What's in store in dimension 4

Sjoerd Crans, CTRC Seminar 30 March 1999

This is a talk outlining some open problems and possible directions for research.
  1. 4-categories. The pasting theorem says that every pasting diagram has a unique composite, i.e., every two ways to actually carry out this composition have equal value in a 4-category because of the axioms for 4-categories. Viewing these steps as rewrites, the pasting theorem says that the graph whose vertices are ways of composing a certain pasting diagram and whose edges are rewrites is connected. Now one could introduce rewrites of rewrites, etc. Question: is the resulting 4-graph 4-connected?
  2. 4-dimensional teisi. Question: is strict or weak functoriality better? The case for strict is that with weak functoriality composition is not a functor; the case for weak is that with strict functoriality the collection of 3D teisi does not form a 4D tas. Question: how to formalize the process simple graph ~> axiom? Question: do the axioms always `close up'? Question: is there a contractibility theorem, saying that given all axioms stating the equality of certain 4-dimensional composites all the other possible `axioms' follow?
  3. Tetracategories. Question: is there a coherence theorem saying that they are equivalent (in some sense) to 4D teisi (strict, or weaker)? If one wants to do that via the method of ``Coherence for tricategories'' [R. Gordon, A. J. Power, and R. Street, Memoirs Amer. Math. Soc. 117 (1995), no. 558] then one needs a functor st: Tricat -> Gray-Cat, with certain properties for which one needs a development of Gray-dimensional universal algebra, and a representability theorem relating a tetracategory T to (some sort of) Rep (T) = Tricat (T, Gray-Cat).
  4. Operadic 4-categories. Question: is there a coherence theorem?
  5. Nerves. There is an adjoint pair of functors
    Pi: sSets    4D-Teisi :N
    defined via free 4D teisi on n-simplices. Question: for iso-teisi, does this induce an adjunction between the corresponding homotopy categories? Question: do 4D iso-teisi classify homotopy 4-types?
  6. Geometry. One can use surface diagrams in 4 dimensions to do geometrical calculations for 4D teisi. More about this in a later talk.
This work was aided by correspondence with Michael Batanin.

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