Re: Structure Preserving: Definition?

[Barry Jay <cbj@it.uts.edu.au>]

Dear Derek,

since "Matrices, Monads and the Fast Fourier Transform" in
the early 90's I've written a couple of other papers on
semantics of datatypes. "A semantics for shape" considers
more general datatypes than just vectors. "Data categories"
is an attempt to embrace co-datatypes as well as
datatypes. These, and other papers about the implications
for computing, including the array programming language FISh
are available from my web-site

http://www-staff.it.uts.edu.au/~cbj

May I add that we are stabilising a prototype of FISh2 which
is altogether more expressive and simpler than FISh. We hope
to release it shortly. 

Now let me address your particular question.

you are concerned that the morphism #: VA -> N mapping a
vector of A's to its length does not appear to preserve any
structure, and so perhaps should not be a morphism at all.
There are two aspects to the answer. First, the existence of
this morphism is part of the definition of the object of
vectors. Given an arrow A -->I we define the corresponding
vectors by the pullback

VA ---> LA
|       | 
|       | 
NxI --> LI

where L is the list functor, and then # is defined by
composing VA -->NxI with the projection from NxI to N. If
the ambient category is Set then such pullbacks exist and
the function # is a well-defined function. The second point
is that # can be thought of as the upper part of an arrow
between arrows which maps a vector of A's to the
corresponding vector of 1's

VA ---> V1  isom   N
|       |          |
|       |          |
NxI --> Nx1 isom   N


I'd be happy to address any other questions you have
privately.

Yours,
Barry Jay


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| Associate Professor C.Barry Jay,      Phone: (61 2) 9514 1814		|
| Associate Dean                        Fax:   (61 2) 9514 1807	        |
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