The n-Category Café

Thanks, I’ve made the changes.

I vaguely recall there’s a deep relation between cyclic objects and objects equipped with an action of the circle group. But what’s the general theorem?

This may not be the most general, but it’s certainly enlightening:

• W. G. Dwyer, M. J. Hopkins, D. M. Kan, The homotopy theory of cyclic sets, Transactions of the American Mathematical Society 291 (1985), 281–289.

“The aim of this note is to show that the homotopy theory of the cyclic sets of Connes is equivalent to that of $SO(2)$-spaces (i.e. spaces with a circle action) and hence to that of spaces over $K(\mathbb{Z},2)$.”

They start by constructing a model category structure on cyclic sets. They also show something David recently mentioned: the nerve of $\Lambda^{op}$ has the homotopy type of $K(\mathbb{Z},2)$.

For those who are jargon-challenged: $SO(2)$ is the circle, and the space $K(\mathbb{Z},2)$ is defined so that if we take all based loops in this space, we get a space homotopy equivalent to the circle!

So, it seems should think of $\Lambda$ or $\Lambda^{op}$ as a discretized version of the circle.

I almost feel like an idiot for not noticing this before, but that Dog of the Curious Incident is wearing a collar!:

• The cyclic category $\Lambda$ is the category of finite collars and collar-preserving maps (just as the category $\Delta$ is the category of finite linear orders and order-preserving maps).

The notion of ‘collar’ should be almost obvious, but I’ll write down the axioms below and give some consequences. As an additional strong hint as to what we mean by ‘collar’, the relation between $\Delta$ and $\Lambda$ can be partly summarized in the slogan,

• A linear chain is a punctured collar

meaning that the category of pointed collars is equivalent to the category of linearly ordered sets.

A collar is a nonempty set $X$ equipped with a ternary relation $C$, where $C(x, y, z)$ intuitively means “reading in Clockwise order starting from $x$, the point $y$ appears before $z$”. Alternatively, if we imagine stereographic projection of the complement of a chosen point $y$ in a collar onto a linearly (i.e., totally) ordered set, that $x \leq z$ in the induced order. Following these intuitions, here are axioms for a collar structure $C$ (all variables are to be universally quantified):

• ($\mathbb{Z}_3$-invariance) $C(x, y, z)$ iff $C(z, x, y)$.
• (Totality) $C(x, y, z) \vee C(z, y, x)$.
• (Reflexivity) $C(x, y, x)$.
• (Transitivity) $C(x, y, z) \wedge C(z, y, w) \Rightarrow C(x, y, w)$.
• (Antisymmetry) $C(x, y, z) \wedge C(z, y, x) \Rightarrow (x = y) \vee (x = z) \vee (y = z)$.

In the jargon of topos-theoretic logic, the theory of collars is a ‘geometric theory’. If we define the cyclic category $\Lambda$ to be the category of finite models of this theory and their homomorphisms, then the topos $Set^{\Lambda}$ is the classifying topos for the theory of collars. (This answers a question raised here.)

In other words, the topos of cyclic sets $Set^{\Lambda}$ is the ‘walking dog with a collar’ (taking ‘dog’ to be a silly synonym for ‘topos’). The generic collar in the cyclic topos is the underlying-set functor

$$U: \Lambda \to Set$$

(this functor is of course representable: $U \cong \Lambda([1], -)$). The pair $(Set^{\Lambda}, U)$ is universal: given a topos $E$ together with a collar object $X$, there is a geometric morphism

$$g: E \to Set^{\Lambda}$$

(with left exact left adjoint $f: Set^{\Lambda} \to E$) and an isomorphism $\phi: f U \to X$, and the pair $(f, \phi)$ is unique up to unique isomorphism.

For example, a collar $X$ in Set is the union (a filtered colimit) of its finite subcollars, which means that $X$ induces a flat right $\Lambda$-module $\hat{X}: \Lambda^{op} \to Set$, given by the corresponding filtered colimit of representables. Tensoring with this flat module gives the required left exact left adjoint

$$\hat{X} \otimes_{\Lambda} -: Set^{\Lambda} \to Set.$$

It should be additional fun to explore the relationships between simplicial sets and cyclic sets by taking advantage of the topos-theoretic point of view. For example, I presume that the simplicial set with one 0-cell and one non-degenerate 1-cell forms an internal collar in simplicial sets, thus setting up a geometric morphism

$$f \dashv g: Set^{\Lambda} \to Set^{\Delta^{op}}$$

and it might be fun to calculate with this example and see what it might say. For example, is there a relation between the monad $g \circ f$ and $\Lambda$ seen as a $\Delta$-bimodule, as David Ben-Zvi pointed out here?

Why are they called planar since they involve tangles?

Whoops, if ‘canopoly’ mimics ‘monopoly’, it would have meant the ‘selling of cans’, from polein. We want polis – city.

I’d rather say that Berger and Moerdijk have generalized the concept of ‘Reedy category’ to include examples like the cyclic category. A Reedy category is a category of ‘shapes’ that you can use to do homotopy theory.

In the old days, the prototypical Reedy category was $\Delta$, the category of simplices. We all know how to do homotopy theory with simplicial sets. Simplicial sets are presheaves on $\Delta$. A Reedy category $R$ is a more general gadget where we can do homotopy theory in the category of presheaves on $R$.

In the old days, a key property of Reedy categories was that all automorphisms are identity morphisms. This is true in $\Delta$, for example. The cyclic category doesn’t have this property. Berger and Moerdijk’s new ‘generalized Reedy categories’ allow nonidentity automorphisms.

One intriguing thing is that in their abstract, Berger and Moerdijk say the cyclic category is the ‘total category of a crossed simplicial group’. I’m not quite what that means, and right now I’m too tired to peek in their paper and find out. But it sounds important and probably not very complicated.