From: Michael Barr <barr@math.mcgill.ca>
To: Categories list <categories@mta.ca>
Subject: Re: A question on adjoints
Date: Wed, 19 Mar 2008 13:43:23 -0500 (EST) [thread overview]
Message-ID: <E1Jc6pa-0002ac-9e@mailserv.mta.ca> (raw)
Actually, F isn't even a functor. The unique arrow 0 --> Ub has to give a
canonical arrow F0 = 1 --> b, which there isn't. You could choose one, of
course, but it could not be functorial.
Actually, I realized the answer to my question cannot be yes. Here's why.
Let A be some complete category to be specified later. Let d be a fixed
object of A. Let B be set\op and Fa = Hom(a,d). The right adjoint is
given by b |---> d^b. It is not entirely trivial to show this, but if my
answer were "yes", then you could show that the class of objects that were
equalizers of powers of d would be complete. It is obviously closed under
products but, over 40 years ago, Isbell gave an example in which it was
not closed under equalizers.
This much is true: if there is an equalizer of the form a --> UFa ===> Ub,
then a ---> UFa ===> UFUFa is an equalizer.
Michael
On Tue, 18 Mar 2008, Vaughan Pratt wrote:
> 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
>>
>>
>
>
next reply other threads:[~2008-03-19 18:43 UTC|newest]
Thread overview: 4+ messages / expand[flat|nested] mbox.gz Atom feed top
2008-03-19 18:43 Michael Barr [this message]
-- strict thread matches above, loose matches on Subject: below --
2008-03-19 23:41 Vaughan Pratt
2008-03-19 5:43 Vaughan Pratt
2008-03-18 18:11 Michael Barr
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