# What's in store in dimension 4

This is a talk outlining some open problems and possible directions for
research.
- 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?
- 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?
- 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**).
- Operadic 4-categories. Question: is there a coherence theorem?
- Nerves. There is an adjoint pair of functors
<--
: s**Sets** 4**D-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?
- 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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