There were six of us. We discussed the connection between monads and applicative functors, based on
the article mentioned before, http://www.soi.city.ac.uk/~ross/papers/Applicative.html
While it is not clear what would be the exact condition for a monad to be an applicative functor, there are some
hints and conditions.
First, for a monad to be an applicative functor
, it has to be strong; strength is discussed in the aforementioned paper and in Eugenio Moggi, "Computational lambda-calculus and monads".
The question is, on which occas ion any monad is strong? And which monads are not?
I had asked this question on mathoverflow
. Turned out there are very interesting examples, the source of which can be found in John Power, "Unicity of Enrichment over Cat or Gpd".
Moggi, by the way, shows that if the category has enough points, every monad is an applicative functor:
Eugenio Moggi, "Computational lambda-calculus and monads".
So, we could probably dig deeper next time.
And by the way, based on the information mentioned above, I believe I can bring an example of a topos and a monad over it that is not strong, and (I believe) not an applicative functor either.