about some new piece of mathematics, be assured that you are likely to be in the presence of something important. In my time, I have heard it repeated for random variables, Laurent Schwartz' theory of distributions, ideles and Grothendieck's schemes, to mention only a few. A proper retort might be: "You are right. There is nothing in yesterday's mathematics that could not also be proved without it. Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectures. (Indiscrete thoughts, p.48, Birkhauser, 1997).

For a couple of new worlds made possible by the species concept see:

1) M. Fiore, N. Gambino, M. Hyland and G. Winskel. The cartesian closed bicategory of generalised species of structures. Journal of the London Mathematical Society, 77(2) (2008), 203-220.

2) J. Baez et al. on stuff types (note, they call species 'structure types').

Answer by David Corfield for Relation between Hecke Operator and Hecke Algebra

2010-03-29T07:41:17Z

Regarding your second question, the relationship between Hecke operators and algebras was discussed in Baez's This Week's Finds. For instance, take a look at David Ben-Zvi's comment on Week 254.

Answer by David Corfield for Ubiquity of the push-pull formula

2010-03-20T13:30:34Z

Isn't this about the Beck-Chevalley condition? Some of its manifestations were discussed in this thread.

Answer by David Corfield for What is "rich structure", actually?

2010-02-22T08:50:39Z

A good case to look at would be Michiel Hazewinkel's 'star example' of a rich structure Symm, the ring of symmetric functions in a countably infinite number of indeterminates.

Symm, the Hopf algebra of the symmetric functions is a truly amazing and rich object. It turns up everywhere and carries more extra structure than one would believe possible. For instance it turns up as the homology of the classifying space BU and also as the cohomology of that space, illustrating its self-duality. It turns up as the direct sum of the representation spaces of the symmetric group and as the ring of rational representations of the infinite general linear group. This time it is Schur duality that is involved. It is the free $\lambda$-ring on one generator. It has a nondegenerate inner product which makes it self-dual and the associated orthonormal basis of the Schur symmetric functions is such that coproduct and product are positive with respect to these basis functions...Symm is also the representing ring of the functor of the big Witt vectors and the covariant bialgebra of the formal group of the big Witt vectors (another manifestation of its auto-duality)...

As the free $\lambda$-ring on one generator it of course carries a $\lambda$-ring structure. In addition it carries ring endomorphisms which define a functorial $\lambda$-ring structure on the rings $W(A) = CRing(Symm, A)$ for all unital commutative rings $A$. A sort of higher $\lambda$-ring structure. Being self dual there are also co-$\lambda$-ring structures and higher co-$\lambda$-ring structures (whatever those may be).

Of course, Symm carries still more structure: it has a second multiplication and a second comultiplication (dual to each other) that make it a coring object in the category of algebras and, dually, (almost) a ring object in the category of coalgebras.

The functor represented by Symm, i.e. the big Witt vector functor, has a comonad structure and the associated coalgebras are precisely the $\lambda$-rings.

All this by no means exhausts the manifestations of and structures carried by Symm. It seems unlikely that there is any object in mathematics richer and/or more beautiful than this one, and many more uniqueness theorems are needed. (Witt vectors. Part 1: 7)

Answer by David Corfield for Analogies between analogies

2010-02-03T09:37:47Z

This question was posed here followed by some attempted answers.

Answer by David Corfield for Is there a relationship between model theory and category theory?

2010-01-18T11:55:55Z

We had a chat about this topic over here, prompted by remarks by David Kazhdan.

Answer by David Corfield for What is the "right" universal property of the completion of a metric space?

2010-01-13T09:49:36Z

Mike Shulman give the impression here that understanding the completion of a uniform space is a little trickier than for a metric space:

Cauchy completion of a metric space is, of course, an instance of Cauchy completion of enriched categories. I believe that Cauchy completion of a uniform space is actually also an instance of a general categorical notion of Cauchy completion, but in the more general setting of an equipment (namely, the equipment of sets and filters). See "Categorical interpretation" at uniform space for a too-brief summary of this point of view.

Answer by David Corfield for categorification of logic

2009-12-16T13:53:15Z

Try Mike Shulman's page.

Comment by David Corfield

2013-04-22T11:49:00Z

Have you checked to see how far the nLab entry on Tannaka Duality (ncatlab.org/nlab/show/Tannaka+duality) takes you?

Comment by David Corfield

2013-04-02T14:47:52Z

Differential geometry using the infinitesimals Todd mentions is known as synthetic differential geometry: ncatlab.org/nlab/show/…. There are a couple of online textbooks by Anders Kock listed there.

Comment by David Corfield

2013-02-11T10:14:16Z

Do you mean by "very serious no-nonsense mathematicians" those whose dial is turned towards requiring strong motivation from more concrete problems before looking for more general, abstract settings? But then how to describe those who need a little less of a push?

Comment by David Corfield

2013-01-16T10:09:17Z

Todd, as I discussed at the nCafe - golem.ph.utexas.edu/category/2012/01/… - the idea you attribute to Tarski of distinguishing the logical from the non-logical is there in Mautner. He was extending Weyl's Kleinian treatment of algebraic groups to symmetric groups. By the way, any chance that you and James might pursue that very interesting work?

Comment by David Corfield

2012-11-26T10:53:15Z

In a discussion on miracles and laws in mathematics, André Joyal said that he considered it a miracle that the complex numbers are algebraically closed after merely adjoining the roots of one simple equation to the reals.

Comment by David Corfield

2012-09-23T14:34:00Z

Isham addresses the topic specifically in 'Topos Theory and Consistent Histories' arxiv.org/abs/gr-qc/9607069.

Comment by David Corfield

2012-05-11T08:26:09Z

A huge number of references are recorded here empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/…, especially concerning the relation of RH to physics.

Comment by David Corfield

2012-05-08T09:13:05Z

Andre Henriques' answer mathoverflow.net/questions/44045/… to the question Neil mentioned points to what may be specific about hyperfinite type III factors here.

Comment by David Corfield

2012-05-07T09:49:34Z

John Baez was wondering about a geometric model for it in Week 149 - math.ucr.edu/home/baez/week149.html - but I don't think ever found one. He discusses its role in classifying principal U(1) 2-bundles here - math.ucr.edu/home/baez/calgary/calgary.pdf.

Comment by David Corfield

2012-04-24T12:34:27Z

I briefly consider mathematical naturalness in section 9.8 of my book books.google.co.uk/books?id=a0U98qfx4TQC

Comment by David Corfield

2012-04-18T15:52:24Z

Thanks for the mention, but linking to a negative review might not be the best way to proceed. How about Brian Davies' review in Notices of the AMS ams.org/notices/201110/rtx111001454p.pdf.

Comment by David Corfield

2012-03-30T11:10:48Z

Perhaps the German term 'Modell' came first, maybe deriving from the nineteenth century idea of models of non-Euclidean geometry, such as the Beltrami-Klein model.

Comment by David Corfield

2012-02-11T14:46:04Z

"note that this Wikipedia entry rather stupidly says..." Why not edit it then?

Comment by David Corfield

2012-01-21T12:13:32Z

Try this post golem.ph.utexas.edu/category/2008/12/… for some category theoretic discussion of Tao's description of cohomology in dynamic systems.

Comment by David Corfield

2011-12-07T12:15:14Z

@Faisal: so the species of finite sets concerns collections of finitely many elements coloured by 1 colour. The cardinality of the groupoid of 2-coloured finite sets is $e^2$, etc. See p. 9 of math.mq.edu.au/~street/ByrneHons.pdf.