From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1159 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: an early exercise in Mac Lane Date: Sat, 10 Jul 1999 14:02:48 -0400 (EDT) Message-ID: <3.0.16.19990710140340.7a0f4d1c@wesleyan.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: ger.gmane.org 1241017606 29752 80.91.229.2 (29 Apr 2009 15:06:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:06:46 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Sat Jul 10 16:29:00 1999 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id PAA16888 for categories-list; Sat, 10 Jul 1999 15:32:21 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Sender: flinton@wesleyan.edu X-Mailer: Windows Eudora Pro Version 3.0 (16) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 44 Xref: news.gmane.org gmane.science.mathematics.categories:1159 Archived-At: 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