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December 21, 2009

Cobordism and Topological Field Theories Weeks 6 & 7

Posted by Alexander Hoffnung

Welcome back to the cobordism and TFT seminar at UCR! We are posting weeks 6 and 7 of the seminar simultaneously because it flows nicely and the new quarter will start before we know it. So we want to be all caught up by the first week of the second quarter.

Week 6 starts back with the regular lecture series by Julie Bergner. These lectures are especially nice as they begin to introduce nn-dimensional topological field theories in Lecture 6 and the higher category theory needed to formulate these concepts in Lecture 7.

You can click here for the previous week’s lecture and keep reading below the fold for a link to the notes for Weeks 6 and 7…

In Lecture 6, Julie reminds us of some of facts about 22-dimensional topological field theories stated in Chris Carlson’s lecture in Week 4. She comes to a sketch of a 22-extended TFT by chopping up manifolds into pieces and considering the problems encountered when one considers TFT’s of dimension higher than 2.

Following the realization that we no longer have a symmetric monoidal category of cobordisms, in Lecture 7, Julie then introduces 22-categories of cobordisms. She then hints at the coming need for (,2)(\infty,2)-categories and gives a definition for extended 𝒞\mathcal{C}-valued TFT’s.

Here are the lecture notes for Weeks 6 and 7.

Next week, John Baez will fill in some details regarding symmetric monoidal weak 22-categories…

Posted at December 21, 2009 4:15 PM UTC

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8 Comments & 0 Trackbacks

Re: Cobordism and Topological Field Theories Weeks 6 & 7

It is a very minor point, but I suggest that the arrows labeled “left” be put into mboxes.

Posted by: Scott Carter on December 22, 2009 9:22 PM | Permalink | Reply to this

Re: Cobordism and Topological Field Theories Weeks 6 & 7

Hi Scott,

There is certainly no point too minor for me, except that I do not see the arrows you are referring to. I am pretty sure I will regret this comment later when I am being less dense, but can you point the way for me?

Posted by: Alex Hoffnung on December 22, 2009 10:14 PM | Permalink | Reply to this

Re: Cobordism and Topological Field Theories Weeks 6 & 7

Alex, Yes I see why. I was speaking about lecture 4 about a 3rd of the way down. Beta has a subscript that is not boxed. The subscript is “left.” In definition 4, it looks to me as if “Map” is dollar M a p dollar, rather than “dollar left bracket backslash mbox left bracket backslash it Map right bracket right bracket […] dollar.” At the bottom of that page, there is a subscript on hom that looks kind of weird.

Right before lecture 7, morphisms has an “n” at the end.

This is not to say I am reading carefully — quite the contrary. Just these things stick out to me.

Posted by: Scott Carter on December 23, 2009 12:17 AM | Permalink | Reply to this

Re: Cobordism and Topological Field Theories Weeks 6 & 7

Thanks Scott.

I will go in and clean up shortly. I also noticed that extra `n’ while looking for the arrows.

Posted by: Alex Hoffnung on December 23, 2009 12:27 AM | Permalink | Reply to this

Re: Cobordism and Topological Field Theories Weeks 6 & 7

Actually, I have never used \mbox. I gave it a quick try and it seemed to make matters worse. I will experiment with it a bit more later.

Posted by: Alex Hoffnung on December 23, 2009 2:38 AM | Permalink | Reply to this

Re: Cobordism and Topological Field Theories Weeks 6 & 7

Try

\beta_{\mathit{left}}

to get

β left\beta_{\mathit{left}}

instead.

Posted by: Toby Bartels on December 23, 2009 9:11 PM | Permalink | Reply to this

Re: Cobordism and Topological Field Theories Weeks 6 & 7

Ah, thanks! That seems to work well.

I also cleaned up Lecture 7 as Scott suggested. Instances of “Map” and “Cob” should look much better now. Now that I have seen the nicer version the old one sticks out like a sore thumb.

Posted by: Alex Hoffnung on December 23, 2009 9:48 PM | Permalink | Reply to this

Re: Cobordism and Topological Field Theories Weeks 6 & 7

I’m not sure whether this is still active but were there ever any lecture notes for the seminars after this one? I’m trying to rap my head around the cobordism hypothesis and so they would be very useful to me.
Regards

Posted by: Tom on February 23, 2012 6:02 PM | Permalink | Reply to this

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