What categorification is?
In categorification sets are replaced with categories and elements of sets are replaced with objects. Equations between elements are replaced with isomorphisms between objects: the right and left hand sides of equations are not the same thing but only related by an isomorphism so that they are not tautologies anymore. Functions between sets are replaced with functors between categories taking objects to objects and morphisms to morphisms and respecting the composition of morphisms. Equations between functions are replaced with natural isomorphisms between functors, which must satisfy certain coherence laws representable in terms of commuting diagrams expressing conditions such as commutativity and associativity.
The isomorphism between objects represents equation between elements of set replaces identity. What about isomorphisms themselves? Should also these be defined only up to an isomorphism of isomorphism? And what about functors? Should one continue this replacement ad infinitum to obtain a hierarchy of what might be called n-categories, for which the process stops after n:th level. This rather fuzzy buisiness is what mathematicians like John Baez are actually doing.
There are good motivations for the categofication. Consider the fact that natural numbers. Mathematically oriented person would think number '3' in terms of an abstract set theoretic axiomatization of natural numbers. One could also identify numbers as a series of digits. In the real life the representations of three-ness are more concrete involving many kinds of associations. For child '3' could correspond to three fingers. For a mystic it could correspond to holy trinity. For a Christian "faith,hope,love". All these representations are isomorphic representation of threeness but as real life objects three sheeps and three cows are not identical.
We have however performed what might be called decategorification: that is forgitten that the isomorphic objects are not equal. Decatecorification was of course a stroke of mathematical genius with enormous practical implications: our information society represents all kinds of things in terms of numbers and simulates successfully the real world using only bit sequences. The dark side is that treating people as mere numbers can lead to a rather cold society.
Equally brilliant stroke of mathematical genius is the realization that isomorphic objects are not equal. Decategorization means a loss of information. Categorification brings back this information by bringing in consistency conditions known as coherence laws and finding these laws is the hard part of categorization meaning discovery of new mathematics. For instance, for braid groups commutativity modulo isomorphisms defines a highly non-trivial coherence law leading to an extremely powerful notion of quantum group having among other things applications in topological quantum compuatation.
No-one would have proposed categorification unless it were demanded by practical needs of mathematics. In many mathematical applications it is obvious that isomorphism does not mean identity. For instance, in homotopy theory all paths deformable to each other in continuous manner are homotopy equivalent but not identical. Isomorphism is now homotopy. These paths can be connected and form a groupoid. The outcome of the groupoid operation is determined up to homotopy. The deformations of closed path starting from a given point modulo homotopies form homotopy group and one can interpret the elements of homotopy group as copies of the point which are isomorphic. The replacement of the space with its universal covering makes this distinction explicit. One can form homotopies of homotopies and continue this process ad infinitum and obtain in this manner homotopy groups as characterizes of the topology of the space.
Cateforification as a manner describe finite measurement resolution?
In quantum physics gauge equivalence represents a standard example about equivalence modulo isomorphisms which are now gauge transformations. There is a practical strategy to treat the situation: perform a gauge choice by picking up one representative amongst infinitely many isomorphic objects. At the level of natural numbers a very convenient gauge fixing would correspond the representation of natural number as a sequence of decimal digits rather than image of three cows.
In TGD framework a excellent motivation for categorification is the need to find an elegant mathematical realization for the notion of finite measurement resolution. Finite measurement resolutions (or cognitive resolutions) at various levels of information transfer hierarchy imply accumulation of uncertainties. Consider as a concrete example uncertainty in the determination of basic parameters of a mathematical model. This uncertainty is reflected to final outcome as via a long sequence of mathematical maps and additional uncertainties are produced by the approximations at each step of this process.
How could onbe describe the finite measurement resolution elegantly in TGD Universe? Categorification suggests a natural method. The points equivalent with measurement resolution are isomorphic with each other. A natural guess inspired by gauge theories is that one should perform a gauge choice as an analog of decategorification. This allows also to avoid continuum of objects connected by arrows: reader can easily imagine what a mess results when one tries to do this;-)!
- At space-time level gauge choice means discretization of partonic 2-surfaces replacing them with a discrete set points serving as representatives of equivalence classes of points equivalent under finite measurement resolution. An especially interesting choice of points is as rational points or algebraic numbers and emerges naturally in p-adicization process. One can also introduce what I have called symplectic triangulation of partonic 2-surfaces with the nodes of the triangulation representing the discretization and carrying quantum numbers of various kinds.
- At the level of "world classical worlds" (WCW) this means the replacement of the sub-group if the symplectic group of δ M4× CP2 -call it G- permuting the points of the symplectic triangulation with its discrete subgroup obtained as a factor group G/H, where H is a normal subgroup of G leaving the points of the symplectic triangulation fixed. One can also consider subgroups of the permutation group for the points of the triangulation. One can also consider flows with these properties to get braided variant of G/H. It would seem that one cannot regard the points of triangulation as isomorphic in the category theoretical sense. This because, one can have quantum superpositions of states located at these points and the factor group acts as the analog of isometry group. One can also have many-particle states with quantum numbers at several points. The possibility to assign quantum numbers to a given point becomes the physical counterpart for the axiom of choice.
What is so fantastic is that finite measurement resolution leads to a replacement of the infinite-dimensional world of classical points with a discrete structure. Therefore operation like integration over entire "world of classical worlds" is replaced with a discrete sum. This makes things much easier- believe or not - and if not try yourself;-).
- What suggests itself strongly is a hierarchy of n-categories as a proper description for the finite measurement resolution. The increase of measurement resolution means increase for the number of braid points. One has also braids of braids of braids structure implied by the possibility to map infinite primes, integers, and rationals to rational functions of several variables and the conjecture possibility to represent the hierarchy of Galois groups involved as symplectic flows. If so the hierarchy of n-categories would correspond to the hierarchy of infinite primes having also interpretation in terms of repeated second quantization of an arithmetic SUSY such that many particle states of previous level become single particle states of the next level.
- The included algebra represents finite measurement resolution in the sense that its action generates states which are cannot be distinguished from each other within measurement resolution used. The natural conjecture is that this indistuinguishability corresponds to a gauge invariance for some gauge group and that TGD Universe is analogous to Turing machine in that almost any gauge group can be represented in terms of finite measurement resolution.
- Second natural conjecture inspired by the fact that symplectic groups have enormous representabive power is that these gauge symmetries allow representation as subgroups of the symplectic group of δ M4× CP2. A nice article about universality of symplectic groups is the article The symplectification of science by Mark. J. Gotay.
- An interesting question is whether there exists a finite-dimensional space, whose symplecto-morphisms would allow a representation of any gauge group (or of all possible Galois groups as factor groups) and whether δ M4× CP2 could be a space of this kind with the smallest possible dimension.
Arrows are not all that is needed
There have been proposals that categories could be fundamental and space-time, symmetries, and particles could emerge in some sense. Personally I do not find this idea sound.
- Categories consist of discrete objects and on basis of above arguments provide indispensable tool for physicist and consciousness theorist since both measurement resolution and cognitive resolution are always finite. In fact, finite resolution is not at all a negative thing since it forces abstraction process by forming equivalence classes from objects not distinguishable from each other. Written language is one of the victories of abstraction process: just a sequence of letters "human" becomes are representation for entire species. It would be however nonsense to assume that the world is actually discrete. Practically all physics would be lost and only manner to get it would be by effectively replacing the discrete structures with continuum. Mathematics would suffer the same fate and there would be very little use for category theory.
- I find also the idea of throwing away group theory as very weird. Isomorphisms between objects form groups and are the corner stone of category theory: why should one throw them away? 90 per cent of recent day quantum physics is group theory and the above arguments suggest that category theory is natural in the description of finite measurement resolution reducing the infinite-dimensional groups involved to discrete groups and giving also a profound connection with number theory. Without symmetries we do not have observables which in quantum theory correspond to Lie algebras for continuous groups. As a matter fact, in TGD framework the role of symmetries is taken to extreme: zero energy states correspond to Lie algebra for an infinite-dimensional Yangian. The world of quantum worlds is Lie algebra.
- It has been also suggested that so called associahedrons emerging in n-category theory could replace space-time and space as fundamental objects. Associahedrons are polygons used to represent geometrically associativity or its weaker form modulo isomorphism for the products of n objects bracketed in all possible manners. The polygon defines a hierarchy containing sub-polygons as its edges containing.... Associativity states the isomorphy of these polygons. According to John Baez associahedrons indeed allow a beautiful geometric realization of the coherence laws.
One must however not forget that the very notion of associahedron is an auxiliary tool which assumes the notion of Euclidian space so that the claim about emergence of space from category theory is an illusion just as the claims that continuous space-time can emerge from a discrete lattice at infrared limit.
One should also remember that the notion of n-category has its roots in homotopy theory which describes topological invariants of various spaces. Only non-sense with arrows remains if one throws away all those structures whose description has motivated the development of the category theoretical approach. This kind of emergence is also in conflict with the very idea of categorification since it would identify the isomorphic points- say points of continuum equivalent within measuremet resolution- to get discrete structure and then conclude that this discrete structure is all that exists.