**Background**: A [preorder](http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/Cyclic_order) 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.


This question was [previously asked on Math.SE](https://math.stackexchange.com/questions/456065/what-kind-of-category-is-a-cyclically-ordered-set), but it hasn't prompted an answer. I'm hoping someone here might be able to help, even though it's not a research question for me. I'm just curious. Thanks!



What kind of category is a cyclically ordered set?
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](http://ncatlab.org/nlab/show/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](http://ncatlab.org/nlab/show/cyclic+set)** in the sense of Connes. 


Cyclic sets lie intermediate between [simplicial sets](http://ncatlab.org/nlab/show/simplicial+set) (which is where the nerve construction takes its values for (higher) categories) and [symmetric sets](http://ncatlab.org/nlab/show/symmetric+set) (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](http://en.wikipedia.org/wiki/Poset_topology) of a poset (which is the nerve (regarded as an abstract simplicial complex) of the poset (regarded as a category)). 


