Apropos of Peter's kind remarks, the journal version of our paper should be in TCS soon, entitled The continuum as a final coalgebra. Meanwhile it's on the web as http://boole.stanford.edu/pub/continuum.ps.gz Abstract: We define the continuum up to order isomorphism, and hence up to homeomorphism via the order topology, in terms of the final coalgebra of either the functor NxX, product with the set of natural numbers, or the functor 1 + NxX. This makes an attractive analogy with the definition of N itself as the initial algebra of the functor 1 + X, disjoint union with a singleton. We similarly specify Baire space and Cantor space in terms of these final coalgebras. We identify two variants of this approach, a coinductive definition based on final coalgebraic structure in the category of sets, and a direct definition as a final coalgebra in the category of posets. We conclude with some paradoxical discrepancies between continuity and constructiveness in this setting. PF>So restrict to the category of posets with _distinct_ top and bottom. ...which has no final object, making the following all the nicer: PF>The functor X v X again has a final coalgebra: The bipointed-set version of this seems like *the* right way to generate the topology of the continuum. Verry nice. PF>PS PF>Just for comparison, consider the category of posets and the functor PF>that sends X to X;1;X. The open interval is an invariant object PF>for this functor but it is not the final coalgebra. For that we need PF>-- as we called it in Cats and Alligators -- Wilson space. Actually, PF>not the space but the linearly ordered set, most easily defined as the PF>lexicographically ordered subset, W, of sequences with values in PF>{-1, 0, 1} consisting of all those sequences such that for all n PF>a(n) = 0 => a(n+1) = 0 (take a finite word on {-1,1} and pad it PF>out to an infinite sequence by tacking on 0s). In other words the continuum plus *two* additional copies of each rational (as opposed to Cantor space's one additional copy). PF>It takes more work, but not an infinite amount, to construct a PF>coalgebra structure on I x I so that the induced binary operation PF>I x I --> I is the midpoint operation, the values of which will be PF>denoted here as x|y. It is pretty much characterized by the PF>equations: idempotence, x|x = x; commutativity, x|y = y|x; and middle- PF>two-interchange, (u|v)|(x|y) = (u|x)|(v|y); together with cancelation: PF>a|x = a|y => x = y. If one chooses a zero, then one may prove that PF>there's an ambient abelian group with unique division by 2, such that PF>the given midpoint algebra is a subset closed under the operation PF>x|y = (x + y)/2. (There must be an existent reference for this.) Here I should be understood as [-1,1]. Peter's implicit definition of this coalgebra structure on I x I has the following equivalent five-equation explicit definition (or seven if you count each of (3) and (4) as two equations). My apologies to anyone reading this with a variable-width font. x + y y + x (1) ----- = ----- 2 2 0 + 0 (2) ----- = 0 2 xo1 x + t --- + 0 -----o1 where the 2 o's are + and t is -1 (3) 2 = 2 or the 2 o's are - and t is 1 -------- ------- (o = operator, t = terminator) 2 2 xo1 yo1 x + y --- + --- -----o1 where the 3 o's are either (4) 2 2 = 2 all - or all + ------- ------- 2 2 x-1 y+1 x + y --- + --- ----- + 0 (5) 2 2 = 2 ------- --------- 2 2 All these equations clearly hold on I; the question is, what is their content? Well, spacing around + and - is significant: without a space the form x-1 x+1 --- (resp. --- ) denotes a non-middle (nonzero) element of I which when 2 2 represented as a string over {-1,0,1} has head -1 (resp. 1) and tail x, x + y while with a space the form ----- denotes a pair (x,y) of I x I. 2 Finally, if -1, 0, or 1 (which includes t) are preceded by a space then they denote the respective infinite constant sequences -1,-1,-1,... or 0,0,0,.. or 1,1,1... (each of which has that constant as its value). (Once a sequence hits a zero it stays zero.) The left hand side always has exactly one spaced +, and at the outermost level, since that's what we're trying to define. On the right hand side, each occurrence of a spaced + denotes a corecursive call (so two corecursive calls in the last equation --- it might seem like a lot of bother to do a corecursive call merely to add 0, but the real point of the second call is of course to divide the result of the first call by 2, which we can do by taking the mean with 0.) Equation (1) merely ensures that any pair (x,y) will match the left hand side of one of (2)-(5) one way round or the other. Equations (2)-(4) provide instant gratification inasmuch as they allow immediate production of the first "trit" (ternary digit) of the mean of x and y given only the first trit of each of them, thereby explicitly defining the desired coalgebra structure on I x I. But if you hit equation (5) you may be in for a long or even infinite wait (consider taking the mean of -1,1,-1,1,... (= -2/3) with 1,-1,1,-1,... (= 2/3), which to us is obviously zero but not so obviously to equation (5)). So equation (5) does not give the coalgebra structure at this part of I x I explicitly, one must regrettably deduce it by corecursion. Vaughan