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September 30, 2012

Where Do Ultraproducts Come From?

Posted by Tom Leinster

I move jobs tomorrow: from Glasgow to Edinburgh, city of James Clerk Maxwell, Arthur Conan Doyle, Robert Louis Stevenson and Dolly the Sheep. But before I go, I want to give you the fourth and final post on my new paper, Codensity and the ultrafilter monad.

It’s not like the first three. There, I explained some definite facts: what codensity monads are, and how they naturally throw up various interesting mathematical structures. But this post contains nothing new except speculation. Specifically, I’ll speculate that codensity monads naturally throw up the notion of ultraproduct — though at present, I can’t see how.

Posted at 5:11 AM UTC | Permalink | Followups (37)

September 25, 2012

Where Do Linearly Compact Vector Spaces Come From?

Posted by Tom Leinster

Where do what come from?

Linearly compact vector spaces are what fill in this blank:

                       sets   are to   vector spaces
                                as
   compact Hausdorff spaces   are to   ???         

I’ll explain the exact sense in which that’s true, as part of the continuing story of codensity monads. Of course, I’ll also give you the actual definition.

Here’s the idea. Compactness of a topological space is something like finiteness of a set; indeed, the archetypal example of a compact space is a finite set with the discrete topology. Can we find a decent notion of “compactness” for topological vector spaces, such that the archetypal example of a “compact” vector space is a finite-dimensional vector space with the discrete topology?

Compactness itself won’t do: for example, 2\mathbb{R}^2 is finite-dimensional but not compact (with either the euclidean or the discrete topology). But linear compactness will fulfil all our desires.

Posted at 11:17 PM UTC | Permalink | Followups (56)

September 23, 2012

Where Do Ultrafilters Come From?

Posted by Tom Leinster

Here’s the central point of my new paper:

There is a standard piece of categorical machinery which, when fed the notion of finiteness of a set, produces as output the notion of ultrafilter.

More snappily:

Ultrafilters are inevitable.

The categorical machine I’m referring to is the one that makes codensity monads (which I explained last time). The result isn’t mine: it seems to have first appeared in a paper published the year I was born. But it deserves to be better known.

I’ll tell you roughly how the theorem works — and, perhaps more importantly, I’ll tell you what it means to integrate against an ultrafilter.

Posted at 3:13 PM UTC | Permalink | Followups (29)

September 20, 2012

Where Do Monads Come From?

Posted by Tom Leinster

If you know some category theory, you probably know that every functor with a left adjoint induces a monad. But much less well known — and undeservedly so — is that you don’t need your functor to have an adjoint in order for it to induce a monad! Even a functor without a left adjoint induces a monad, just as long as certain limits exist.

This is called the codensity monad of the functor. I’ve just arXived a paper about them (and ultrafilters, which I won’t get to today). This is the first of a short series of posts explaining what’s in my paper.

Today I’ll explain what a codensity monad is.

Posted at 11:55 AM UTC | Permalink | Followups (36)

September 14, 2012

Gauge Spaces and the Stone-Cech Compactification

Posted by Mike Shulman

If you’d like a change of pace from category theory, physics, and philosophy, have a look at these notes that I just posted to the arXiv:

Probably the best way to introduce them is to quote from the second introduction (the one addressed to people who know what a topological space is):

The goal of these notes is to develop the basic theory of the Stone-Cech compactification without reference to open sets, closed sets, filters, or nets. In particular, this means we cannot use any of the usual definitions of topological space. This may seem like proposing to run a marathon while hopping on one foot, but I hope to convince you that it is easier than it may appear, and not devoid of interest.

Posted at 1:02 AM UTC | Permalink | Post a Comment

September 13, 2012

Introducing the GR & QFT seminar

Posted by John Huerta

Here at the Australian National University, I’ve started running a seminar with Mathew Langford. Mat is a PhD student working on geometric analysis, specifically something called extrinsic curvature flow, but we both love mathematical physics, so we decided to teach some to each other, and whomever else wanted to listen. As it turns out, that’s a lot of people! I guess I shouldn’t be surprised; there are a lot of mathematicians who want to learn some more physics, and quite a few physicists who want to learn more mathematics. Now, through the magic of the Internet, I’d like to invite you to join in our discussion. So, over the next few weeks, as Mat lectures about general relativity and I lecture about quantum field theory, I’ll blog about it here. I’ll also keep a website for the seminar, complete with exercises:

In Mat’s opening lecture, about GR, he discussed three different views of spacetime, corresponding to the physics of Aristotle, Galileo, and Einstein. Bizarrely, it’s Galilean spacetime that’s the hardest to define: this turns out to be a fiber bundle, the bundle of spaces over times, equipped with a little extra structure to tell us what it means to be “inertial”.

You can find the notes for Mat’s lecture here:

Below the fold, I’ll give you a summary of these three kinds of spacetime, and how these mathematical ideas relate to the treatment you would meet in a physics class.

Posted at 8:25 AM UTC | Permalink | Followups (20)

September 10, 2012

The Ax-Grothendieck Theorem According to Category Theory

Posted by David Corfield

The online world of mathematics has taken a considerable interest in Mochizuki’s proposed proof of the ABC conjecture. The best exposition, including some by Minhyong Kim, is in answer to this MO question. An occasional visitor here, Minhyong mentioned Mochizuki’s interest in anabelian geometry to us back here.

I’m bringing this up because at Quomodocumque Terry Tao comments in relation to the announcement of the proof:

I have always been fond of the idea that model-theoretic connections between objects (e.g. relating two objects by comparing the sentences that they satisfy) are at least as important in mathematics as the more traditional category-theoretic connections (where morphisms are the fundamental connective tissue between objects) or topological connections (where the objects are gathered into some common topological space or metric space in order to compare them)…

Posted at 10:24 AM UTC | Permalink | Followups (16)

Every Functor is a (Co)Reflection

Posted by Mike Shulman

Recall that a reflection is a left adjoint to the inclusion of a full subcategory, and a coreflection is a right adjoint to such an inclusion. Of course, it is not literally true that every functor is a (co)reflection! However, it is true that every functor can be made into a (co)relection by changing its domain. More precisely, we can prove:

Theorem: Let F:CDF:C\to D be any functor. Then there exists a category EE, which contains CC as a full subcategory and DD as a full reflective subcategory, such that when the reflector is restricted to CC it agrees with FF.

You might try thinking about how to prove this; I’ll give an answer below the fold.

Posted at 5:46 AM UTC | Permalink | Followups (1)

September 8, 2012

General Covariance in Homotopy Type Theory

Posted by Urs Schreiber

In physics, the term general covariance is meant to indicate the property of a physical system or model (in theoretical physics) whose configurations, action functional and equations of motion are all equivariant under the action of the diffeomorphism group on the smooth manifold underlying the spacetime or the worldvolume of the system. The archetypical example of a generally covariant system is of course Einstein-gravity / “general relativity”.

I indicate here how general covariance has a natural formalization in homotopy type theory, hence internal to any \infty-topos. For background and all details see at general covariance on the nnLab, and the links given there.

Posted at 11:12 PM UTC | Permalink | Followups (5)

September 5, 2012

The Spread of a Metric Space

Posted by Simon Willerton

Given a finite metric space XX we can define the spread E 0(X)E_0(X) by

E 0(X) x1 ye d(x,y).E_0(X)\coloneqq \sum_x \frac{1}{\sum_{y} e^{-d(x,y)}}.

This turns out to be a nice measure of the ‘size’ of the metric space. I’ve just finished a paper on this:

In this post I’ll give a quick overview of the paper, mentioning connections to biodiversity; magnitude; volume and total scalar curvature; and Hausdorff dimension.

A few of these ideas are looked at in a slightly more dynamic way in my recent talk at the CRM Exploratory programme on the mathematics of biodiversity: Magnitude and other measures of metric spaces.

I should say that Tom Leinster convinced me to switch to using the word ‘spread’ as prior to that I had been using the much more uncouth word ‘bigness’.

Posted at 5:09 PM UTC | Permalink | Followups (10)

September 4, 2012

From Poisson To String Geometry

Posted by Urs Schreiber

The next event organized by our Research Network String Geometry is next week the conference:

  • From Poisson to String Geometry

    Erlangen, September 11 - 14 2012

    (webpage)

First I didn’t plan to go myself, because I am teaching an intensive course and have some other things to look after. But after being pressed now I agreed to come just on Friday, and then talk about this:

This is joint work with Chris Rogers which we will have written up by end of the year.

Roughly, I’ll be presenting the content of sections 2.6.1 and 4.4.17 of differential cohomology in a cohesive topos.

Posted at 11:39 PM UTC | Permalink | Followups (8)