categories: re: Exact adjunctions

["Walter Tholen" <tholen@pascal.math.yorku.ca>]

Hi Marco -

I learned about the fact that you are alluding to when I took my first course
on category theory with Nico Pumpluen in 1969 at the University of Muenster,
and this is documented in his extended and mimeographed lecture notes (as "Satz
4.4") that I was in charge of producing when he gave the course again in 1972.
It may have been "folklore" knowledge already at that time. Nico called such
adjunctions Galois adjunctions, but I agree with Peter that idempotent
adjunction is the better name.
Well, all this may not really be an answer to your question, depending on what
you call "published", in which case you can take this as background
information.

Best regards,
Walter.


On Nov 21,  7:16pm, Marco Grandis wrote:
> Subject: categories: Exact adjunctions
> There is a nice lemma on adjoint functors, purely 2-categorical, and
> certainly known to various colleagues.
>
> As I need it in a paper on homotopy, I would like to know if it is
> *published with proof*, somewhere.
> I also like to "advertise" it here, because I think it deserves to be known
> more widely.
>
> LEMMA.
>
> If,in an adjunction, any one of the four natural transformations which
> appear in the triangle identities is invertible, so are the other three.
> [Proof below.]
>
> A few years ago I was considering such adjunctions, which I was calling
> "connections" because adjunctions between ordered sets ("covariant Galois
> connections") are always of this type.