co-exponential question

co-exponential question

[Bill Halchin]

This is actually a "dual" question.
Basically I want to do the dual of the construction
gives the notion of an exponential or map object.

Suppose we have a category C with sums. Then we build the
following category from C.

object:  T+X<-----Y

map: from T+X<-----Y  to T+X'<-------Y is a C map "alpha"  such
that we have the following diagram:

                   I-sub-T+alpha
         T+X---------------------------->T+X'
           ^                              ^
            \                            /
             \                          /
              \                        /
               \                      /
                \                    /
                 \                  /
                  \                /
                   \              /
                    \            /
                     \          /
                      \        /
                       \      /
                        \    /
                         \  /
                          \/
                          Y


   Then suppose there exists a C-object called Y**T such
that  T+Y**T<-------Y is the initial object of the category
just built above. What significance does Y**T have opposed
to the concept of an exponential???? If I did everything
correctly it (Y**T) should be the dual of T**Y.


Regards,

Bill Halchin

Re: co-exponential question

[William James]

Bill Halchin wrote:
> 
> This is actually a "dual" question.
> Basically I want to do the dual of the construction
> gives the notion of an exponential or map object.
> 
> Suppose we have a category C with sums. Then we build the
> following category from C.
> 
> object:  T+X<-----Y
> 
> map: from T+X<-----Y  to T+X'<-------Y is a C map "alpha"  such
> that we have the following diagram:
> 
>                    I-sub-T+alpha
>          T+X---------------------------->T+X'
>            ^                              ^
>             \                            /
>              \                          /
>               \                        /
>                \                      /
>                 \                    /
>                  \                  /
>                   \                /
>                    \              /
>                     \            /
>                      \          /
>                       \        /
>                        \      /
>                         \    /
>                          \  /
>                           \/
>                           Y
> 
>    Then suppose there exists a C-object called Y**T such
> that  T+Y**T<-------Y is the initial object of the category
> just built above. What significance does Y**T have opposed
> to the concept of an exponential???? If I did everything
> correctly it (Y**T) should be the dual of T**Y.

Forgive my ignorance: is T+Y**T<----Y to be initial because
the usual construction for the exponential has the relevant
arrow as terminal? Can you suggest to me a reference for that
construction of exponentials?

As for significance: my own research into co-exponentials is
in terms of them as characteristic of lattices dual to
Heyting algebras. These dual-Heyting algebras work well
as algebras for a brand of paraconsistent logic (they have
a complement operator which has in general that an element
and its complement overlap). Alternatively, you can
think of co-exponentials as productive of the interesting
topological notions associated with closed sets, like boundary.

My feeling is that co-exponentials count as useful in more
interesting kinds of maths than turn up simply in those
categories dual to toposes.

William James

Re: co-exponential question

[Paul Levy]

I don't know if this is relevant to your question, but there is an example in
programming semantics where the dual of a cartesian closed
category has independent significance, due to Lafont, Streicher, Reus
and Hofmann.  (Some further work was done by Selinger.)  It is found in the following paper:
		  
@Article{StreicherReus:continuations,
  title =        "Classical logic, continutation semantics and abstract
                 machines",
  author =       "Th. Streicher and B. Reus",
  pages =        "543--572",
  journal =      "Journal of Functional Programming",
  month =        nov,
  year =         "1998",
  volume =       "8",
  number =       "6",
}

The construction is as follows.

Let C be a distributive
category, and let Ans (the "answer type") be an object in C, such that the exponential X -> Ans exists for
each object X.  Define two categories K and
N as follows. Both have the same objects as C.  In K, a morphism from
X to Y is a C-morphism from X x (Y -> Ans) to Ans.  In N, a morphism
from X to Y is a C-morphism from (X -> Ans) x Y to Ans. 

 Clearly these two
categories are dual.  Streicher and Reus' paper makes it clear that just as K can be
used to interpret a typed call-by-value language with control effects (which
was well-known), N can be used to interpret a typed call-by-name language
with control effects.  Like any call-by-name model, N is cartesian closed:

the product of X and Y in N is given by X+Y

the exponential from X to Y in N is given by (X -> Ans) x Y

K is certainly an important category, but I wouldn't say that the fact
that it has coexponentials is significant.  

Paul

===========================================================================
Paul Blain Levy, Department of Computer Science,
Queen Mary and Westfield College, LONDON E1 4NS
http://www.dcs.qmw.ac.uk/~pbl/
===========================================================================

Re: co-exponential question

[Andrzej Filinski]

Paul Levy writes:

> I don't know if this is relevant to your question, but there is an
> example in programming semantics where the dual of a cartesian closed
> category has independent significance, due to Lafont, Streicher, Reus
> and Hofmann.  (Some further work was done by Selinger.)  It is found
> in the following paper:  [Streicher & Reus 1998]

Indeed, co-exponentials are closely tied the semantics of call/cc-like
control operators. This connection is perhaps expressed more directly in
the following (regrettably somewhat unpolished) work:

@MastersThesis{Filinski:89a,
  author =      "Andrzej Filinski",
  title =       "Declarative Continuations and Categorical Duality",
  school =      "Computer Science Department, University of Copenhagen",
  year =        1989,
  month =       Aug, 
  note =        "DIKU Report 89/11",
  URL =         "http://www.brics.dk/~andrzej/papers/DCaCD.ps.gz"
}

@InProceedings{Filinski:89b,
  author =      "Andrzej Filinski",
  title =       "Declarative Continuations: An Investigation of
                 Duality in Programming Language Semantics",
  booktitle =   "Category Theory and Computer Science",
  series =      LNCS,
  number =      389,
  address =     "Manchester, UK",
  month =       Sep,
  pages =       "224-249"
}

Specifically, in the category induced by the types and terms of a
call-by-value language with first-class continuations, the coproduct
functor - + A actually has a left adjoint - x (A -> 0), where 0 is the
empty type.  The operational intuition behind this construction is that
a function f : X -> Y + A returning either a "normal" (Y) or an
"exceptional" (A) result corresponds to a function f' : X x (A -> 0) ->
Y, returning only the normal result, but passing any exceptional results
to an additional, non-returning-function parameter.

(In a purely functional language, the type A -> 0 would of course
contain at most one value; but in the presence of control effects, such
as exceptions, there can be many distinct "functions" of this type. And
with a sufficiently powerful control operator, it actually becomes
possible to define co-application and co-currying with equational
properties exactly mirroring those of CCC exponentials.)

Still, as Paul notes,

> K is certainly an important category, but I wouldn't say that the fact
> that it has coexponentials is significant.  

constructing a categorical semantics of a language with control operators
_directly_ in terms of coexponentials is somewhat awkward, and the
formulation used by Streicher and Reus is almost certainly nicer to work
with in practice.

-- Andrzej

Re: co-exponential question

[Peter Selinger]

I agree completely with Paul and Andrzej, except possibly for this:

> Still, as Paul notes,
> 
> > K is certainly an important category, but I wouldn't say that the fact
> > that it has coexponentials is significant.  
> 
> constructing a categorical semantics of a language with control operators
> _directly_ in terms of coexponentials is somewhat awkward, and the
> formulation used by Streicher and Reus is almost certainly nicer to work
> with in practice.

The _direct_ categorical semantics of languages with control operators
is actually not too unnatural and leads to some interesting category
theory. Let me first clarify the difference between direct and
indirect semantics. To give a direct semantics, one starts with a
class of categories with algebraic structure (for our purposes:
structure that is given by object constructors, morphism constructors,
and equations on morphisms). In the direct interpretation of a
language, types are interpreted as objects, type constructors as
object constructors, and typing judgments as morphisms, in such a way
that the language forms an _internal language_ for the class of
categories at hand. This means, among other things, that one can
construct a syntactic category from a theory of the language. Thus,
all structure of the category (on objects and morhpisms) is denotable
in the language. The usual interpretation of the simply-typed lambda
calculus in a cartesian-closed category is perhaps the best-known
example of a direct semantics. (Direct semantics of higher order can
be given in fibered categories, but this is not necessary for our
purposes here).

In an indirect semantics, one does not interpret the language directly
in a category with algebraic structure, but in a derived auxiliary
category. Moggi's monadic semantics is an example of an indirect
semantics.  Given a category C with a monad T, one interprets the
language (for instance, a call-by-value lambda calculus) in the
Kleisli category of the monad. Notice that the structure of the
Kleisli category, per se, is not necessarily algebraic in the above
sense. The Reus-Streicher-Lafont-Hofmann semantics is indirect; in
fact it is the monadic semantics for the continuations monad
R^(R^(-)).

Indirect semantics are very successful in practise. Among other
things, they have proven excellent tools for reasoning about and
implementing control effects. This point is made convincingly in
Andrzej Filinski's PhD thesis [1]. However, the correspondence between
syntax and semantics is not as close as in the direct case: one cannot
usually speak of internal languages in the context of indirect
semantics.  The reason is that, from a syntactic theory of the
language, one can reconstruct only the Kleisli category of T, but not
the underlying category C or the monad T itself. Thus, direct
semantics might be more satisfying to the theorist, because they
capture the essence of the language precisely, and without resorting
to a translation to a meta-language.

The difference between direct and indirect semantics is explained very
nicely in a recent paper by Carsten Fuhrmann [2], where he also gives
a general method of constructing direct semantics from indirect ones.
It seems that direct semantics are often characterized by the presence
of certain _non-natural_ transformations, and that indirect semantics
are a way of hiding this non-naturality.

The first direct semantics for a call-by-value language with control
effects are the tensor-not categories of Hayo Thielecke [3]. His work
was later extended by me to a language with finite coproducts (this is
where co-exponentials enter the picture) and to the call-by-name case
[4]. I called the resulting class of categories "control
categories". The connection between the direct and indirect semantics
can be established via categorical representation theorems, and such
theorems were shown by Fuhrmann for tensor-not categories [5], and
independently by myself for control categories [4].

It is interesting that a call-by-name language is an internal language
for control categories, and the corresponding call-by-value language
is an internal language for the dual co-control categories. As a
consequence, one obtains a syntactic duality between call-by-name and
call-by-value [4]. Such a duality was conjectured by Streicher and
Reus [6], and certainly anticipated by Filinski's work on the
symmetric lambda calculus [7].

Best wishes, -- Peter

[1] A. Filinski. Controlling effects. PhD Thesis in Computer Science,
Carnegie Mellon University, 1996.

[2] C. F\"uhrmann. Direct models for the computational
lambda-calculus. Technical Report ECS-LFCS-98-400, Edinburgh, 1998.

[3] H. Thielecke. Categorical structure of continuation passing
style. PhD Thesis in Computer Science, Edinburgh, 1997.

[4] P. Selinger. Control categories and duality: on the categorical
semantics of the lambda-mu calculus. Submitted 1999. Available from
Hypatia or http://www.math.lsa.umich.edu/~selinger/papers.html

[5] C. F\"uhrmann. Exponential categories and semantics of
continuations. Presented at MFPS, 1998.

[6] T. Streicher, B. Reus. Classical Logic, continuation semantics and
abstract machines. Journal of Functional Programming 8(6)543-572, 1998.

[7] A. Filinski. Declarative continuations and categorical duality.
Masters Thesis in Computer Science, DIKU, University of Copenhagen,
1989. DIKU Report 89/11.