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