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October 13, 2010

The Scottish Category Theory Seminar

Posted by Tom Leinster

We’re pleased to announce the third meeting of the Scottish Category Theory Seminar, from 2.00 to 5.30 on Thursday 2 December, at the University of Strathclyde (in Glasgow city centre). All are welcome to attend. We have three invited speakers:

Abstracts are below. The speakers are chosen to represent category theory in, respectively, computer science, pure mathematics, and physics.

The meeting is generously supported by the Edinburgh Mathematical Society.

Information is on the Scottish Category Theory Seminar home page. Abstracts are there and, for your convenience, below.

If you’re planning to come, it would help us if you sent a quick email saying so. Our email address is scotcats#cis,strath,ac,uk (making the obvious substitutions). This reaches the organizers: Neil Ghani, Alex Simpson, and me.

Here are the abstracts.

Marcelo Fiore: On higher order algebra

The purpose of this talk is to give an overview of recent work on a mathematical theory of higher-order algebraic structure.

Specifically, I will introduce a conservative extension of universal algebra and equational logic from first to second order that provides a model theory and formal deductive system for languages with variable binding and parameterised metavariables. Mathematical theories encompassed by the framework include the (untyped and typed) lambda calculus, predicate logic, integration, etc.

Subsequently, I will consider the subject from the viewpoint of categorical algebra, introducing second-order algebraic theories and functorial models, and establishing correspondences between theories and presentations and between models and algebras. The concept of theory morphism leads to a mathematical definition of syntactic translation that formalises notions such as encodings and transforms in the context of languages with variable binding.

References

M.Fiore, Second-order and dependently-sorted abstract syntax. In Proceedings of the 23rd Logic in Computer Science Conference (LICS’08), pages 57-68, 2008.

M.Fiore and C.-K.Hur. Second-order equational logic. In Proceedings of the 19th EACSL Annual Conference on Computer Science Logic (CSL 2010), LNCS 6247, pp. 320-335, 2010.

M.Fiore and O.Mahmoud. Second-order algebraic theories. In Proceedings of the 35th International Symposium on Mathematical Foundations of Computer Science (MFCS 2010), LNCS 6281, pp. 368-380, 2010.

Peter Johnstone: Hochas and minimal toposes

Recently D. Pataraia introduced the notion of hocha (higher-order cylindric Heyting algebra) in connection with his solution of the problem “does every Heyting algebra occur as Sub(1) in a topos?”. In this talk we focus on the relationship between hochas and toposes: we show that there is an equivalence between (finitary) hochas and toposes satisfying a natural “minimality” condition, and in particular that the category of finitary hochas is nothing other than the ind-completion of the dual of the free topos.

Bas Spitters: Bohrification: topos theory and quantum theory

The recently developed technique of Bohrification associates to a (unital) C *C^*-algebra

  • the Kripke model, a presheaf topos, of its classical contexts
  • in this Kripke model a commutative C *C^*-algebra, called the Bohrification
  • the spectrum of the Bohrification as a locale internal in Kripke model.

We will survey this technique, provide a short comparison with the related work by Isham and co-workers, which motivated Bohrification, and use sites and geometric logic to give a concrete external presentation of the internal locale. The points of this locale may be physically interpreted as (partial) measurement outcomes.

Posted at October 13, 2010 8:32 PM UTC

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Re: The Scottish Category Theory Seminar

New news: Benno van den Berg will give a series of three lectures on algebraic set theory at Strathclyde the day after our meeting, Friday 3 December.

This isn’t part of the Scottish Category Theory Seminar itself, but it’s an extra reason to make the trip. Details will appear, I believe, on the Seminar web page.

Posted by: Tom Leinster on October 20, 2010 9:34 AM | Permalink | Reply to this

Re: The Scottish Category Theory Seminar

Titles and abstracts are now available. I’ve updated this post to include them. Further information is here.

Posted by: Tom Leinster on November 4, 2010 10:10 PM | Permalink | Reply to this

Re: The Scottish Category Theory Seminar

Programme change!

Unfortunately, due to circumstances beyond anyone’s control, it won’t be possible after all for Ieke Moerdijk to come and speak to us.

Fortunately, Peter Johnstone has been kind enough to step in at the last moment. Come and learn from him about hochas and minimal toposes: abstract above.

Posted by: Tom Leinster on November 25, 2010 2:51 PM | Permalink | Reply to this

Re: The Scottish Category Theory Seminar

hochas? what be they? form ochas and cochas I know
but hochas?

Posted by: jim stasheff on November 27, 2010 12:51 PM | Permalink | Reply to this

Re: The Scottish Category Theory Seminar

The abstract reveals all… In a topos, the subobjects of the terminal object 11 form a poset Sub(1)Sub(1), where the order relation is inclusion. For example, in the topos SetSet, the terminal object is the one-point set 11, and its poset of subsets has just two elements: \emptyset and 11 itself. For another exampe, fix a set II and take the topos Set ISet^I of II-indexed families of sets. There the terminal object is the constant family (1) iI(1)_{i \in I}, and in fact Sub(1)Sub(1) is the power set of II.

The poset Sub(1)Sub(1) is always a Heyting algebra, which (as you may know) is a poset with certain further properties. A snazzy way to say it is that a Heyting algebra is a poset that, when regarded as a category, has finite coproducts and is cartesian closed.

So there’s a natural question: which Heyting algebras arise from a topos in this way? (Well, I see now that it’s natural, though it hadn’t occurred to me before I read Peter’s abstract.) Apparently this question was asked and answered by D. Pataraia, and the answer has something to do with “higher order cylindric Heyting algebras”—hochas.

(Certainly any complete Heyting algebra arises in this way, since a complete Heyting algebra is the same thing as a frame or locale, and in the topos of sheaves on a locale XX, the poset Sub(1)Sub(1) is XX itself. But there must be non-complete Heyting algebras of the form Sub(1)Sub(1) too.)

On the other hand, I don’t know about ochas and cochas.

Posted by: Tom Leinster on November 27, 2010 5:45 PM | Permalink | Reply to this

Re: The Scottish Category Theory Seminar

If I go to the talk on Bohrification, will I get Bohred?

Posted by: John Baez on November 26, 2010 11:07 AM | Permalink | Reply to this

Re: The Scottish Category Theory Seminar

The only possible response to that is an even worse pun—and trust me, I can think of one—but I’ll bite my tongue.

Posted by: Tom Leinster on November 26, 2010 3:08 PM | Permalink | Reply to this

Re: The Scottish Category Theory Seminar

Thanks, Tom, for the restraint as I can probably guess what it might be!

Posted by: Tim Porter on November 26, 2010 4:06 PM | Permalink | Reply to this

Re: The Scottish Category Theory Seminar

I should have bit mine too.

In penance for my sin, I should probably say a word about ‘Bohrification’. As far as I understand, it’s a way of making precise Bohr’s idea that while the world is quantum-mechanical, people living in the world can only see it through classical ‘views’: each view allows us to see some things but not others. So, mathematically, you start with noncommutative C*-algebra AA and construct a topos with a commutative C*-algebra A̲\underline{A} internal to it, as shown in this charming picture:

in a paper by Heunen, Landsman and Spitters.

Posted by: John Baez on November 26, 2010 9:03 PM | Permalink | Reply to this

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