Categorification and the periodic table of categories

Periodic Pattern of Human Knowing (Part #9)

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Within this framework, the articulation of John Baez (The Dimensional Ladder, 2005) regarding the "Periodic Table" (of n-categories) in the light of a "Ladder of n-Categories" is of particular interest, notably with respect to the relationships to physics (see also John Baez, Categories, Quantization, and Much More, 2006). .

The appropriateness of the "ladder" metaphor has of course been challenged from some perspectives, notably by feminists. The same might be said of the "table" metaphor, as discussed separately (Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori, 2006). For example, Alison Bailey (Locating Traitorous Identities: toward a view of privilege-cognizant white character, In: Deborah Orr, et al. Feminist Politics, 2007) uses language that could appropriately inform a richer understanding of a Periodic Table of Life:

Not a climb up the ladder, but a discarding of the hierarchies and rigidites implicit in the ladder metaphor. It involves recognition of the plasticity and the thickness of identities, a new understanding of the ways in which identities interlock, being freed from old limitations, and the emergence of new possibilities.... When we pay attention only to statistical ladders, we tend to substitute two-dimensional markers for the multidimensional situations whose changes need to be evaluated. We fail to see the ways in which the ladder conceals the composition of the masses struggling at the bottom. The trick is to see "rising" more multidimensionally: not as progress up defined ladders, but as the yeast that allows the dough to spring back against the hands that knead it -- the pressure that expands. It empowers through change in the structure of our identities and the possibilities inherent in the categories that locate us; change in the categories that we locate; change in our relations to one another. (p. 181) .

Categorification might then be understood as a move towards a meta-process through which disparate schemas may be organized at a higher level of abstraction -- posing the question of whether their advocates subscribe to their being subsumed in this way.

Although seemingly (and perhaps necessarily) abstruse, categorification is currently a focus for interrelating disparate concerns as indicated by the preoccupations of a workshop on Categorification and Geometrisation from Representation Theory (Glasgow, 2009). This event is premised on the recognition that:

For a long time the idea of categorification has been in the background of many ideas in algebraic Lie theory and its connections to geometry. Several hard questions in Lie theory have been solved by translation (often via geometry) into combinatorics. For example, irreducible modules are labelled by combinatorial data and multiplicity formulas can be computed via combinatorially defined polynomials. On the other hand, topological questions are sometimes transferred into combinatorics in order to produce a clean answer: combinatorially defined knot invariants via polynomials; changing of coordinate systems via mutation rules; etc. It is becoming increasingly clear that the connecting principle of many such results in both Lie theory and topology is the idea of categorification. The notion of "ecategorification" goes back to Crane and Frenkel, motivated by mathematical physics, and in particular by the hope to construct higher dimensional topological quantum field theories.

Current concerns are to clarify the notion of categorification and its appearance in three different areas of mathematics: algebraic geometry, symplectic geometry and representation theory.

The further development of a generalized periodic table of n-categories is currently an active concern in relation to homology and cohomology, notably as a generalization of the Witt group. This is an abelian group whose elements are represented by symmetric bilinear forms over the field. Two symmetric bilinear forms are equivalent if one can be obtained from the other by adding zero or more copies of a hyperbolic plane (the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector). The Witt group is considered to be the simplest case of the cohomology of the periodic table of n-categories as recently discussed (Noah Snyder, The Witt group, or the cohomology of the periodic table of n-categories 30 March 2009). Homology in mathematics is a procedure to associate a sequence of abelian groups or modules with a given mathematical object. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Both might best be understood as methods for the detection of degrees of isomorphism.

Such developments are of course a delightful exploration of abstractions -- to the extent that anyone (especially this writer) can rise to the challenge of comprehension they represent. It is to be hoped that they give rise to greater insight into how the dramatic differences in society can be related, if this remains of relevance. Unfortunately it remains unclear as to whether such developments are capable of organizing the representation of the field of mathematics itself in more meaningful and accessible ways -- the challenge to which Corfield pointed (above) -- especially if there are "competing" views on how this is to be achieved and whether the formulations of such disparate views can themselves be integrated into an overarching theory, through some understanding of "complementarity".

Essentially it would appear that there is a capacity to discover and organize categories more coherently at higher orders of abstraction. The Periodic Table, as commonly known, is then potentially to be understood as a relatively simple instance of this. The question is whether it is the most readily comprehensible pattern which may be indicative of the nature of a possible Periodic Pattern of Human Life -- or whether such a pattern only emerges at yet higher orders of abstraction, beyond average capacity to comprehend it in any useful way.

Whether the Periodic Table can fruitfully be considered as a metaphor of human life, Kenneth Boulding, as cofounder of general systems theory, offers the following insight relating to such use of metaphor in providing an integrative understanding of human life:

Our consciousness of the unity of self in the middle of a vast complexity of images or material structures is at least a suitable metaphor for the unity of group, organization, department, discipline or science. If personification is a metaphor, let us not despise metaphors -- we might be one ourselves. (Ecodynamics; a new theory of societal evolution, 1978)

If "cohomology" is the key to the way forward (or "upward"), then it is appropriate to point out that the provocative checklist of "co-" terms (above) includes terms that bear on the self-reflexive comprehensiveness of any such pattern. Most of the relevant literature is subject to "copyright" and there is a high degree of (deniable) "competition" between mathematicians seeking to affirm their identities through claims on new conceptual territory -- replicating, without addressing, challenges faced by society, as discussed elsewhere (And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000; Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007). It might be asked whether there is any greater irony than efforts to copyright the ultimate periodic table of categories.

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