categories: Time for functors to grow up; three queries

["David Benson" <dbenson@eecs.wsu.edu>]

Peter Freyd recently reminded us that the term <<functor>> is now
sixty years old.  So it is time for this concept to grow up.

Early on, John Gray and others noticed that functors ought not
compose on-the-nose, but up-to-isomorphism, and (I believe)
called the result <<pseudo-functors>>.  Well, there is nothing
<<pseudo>> about it and it is these latter entities that deserve
the name <<functor>>, with the first 50 years of use then becoming
     XXX functor
where XXX might be <<young>> or <<nasal>> or ...

Now that I have caught your attention, we need to consider
image factorization systems for <<grown-up>> functors, and
appropriate names along the way.  I want to thank

Jack Duskin, Jonathan Kirby, Ronnie Brown, Peter May, Frank Atanassow,
David Carlton, John Baez and Peter Freyd

for responses to my rfn reminding me of terminology and where to look for it in print.

Since I attempt to write for computer scientists, I am reluctant to
overload various declensions of the verb

                   <<to represent>>

for categorical terms, since these may conflict with various already
overloaded variations within computer science.  So I will follow
a suggestion of Charles Wells and then
Paul Taylor's ``Practical Foundations of Mathematics'', to write

Definitions.  
(1)
A category D is said to be an \emph{essential subcategory}
of category C if every object of C is isomorphic to some object of D.

(2)
A category D is said to be a \emph{replete subcategory}
of category C if every object of C which is isomorphic
to an object of D is already in D, [1].

(3)  
A functor F: A --> B is said to be \emph{eso}, essentially surjective on objects,
when $\obj(F)$ is surjective up to an isomorphism. (Follows [1]).

(4)  
A functor F: A --> C is said to
factor through its \emph{essential image}, E, as

                 e      i
              A ---> E ---> C

when e is eso and full, i is injective on objects and also faithful,
and under the action of i, E is a replete subcategory of C.

QUERY (1):  Is (4) stated properly?  Is there any reference into the
            literature at all for this definition?  I looked fairly
            intensively this past weekend and could locate nothing...


Propositions.
(1) Each functor in the category of categories has an essential image factorization.
(2) The composition of two eso functors is again eso.
(3) For each eso functor F: A ---> C
    with image factorization through its essential image

                 e      i
              A ---> E ---> C

    the functor  i  makes E an essential subcategory of C
    and indeed,  i  is a bijection on objects.
    Furthermore  e  is both full and faithful.

QUERY (2):  These seem so obvious that I shouldn't bother to provide
            proofs in a paper that is already 47 pages long, except
            that many computer scientists may desire to see a proof.
            Is there any previously appearing paper, no matter how
            obscurely printed, that I could refer doubting readers to?

Proposition.
(4)
The essential image factorization of functors forms a
                 stable factorization system, [1],
That is, given that (e,m) is an essential image factorization
of some functor F, in the two strong Pullback ([1] calls these pseudo-pullbacks)
diagrams just below, which pullback against some functor G,
 e_1  is eso and full while  m_1  is injective on objects and faithful:

                   e_1         m_1
             A_1 ------> E_1 -------> D
              |           |           |
              |           |           | G
              |           |           |
              v           v           v
              A --------> E --------> C
                    e           m

QUERY (3): This seems to me to be obvious in the large category of
           small categories, but for the same reasons as for the
           previous query, I am either going to have to include a
           proof or a citation.  Having looked fairly hard for the
           later, I am again requesting assistance.

[1] P. Taylor, Practical Foundations of Mathematics,
    Cambridge University Press, 1999.

Thank you all very much indeed!

Best regards,
David
--
Professor David B. Benson                                (509) 335-2706
School of EE and Computer Science (EME 102)              (509) 335-3818 fax
PO Box 642752, Washington State University               office: Sloan 308 and 307
Pullman WA 99164-2752   U.S.A.                           dbenson@eecs.wsu.edu
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