Operadification III

The multicategorical analogue of the natural number diagram

$1 \rightarrow N \rightarrow N$

looks like

$\Delta^{\cap} \rightarrow \textrm{Tree} \rightarrow \textrm{Tree}$

where the category $\Delta^{\cap}$ (dimension not specified) has objects $n$ represented by single level trees, the associahedra trees. That is, since we are allowed any number of input identity arrows, the simplest one object category has an arrow for each $n$. The category Tree, by definition, extends these single level trees to $k$-ordinal trees of $k$ levels. In other words, the ordinals $N$ in Set are replaced by the levels of the $k$-ordinal trees. This is how we wanted to represent $n$ in the quantum world, in association with dimension.

Now recall that the $k$-ordinal trees can represent Batanin's polytopes, which are topological spaces. The successor map simply adds a leaf to every top branch. For example, the sequence of $k$ dimensional spheres arises as a version of the ordinals in this sense.

Multicategorical arithmetic therefore compares an ordinary cosimplicial object in $C$ with a weakened kind of
cohomological object Tree $\rightarrow C$. By truncating the categories at level $k$, one obtains a multicategorical analogue of modular number objects. There are many motives for studying this kind of arithmetic.