I agree with everything George has said here. Mac Lane's paper was amazing and I meant no disrespect. Michael On Fri, 21 May 2010, George Janelidze wrote:

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

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