Re: "injective" terminology

[Michael Barr <barr@math.mcgill.ca>]

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

On Wed, 19 May 2010, Prof. Peter Johnstone wrote:

> Like Michael, I've occasionally been bothered by the conflict between
> the two uses of "injective". However, for me it's the use of the word
> as a dual for "projective" that feels wrong; the opposite of "pro" is
> not "in" but "con" (or "contra"). Also, the use of "injective"
> and "surjective" for maps is so well established throughout
> mathematics that I don't think there is any chance of changing it.
> I've thought of using "coprojective" for the dual of "projective";
> but for anyone with a classical education that word means
> "shit-throwing".
>
> Peter Johnstone

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