Left and right

Left and right

[Charles Wells]

>If it were possible to start afresh with the terminology of category
>theory (of course it isn't, as Mike pointed out), I'd be in favour of
>using "left" and "right" as much as possible, and eliminating the "co-"s.
>(But even this is not guaranteed free from ambiguity. Has anyone apart
>from me (and, I suppose, the authors) noticed that the usage of the
>terms "left coset" and "right coset" in Mac Lane & Birkhoff's Algebra
>is the opposite of that in Birkhoff & Mac Lane?)

Not all lefts and rights are the same.  Left adjoint refers to the fact
that arrows FROM a value of the left adjoint into an object correspond to
arrows INTO the value of the right adjoint at that object.  Since English
is written left to right, Hom(A,B) means arrows from A to B, so in the
equation Hom(FA, B) = Hom(A,UB) the F winds up on the left side of the hom
set.  This is a natural name given the way we write our language, and so it
is not hard to reconstruct what the phrases left and right adjoint mean.

On the other hand the left and right in "left inverse" and "right inverse"
depend on the order in which we write composition, and that is independent
of the way we write our language.  I for one can never remember which is
which, a learning disability no doubt accounted for by the fact that I
worked on semigroups before I became a category theorist, leaving me
without a default way to write composition.  In any case, I would like
names such as retraction and split.  


Charles Wells, Department of Mathematics, Case Western Reserve University,
10900 Euclid Ave., Cleveland, OH 44106-7058, USA.
EMAIL: charles@freude.com. OFFICE PHONE: 216 368 2893.
FAX: 216 368 5163.  HOME PHONE: 440 774 1926.  
HOME PAGE: URL http://www.cwru.edu/artsci/math/wells/home.html

Re: Left and right

[James Stasheff]

Not to mention the notation Ext(A,B) in which B is extended BY A

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On Mon, 6 Jul 1998, Charles Wells wrote:

> 
> >If it were possible to start afresh with the terminology of category
> >theory (of course it isn't, as Mike pointed out), I'd be in favour of
> >using "left" and "right" as much as possible, and eliminating the "co-"s.
> >(But even this is not guaranteed free from ambiguity. Has anyone apart
> >from me (and, I suppose, the authors) noticed that the usage of the
> >terms "left coset" and "right coset" in Mac Lane & Birkhoff's Algebra
> >is the opposite of that in Birkhoff & Mac Lane?)
> 
> Not all lefts and rights are the same.  Left adjoint refers to the fact
> that arrows FROM a value of the left adjoint into an object correspond to
> arrows INTO the value of the right adjoint at that object.  Since English
> is written left to right, Hom(A,B) means arrows from A to B, so in the
> equation Hom(FA, B) = Hom(A,UB) the F winds up on the left side of the hom
> set.  This is a natural name given the way we write our language, and so it
> is not hard to reconstruct what the phrases left and right adjoint mean.
> 
> On the other hand the left and right in "left inverse" and "right inverse"
> depend on the order in which we write composition, and that is independent
> of the way we write our language.  I for one can never remember which is
> which, a learning disability no doubt accounted for by the fact that I
> worked on semigroups before I became a category theorist, leaving me
> without a default way to write composition.  In any case, I would like
> names such as retraction and split.  
> 
> 
> Charles Wells, Department of Mathematics, Case Western Reserve University,
> 10900 Euclid Ave., Cleveland, OH 44106-7058, USA.
> EMAIL: charles@freude.com. OFFICE PHONE: 216 368 2893.
> FAX: 216 368 5163.  HOME PHONE: 440 774 1926.  
> HOME PAGE: URL http://www.cwru.edu/artsci/math/wells/home.html
>