From: Galchin Vasili <vngalchin@yahoo.com> I have been reading some of Prof. Pratt's papers on Chu spaces. I am just trying to understand Topos. Both seem to generalize the notion of topological spaces. Is there any relationship beteween Chu spaces and Toposes?
About all that a topos has in common with Chu(V,k) is that both are symmetric monoidal closed categories with all finite limits. Among the conditions that hold for every topos but only for degenerate Chu categories is cartesian closedness. Conversely, all Chu categories but only degenerate toposes are self-dual. Not a strong basis for a relationship. Marriage would seem out of the question, maybe half a game of correspondence chess. There are a couple of things worth mentioning about Chu spaces vs. topological spaces that may help in understanding why that connection does not extend nicely to toposes. First, Chu spaces are to topological spaces as (universal) algebras are to groups. The former (in each fiefdom) relaxes the specific signature and axioms of the latter while retaining its underlying machinery: open sets and continuous functions in the first fiefdom, n-ary structures and homomorphisms in the second. As noted by Lafont and Streicher [LICS'91], Chu(Set,2) embeds Top, much as the category of ternary structures and their homomorphisms embeds Grp (by "embeds" and "subcategory" I mean "fully embeds" and "full subcategory" throughout). A topological space may be understood as a partial algebra equipped with a "signature" consisting of an operation (or proper class of operations if one takes arity into account) for taking the limit of a sequence. This operation is the only one to survive the requirement that the family of open sets of a topological space be closed under arbitrary union and finite intersection. Strengthening "finite" to "arbitrary" restricts the domain of the limit operation to finite sequences, the "signature" of a preordered set. Further requiring closure under complement removes (discretizes) the partial order. Conversely, when the closure conditions are removed, all heaven breaks loose with even finitary partial operations being admitted. Note the contravariance between strength of condition on the open sets and size of language: more restrictions means less signature. An altogether different connection with topology involves Stone duality. The easy part of this duality is to match up a Boolean algebra B to a quasitopological space (one whose open sets need only be closed under finite union and finite intersection, with continuity defined standardly) whose points are the ultrafilters of B (Boolean homomorphisms from B to the 2-element Boolean algebra) and whose open sets are the elements of B (taking membership of such a point in such an open set to be the converse of membership of elements of B in ultrafilters of B). The Lafont-Streicher embedding of Top in Chu(Set,2) extends without change of wording to embed Qtop (quasitopolical spaces, no connection with Qtips). Furthermore Bool embeds in Chu(Set,2) in an obvious way, such that the corresponding representation of the above dualization of B is then nothing more than the transposition of this representation of B yielding the Lafont-Streicher representation of the dual of B. The hard bit in Stone's theorem is to show that the process of generating a topological space from this dual (i.e. the result of closing the open sets under arbitrary union) embeds these dualizations in Top (fully as always), and moreover as totally disconnected compact Hausdorff spaces. The question then arises, what is Stone's theorem morally? On an Arcturan planet teeming with mathematicians, the fundamental theorem of arithmetic is surely as it is on Earth (not necessarily under the same name), but what about Stone duality for Boolean algebras? What useful purpose is served by the hard bit of Stone's theorem? Why take the (hard) second step when one could as usefully stop after performing the (very easy) transposition? It is sometimes said that Stone's theorem makes a precise connection between geometry and algebra. It seems to me that the connection between geometry and totally disconnected topological spaces is a much bigger stretch than between topological spaces and quasitopological spaces. If connecting algebra to geometry were the primary motivation behind Stone duality, one would only go to the considerable bother of representing a perfectly good and very easily obtained quasitopological space as a topological space if one's religion--or mathematics--forbade entertaining the former as a geometrical notion. While acknowledging the weight of a century of tradition underlying this proscription here on Earth, interplanetary mathematical anthropology needs to be able to interpret more neutrally the morality of mathematical decisions made on other planets. My own view is that the connection between algebra and geometry is not the most important lesson of Stone duality. The important point of the theorem Stone proved was that any Boolean algebra B could not only be represented (via Birkhoff's more general representation of distributive lattices) as a field of sets (one closed under finite union, finite intersection, and complement), but that there was a canonical, even universal, such representation. Later (I don't know when) it became clear that Stone's representation applied not just to the objects but the morphisms, showing that this representation was in fact a duality. And a strikingly beautiful duality it is, independently of its connection with geometry.
From this perspective the better version of Stone's theorem would, I think, be the one more easily grasped. What I like so much about Chu spaces here is that it reduces Stone duality for Boolean algebras to matrix transposition. This makes it something one can present in an introductory text on algebras, lattices, and varieties, or proudly take home to show to one's loved ones, which is more than can be said of a lot of mathematics.
A pedagogically ideal illustration of Stone duality is that between finite distributive lattices with top and bottom and finite posets, observed explicitly by Birkhoff and implicitly by Stone, both in 1937. The 5 (16,63...) distributive lattices of height 3 (4,5,...) correspond to the 5 (16,63,...) posets with 3 (4,5,...) elements. This duality further pairs up distributive lattice homomorphisms (respecting 0 and 1) with monotone functions between posets, a correspondence with a terrific visual appeal when illustrated with embeddings (which dualize as surjections) between Hasse diagrams of both the lattices and the posets. This example can be further extended to a great introduction to Chu spaces. Here are the five Chu spaces over 2 corresponding to the five dual pairs of 3-element posets and height-3 distributive lattices. stuvwxyz stvwxz stvxz stwxz stxz a 10101010 100100 10000 10100 1000 b 11110000 111000 11100 11000 1100 c 11001100 110110 11010 11110 1110 The rows of the first represent the elements of the discrete tripleton {a,b,c} while its columns represent the elements of the 3-cube (8-element Boolean algebra) {s,t,u,v,w,x,y,z} with s at the top and z at the bottom. Deleting the columns violating a <= c yields the second, in which it can be seen at a glance that row a <= row c; its columns form the 6-element distributive lattice dual to the poset ({a,b,c},a<=c). The remaining three matrices are obtained by further deletions to satisfy a <= b, b <= c, or both, in that order. In every case their columns form a distributive lattice, the last being the 4-chain (qua lattice) dual to the 3-chain (qua poset). Deleting being the opposite of adjoining, these rightward-moving transformations can be understand forwardly as monotone functions acting as the identity on the underlying set, or, dually, backwardly as lattice embeddings. Easy but fun, whence pedagogically ideal. Talking of long-standing tradition, Mike Barr and I decided on the phone one day in 1992 to call the objects of the Chu construction Chu spaces. (In our respective published accounts of this conversation I've attributed the actual suggestion to Mike and vice versa.) I now regret that the term "couple" did not occur to either of us. An apt description of a Chu space is as a related pair of complementary objects. "Couple" seems a fine word for such a notion. I've started using it in my papers as a synonym for Chu space. Now if only people would realize that couples are just as important as triplesxxxxxxxmonads. Vaughan Pratt