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John Power made a second visit to AIST, leaving Bath on 28 November 2009 and working at AIST until 28 December 2009, when AIST ended work for the New Year holiday. While in AIST, he extended his discussions with various researchers from a year ago, also exending those of previous meetings in Bath.

Specifically, that included discussion with Kitamura-san, whose research has developed over the past year and who hopes to visit Bath around March 2010 as suggested in the report on John's first visit. He also made plans with Takeyama-san about a proposed task for Martin Churchill on the latter's second visit to AIST scheduled for January 2010. And he spoke with a few other researchers, notably Hitoshi Ohsaki, about the possibility of final visits by AIST personnel to Bath to round off the research before the end of the grant.

That said, a focus for John's research interaction on this trip has been a series of talks on locally finitely presentable categories. The relevant work is fundamental to the ideas in the two papers he has written in connection with the grant, but it is subtle, and the AIST-based researchers sought a deeper understanding of the ideas than written in those papers, which simply assumed knowledge of fundamental facts about them. The effort involved with preparing and presenting those talks occupied the bulk of his time in Japan.

He also put effort into rewriting the second of the two papers he has written in connection with the grant, as well as doing work on category theoretic semantics of logic programming, which extends earlier work he did in Japan and which relates to the two papers produced here.

Locally Finitely Presentable Categories seminar

Topic 1: The definition of locally finitely presentable category
Topic 2: The definition of weighted colimit and its representations in terms of conical colimits and coends
Topic 3: The definition and calculation of Kan extensions, and the implications of fully faithfulness
Topic 4: The Yoneda lemma as the free cocompletion
Topic 5: Finite limits commute with filtered colimits in Set
Topic 6: The definition of flatness and proof that a functor is flat if and only if it preserves finite limits
Topic 7: Gabriel-Ulmer duality 1: all locally finitely presentable categories are of the form FL(T,Set)
Topic 8: Gabriel-Ulmer duality 2: FL(T,Set) is locally finitely presentable, in particular cocomplete

Future plans

Plans for the development of the relationship between Bath and AIST have been disrupted by the imminent restructuring within AIST. The situation will not become clear for some time yet, but it would seem sensible to further the interactions during the remaining period of this grant as follows:

proceed with Martin Churchill's proposed visit to AIST in January to continue his work relating game semantics (a Bath specialty) with Agda (an AIST specialty) in collaboration with AIST's Makoto Takeyama

proceed with Yoshiki Kinoshita'a proposed second visit to Bath in April: with his directorship responsibilities, he could not visit for as long as planned in October

have Makoto Takeyama visit Bath around March or April to continue his joint work with Martin Churchill

have some combination of Hitoshi Ohsaki, Tang Nguyen and Takashi Kitamura (as planned last year) visit Bath around March or April to try to broaden the relationship between Bath and AIST in preparation for the restructuring within AIST.

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