FP Lunch

Lots of people away. I used the occassion to give an educational introduction to my latest obsession: classifying functors by the limits they preserve. The list functor preserves pullbacks, and indeed I showed on the whiteboard that all containers preserve pullbacks. The question: do you know a functor that does not preserve pullbacks created some silence, which i used to finish my sandwhich. I then pulled out my two favorite examples: the continuation functor and the unordered pair functor where and and not wanting to bore the audience too much, I handwaved an explanation why unordered pairs do neither preserve pullbacks nor equalizers. The question remains what do the preserve and why are they more civilized than continuations…

After an exhausting Fun in the afternoon which turned into Fun with beer in the evening, we had Tarmo Uustalu around. I asked whether he had any idea how to say “generalized monad”, i.e. what we called Kleisli structure in that old paper with Bernie Reus in CatSpeak. Tarmo had a brilliant proposal, namely to view it as a monoid in the category of endofunctors (I am still working on the details of this proposal).

Later Conor talked about div (divergence) as a comonad and we entered the abyss of differential calculus of functors again, only to discover that we still don’t know how to derve Hank’s combinator from the defining adjunction. More explanation to be added at a later stage…

We had Bob Coecke from Oxford as a visitor. I used the lunch to introduce people to the concrete construction of a dagger compact closed category as a Kleisli category of a (generalized) monad, namely with and to ask Bob some questions about this construction. In particular, I was interested in the relation to the “official” category of vector spaces. Rather surprisingly, Bob seems to prefer to work in this Kleisli category anyway, which identifies a space and its dual.