On Mon, 29 Mar 2004, Peter Freyd wrote:
There's a particular operator that keeps popping up for me.
In an arbitrary heyting algebra define x << y to mean that not only is x less than or equal to y, but the value of y -> x is as small as it can be, that is, y -> x = x. In a complete heyting algebra define an order-preserving, inflationary unary operation s by
sx = inf{ y | x << y }.
E.g.: on a linearly ordered set if { y | x < y } has a least element then that's what sx is. If there is no smallest element above x, then sx = x (even without the completeness hypothesis). In particular, note, there no assertion that x << sx.
The subobject classifier in an elementary topos is complete in the relevant sense: s is definable. A quick description of the construction to follow is that we're going to turn s into the successor operation on an NNO.
DIVERSION: The definition I just gave is the first I came across. The next incarnation for me was when I wanted a measure of the failure of booleaness. In any topos, *A*, there's a largest subterminator B with the property that the slice category *A*/B is boolean. But given any subterminator, U, we have its "closed sheaves", *A*_(U), the full subcat of objects A such that AxU --> U is an iso. (This is a subcategory of sheaves for a Lawvere-Tierney topology. Starting with a space X then Sheaves(X)_(U) may be identified with Sheaves(U'), where U' denotes the complement of U.) Note that the lattice of subterminators in *A*_(U) is isomorphic to the interval of subterminators in *A* from U up. We can define BU to be the largest subterminator in *A*_(U) such that *A*_(U)/BU is boolean. The interval of subterminators in *A* from U up to BU is boolean and in the relevant internal sense, BU is the largest such subterminator. We can, of course, translate this all to a unary operation on Omega.
It's the same operator s.
When one specializes this to a space X it becomes historically familiar if we dualize it it to a deflationary operator on closed subsets. It's the operation that removes isolated points. The very operation that got Cantor started. Hence the word "historically".
I haven't yet digested the rest of Freyd's post, but all of the above, including the notation x << y, the connection with collapsing maximal Boolean intervals, the "historical" connection with Cantor, and a lot more, can be found in a series of papers of Harold Simmons: H. Simmons, "The Cantor-Bendixson analysis of a frame", Seminaire de mathematique pure, Rapport no. 92, Institut de Mathematique Pure, Universite Catholique de Louvain, January 1980. H. Simmons, "An algebraic version of Cantor-Bendixson analysis", in Categorial Aspects of Toplogy and Analysis, pp. 310-323, Springer LNM 915, 1982. H. Simmons, "Near-discreteness of modules and spaces as measured by Gabriel and Cantor", J. Pure and Appl. Alg. 56 (1989), 119-162. H. Simmons, "Separating the discrete from the continuous by iterating derivatives", Bull. Soc. Math. Belg. 41 (1989), 417-463. The operation Freyd is calling s (and the associated relation <<) arose in connection with the so-called Reflection Problem for Frames, namely to characterize those frames that have a reflection into the category of complete Boolean algebras. When such reflections exist, they can be found by iterating the functor A |-> N(A), which freely complements the elements of A (and is also the frame of nuclei on A, ordered pointwise), until it "terminates": A -> N(A) -> N^2(A) -> ... -> N^a(A) -> ... (a in ORD). (These maps are all both mono and epi and are components of natural transformations between iterates of N). A basic result here is that N(A) is Boolean iff x << sx for all x in A. The general reflection problem remains open. -- Todd Wilson A smile is not an individual Computer Science Department product; it is a co-product. California State University, Fresno -- Thich Nhat Hanh