Re: Further to my question on adjoints

["George Janelidze" <janelg@telkomsa.net>]

Dear Michael,

Let C be the category of commutative rings (with 1), let t be the unique
positive real number with tttt = 2 (if I knew TeX better, I would probably
write t^4 = 2), and E be the smallest subfield in the field of real numbers
that contains t. Then:

(a) Every power of E has exactly one element x such that xx = 2 and there
exists y with x = yy. Let us call this x the positive square root of 2.

(b) Every morphism between powers of E preserves the positive square root of
2.

(c) Therefore every equalizer of two arrows between powers of E has an
element x with xx = 2 (note that I am not saying anything about the
existence of y, since y above is not determined uniquely!).

(d) Therefore the field Q of rational numbers cannot be presented as an
equalizer of two arrows between powers of E.

(e) On the other hand Q can be presented as an equalizer of two arrows
between two objects in C that are equalizers of two arrows between powers of
E. Indeed: the equalizer of the identity morphism of E and the unique
non-identity morphism of E is the subfield D in E generated by tt (which is
just the square root of 2); and the equalizer of the identity morphism of D
and the unique non-identity morphism of D is Q.

(f) This also gives negative answer to the question about "internally
complete", since no arrow of our subcategory composed with the two morphisms
D ---> D above will give the same result.

This story is of course based on the fact that there are Galois field
extensions L/K and M/L, for which M/K is not a Galois extension.

Best regards, George

----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "Categories list" <categories@mta.ca>
Sent: Monday, May 12, 2008 2:34 PM
Subject: categories: Further to my question on adjoints


> In March I asked a question on adjoints, to which I have received no
> correct response.  Rather than ask it again, I will pose what seems to be
> a simpler and maybe more manageable question.  Suppose C is a complete
> category and E is an object.  Form the full subcategory of C whose objects
> are equalizers of two arrows between powers of E.  Is that category closed
> in C under equalizers?  (Not, to be clear, the somewhat different question
> whether it is internally complete.)
>
> In that form, it seems almost impossible to believe that it is, but it is
> surprisingly hard to find an example.  When E is injective, the result is
> relatively easy, but when I look at examples, it has turned out to be true
> for other reasons.  Probably there is someone out there who already knows
> an example.
>
> Michael
>
>
>