I have been trying to understand whether category theory might provide some deeper understanding about quantum TGD, not just as a powerful organizer of fuzzy thoughts but also as a tool providing genuine physical insights. Kea is also interested in categories but in much more technical sense. Her dream is to find a category theoretical formulation of M-theory as something, which is not the 11-D something making me rather unhappy as a physicist with second foot still deep in the muds of low energy phenomenology.
Kea talks about topos, n-logos,... and their possibly existing quantum variants. I have used to visit Kea's blog in the hope of stealing some category theoretic intuition. It is also nice to represent comments knowing that they are not censored out immediately if their have the smell of original thought: this is quite too often the case in alpha male dominated blogs. It might be that I had luck this morning!
1. Locales, frames, Sierpinski topologies and Sierpinski space
Kea mentioned the notions of locale and frame . In Wikipedia I learned that complete Heyting algebras, which are fundamental to category theory, are objects of three categories with differing arrows. CHey, Loc and its opposite category Frm (arrows reversed). Complete Heyting algebras are partially ordered sets which are complete lattices. Besides the basic logical operations there is also algebra multiplication. From Wikipedia I learned also that locales and the dual notion of frames form the foundation of pointless topology. These topologies are important in topos theory which does not assume the axiom of choice.
So called particular point topology assumes a selection of single point but I have the physicist's feeling that it is otherwise rather near to pointless topology. Sierpinski topology is this kind of topology. Sierpinski topology is defined in a simple manner: set is open only if it contains a given point p. The dual of this topology defined in the obvious sense exists also. Sierpinski space consisting of just two points 0 and 1 is the universal building block of these topologies in the sense that a map of an arbitrary space to Sierpinski space provides it with Sierpinski topology as the induced topology. In category theoretical terms Sierpinski space is the initial object in the category of frames and terminal object in the dual category of locales. This category theoretic reductionism looks highly attractive to me.
2. Particular point topologies, their generalization, and finite measurement resolution
Pointless, or rather particular point topologies might be very interesting from physicist's point of view. After all, every classical physical measurement has a finite space-time resolution. In TGD framework discretization by number theoretic braids replaces partonic 2-surface with a discrete set consisting of algebraic points in some extension of rationals: this brings in mind something which might be called a topology with a set of particular algebraic points.
Perhaps the physical variant for the axiom of choice could be restricted so that only sets of algebraic points in some extension of rationals can be chosen freely. The extension would depend on the position of the physical system in the algebraic evolutionary hierarchy defining also a cognitive hierarchy. Certainly this would fit very nicely to the formulation of quantum TGD unifying real and p-adic physics by gluing real and p-adic number fields to single super-structure via common algebraic points.
There is also a finite measurement resolution in Hilbert space sense not taken into account in the standard quantum measurement theory based on factors of type I. In TGD framework one indeed introduces quantum measurement theory with a finite measurement resolution so that complex rays becomes included hyper-finite factors of type II1 (HFF, see this).
This program, which I formulated only after this section had been written, might indeed make sense (ideas never learn to emerge in the logical order of things;-)). The lucky association was to the ideas about fuzzy quantum logic realized in terms of quantum 2-spinor that I had developed a couple of years ago. Fuzzy quantum logic would reflect the finite measurement resolution. I just list the pieces of the argument.
Spinors and qbits: Spinors define a quantal variant of Boolean statements, qbits. One can however go further and define the notion of quantum qbit, qqbit. I indeed did this for couple of years ago (the last section in Was von Neumann Right After All?).
Q-spinors and qqbits: For q-spinors the two components a and b are not commuting numbers but non-Hermitian operators. ab= qba, q a root of unity. This means that one cannot measure both a and b simultaneously, only either of them. aa+ and bb+ however commute so that probabilities for bits 1 and 0 can be measured simultaneously. State function reduction is not possible to a state in which a or b gives zero! The interpretation is that one has q-logic is inherently fuzzy: there are no absolute truths or falsehoods. One can actually predict the spectrum of eigenvalues of probabilities for say 1. q-Spinors bring in mind strongly the Hilbert space counterpart of Sierpinski space. One would however expect that fuzzy quantum logic replaces the logic defined by Heyting algebra.
Q-locale: Could one think of generalizing the notion of locale to quantum locale by using the idea that sets are replaced by sub-spaces of Hilbert space in the conventional quantum logic. Q-openness would be defined by identifying quantum spinors as the initial object, q-Sierpinski space. a (resp. b for dual category) would define q-open set in this space. Q-open sets for other quantum spaces would be defined as inverse images of a (resp. b) for morphisms to this space. Only for q=1 one could have the q-counterpart of rather uninteresting topology in which all sets are open and every map is continuous.
Q-locale and HFFs: The q-Sierpinski character of q-spinors would conform with the very special role of Clifford algebra in the theory of HFFs, in particular, the special role of Jones inclusions to which one can assign spinor representations of SU(2). The Clifford algebra and spinors of the world of classical worlds identifiable as Fock space of quark and lepton spinors is the fundamental example in which 2-spinors and corresponding Clifford algebra serves as basic building brick although tensor powers of any matrix algebra provides a representation of HFF.
Q-measurement theory: Finite measurement resolution (q-quantum measurement theory) means that complex rays are replaced by sub-algebra rays. This would force the Jones inclusions associated with SU(2) spinor representation and would be characterized by quantum phase q and bring in the q-topology and q-spinors. Fuzzyness of qqbits of course correlates with the finite measurement resolution.
Q-n-logos: For other q-representations of SU(2) and for representations of compact groups (see appendix of this) one would obtain something which might have something to do with quantum n-logos, quantum generalization of n-valued logic. All of these would be however less fundamental and induced by q-morphisms to the fundamental representation in terms of spinors of the world of classical worlds. What would be however very nice that if these q-morphisms are constructible explicitly it would become possible to build up q-representations of various groups using the fundamental physical realization - and as I have conjectured (see this) - McKay correspondence and huge variety of its generalizations would emerge in this manner.
The analogs of Sierpinski spaces: The discrete subgroups of SU(2), and quite generally, the groups Zn associated with Jones inclusions and leaving the choice of quantization axes invariant, bring in mind the n-point analogs of Sierpinski space with unit element defining the particular point. Note however that n≥3 holds true always so that one does not obtain Sierpinski space itself. Could it be that all of these n preferred points belong to any open set? Number theoretical braids identified as subsets of the intersection of real and p-adic variants of algebraic partonic 2-surface define second candidate for the generalized Sierpinski space with set of preferred points. Recall that the generalized imbedding space related to the quantization of Planck constant is obtained by gluing together coverings of M4×CP2→ M4×CP2/Ga×Gb along their common points. The topology in question would mean that if some point in the covering belongs to an open set, all of them do so. The interpretation could be that the points of fiber form a single inseparable quantal unit.
For more details see the chapter Was von Neumann Right After All?.