Re: A question on adjoints

[Vaughan Pratt <pratt@cs.stanford.edu>]

Isn't the following a counterexample?

Let A = Set and let B = A\{0} (the category of nonempty sets).  Let F
send the empty set in A to the singleton set in B, and otherwise let F
and U be the evident identity functors between A and B.  Similarly let
\eta and \epsilon be the identity natural transformations, except for
\eta_0 which can only be the unique function from 0 to 1.   Naturality
of \eta and \epsilon depends on both being the identity, except for
\eta_0 but that's from the initial object so all its diagrams commute.

Then 0 equalizes the two arrows from U1 to U2 but \eta_0 does not
equalize UF\eta a and \eta UFa since the latter two are both 1_1 in A
whence they are equalized by 1.

Vaughan

Michael Barr wrote:
> I guess I am getting old and dumb.  This question should have been a snap
> for me years ago.  It is old fashioned, only a 1-categorical question and
> not about internal vs. external.
>
> Suppose F: A --> B is left adjoint to U: B --> A.  Suppose a is an object
> of A and b, b' objects of B such that there is an equalizer
>   a ---> Ub ===> Ub'.  (The two arrows Ub to UB' are not assumed to be U
> of arrows from B.)  Does it follow that a ---> UFa ===> UFUFa is an
> equalizer?  The arrows are \eta a, UF\eta a and \eta UFa of course.
>
> Michael
>
>