From: "Fred E.J. Linton" <FLinton@mail.wesleyan.edu>
To: categories@mta.ca
Subject: Re: an early exercise in Mac Lane
Date: Sat, 10 Jul 1999 14:02:48 -0400 (EDT) [thread overview]
Message-ID: <3.0.16.19990710140340.7a0f4d1c@wesleyan.edu> (raw)
At 01:50 PM 7/8/99 -0700, Lyle Ramshaw was
> ... puzzled by an exercise on page 15 of Mac Lane's classic text ...
I have three remarks to add to this discussion.
The first is more of a confession: for nigh onto thirty years I seem to have
misread this problem, thinking what it sought was a functor Grp ---> Grp
commuting with the underlying-set functors, like the passage from a group
to its opposite. I now stand, belatedly, corrected.
Second (and surely others will be chiming in on this point too),
Baez's suggestion to trivialize all homomorphisms can't work -- you can't be
a functor if you trivialize identity maps.
Finally, the sort of example Mac Lane probably had in mind for an exercise
on such an early page as p. 15: writing t for the only non-identity
automorphism (an central involution, actually) of, say, the chosen group G
= Z
(or of G = Z/3Z , if you prefer), define T: Grp ---> Grp as follows:
T(X) = X , whatever the group X , and, for f: X --> Y in Grp , set
+ -
| f , if either X = G = Y or neither X nor Y is G ;
T(f) = | ft , if only X = G ;
| tf , if only Y = G .
+ -
(This ASCII graphic works best if you display it in a fixed-width font.)
I'll spare CATEGORIES readers the straightforward details of the checking
that T really is a functor. But I'll add the aside that there's nothing
special about the choices of G and t above, beyond G being a group
and t being a central, involutive automorphism of G . Even "involutive"
isn't really needed, except for the typographical convenience, here in
ASCII-land, of not having to compose a " t-inverse " with f in *one*
(but, please, *not* in the other) of the last two lines of the definition
of T(f) :-) .
Cheers,
-- Fred [E.J. Linton, aka <FLinton@Wesleyan.edu>
next reply other threads:[~1999-07-10 18:02 UTC|newest]
Thread overview: 4+ messages / expand[flat|nested] mbox.gz Atom feed top
1999-07-10 18:02 Fred E.J. Linton [this message]
-- strict thread matches above, loose matches on Subject: below --
1999-07-09 20:05 Michael Barr
1999-07-09 13:39 john baez
1999-07-08 20:50 ramshaw
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