Construction Site
What is an n-category? Something indexed by an ordinal n? But if we want to talk about higher toposes we can't begin by assuming that we know what an ordinal is! The 1-topos Set conveniently has an object of ordinals, characterised by a successor function. But n-logoses, being fundamental entities, cannot be based on diagrams in Set.
Another place where ordinals are found is in simplicial sets, which are Set valued functors from a category Ord of finite ordinals and order preserving functions. Think of real geometric n-simplices on n vertices, like the tetrahedron for the ordinal 3. But once again, in defining Ord, we assumed that we knew what an ordinal was. The only way out of this dilemma seems to be to construct ordinals as we define n-logoses. For an elementary topos, for example, we certainly need the concepts 0 and 1 (empty set and one point set), but the concept of the ordinal 2 could wait until we need it for the triangle of ternary logic. This suggests using this triangle whenever we might be tempted to use the ordinal 2. For example, a simplicial category Ord can't be defined until we have a notion of $\omega$-logos. Sounds like an awful lot of work!
The ancients knew that for a set theoretic cardinal n, there was always a decomposition into prime factors. This is the fundamental theorem of arithmetic, but it does not hold for general fields. Anyway, there's a problem with Euclid's original proof by contradiction: it's not constructive. Just because a statement is false, doesn't mean we can assume that its converse is true! And what is a converse? We've seen that 3-logoses require a notion of ternary complementation. Somewhere in the land of $\omega$-logoses there is an awfully complicated notion of primeness.
If any of this makes sense, it suggests that the infinite branchings of the surreal tree follow the process of logosification. Looking at the positive half only, we see a simple Y tree at the level of 2. At the branches, the number 2 has been defined along with the number 1/2. We conclude that the concept of 2-category must be accompanied by a T-dual concept of 1/2-category!
Maybe the appearance of associahedra as coherence laws for n-categories would be a little less surprising in this setup. That is, an (n+2)-leaved 1-tree polytope appears in the definition of n-category, as n goes to infinity.
Another place where ordinals are found is in simplicial sets, which are Set valued functors from a category Ord of finite ordinals and order preserving functions. Think of real geometric n-simplices on n vertices, like the tetrahedron for the ordinal 3. But once again, in defining Ord, we assumed that we knew what an ordinal was. The only way out of this dilemma seems to be to construct ordinals as we define n-logoses. For an elementary topos, for example, we certainly need the concepts 0 and 1 (empty set and one point set), but the concept of the ordinal 2 could wait until we need it for the triangle of ternary logic. This suggests using this triangle whenever we might be tempted to use the ordinal 2. For example, a simplicial category Ord can't be defined until we have a notion of $\omega$-logos. Sounds like an awful lot of work!
The ancients knew that for a set theoretic cardinal n, there was always a decomposition into prime factors. This is the fundamental theorem of arithmetic, but it does not hold for general fields. Anyway, there's a problem with Euclid's original proof by contradiction: it's not constructive. Just because a statement is false, doesn't mean we can assume that its converse is true! And what is a converse? We've seen that 3-logoses require a notion of ternary complementation. Somewhere in the land of $\omega$-logoses there is an awfully complicated notion of primeness.
If any of this makes sense, it suggests that the infinite branchings of the surreal tree follow the process of logosification. Looking at the positive half only, we see a simple Y tree at the level of 2. At the branches, the number 2 has been defined along with the number 1/2. We conclude that the concept of 2-category must be accompanied by a T-dual concept of 1/2-category!
Maybe the appearance of associahedra as coherence laws for n-categories would be a little less surprising in this setup. That is, an (n+2)-leaved 1-tree polytope appears in the definition of n-category, as n goes to infinity.
posted by Kea | 12:39 PM
4 Comments:
Interesting problems. For some time ago I wrote about quantum physicist's point of view concerning the construction of integers, rationals and algebraics.
I believe that the philosophy underlying axiomatization should reflect very closely the process of how we become conscious about, say prime decomposition of integer. The set theoretic definition of integers does not do this.
It might be that category theory creates these problems by giving up totally the Platonistic notion of number theoretical anatomy in trying to reduce everything to arrows.
In the construction of infinite primes the number theoretic anatomy becomes fundamental and implies also generalization of the number concept.
One very interesting finding is that the quantum states corresponding to infinite primes at the first level of the hierarchy provide representations for ordinary integers, rationals and algebraics and can be mapped to them. Quite generally, higher level represents the level below it (successor again).
The basic conscious experiences about decomposition into factors are assigned with quantum jumps between infinite integers intepreted as many particle states. Number theoretical particle reactions are in question. There is single conservation law: the algebraic number assignable to the state is a multiplicative conserved quantum number. Conservation law states that total numbers of bosons and fermions in modes the labelled by ordinary primes are conserved.
Decomposition into primes becomes the primary concept in this approach. In this representation successor axiom would physically correspond to the possibility of adding arbitrary number of bosons to a given bosonic mode. Successor axiom would be formulated in terms of multiplication by saying that all powers of given prime exist.
Perhaps one could build axiomatization using super-symmetric arithmetic quantum field theory approach as starting point with emphasis on product and co-product aspects of number theoretic particle reactions.
I believe that the philosophy underlying axiomatization should reflect very closely the process of how we become conscious about, say prime decomposition of integer.
Yes, that's it exactly. But I'm convinced the only solid axioms we have as a guide are the elementary topos ones (not counting NNO and successor). And the product/coproduct structure there is highly non-trivial. To me SUSY is a highly derived concept.
I agree that 'factorization' is fundamental. Please, if there's anybody out there that knows of any interesting work on 'number theory topos axioms', TELL us!
I have no precise view about the axioms of topos but understand that you have kind of minimalism in mind. One could argue that there is not much point in minimalism since Goedel in any case says that there is no bound for primary mathematical truths.
I am accustomed to divide mathematics into algebra on one hand and set theory-->topology-->geometry on the other hand, and to freely use physicist's intuition for both separately. I wonder how the axioms of topos relate to this division.
In set theory algebraic operations can be seen as maps from Cartesian square of algebra as linear space to algebra. In more algebraic context one would Cartesian square with tensor square. The latter picture would correspond to the physicist's view about algebraic operation as a fusion of particles. Arithmetic SUSY emphasizes algebraic aspect and is certainly highly derived in an axiomatics starting from set theory view. In my mathematical innocence I am just wondering whether the SUSY picture could define a morning exercise in axiomatization;-).
Hi Kea,
Using 'Category Theory' with the most liberal meaning:
Category: Error-correcting codes
a. Golay
b. Genetic
This is not to say that the genetic code is a Golay code, but apparently some, but not all genetic code errors can be corrected.
1 - Manish Gupta [Queen's U, CA] reviews in IEEE publication 'The Quest for Error Correction in Biology' 2006
http://www.mast.queensu.ca/~mankg/public_html/public_html/home/doc/my_papers/ieeeemb.pdf
2 - Daniels DS, Woo TT, Luu KX, Noll DM, Clarke ND, Pegg AE, Tainer JA.
Skaggs Institute for Chemical Biology, The Scripps Research Institute
'DNA binding and nucleotide flipping by the human DNA repair protein AGT' 2004
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=pubmed&cmd=Retrieve&dopt=AbstractPlus&list_uids=15221026&query_hl=3&itool=pubmed_docsum
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