# What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms that give us partial orders etc., so plenty of everyday "order" concepts can be modeled with such a binary relation.

Similarly, a category has a composition of morphisms $\circ$, which has identities and sends $C(a,b)\times C(b,c)\to C(a,c)$. So we can also model an preordered set as a category in which every hom-set has at most one morphism. I haven't really studied this perspective, but I gather that it's useful.

Now, a cyclic ordering is more naturally a ternary relation! Its version of transitivity states that $(a,b,c)\wedge(a,c,d)\to(a,b,d)$. A cyclic ordering is also cyclic: $(a,b,c)\to(b,c,a)$. If you hunt around Wikipedia you can find a few different kinds of cyclic orders with more or less restrictive axioms. For this question, let's understand "cyclic order" broadly, so it might be strict or not, and it might be total or not: whatever is convenient!

Question: Would it be useful to model cyclically ordered sets as categories? How would you do it?

I'm guessing that higher categories might be useful, so that you can replace the ternary relation $(a,b,c)$ with a 2-morphism like $(a\rightarrow b)\Rightarrow(a\rightarrow c)$, and there could be a composition of 2-morphisms like $$\left[(a\rightarrow b)\Rightarrow(a\rightarrow c)\right]\times\left[(a\rightarrow c)\Rightarrow(a\rightarrow d)\right]\mapsto\left[(a\rightarrow b)\Rightarrow(a\rightarrow d)\right].$$ But I'm getting way out of my depth here! I've skimmed over some higher categories on nLab, and I don't see a kind of 2-morphism that does quite what I want. Simplicial 2-morphisms looked promising at first, but they don't seem to compose in the right way. Is that idea a dead end?

Note that I'm not asking about any category of monotone functions between cyclically ordered sets. At least, I don't think that's what I'm asking.

Edits: I've revised the wording of the question slightly. Also, I've gone ahead and cross-posted the question on MO.

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## 1 Answer

From Qiaochu Yuan's answer on MO:

It seems like cyclically ordered sets ought to be regarded as cousins of categories rather than as categories themselves. More precisely, a cyclically ordered set $C$ has a nerve $N(C)$ analogous to the nerve of a (higher) category or (higher) groupoid. The nerve is first of all a sequence $N(C)_n$ of sets, namely the set of $n$-tuples $(a_1, ... a_n)$ such that the relation $(a_i, a_j, a_k)$ holds for all $1 \le i < j < k \le n$ (or two of $a_i, a_j, a_k$ are equal). There are various natural maps between the $N(C)_n$ which make $N(C)$ a cyclic set in the sense of Connes.

Cyclic sets lie intermediate between simplicial sets (which is where the nerve construction takes its values for (higher) categories) and symmetric sets (which is where the nerve construction takes its values for (higher) groupoids). They are in particular simplicial sets, so you can think of cyclically ordered sets as modeled by simplicial sets with extra data. This construction is a beefed-up version of the construction of the order complex of a poset (which is the nerve (regarded as an abstract simplicial complex) of the poset (regarded as a category)).

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