Re: A small cartesian closed concrete category

[Andrej Bauer <Andrej.Bauer@fmf.uni-lj.si>]

PETER EASTHOPE wrote:
> Is there a cartesian closed concrete category which
> is small enough to write out explicitly?  It would be
> helpful in learning about map objects, exponentiation,
> distributivity and etc.   Can such a category be made
> with binary numbers for instance?

A Heyting algebra, viewed as a category (every poset is a category), is
a CCC. If you take a small Heyting algebra, e.g. the topology of a
finite topological space, you can write it out explicitly.

If you would like a CCC made from n-bit binary numbers, here is how you
can do it:

The two-point lattice B = {0, 1} is a Boolean algebra with the usual
logical connectives as the operations. Because B is a poset with 0<=1,
it is also a category (with two objects 0, 1 and a morphism between
them). Since every Boolean algebra is a Heyting algebra, B is cartesian
closed, with the following structure:
- 1 is the terminal object
- the product X x Y is the conjuction X & Y
- the exponential Y^X is the implicatoin X => Y

The product of n copies of B is the same thing as n-tuples of bits,
i.e., the n-bit numbers. This is again a CCC (with coordinate-wise
structure).

At this late hour I cannot see what can be said about finite CCC's which
are not (eqivalent to) posets.

Andrej