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From: Todd Wilson <twilson@csufresno.edu>
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Subject: categories: Re: on the axiom of infinity
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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