categories: Re: Time for functors to grow up; three queries"

[Steve Lack <stevel@maths.usyd.edu.au>]

David Benson writes:

 > 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

Like Tom Leinster, I assume that David actually means to treat
Cat as a bicategory rather than a 2-category; the rest of the
posting seems to contain nothing about pseudofunctors, but rather
about factorizations for ordinary functors.

Here are a few references about factorization systems in 2-categories
and bicategories:

[1] Ross Street, Two-dimensional sheaf theory, JPAA 23(1982) 251-270
[2] Ross Street, Characterization of bicategories of stacks, SLN 962
[3] Carboni, Johnson, Street, & Verity, Modulated bicategories, 
JPAA 94(1994) 229-282.

In [1] there is defined a notion of regular 2-category, based on what
Ross calls the (acute, chronic) factorization system. In [2] a (very similar) 
notion of regular bicategory is defined. Both are based on the
factorization (eso,inj.obj&fully faithful) on Cat. In [3], factorization
systems (strong liberal, conservative) and (liberal, strong conservative)
are described which may or may not exist on a bicategory. (Neither exists
on Cat, although the second exists the bicategory Cat_cc of Cauchy compete
categories.) There is also, implicitly, a general notion of factorization
system on a bicategory.

A few other small points:

 > 
 > (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].

Presumably what is really needed is: a subcategory D of C
is said to be replete if whenever an object c of C is isomorphic
to an object d of D, via an isomorphism gamma:c-->d, then BOTH 
c and gamma lie in D.

Often one considers repleteness only when the subcategory is full,
and then the extra condition that the isomorphism lies in the
subcategory is not needed.

 > 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.
 > 

 > (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.

The last sentence is clearly false in general: if i and e were both
faithful, then so would F be.

 > 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

I don't have a copy of Paul's book, but under any of the usual
meaning of pseudopullback this would be false. While it is true
that the eso&full functors are stable under pseudopullback (although
not ordinary pullback) it is not true that the io&faithful ones are
so.

If by the psudopullback of m and G you mean the category E_1 whose
objects are triples (e,gamma,d), where e and d are objects of E and
D, and gamma:me-->Gd is an isomorphism in C. To see this, let E=D=C,
with m and G the identity functors. Then an object of E_1 is just an
isomorphism in C, so if C has non-identity isomorphisms, m_1 will not
be injective on objects.

If, on the other hand, by pseudopullback you mean ``anything equivalent
to what I just described'', then the question of whether m_1 is injective
on objects is not even well-posed, since E_1 is determined only up to
equivalence.

(There is also a third meaning, but it once again is equivalent to 
the first, and once again the pseudopullback of an injective on objects
functor will not in general be injective on objects.)

Steve.