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Fraisse Amalgams as Limits November 5, 2009

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Our very first post here spoke about the Fraïssé amalgam, a way of constructing a universal object out of a countable set of similar objects satisfying certain conditions. For example the amalgam of the set of finite graphs is the infinite random graph.

Alexandre was asking there why the results of such amalgamation should be the kinds of entity we encounter through different routes. I should imagine that the answer to this has much in common with answers to the questions Michiel Hazewinkel is posing in Niceness Theorems:

Many things in mathematics seem almost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the rings of representations of the symmetric groups, the free lambda ring on one generator, the homology and cohomology of the classifying space BU, … . In addition attention is paid to the phenomenon that solutions to universal problems (adjoint functors) tend to pick up extra structure.

Evidently Hazewinkel sees category theory as the right tool for the problem. So might Fraïssé amalgamation be given a category theoretic gloss? Here are a few attempts.

Trevor Irwin, Fraisse limits and colimits with applications to continua:

The classical Fraïssé construction is a method of taking a direct limit of a family of finite models of a language provided the family fulfills certain amalgamation conditions. The limit is a countable model of the same language which can be characterized by its (injective) homogeneity and universality with respect to the initial family of models. A standard example is the family of finite linear orders for which the Fraïssé limit is the rational numbers with the usual ordering.

We present this classical construction via category theory, and within this context we introduce the dual construction. This respectively constitutes the Fraïssé colimits and limits indicated in the title. We provide several examples.

We then present the projective Fraïssé limit as a special case of the dual construction, and as such it is the categorical dual to the classical (injective) Fraïssé limit. In this dualization we use a notion of model theoretic structure which has a topological ingredient. This results in the countable limit structures being replaced by structures which are zero-dimensional, compact, second countable spaces with the property that the relations are closed and the functions are continuous.

We apply the theory of projective Fraïssé limits to the pseudo-arc by first representing the pseudo-arc as a natural quotient of a projective Fraïssé limit. Using this representation we derive topological properties of the pseudo-arc as consequences of the properties of projective Fraïssé limits. We thereby obtain a new proof of Mioduszewski’s result that the pseudo-arc is surjectively universal among chainable continua, and also a homogeneity theorem for the pseudo-arc which is a strengthening of a result due to Lewis and Smith. We also find a new characterization of the pseudo-arc via the homogeneity property.

We continue with further applications of these methods to a class of continua known as pseudo-solenoids, and achieve analogous results for the universal pseudo-solenoid.

Wieslaw Kubiś, Fraisse sequences – a category-theoretic approach to universal homogeneous structures:

We present a category-theoretic approach to universal homogeneous objects, with applications in the theory of Banach spaces and in set-theoretic topology.

Olivia Caramello, Fraïssé’s construction from a topos-theoretic perspective:

We present a topos-theoretic interpretation of (a categorical generalization of) Fraisse’s construction in model theory, with applications to countably categorical theories.

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Two Streams in Hatfield July 6, 2009

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Brendan Larvor and I ran a conference – Two Streams in the Philosophy of Mathematics from 1-3 July. I thought I’d put up a post here for post-conference discussion.

In accordance with one stream’s policy of encouraging dialogue with real mathematicians, we invited Yehuda Rav (Paris-Sud), Michael Harris (Jussieu) and this blog’s very own Alexandre Borovik. For me two of the most interesting issues to emerge during the conference was Borovik’s ‘phantoms’ and Harris’s ‘avatars’. The first of these may occur when there is a question as to whether a certain entity exists. Even if it does not, it may transpire that some counterpart of this nonexistent entity exists elsewhere. The setting of finite simple groups is a rich environment for this phenomenon.

In the case of avatars, on the other hand, they all exist, but they indicate the existence of a not yet expressible universal object. Grothendieck’s theory of motives is the classic example, and indeed it was here that he coined the term ‘avatar’ to describe an instantiation of a motive in a particular cohomological setting.

What I’d like to know is what can be said about these phenomena. What is the right language to formulate them? Do we have earlier cases of avatars or phantoms which we now know how to express? Might it be possible to understand both phenomena in the same framwork? I.e., perhaps there may be avatars which happen not to exist, but for which existing fellow avatars act as phantoms.

So that’s a small taste of two of the talks. There were fourteen others. Personally, I was very pleased to hear Ivor Grattan-Guinness speak about ‘notions’, such as symmetry, convexity, and linearity, continually reappearing in mathematics. My own talk focused on duality, but I gave it a Cassirerian gloss as a ‘principle’.

If anyone would like to share their thoughts on the conference, please feel free.

The Invisible Dialog Between Mathematics and Theology April 29, 2009

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An interesting paper by Ladislav Kvasz — The Invisible Dialog Between Mathematics and Theology, in Perspectives on Science and Christian Faith, Vol. 56, pp. 111-116.

The thesis of the paper is that monotheistic theology with its idea of the omniscient and omnipotent God, who created the world, influenced in an indirect way the process of this mathematicization. In separating ontology from epistemology, monotheistic theology opened the possibility to explain all the ambiguity connected to these phenomena as a result of human finitude and so to understand the phenomena themselves as unambiguous, and therefore accessible to mathematical description.

This thesis is explored through five notions: infinity, chance, the unknown, space and motion.

What we refer to today as infinite was in Antiquity subsumed under the notion of apeiron (\alpha \pi \epsilon i \rho o \nu).  Nevertheless, compared with our modern notion of infinity, the notion of apeiron had a much broader meaning. It applied not only to that which was infinite, but also to everything that had no boundary (i.e. no peras), that was indefinite, vague or blurred. According to ancient scholars apeiron was something lacking boundaries, lacking determination, and therefore uncertain. Mathematical study of apeiron was impossible, mathematics being the science of the determined, definite and certain knowledge. That which had no peras, could not be studied using the clear and sharp notions of mathematics.

Modern mathematics, in contrast to Antiquity, makes a distinction between infinite and indefinite. We consider the infinite, despite the fact that it has no end (finis), to be determined and unequivocal, and thus accessible to mathematical investigation. Be it an infinitely extended geometrical figure, an infinitely small quantity or an infinite set, we consider them as belonging to mathematics. The ancient notion of apeiron was thus divided into two notions: the notion of the infinite in a narrow sense, which became a part of mathematics, and the notion of the indefinite, which, as previously, has no place in mathematics.

So

While for the Ancients apeiron was a negative notion, associated with going astray and losing the way, for the medieval scholar the road to infinity became the road to God. God is an infinite being, but despite His infiniteness, He is absolutely perfect. As soon as the notion of infinity was applied to God, it lost its obscurity and ambiguity. Theology made the notion of infinity positive, luminous and unequivocal. All ambiguity and obscurity encountered in the notion of infinity was interpreted as the consequence of human finitude and imperfection alone. Infinity itself was interpreted as an absolutely clear and sharp notion, and therefore an ideal subject of mathematical investigation.

Evidence for the change from the Ancients is provided by Kvasz in his book Patterns of Change where he quotes Nicholas of Cusa on page 77:

It is already evident that there can be only one maximum and infinite thing. Moreover, since any two sides of any triangle cannot, if conjoined, be shorter than the third: it is evident that in the case of a triangle whose one side is infinite, the other two sides are not shorter. And because each part of what is infinite is infinite: for any triangle whose one side is infinite, the other sides must also be infinite. And since there cannot be more than one infinite thing, you understand transcendently that an infinite triangle cannot be composed of a plurality of lines, even though it is the greatest and truest triangle, incomposite and most simple… (Nicholas of Cusa 1440, p. 22) De Docta Ignorantia, trans. J. Hopkins.

It may seem odd to us that Nicholas could not imagine the limit as an isosceles triangle of fixed base is extended, but the point is that such a discussion of an infinitely large object would have been unthinkable for the Greeks.

Ultraproducts the Category Theoretic Way January 21, 2009

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Following Alexandre’s two posts on ultraproducts of fields (here and here), I was wondering about the category theoretic view on ultraproducts. From Michael Barr’s Models of Sketches we read

Unlike limits and colimits, an ultraproduct is not defined by any universal mapping property. Of course, if the category has limits and (filtered) colimits, then it has ultraproducts constructed as colimits of products…But usually the category of models of a coherent theory (such as the theory of fields) lacks products and hence does not have categorical ultraproducts.

In fact the category theoretic definition is slightly different from the usual one, and comes with an advantage, according to Barr:

The only argument for banning the empty model that has any force comes from the observation that if (M_i) is a collection of models and M is a nonprincipal ultraproduct of the M_i, then one wants and expects that M(s) will be empty if and only if the set of  i for which M_i(s) is null belongs to the ultrafilter. If one takes the traditional definition of an ultraproduct as a quotient of the product, the ultraproduct will be empty as soon as one factor is.

Defined in terms of a colimit of products this problem goes away.

But how to cope with the problem Barr mentions with fields?

Born Free January 12, 2009

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Continuing our discussion of free entities, from Fiore and Leinster’s A simple description of Thompson’s group F we read

…many entities of interest can be described as free categories with structure. For instance, the braided monoidal category freely generated by one object is the sequence (B_n)_{n \geq 0} of Artin braid groups; the monoidal category freely generated by a monoid consists of the finite ordinals; the symmetric monoidal category freely generated by a commutative monoid consists of the finite cardinals; the symmetric monoidal category freely generated by a commutative Frobenius algebra consists of 1-dimensional smooth oriented manifolds and diffeomorphism classes of 2-dimensional cobordisms.

You can find definitions of many terms used in higher category theory at the exciting new wiki nLab. For example, see braided monoidal category.

Fiore and Leinster continue,

In this vein, our result is that the monoidal category freely generated by an object A and an isomorphism A \otimes A \longrightarrow A is equivalent to the groupoid 1 + F, where 1 is the trivial group and + is coproduct of groupoids.

and for the related Thompson’s group V,

just replace ‘monoidal category’ by ‘symmetric monoidal category’, or equally ‘finite-product category’.

Coalgebra December 8, 2008

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I posted on coalgebra over at my other blog. I won’t rehash the material of the discussion over here. But I’m still searching for an answer to the question I had of whether coalgebra has been formalised slowly due to algebraic blinkers.

A highlight for me was this comment from Dan Piponi, where he explains about how one needs to employ guarded recursion to work with ‘codata’. He says

In the real world of open ended loops like OSes and word processors, it’s often not computability we need, but productivity (absolutely no pun intended). And that makes coalgebraic reasoning an important topic.

Operating systems don’t require the totality of input in order to calculate a response by recursively breaking it into its atomic parts and deriving a function’s value by recomposition. They respond as each piece of new input arrives.

Coalgebra throws up a whole load of infinitely large entities.

New Directions in the Philosophy of Mathematics October 8, 2008

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To celebrate the founding of MIMS, the mathematics department of the recently unified Manchester University, it was proposed that various workshops named ‘New Directions in…’ be run. They kindly agreed to allow Alexandre Borovik and me to organise one of these workshops on the Philosophy of Mathematics.

So, on Saturday 4 October, we began with Mary Leng, a philosopher at Liverpool, talking about whether the creation of mathematical theories, e.g., Hamilton’s quaternions, gives us any more reason to think mathematical entities exist than does the discovery of new consequences within existing theories. She concluded that it does not — both concern the drawing of consequences from suppositions, e.g., “Were there to be a 3 or 4-dimensional number system sharing specified properties with the complex numbers, then…”.

(more…)

Trimble on Category Theoretic Foundations September 1, 2008

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Todd Trimble has an excellent post on the Elementary Theory of the Category of Sets. He promises us more and asks for feedback.

Model Theory and Category Theory July 29, 2008

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David Kazhdan has some interesting things to say about model theory, and in particular its relationship to category theory, in his Lecture notes in Motivic Integration.

In spite of it successes, the Model theory did not enter into a “tool box” of mathematicians and even many of mathematicians working on “Motivic integrations” are content to use the results of logicians without understanding the details of the proofs.

I don’t know any mathematician who did not start as a logician and for whom it was “easy and natural” to learn the Model theory. Often the experience of learning of the Model theory is similar to the one of learning of Physics: for a [short] while everything is so simple and so easily reformulated in familiar terms that “there is nothing to learn” but suddenly one find himself in a place when Model theoreticians “jump from a tussock to a hummock” while we mathematicians don’t see where to “put a foot” and are at a complete loss.

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The Sublime June 11, 2008

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There’s an interesting post – Whatever Happened to Sublimity? – at the blog Siris. It includes a quotation from Edmund Burke

But let it be considered that hardly any thing can strike the mind with its greatness which does not make some sort of approach toward infinity; which nothing can do while we are able to perceive its bounds; but to see an object distinctly, and to perceive its bounds, are one and the same thing. A clear idea is, therefore, another name for a little idea. (A Philosophical Inquiry into the Origin of Our Ideas of the Sublime and the Beautiful, Part II, Section V.)

A natural question to ask, then, is where do we encounter the sublime in mathematics? And an obvious answer, you might think, would be the mathematical infinite.

Joseph Dauben has an interesting section in his book – Georg Cantor: his mathematics and philosophy of the infinite, Harvard University Press, 1979 – on how Cantor, receiving such discouragement from his mathematical colleagues, found an audience in certain thinkers within the Catholic church. Where earlier in the nineteenth century any attempt to describe a completed infinity was viewed as a sacrilegious attempt to circumscribe God, some theologians were open to Cantor’s new hierarchy of infinities, with its unreachable Absolute Infinite leaving room for the divine.

Personally, set theory has rarely invoked in me a sense of the sublime. On the other hand, the following comment by Daniel Davis does:

Behrens and Lawson use stacks, the theory of buildings, homotopy fixed points, the above model category, and other tools to make it possible to use the arithmetic of Shimura varieties to help with understanding the stable homotopy groups of spheres.

There’s plenty about the infinite in that statement, it’s true, but there’s so much more to it than that.

Brandon Watson, the blogger at Siris, writes

…what often strikes me when I look around at the philosophical scene today is how foreign this has all become. There are a few exceptions, but sublimity has vanished as a serious concern.

With regard to the philosophy of mathematics in particular I couldn’t agree more.