Subject: categories: two questions about universal quantification

From: Barry Jay <[email-masked]>
Date: Thu, 28 Jan 1999 18:40:37 +1100 (EST)

Can anyone help with the following two questions? 
Barry Jay

Question 1 
~~~~~~~~~~

Has the following lemma been seen before? 

Lemma: If D is a cartesian closed category having all finite limits
then so is its arrow category.

Proof

Finite limits in the arrow category are constructed pointwise. 
Arrows from a: A -> I to b: B-> J In the arrow category are given by
commuting squares (all vertical lines are arrows pointing down)

      f
  A -----> B
  |        |
 a|        |b
  |        |
  I -----> J
      u

The object of such appropriate pairs (f,u) is given by the pullback

  E -----> A->B
  |         |
  |         | A->b
  |         |
 I->J ---> A->J
     a->J

The universal property follows directly. QED

The proof uses a form of universal quantification over A.
Similar constructions arise with the use of ends and dinatural
transformations. I also used them aggresively in my paper on data
categories (http://linus/~cbj/Publications/data_categories.ps.gz) to
quantify over all paths through a tree. The lemma generalises to other
functor categories over D provided there are sufficient limits in D to
express all of the constraints.

Question 2 
~~~~~~~~~~

The proof above shows how pullbacks and cartesian closure interact to
provide a form of universal quantification.  Call this new
construction quantification by exponentiation. What is its expressive
power?

Discussion
~~~~~~~~~~

Universal quantification wrt an arrow f : X -> Y is usually given as a
right adjoint to the functor D/Y --> D/X which pulls back along f.
Call this quantification by adjunction. When it is specialised to a
projection XxY ---> X then it can be thought of as quantification over
the object Y. Quantification by exponentiation can be seen as
supplying some of the ingredients necessary for this special case. For
example, consider the equaliser E of two functions into an
exponential:

  f,g : X --> (A->B)

If the quantifier by adjunction exists for the projection XxA ---> X
then E is given by applying it to the equaliser of the uncurried forms
of f and g from XxA to B. 
  
Motivation
~~~~~~~~~~

One the one hand, this structure is weaker than that normally assumed,
which may sometimes be a good thing. On the other hand, the power
resulting from this combination of features may still be more than
desired.

Many models of computation assume both cartesian closure and and all
finite limits but this is already more than we need to model the types
of most languages, i.e. a type corresponding to one of the objects
produced by quantification by exponentiation would have undecidable
membership, so that values would need to come equipped with a proof of
their membership. So (assuming that cartesian closure is a given) the
question becomes whether there is a smaller class of limits which are
good enough to model programming languages. 

One candidate is to work in an extensive category, in which only
pullbacks along coproduct inclusions are presumed. A short argument
shows that if Z has decidable equality then any cospan over it has a
pullback. Actually, there is a rich, but under-explored class of
pullbacks implied by this innocuous assumption. For example, the list
functor preserves such pullbacks (this is not quite trivial). There
may well be other, better candidates. I would be glad to hear of them.

*************************************************************************
| Associate Professor C.Barry Jay, 					|
| Reader in Computing Sciences		Phone: (61 2) 9514 1814		|
| Head, Algorithms and Languages Group,	Fax:   (61 2) 9514 1807		|
| University of Technology, Sydney,	e-mail: [email-masked]	|
| P.O. Box 123 Broadway, 2007,	  http://www-staff.socs.uts.edu.au/~cbj	|
| Australia.			        FISh homepage ... ~cbj/FISh     |
*************************************************************************

---

From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Thu, 28 Jan 1999 09:45:31 +0000 (GMT)

Barry Jay asked

> then so is its arrow category.

This is a special case of the cartesian closedness of a category 
obtained by Artin glueing, which was covered in the Carboni--Johnstone
paper (Math. Struct. Comp. Sci. 5 (1995), 441--459). (However, we don't
claim any originality for it; it is implicit in Gavin Wraith's 1974
paper on Artin glueing.) Of course, the "glueing functor" in Barry's case
is the identity on D.

Peter Johnstone

---

From: "Roy L. Crole" <[email-masked]>
Date: Thu, 28 Jan 1999 09:36:34 +0000 (GMT)

>then so is its arrow category.

If I've not misunderstood Barry's question 1, this is surely just the 
gluing or sconing lemma for the particular functor id_D : D -> D, 
where

     Glue(id_D) = arrow(D)

Gluing along the identity gives a pleasant form for exponentials; the
glued exponentials in Glue(G) for some G : C -> D collapse to the diagram
in Barry's note.