October 26, 2012

The Zorn Identity

Posted by Tom Leinster

(Note: the Café went down for a few days in early November 2012, and when Jacques got it back up again, some of the comments had been lost. I’ve tried to recreate them manually from my records, but I might have got some of the threading wrong.)

My aim here is to make one simple point:

Zorn’s lemma has almost nothing to do with the axiom of choice.

It hasn’t got much to do with set theory, either.

Posted at 11:33 AM UTC | Permalink | Followups (35)

October 18, 2012

Object Classifiers and (∞,1)-Quasitoposes

Posted by Mike Shulman

I’m spending this year at the Institute for Advanced Study in Princeton with a bunch of homotopy type theorists and fans. So far things have been mostly ramping up slowly, but there’s significant progress already being made in a few areas (mainly technical ones), and indications of more exciting things down the line. I intend to have some posts soon about what we’ve been up to, but first I want to advertise this recent preprint:

Although the title refers to Voevodsky’s univalence axiom for homotopy type theory, the paper is primarily about the $(\infty,1)$-categorical structure that models it, namely object classifiers.

Posted at 1:35 AM UTC | Permalink | Followups (16)

October 13, 2012

The Curious Dependence of Set Theory on Order Theory

Posted by Tom Leinster

(Note: the Café went down for a few days in early November 2012, and when Jacques got it back up again, some of the comments had been lost. I’ve tried to recreate them manually from my records, but I might have got some of the threading wrong.)

This is the first of a series of posts on set theory, order theory, and how they are intertwined.

I won’t assume you know any more set theory than the average pure mathematician: in other words, no axiomatic set theory, just the ordinary “naive” principles that mathematicians use every day. (And I’ll steer clear of the debate about which system of axioms is “best”. Important as that is, it tends to suck the oxygen out of a room.) As far as order theory is concerned, I’ll assume just the very basic definitions, such as “partially ordered set”.

In this first installment, I’ll ask: why is it that some of the fundamental theorems of set theory do not mention order in their statements, but apparently need substantial order theory in their proofs?

Posted at 4:22 PM UTC | Permalink | Followups (71)

October 11, 2012

Analysis Fellowship in Edinburgh

Posted by Tom Leinster

Youngish analysts might be interested in this:

It’s a sweet deal: you get a reduced teaching load for several years, after which the position turns into an ordinary “permanent” job.

If you have questions, feel free to get in touch. I’ve only just arrived there myself, but I’ll tell you what I can. My address is Firstname.Lastname@ed.ac.uk.

Posted at 5:29 PM UTC | Permalink | Post a Comment

October 9, 2012

Tangency

Posted by Urs Schreiber

$\,$

This is a guest post by Andrew Stacey (NTNU, currently on sabbatical at Oxford .

$\,$

Quiz Time

Here’s a question for you all. What is the tangent space at $0$ of the unit interval, $T_{0} [0,1]$? To make it easier, I’ll make it multiple choice:

1. $\{ 0\}$,

2. $[0,\infty )$,

3. $\mathbb{R}$,

4. None of the above.

What about $\mathbb{R} \vee \mathbb{R}$, the space formed by gluing two copies of $\mathbb{R}$ together at their origins (you can think of this as the union of the $x$ and $y$ axes if it helps). Here are your options for this space.

1. $\{ 0\}$,

2. $\mathbb{R} \vee \mathbb{R}$,

3. $\mathbb{R} ^{2}$,

4. None of the above.

There’s no particular “right answer” to these (though your answers to the two questions should match up). I can justify all three of the concrete answers. I shan’t, yet, because I want to know what others think and why without tainting the survey.

Background

Those who know me will know that I like loop spaces. I’m pretty happy to meet them in any guise, but if I had to express a preference then it would be as a differential topologist (me, that is, not the loop spaces). This means that I’m considering them as some sort of infinite dimensional manifold.

It’s not a long, nor a difficult, path (ha ha) from loop spaces to more general smooth spaces. Spaces that are almost, but not completely, unlike manifolds. We’ve had many discussions here about what a generalised smooth space should look like. Right now, I don’t want you to think too deeply about that. I just want you to be aware of the fact that there are smooth spaces beyond manifolds. They don’t have charts, but they have a strong family resemblance to manifolds so a lot of intuition and ideas can be extended from manifolds to these more general spaces.

This is what I’m trying to do with tangent spaces: extend them from manifolds to generalised smooth spaces.

Tangent Spaces

The problem is that there is not a unique definition of “tangent space” in differential topology. There are several equivalent definitions, but they do not remain equivalent when one generalises them. That’s okay because actually I’m not after a unique definition. I’m after a characterisation. Thus the question I really want to ask is the following:

Suppose I gave you two smooth spaces, $X$ and $Y$, and told you that $Y$ was a tangent space for $X$ (I’d probably better give you the projection $\pi \colon Y \to X$ as well). What would you expect that to tell you about $Y$?

Note that I’m using “tangent space” here to mean all the pointwise tangent spaces put together into a new smooth space. I can’t say tangent bundle because they may not form a bundle. Note also that, following from what I said about the different definitions, I’m using the indefinite article: a tangent space.

Conclusion

My desired conclusion from this is to be able to give a characterisation of a tangent structure on a category of generalised smooth spaces. It will consist of an endofunctor, and one or two natural transformations, where the functor assigns to a smooth space a tangent space. But before I can characterise such functors, I need to know what characterises a tangent space. Hence the question.

The longer term goal is that I want to use tangent spaces as a tool to study smooth spaces. A finite dimensional manifold is actually modelled on its pointwise tangent spaces and this turns out to be a very important property in studying mapping spaces. More general smooth spaces will not have as close a relationship, but nonetheless there might still be enough of a relationship to be able to exploit it.

Posted at 5:59 PM UTC | Permalink | Followups (33)

October 4, 2012

Symmetric Monoidal Bicategories

Posted by Tom Leinster

Guest post by Nick Gurski

Angélica Osorno and I recently posted a preprint of our paper

on the arXiv. This is a project that she and I have been collaborating on for almost a year and a half, and I am very excited that we finally were able to prove the big coherence result:

Theorem: In a symmetric monoidal bicategory, every* diagram of constraint 2-cells commutes.

Below the fold, I will tell you a little bit about what a symmetric monoidal bicategory is, what this coherence theorem means if you are working with one, and why we were interested in proving this theorem in the first place.

Posted at 4:15 PM UTC | Permalink | Followups (8)