This post consists of some simple, possibly incorrect, thoughts that I had on surreal numbers and $\infty$-operads. Let’s begin with a general statement. There are many equivalent models for $\infty$-operads. All are Quillen equivalent, and in particular, we can state:

Theorem: There is a chain of Quillen equivalences between model categories presenting $\infty$-operads: $(\mathrm{Set}_{\mathbf{\Delta}})_{/\mathrm{N}\mathbf{\Delta}^{op}_{\mathrm{Fin}_\ast}}$ is Quillen equivalent to $\mathrm{Op}_\infty\simeq_Q\mathrm{Set}_\mathbf{\Omega}$.

(One question which I have: how can we prove, directly, the equivalence between the first and the last model category in the chain of Quillen equivalences above? I only know of a way of proving it by showing that both are equivalent to Lurie’s model of $\infty$-operads.)

Here, $\mathbf{\Omega}$ is the dendroidal category, i.e., the category of finite rooted trees considered as operads. Our interest in this section will be to motivate the study of $(\infty,2)$-operads by relating the theory of $\infty$-operads to the surreal construction; a study of $(\infty,2)$-operads could possibly provide input into how this construction generalizes, and hence into the surreal numbers themselves. We will provide a brief review of the surreal construction, and state how this can be generalized to the “unique extension property”, a term coined by Lurie in a talk given in 2000 at an MSRI conference.

A surreal number corresponds to two sets of numbers constructed previously, i.e., $x=(X_L,X_R)$, such that if $x_L\in X_L$ and $x_R\in X_R$, then $x_L\not\geq x_R$. For example, when $X_L=X_R=\emptyset$, then $x=:0$. Similarly, we can define $1:=(\emptyset,\{0\})$, and $-1:=(\{0\},\emptyset)$. Infinitely many numbers can be created in this construction, because if $X_L$ or $X_R$ is $\emptyset$, then the required condition is trivially satisfied. A naïve way to look at this construction would be to say that the surreal numbers are constructed in stages, when in stage $n$ we use whatever is constructed thus far, and we consider all possible ways to adjoin a single element to the constructed set. The unique extension property says that certain sets $S$ satisfy the following condition: given $S\cup \{x\}\supseteq S\subseteq S\cup\{y\}$, either $S\cup \{x\}\simeq S\cup\{y\}$ by an isomorphism fixing $S$, or there is a unique set $S\cup \{x,y\}$ such that $S\cup \{x,y\} \supseteq S\cup \{x\}\supseteq S\subseteq S\cup\{y\}\subseteq S\cup \{x,y\}$. Linearly ordered sets are examples; in fact, each stage of the surreal construction yields a linearly ordered set. The class of sets $S$ satisfying the unique extension property also includes “circular sets”.

However, what we are interested in is the collection of finite rooted trees: any rooted tree satisfies the unique extension property. In particular, given a trivial tree, i.e., a marked vertex in the trivial graph, we can adjoin a single edge in only one way. This is the first stage of the construction. In the second stage, we can choose to either attach an edge to the root or to the vertex on the boundary of the edge. The unique extension property allows us to adjoin these two edges in one stage, which yields the second stage of the construction. Iterating this gives us the collection of finite rooted trees, which is simply the set of objects of the dendroidal category!

This provides a definitive program for generalizing $\infty$-operads. Let $\mathscr{C}$ be a category whose underlying set of objects, $S$, satisfies the unique extension property. We can then iterate the surreal construction of adjoining elements in stages to the set, which allows us to expand $S$ to its “surreal-ification”, denoted $\overline{S}$. This allows us to expand $\mathscr{C}$, too, into a category $\overline{\mathscr{C}}$, by specifying, for every object constructed in each stage of the surreal construction, maps to and from elements of $S$. We may then consider presheaves on $\overline{\mathscr{C}}$, and ask for a model structure on the resulting category which reduces to the ordinary model category of dendroidal sets in the case where $\overline{\mathscr{C}}=\mathbf{\Omega}$, i.e., $\mathscr{C}$ is the category containing the trivial tree regarded as an operad. This leads to surreal $\infty$-operads over $\mathscr{C}$, and these in turn should provide deep combinatorial information about the surrealification $\overline{\mathscr{C}}$ of $\mathscr{C}$.

What do you think?
SKD