Re: "injective" terminology

Re: "injective" terminology

[Fred E.J. Linton]

On Tue, 18 May 2010 07:08:42 PM EDT, Michael Barr <barr@math.mcgill.ca>
wrote:

> ... There are too many contexts in which you
> have a concrete category (say of compact hausdorff spaces) in which you
> are dealing with both injective objects and 1-1 maps that I feel we need a
> better word for the latter than "injective".  Of course, I could just
> revert to 1-1 and perhaps I will.  But we have "projective" and
> "surjective" for the dual.  This suggests "superjective", except that
> that is so ugly.
> 
> Any thoughts?

Mon[omorph]ic?

-- Fred




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Re: "injective" terminology

[Fred E.J. Linton]

> The dual to this concept is unimaginatively called "coinductive".)

Causing the unwary or inexperienced to wonder either 

(a) how "coin" entered the picture, or
(b) whether that's a typo for "conductive".

Cheers, -- Fred



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"injective" terminology

[Michael Barr]

Since there has been such a lively discussion of language (which I have
kept out of because I have seen too many papers start out by saying, "By
ring, we mean a commutative ring with unit"), I though I would bring up
one that has long bothered me. There are too many contexts in which you
have a concrete category (say of compact hausdorff spaces) in which you
are dealing with both injective objects and 1-1 maps that I feel we need a
better word for the latter than "injective".  Of course, I could just
revert to 1-1 and perhaps I will.  But we have "projective" and
"surjective" for the dual.  This suggests "superjective", except that
that is so ugly.

Any thoughts?

Michael

P.S. I originally used *-autonomous to mean symmetric and then I wrote a
paper called, "Non-symmetric *-autonomous categories", so I am just as
guilty.


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Re: "injective" terminology

[Prof. Peter Johnstone]

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
----------------------
On Tue, 18 May 2010, Michael Barr wrote:

> Since there has been such a lively discussion of language (which I have
> kept out of because I have seen too many papers start out by saying, "By
> ring, we mean a commutative ring with unit"), I though I would bring up
> one that has long bothered me. There are too many contexts in which you
> have a concrete category (say of compact hausdorff spaces) in which you
> are dealing with both injective objects and 1-1 maps that I feel we need a
> better word for the latter than "injective".  Of course, I could just
> revert to 1-1 and perhaps I will.  But we have "projective" and
> "surjective" for the dual.  This suggests "superjective", except that
> that is so ugly.
>
> Any thoughts?
>
> Michael
>
> P.S. I originally used *-autonomous to mean symmetric and then I wrote a
> paper called, "Non-symmetric *-autonomous categories", so I am just as
> guilty.
>

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Re: "injective" terminology

[Michael Barr]

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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Re: "injective" terminology

[George Janelidze]

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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Re: "injective" terminology

[Toby Bartels]

The term "projective limit" contrasts with "inductive limit",
so I have sometimes felt like saying "inductive object".

However, I've never actually done so; besides having no precedent,
the term "inductive object" already has an established meaning:
an inductive object is an initial algebra of a polynomial endofunctor.
(Example: A natural-numbers object is an initial algebra of X + 1.
The dual to this concept is unimaginatively called "coinductive".)
This is used in logic and computer science.


--Toby


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Re: "injective" terminology

[Timothy Porter]

I always thought that ind-object in the work of Grothendieck and the SGA
seminars was short for inductive object.

Tim



On 21/05/2010 18:35, Toby Bartels wrote:
> The term "projective limit" contrasts with "inductive limit",
> so I have sometimes felt like saying "inductive object".
>
> However, I've never actually done so; besides having no precedent,
> the term "inductive object" already has an established meaning:
> an inductive object is an initial algebra of a polynomial endofunctor.
> (Example: A natural-numbers object is an initial algebra of X + 1.
> The dual to this concept is unimaginatively called "coinductive".)
> This is used in logic and computer science.
>
>
> --Toby
>



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Re: "injective" terminology

[Toby Bartels]

Timothy Porter wrote:

>I always thought that ind-object in the work of Grothendieck and the SGA
>seminars was short for inductive object.

The "ind-" in "ind-object" does stand for "inductive",
which is dual to "projective" for the "pro-" in "pro-object",
but an ind-object is not the same thing as an injective object
(and also a pro-object is not the same thing as a projective object).
http://ncatlab.org/nlab/show/ind-object
http://ncatlab.org/nlab/show/injective+object
http://ncatlab.org/nlab/show/pro-object
http://ncatlab.org/nlab/show/projective+object


--Toby


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Re: "injective" terminology

[Michael Barr]

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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