Casual Category Theory - Spring 2000
Events in time-reversed order
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Tuesday 14:15 04/07/2000 (Daniele Varacca) |
Second Continuaton: On enriched Categories (again see Kelly's book [GMK82], sections 1.4-1.9. To celebrate the first six months of CCT, Oliver and I will offer you bier/beer and cookies. (You are also welcome to come also AT THE END of the talk!) Otherwise see you after the holidays for a new exciting semester into the magical world of categories. |
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Friday 11:00 (sharp) 30/06/2000 (Daniele Varacca) |
Continuaton: On enriched Categories Now we will see the definition of symmetric monoidal category and monoidal closed category V, and the properties of the correspondent V-categories. We will then introduce the notion of extraordinary V-naturality. If time permits we will state the weak version of Yoneda Lemma for V-categories. (again see Kelly's book [GMK82], sections 1.4-1.9. |
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Friday 11:15 23/06/2000 (Daniele Varacca) |
On enriched Categories A V-enriched category, in few words, is a category having hom-objects in V, where V is a monoidal category. If V is Set, we have a locally small category. If V is Cat we have a 2-category. See Kelly's book [GMK82], first chapter. |
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Friday 14:15 9/06/2000 (Mario Jose Caccamo) | Proofs of some well-known properties of presheave categories, namely: presheave categories are free colimit completions or any presheave functor is isomorphic to a colimit of representables. |
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23/05/2000 (Mario Jose Caccamo) |
Continuation of last week's talk This is supposed to cover Yoneda, limits (ends) and adjunctions. |
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16/05/2000 (Mario Jose Caccamo) |
The talk will consist in a condensed presentation of the material
taught by Glynn last semester. I will concentrate on the theory of representable functors and its application to the presentation of limits (ends) and adjunctions. I will approach these concepts in the light of an adapted (and more legible) version of the Proposition II.1 (pg. 59) in [SML] which supplies a nice tool to reason about representables. Part of the material can be found in http://www.brics.dk/~mcaccamo/cat/yoneda.ps. The session is planned to last 45 minutes -- on demand to be continued in the subsequent week. |
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18/04/2000 and 02/05/2000 (Zhe Yang) | The basic (mathematical) notion of monads and some application in
programming languages References: Chapter VI in [SML] Eugenio Moggi: [EM89] and [EM91]. |
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11/04/2000 (Daniele Varacca) | Weak Bisimulation and Open Maps (continued) We will start with the saturation monad and the unfolding functor U. The examples will help to clear up the [slightly confused] notion. |
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04/04/2000 (Daniele & Mario) | Weak Bisimulation and Open Maps According to a LICS'99 paper of Fiore, Cattani, Winskel (the BRICS report version is [FCW99]) |
| 21/03/2000 (Mikkel) | Just Preasheaves |
| 14/03/2000 (Mikkel Nygaard) | The talk will be based on sections 2 (open maps, bisimulation) and 4
(presheaf models) of [JNW96]. I'll include detailed proofs on your demand, and depending on how many you wish to see, the session will last 1 or 2 hours. |
| 7/03/2000 (Mikkel Nygaard) | [WN94], sections 4, 5, and 6 (continued). They are all quite readable and the material serves only as background to open maps and presheave models. |
| 29/02/2000 (Mikkel Nygaard) | Section 2 of [WN94].
It introduces the category
of transition systems, T, and shows how CCS-like constructs such as
restriction, relabelling, parallel composition, and nondeterministic sum
arise as universal constructions. The idea is that as such, some of them will be preserved by adjoints relating T categories of other models (synchronisation trees, event structures, petri nets,...). The approach has 4 objectives: 1. to obtain a guide away from ad hoc definition of process language constructs (eg. use categorical product as a general form of parallel composition instead of a less canonical form such as the one found in CCS) 2. to uncover formal relationships between different models (eg. adjunction between T and the category of synchronisation trees) 3. to enable transfers of technology between models (by preservation properties of adjoints, eg. limits and so product and so generalised parallel composition is preserved by right adjoints) 4. (long term) to unify concurrency and traditional denotational semantics |
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Daniele Varacca - Last modified: Mon Jul 23, 2001