Re: Help!

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

George Janelidze wrote:
> (a) a topological space (defined via open sets) has no constant sets, one
> variable set X, and its structure is an element t in PP(X) that is closed
> under finite intersections and arbitrary unions.

The point of categories being (presumably) to shift the burden of
structure from the objects to the morphisms, one would illustrate this
point using your example by pointing out that the topological structure
imputed to a space by the above definition is at least as well imputed
with the definition of a space as the set of continuous functions to the
space from the one-point space together with the set of continuous
functions from it to the Sierpinski space.  It sounded like you were
headed in roughly that direction but then moved on to other points
before getting there.

> (b) a vector space has one constant set A ("the set of scalars"), one
> variable set X ("the set of vectors"), and its structure can be defined as
> element (a,b,c,d) of P(AxAxA)xP(AxAxA)xP(AxXxX)P(XxXxX), where a, b, c, d
> are addition of scalars, multiplication of scalars, scalar (-on-vector)
> multiplication, and addition of vectors respectively; that (a,b,c,d) should
> satisfy familiar conditions of course.

Ditto with the one-dimensional space in place of the one-point and
Sierpinski space (which itself is a kind of one-dimensional space for
topology).

> 4. Linear algebra tells us that instead of working with linear
> transformations of finite-dimensional vector spaces we can work with
> matrices, but one cannot formulate this properly without using the concept
> of equivalence of categories (the category of finite-dimensional K-vector
> spaces is equivalent to the category of natural numbers with matrices with
> entries from K as morphisms).

You may be setting the bar for "proper" higher than necessary to satisfy
us engineers.  I'm currently involved in a computer project where the
question of the proper formulation of matrices came up.  One team had
formulated them in terms of the Kleisli construction for monads as
defined in CTWM, the monad in question being the one that you yourself
would surely come up with for the variety Vct_C, C the complex numbers.

Unfortunately that formulation was giving the computer conniptions.
This could have been construed as bearing out your point were it not for
the fact that another team came along with a reformulation of monads
that overcame the difficulty.

Since I know this list is good at keeping secrets (such as the secret of
categories) I'll be happy to share with you all, in my next message, my
confidential report on the current status of this reformulation.  Our
CEO is not convinced of the correctness of the reformulation, the fact
that it fixed the buggy behavior notwithstanding, and has asked me for a
qualified expert second opinion---where better than this list for a
question about monads?

Vaughan