Re: Paper available

[Vaughan Pratt <pratt@cs.stanford.edu>]

Vincent's paper

vs27@mcs.le.ac.uk wrote:
 > Hi Walter, let me advert my paper
 > on a similar and related subject
 > http://arxiv.org/abs/math/0602463
 > "Flatness, preorders and generalized metric spaces"
 > that treats completions of non symmetric spaces.
 > Cheers,
 > V.

reminds me of a question I've been meaning to ask for several years, in
fact since my CT'04 talk on communes over bimodules, but wasn't quite
sure how to formulate.

In any setting, ordinary or enriched, it is possible to introduce
presheaves immediately after defining "category," even before defining
"functor."

Ordinarily one does not do so because functors are more fundamental to
category theory than presheaves, being an essential stepping stone to
the notion of natural transformation, Mac Lane's motivating entity for
the whole CT enterprise.

But just as dessert tends to lose its appeal when complete demolition of
the main course is a prerequisite, so are applications of CT most
effective for a foreign (non-CT) audience when they don't assume that
the whole CT enchilada has been digested.  For applications of
presheaves it is helpful to know what is the absolute minimum of CT
required by the audience.

Just as it is not necessary to understand the principle of the internal
combustion engine when getting one's driver's license by showing that
one can control such an engine, it should not be necessary to know what
a functor, natural transformation, adjunction, or colimit is to freely
construct a presheaf on a small category J as a colimit.  The following
construction should suffice for those who know nothing more about CT
than the definition of category.

Grow a presheaf category C starting with C = J (with Set^{J^op} as the
unstated secret goal) as follows.  Independently adjoin objects x to C.
  For each such x and each object j in J, further adjoin morphisms from
j to x (more generally in the V-enriched case, assign an object of V to
C(j,x)), with composites of the morphisms of C(j,x) with those of J
chosen subject only to the requirement that C remain a category.  For
any objects x,y of C, with x not in J (y in J is ok), populate C(x,y)
with as many morphisms f,g,... as possible (in the V-enriched case, a
suitable limit), again choosing composites with morphisms from any j to
x arbitrarily, subject to the requirements that (i) if for all j and all
morphisms a: j --> x, fa = ga, then f = g, and (ii) again that C remain
a category (which then determines all remaining composites x --> y -->
z).  A pre-question here is, did I inadvertently leave anything out?

My main question is, is there a reference for this process that I can
cite?  Any such reference must make the point that the prerequisites for
this process include categories but exclude the rest of CT (as
prerequisites---obviously some additional parts of CT are directly
derivable, the point is that they're not prerequisites for the student).

Ordinarily one reason for not bothering with such a thing would be that
one can avoid even the categories by talking about equational theories
with only unary operations.  My application however is to communes,
which are trickier to describe from a purely algebraic perspective
(they're chupological rather than coalgebraic), but very natural from
the above colimit-based perspective.

Vaughan