Re: "injective" terminology

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

What makes it worse, there are also "product projections" and "coproduct
injections" that might be "non-surjective" and "non-injective"
respectively...

And, generally speaking, mathematics can contribute a lot to the discussion
in e.g.

http://home.alphalink.com.au/~umbidas/Homonyms_main.htm#cape

I would like, however, to make a comment about "free" and "fascist" used by
Saunders Mac Lane:

I don't think Saunders had ever defined "a free object in a category" to
mean "projective"; if I am wrong, please correct me. What he really did - in
[S. Mac Lane, Duality for groups, Bull. AMS 56, 1950, 485-516] - was:

(a) Theorem 1.1, which, expressed in the modern language, would say that an
abelian group is free if and only if it is a projective object in the
category of abelian groups.

(b) Remark that the same result holds for free (nonabelian) groups (in the
category of groups).

(c) Then he defines "infinitely divisible" abelian groups and proves Theorem
1.1', which, expressed in the modern language, would say that an abelian
group is ("infinitely") divisible if and only if it is an injective object
in the category of abelian groups.

(d) Then he discusses "duality" - very interesting, since it is one of the
first clear suggestions to consider dual properties (although there is
another paper he published in 1948). And, by the way, "onto" is also
mentioned - not "surjection", while later (page 497) there are "injections"
and "projections" with different meanings (reading there about what he calls
a "bicategory" one should essentially think of a factorization system...).

(e) Then in a footnote he says: "Call the dual (in this sense) of a free
(nonabelian) group a fascist group. R. Baer has shown to me a proof of the
elegant theorem: every fascist group consists only of the identity element."

Well, it is clear that "fascist" was ironic, but how seriously would
Saunders Mac Lane think of introducing "a free object in a category" 60
years ago, I don't know...

Finally - for those who had not seen "Duality for groups" - I must mention
that a lot more of categorical algebra was invented there...

George

----- Original Message -----
From: "Michael Barr" <barr@math.mcgill.ca>
To: "Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>
Sent: Wednesday, May 19, 2010 11:59 AM
Subject: categories: Re: "injective" terminology


> Perhaps I didn't make it clear that monomorphism is not always the same as
> 1-1 in a concrete category.  As for Peter's suggestion below, the
> terminology of injective for objects is as well established as the use of
> the same word for maps.  When I was a student, we talked of 1-1 maps and
> onto maps and I never heard the words injective and surjective.  But
> injective and projective objects followed well-established usage,
> certainly by the 1950s and probably well before.  Somebody (Mac Lane?)
> once tried using fascist, dual to free, instead of injective.  But of
> course, not every projective is free and, in any case, this never got any
> traction.
>
> Michael
>

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