User sixwingedseraph - MathOverflow

Answer by SixWingedSeraph for Is there a notion of congruence relation for essentially algebraic structures?

2010-08-13T01:42:08Z

As Finn says, lfp categories have all coequalizers. However, there is a fly in the ointment. In set-models of algebraic theories, the underlying functor preserves the congruences and the quotients of the congruences. This means that a congruence is an equivalence relation on the underlying set and the quotient alegbraic structure has underlying set the set-quotient of the congruence. This is true even in many sorted algebraic theories.

But in models of finite limit sketches the quotient can blow up. The quotient need not be a structure whose underlying set is the quotient of the equivalence relation on the underlying set. An example of this happens in the category of small categories when you merge nonisomorphic objects. All of a sudden arrows may compose that didn't meet each other before, creating new arrows. (So the underlying set of arrows in the quotient is not the quotient of the congruence on the underlying set.)

This is spelled out and proved in Toposes, Triples and Theories, by Michael Barr and Charles Wells, in Theorem 4.1 of Chapter 8. In that theorem, "LE" means "Finite Limits" and "EE" means "effective equivalence relations whose quotients are preserved by the underlying set functor". This book is available for free on the internet -- just google it. Exercises EEPO and ORTHODOX give specific examples of that behavior. I think there is a specific example for small categories somewhere but at the moment I can't find it.

ADDED: The specific example is in section 1.8, exercise CBB. Section 1.8 talks about effective equivalence relations in general and calls them congruences.

Where does the generic triangle live?

2009-11-18T21:22:42Z

This is a reformulation of my question Characterizing triangles unembeddedly.

Motivation 1: There is such a thing as a generic group. In category theory this is done by constructing "theory" of the group, which is a category in a certain doctrine. Functors (in that doctrine) to Set, or more generally to any topos, are groups. The barest such theory (as usually seen) is the Lawverean algebraic theory of groups. This theory is a category containing an object and operations making it a group object in that category, and the theory is the smallest such category that contains all finite limits. There are fancier ones; the fanciest is the classifying topos for groups, which is in some sense the initial topos-with-group object. Since in a topos, you have full-scale first order intuitionistic logic, the classifying topos for groups allows you to reason about the generic group inside the classifying topos and the theorems you prove will be true for all groups. (This is only an approximation of the actual situation.) In particular you can't prove it is abelian and you can't prove it isn't; the logic clearly does not have excluded middle.

Motivation 2: You can prove that a triangle that has two angles that are equal must be isosceles (has two sides that are equal). You can do this with Pappus' proof: Look at the triangle, flip it over the perpendicular from the odd angle to the other side, look at it again, and the side-angle-side theorem shows you that the "two" triangles are congruent, so two sides much be equal. This appears to me to be true without requiring the parallel postulate. So the theorem and the proof must be true not only in Euclidean 2-space but in any surface of constant curvature. (Here I am getting into territory I know very little about, so this particular motivation may be totally misguided.)

So what I want is a classifying space of some sort that contains the generic triangle in such a way that maps of the correct sort to any surface of constant curvature are triangles, and so that Pappus' proof can be carried out in the classifying space. The space doesn't have to be a topos or a category at all. I have no clue as to what sort of structure it would be.

Note 1: Even the Lawvere theory of groups has its own internal logic -- in this case equational logic. You certainly cannot prove the generic group is or is not abelian with equational logic.

Note 2: It does not seem reasonable to me that Pappus' proof would work in a surface with variable curvature. But maybe there is some trick to define "angle mod curvature" that would make it true.

Note 3 added 3 Dec 2009: One way of reformulating my question is: How do you give a suitably general definition of "triangle that allows Pappus' proof". Commenters who asked "which definition of triangle are you using" missed the point: I am asking for a definition. Mathematical research commonly consists of trying to find the right definition to make your intuitive proof work. Questions like that belong in MathOverflow and should not be criticized for not being "well formulated". (Of course many questions of this sort could have been solved by looking in Wikipedia or thinking for five minutes, and they deserve to be criticized.)

Answer by SixWingedSeraph for Depressed graduate student.

2010-01-02T01:27:10Z

Ryan Budney and Andrej Bauer cover most of what I would say, but I would add:

Spend a couple of hours a week reading some Wikipedia article about a piece of math that you have heard of but don't understand. Follow the links when you see an unfamiliar word. You will know more that when you started even if you are still uncertain about parts of it.

I hope you have one or more friends you can shoot the bull about math with. Then talk to them about the article you read, get into arguments about it, talk while taking a walk with them, and so on.

Answer by SixWingedSeraph for Characterising extendable automorphisms

2009-12-26T03:46:55Z

An abstract answer to the question for all groups is given in the papers below. I have not followed the field in recent years. There may be other papers specific to finite groups.

Charles Wells, Automorphisms of Group Extensions, 1970.

Kung Wei Yang Isomorphisms of group extensions. Pacific J. Math. Volume 50, Number 1 (1974), 299-304.

D.J.S. Robinson, Applications of cohomology to the theory of groups, Groups – St. Andrews 1981, London Math. Soc. Lecture Notes vol. 71 (1982), pp. 46–80.

Jin Ping, Automorphisms of groups Journal of Algebra Volume 312, Issue 2, 15 June 2007, Pages 562-569

Answer by SixWingedSeraph for Homological Algebra for Commutative Monoids?

2009-12-09T02:54:45Z

This is an answer to one part of your question. The paper “Extension Theories for Monoids” by Charles Wells, Semigroup Forum 16 (1978), 13-35, gives a precise answer to the specific question: How does the Beck cohomology theory for monoids classify extensions of monoids? (It classifies Leech extensions.) The paper with corrections and a list of subsequent papers related to it may be found here. Beck's thesis is now online here.

Answer by SixWingedSeraph for Origins of Mathematical Symbols/Names

2009-12-09T14:27:54Z

Pat Ballew's blog Math Words has interesting stuff.

Answer by SixWingedSeraph for Famous mathematical quotes

2009-11-29T21:37:31Z

"Mathematics is the art of giving the same name to different things." Henri Poincaré.

(This was in response to "Poetry is the art of giving different names to the same thing.")

Answer by SixWingedSeraph for Famous mathematical quotes

2009-11-29T21:57:22Z

"The price of metaphor is eternal vigilance." Norbert Wiener.

Answer by SixWingedSeraph for Famous mathematical quotes

2009-11-29T21:54:53Z

"[Mathematics consists of] true facts about imaginary objects." Philip Davis and Reuben Hersh.

Answer by SixWingedSeraph for Famous mathematical quotes

2009-11-29T21:40:53Z

"Later mathematicians will regard set theory as a disease from which one has recovered." Henri Poincaré.

Answer by SixWingedSeraph for Why forgetful functors usually have LEFT adjoint?

2009-11-21T17:13:32Z

Forgetful functors usually have a left adjoint because they usually preserve limits. For example, the underlying set of the direct product of two groups is the direct product of the underlying sets, and similarly for equalizers (that gives you all finite limits).

However, functors that preserve limits don't have to have left adjoints, because once in a while what you want to do to construct a free object results in a proper class. An example is complete lattices. Freyd's Adjoint Functor Theorem gives a necessary and sufficient condition for a limit-preserving functor to have a left adjoint. The proof and related results is discussed in section 1.9 of Toposes, Triples and Theories.

Answer by SixWingedSeraph for In what sense are fields an algebraic theory?

2009-10-29T02:47:59Z

As previous answers have said, fields are not algebraic. They are also not essentially algebraic, because categories of models of essentially algebraic theories have an initial object, and the category of fields instead has a set of initial objects -- Z and Z_p for each prime p. (There is no map from a field of one characteristic to a field of a different characteristic, so there can't be a single initial object.)

Fields are models of a a theory which is essentially algebraic plus allows specification of disjunctions. In the language of sketches, fields are the models of a "finite sum sketch." This was proved in Diers' thesis and is spelled out in the paper "The formal description of data types using sketches" by Charles Wells and Michael Barr, in volume 298 of the Springer Lecture Notes in Computer Science, 1988. For a general overview of sketches, see "Sketches: Outline with References" at http://www.cwru.edu/artsci/math/wells/pub/pdf/sketch.pdf .

ADDED 17 November 2009. The category of models of a finite-sum theory is not as nice as the models of an algebraic theory or even an essentially algebraic theory. Generally, the more different kinds of things you can specify in a sketch, the more awkward the category of models is. The category of fields is pretty awkward!
It does have filtered colimits and is closed under ultraproducts. Field theorists have made considerable use of the closure under ultraproducts.

Answer by SixWingedSeraph for Most harmful heuristic?

2009-10-28T01:32:02Z

Writing a proof as a chain of expressions connected by equals signs whether they are appropriate or not.

Answer by SixWingedSeraph for Why do I find Category Theory mostly just a way to make simple things difficult?

2009-10-27T20:41:56Z

I think the other answers miss one aspect of this question. Mathematicians vary in how they do math. Some are "syntactic thinkers" (maybe you), some are "conceptual", and some are "geometric" in the way they think. That is the way Leone Burton's book Mathematicians as Enquirers: Learning about Learning Mathematics., Kluwer, 2004, analyzes it. Others take geometric and conceptual to be variations of the same category, and different names are used for the categories, too.

People are different, and in how they think about abstract ideas they are different in a very deep way. That is my own experience in both my research career in and teaching. I took logic from Joe Shoenfield (which gives me a respectable background!) and did work in abstract algebra and then discovered category theory and thought: Way to go! That is because I think primarily conceptually.

Mike Barr said that to a person with a hammer, everything looks like a nail. I keep translating problems into categorical language. You go the other way. These differences run deep, and should be taken into account when reading other people's stuff.

Characterizing triangles unembeddedly

2009-10-20T00:38:21Z

The mathedu mailing list has a recent longish thread at

http://www.nabble.com/Why-do-we-do-proofs--to25809591.html

which discussed among other things whether we should teach triangles as labeled or unlabeled to high school students (this is a vast oversimplification of the thread). I have long been concerned with how we think (informally and formally) about mathematical objects, so naturally I started to consider how we think about triangles.

Consider circles. Most informal and formal descriptions involve an embedding into R^2, but they can be characterized as manifolds (even as Riemannian manifolds) of dimension 1 with specific properties, independent of any embedding. This sort of thing has turned out to be a major way to think about all sorts of spaces. So can we describe triangles in a similar way?

Unfortunately, manifolds are far removed from my usual mathematical work (category theory). What I think I understand is that there can be piecewise linear manifolds, even Riemannian ones. So perhaps we can say a triangle is a piecewise linear manifold of dimension 1 with certain properties. Now, I want to define a triangle so that it comes complete with information about the lengths of its sides and what the three angles are. Riemannian manifolds have a way to specify length and angles, and I can believe you can make the sides have specific lengths. But the angles? It seems to me that the tangent spaces (like those on a circle) result in all angles being 0 or pi, except at the corners where they don't exist. But I may not understand the situation correctly.

So my question is: Is there a known methodology that allows triangles to be characterized independent of embeddings in such a way that incorporates information about side lengths and angles?

Comment by SixWingedSeraph

2010-08-15T23:03:31Z

A specific technique in category theory for building structures is outlined in Graph Based Logic and Sketches, by Atish Bagchi and Charles Wells, arxiv.org/abs/0809.3023. This was specifically designed to be translated easily into an object-oriented program.

Comment by SixWingedSeraph

2010-08-14T18:54:24Z

Some colimits blow up even with algebraic theories. For example, the underlying set of the coproduct of two groups in the category of groups is not the coproduct (disjoint sum) of the underlying sets.

Comment by SixWingedSeraph

2010-08-13T02:14:36Z

My answer to Peter Arndt's question contains some information about this question.

Comment by SixWingedSeraph

2010-08-13T02:11:44Z

Congruences on categories work very nicely when restricted to bijections on objects. This has attracted a lot of interest. Extension Theories for Categories, by Charles Wells. cwru.edu/artsci/math/wells/pub/pdf/catext.pdf For the following references I thank Peter Webb: An Introduction to the Representations and Cohomology of Categories, by Peter Webb. math.umn.edu/~webb/Publications/… G. Hoff, Cohomologies et extensions de categories, Math. Scand. 74 (1994), 191--207. H.-J. Baues and G. Wirsching, Cohomology of small categories, JPAA 38

Comment by SixWingedSeraph

2010-07-31T16:28:41Z

If you can give a precise statement of "not too many small periodic blocks in a row" that would apparently be a counterexample to Borel's conjecture. But Borel's conjecture may have a more precise statement that rules this out.

Comment by SixWingedSeraph

2010-05-11T21:10:08Z

I wrote about this question here: sixwingedseraph.wordpress.com/2010/05/12/…

Comment by SixWingedSeraph

2010-02-08T03:02:28Z

This is the observation that should have occurred to everyone first! (It didn't to me either.) It is so familiar we forget how amazing it is.

Comment by SixWingedSeraph

2010-01-08T15:13:50Z

Read about sketches in Johnstone's book Sketches of an Elephant (second volume). They give a hierarchy going down from your number (1) -- geometric logic, essentially algebraic logic (finite-limit theories), algebraic logic. There are many variations. There is undoubtedly no end to such classifications, both up and down from first order logic.

Comment by SixWingedSeraph

2010-01-02T01:30:45Z

" I guess we want the category to be additive and the biproduct to be vector addition, but I have no idea whether this actually happens." Make it happen! Mod out the the messy part so you get what you want. This works remarkably often. Of course, sometimes it gives you the trivial object...

Comment by SixWingedSeraph

2009-12-16T20:24:30Z

This is a variant of "Craig's trick" that shows that a first order theory with a r.e. set of axioms has a recursive set of axioms.

Comment by SixWingedSeraph

2009-12-16T20:20:18Z

"You need at least up to first-order logic to define anything of value and interpret any axioms." This is not correct. First order logic is one way of formalizing math. Another way is to work in some topos directly with its objects and arrows. You can work inside an algebraic structure using only equational logic, which is much weaker than first order logic. Linear logic is not even weaker that f.o.l. -- it is simply different. In any case, as I said in my previous comment, any logical system involving types and terms can be turned into a category

Comment by SixWingedSeraph

2009-12-16T18:05:35Z

You can also take a logical theory and turn it into a category. The objects are the types and the arrows are the terms. This is done for toposes in Elementary Categories, Elementary Toposes, by Colin McLarty (and in many other places, but this one is especially clear). The technique has been known for something like thirty years and has been applied in many settings, for example linear logic.

Comment by SixWingedSeraph

2009-12-12T02:33:08Z

I am deleting this answer. I remember (I think) that some category involving unit balls of Banach spaces was proved monadic in the 1970's but what I said was wrong. I should read my own book.

Comment by SixWingedSeraph

2009-12-10T16:12:54Z

The reference to Grillet's and Novikov's papers is in the preamble to the Extension Theories paper on line as in the previous comment.

Comment by SixWingedSeraph

2009-12-10T16:06:21Z

The entire paper “Extension Theories for Monoids” is now available here: cwru.edu/artsci/math/wells/pub/pdf/ExtThMon.pdf Look at Grillet's paper, too. I have not read the Novikov paper.