## June 8, 2007

### The Curious Incident of the Dog in the Night-time

#### Posted by David Corfield

Why is the cyclic category failing to bark on this blog? It has a wonderful pedigree (p. 27), and it turns up in all kinds of interesting places, e.g., Jones (p. 20) and Ben-Zvi and Nadler (p. 20).

Loday has interesting things to say about it here.

Its Leinster-Euler characteristic appears to be infinite, which relates to its sharing the same classifying space as the circle.

Posted at June 8, 2007 8:14 AM UTC

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### Re: The Curious Incident of the Dog in the Night-time

Until I clicked, I assumed that Jones was Jones, (which is also a pretty interesting paper).

Posted by: Simon Willerton on June 8, 2007 3:47 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

David,

Thanks for the post and the links!

The early version of my paper with Nadler you linked to has been removed but the paper is available as http://arxiv.org/abs/0706.0322 — and the cyclic category now rears its lovely head on p.20. (Maybe the post could be updated?)

Posted by: David Ben-Zvi on June 8, 2007 4:00 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Posted by: David Corfield on June 8, 2007 4:17 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Thanks for both the links AND the nice title.

Posted by: Kea on June 8, 2007 11:40 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Now you’ve got me curious, David. I’d like to know more about this cyclic category – it seems to be the gateway to a pretty big world. Maybe someone can answer the following questions:

Is $\Lambda$ the walking something or other?

What does the cyclic topos $Set^{\Lambda}$ classify?

Posted by: Todd Trimble on June 9, 2007 10:20 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Great questions, Todd! The place I’d look for answers is Loday’s book Cyclic Homology. But I’m too lazy to hike down to the library, so I’ll just peek at some pages of the online version.

It looks like the fun starts in section 6.1, with an analysis of the relation between $\Delta$ and $\Lambda$. The first fact: $\Delta$ sits inside $\Lambda$, and every morphism $f: m \to n$ in $\Lambda$ factors uniquely as an automorphism of $m$ followed by morphism in $\Delta$. An automorphism of $m$ in $\Lambda$ is just a ‘rotation’ in the group $\mathbb{Z}_{m+1}$.

But at the end one wants some snappy conceptual interpretation! It must exist…

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?

Posted by: John Baez on June 9, 2007 11:30 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Here’s one thing I’m vaguely thinking. If you have for example a 2-category with duals of 1-morphisms and 2-morphisms (having in particular the ‘pivotal’ property), and a ‘necklace’ of 1-morphisms

$A_0 \stackrel{f_1}{\to} A_1 \stackrel{f_2}{\to} ... \stackrel{f_n}{\to} A_n = A_0$

then there ought to be a definable isomorphism

$hom(1_{A_0}, f_n f_{n-1}... f_1) \cong hom(1_{A_1}, f_1 f_n ... f_2)$

(in string diagrams: grab the string labelled $f_1$, and pass it over the top and down the other side with the help of duality – I’m not high-tech enough to have a good picture here).

In particular, if all the $f_i$’s were a given monad $M$, then one could mimic the construction of the cyclic module associated with an associative algebra, described by Connes in his paper (page 27). I am thinking even more particularly of the case where the monad $M = g f$ comes from an ambidextrous adjunction between two dual 1-morphisms $g$ and $f$.

In almost looks like the cyclic category $\Lambda$ was born to do that sort of operation, but I don’t know how to say it nicely!

Posted by: Todd Trimble on June 10, 2007 12:18 AM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Here’s a thought based on yours.

$\Delta$ is the free monoidal category on a monoid. Could $\Lambda$ be the free monoidal category on a ‘cyclic’ monoid, meaning something like a monoid $A$ equipped with ‘cycling’ maps:

$z: A^{\otimes n} \to A^{\otimes n}$

which are compatible with the various multiplication operations:

$\delta_i : A^{\otimes n} \to A^{\otimes (n-1)}$

and unit operations:

$\iota_i : A^{\otimes n} \to A^{\otimes (n+1)}$

(otherwise known as face and degeneracy maps)?

It seems that if one defines ‘compatible’ correctly, it’s almost bound to be true.

Of course any commutative monoid gives a cyclic monoid, but there seems to a subtle generalization going on here. It must be related to the cyclic property of the trace

$tr(a b) = tr(b a)$

especially since this is how Connes relates traces to cyclic cohomology.

Posted by: John Baez on June 11, 2007 6:28 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Could Λ be the free monoidal category on a ‘cyclic’ monoid…?

Unfortunately, I don’t think $\Lambda$ has a monoidal structure! Certainly none compatible with the monoidal structure on $\Delta$: for instance, how would you take the product of the 2-cycle on [2] with the 1-cycle on [1]?

In general, there may be some delicate questions of compatibility involving the mixture of the cycling maps with monoidal structure. If the compatibility includes a Yang-Baxter condition, then it looks like the free thing here is just finite sets. For if the free thing whose objects are tensor powers of $A$ admits an involutive Yang-Baxter operator, then we get symmetric monoidal structure on the free thing, and $A$ would be commutative with respect to the symmetry, and the free symmetric monoidal category with a commutative monoid is given by the category of finite sets. Or am I making a mistake?

Posted by: Todd Trimble on June 11, 2007 10:13 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

I guess that problem you mention with the monoidal structure on $\Lambda$ doesn’t apply to the related $\Delta S$, introduced on p. 12 of Loday’s notes. Rather than just cyclic permutations being allowed before an order preserving map, any permutation is allowed. On p. 14 he calls this $\Delta S$ the category of “noncommutative sets”.

Posted by: David Corfield on June 12, 2007 9:09 AM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

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.

Posted by: John Baez on June 9, 2007 11:46 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

The way I understand it the category $\Lambda$ is just a way to express the circle and its classifying space in simplicial language. In the world of spaces up to homotopy, we have a space $BS^1$, the classifying space of the circle (aka $K(\Z,2)$), which is the homotopy quotient of the point (aka $ES^1$) by the circle $S^1$. (To be pedantic by homotopy world I mean model category, or quasicategory, or any notion of this kind.) So there’s a natural groupoid, point times point over $BS^1$, expressing this quotient construction (just as for any map $X\to Y$ we have a groupoid $X\times_Y X$ expressing the gluing laws imposed on $X$ by the map).

The cyclic category is a good model for this structure in the simplicial world: the homotopy world of simplicial sets is equivalent to that of spaces, so we may equivalently work with simplicial sets in the above story, but try to find nice ones if we want to be concrete, like we think of the point as $ES^1$ in this context. The simplicial category $\Delta$ has as its classifying space a point, and simplicial sets are equivalent to spaces. Likewise $\Lambda$ has as its classifying space $BS^1$ (we just add some morphisms to $\Delta$ to impose circular symmetry), so a cyclic set should just be a space over $BS^1$, as indeed Dwyer-Hopkins-Kan or various results in Loday’s book express. (p.s. I’m sure I’m mixing up $\Lambda$ and $\Lambda^{op}$, but luckily they’re equivalent so one can ignore the cyclic/cocyclic distinction)

If we want to understand also the categorical structure of $\Lambda$, we have just to remember that $\Lambda$ has the same objects as $\Delta$, so we can think of the Hom spaces $\Lambda(m,n)$ as a $\Delta$-bimodule, in fact an algebra (monoidal object) in $\Delta$-bimodules. If we realize this on either factor we just get the circle $S^1$. This sounds confusing (at least to me) but when you unpack all this you find that all of the information of the cyclic category is just modelling the group(oid) $pt \times_{BS^1} pt = S^1$.

So at the end of the day, if you internalize the belief that simplicial sets and spaces are the same thing (ie do homotopy theory), then $\Lambda$ is just a good concrete model for the circle and its action on a point, which we could of course say easily in other languages.

Posted by: David Ben-Zvi on June 10, 2007 6:55 AM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Simplical sets are equivalent to NICE spaces, e.g. the Warsaw circle is NOT included.

Posted by: jim stasheff on June 10, 2007 4:53 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

David Ben-Zvi writes:

If we want to understand also the categorical structure of $\Lambda$, we have just to remember that $\Lambda$ has the same objects as $\Delta$, so we can think of the Hom spaces $\Lambda(m,n)$ as a $\Delta$-bimodule, in fact an algebra (monoidal object) in $\Delta$-bimodules. If we realize this on either factor we just get the circle $S^1$. This sounds confusing (at least to me) but when you unpack all this you find that all of the information of the cyclic category is just modelling the group(oid) $pt \times_{BS^1} pt = S^1$.

Cool, thanks! I enjoy this sort of abstract nonsense, but Todd Trimble actually eats it for breakfast, so I’m hoping he’ll describe $\Lambda$ as the ‘free $\Delta$-bimodule such that…’ — or something like that — and use this to give a slick proof that the nerve of $\Lambda$ is $BS^1$.

Posted by: John Baez on June 11, 2007 6:17 PM | Permalink | Reply to this

### The Cyclic Category

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?

Posted by: Todd Trimble on June 23, 2007 1:35 PM | Permalink | Reply to this

### Re: The Cyclic Category

So $Set^{\Lambda}$ is a walking something, but can $\Lambda$ itself also be seen as a walking something?

Hmm. $\Delta$ is also a walking something, so the topos of simplicial sets is a walking topos with a linearly ordered set?

I guess I’m wondering about a two way ‘mapping’ for descriptions of category and classifying topos as walking Xs.

Posted by: David Corfield on June 23, 2007 8:04 PM | Permalink | Reply to this

### Re: The Cyclic Category

Yes, we seem to have three edges of a square, but I still don’t know how to complete the fourth in a nice way:

• $\Delta^{op}$ is the category of finite intervals (linear orders with top and bottom) and order-preserving maps that preserve top and bottom, and the topos of simplicial sets $Set^{\Delta^{op}}$ classifies intervals.
• $\Lambda$ is the category of finite collars, and $Set^{\Lambda}$ classifies collars.
• $\Delta^{op}$ is the walking comonoid.
• $\Lambda$ is the walking …?

Of course, we could include the fact that $\Delta$ is the walking monoid, and the topos of cosimplicial sets $Set^{\Delta}$ classifies linear orders. So what’s the two-way connection between monoid and linear order?

I’ll just make some surface-level remarks here. One, the monoidal structure on $\Delta$ is the ordinal sum. If you imagine the linearly ordered set $[m]$ as the (free category on the) graph

$1 \to ... \to m$

then the ordinal sum $[m] + [n]$ is obtained by sticking an arrow between the last element $m$ of $[m]$ and the first element 1 of $[n]$. At the topological level, this monoidal structure corresponds to simplicial join.

Two, the monoidal product in $\Delta^{op}$ of $[m]$ and $[n]$, where $[m]$ this time denotes the $m$-element interval, is the interval obtained by identifying the top of $[m]$ with the bottom of $[n]$.

So in these two cases, there is some relation between the category as a walking something involving monoidal structure, and the topos it generates as a walking something involving order structure. As $\Lambda$ contains both $\Delta$ and $\Delta^{op}$, it would be nice to develop a companion statement for the cyclic category and the cyclic topos which would also ‘contain’ these two cases. Somehow it should be simple!

Posted by: Todd Trimble on June 23, 2007 9:44 PM | Permalink | Reply to this

### Re: The Cyclic Category

I retract the assertion that the cyclic category $\Lambda$ is the category of collars (as I defined them). The category of collars has a terminal object, and this alone is fatal.

There may be a fix which would retain what seemed to be desirable features of the approach I wanted to take, but clearly I need to sleep on it.

Posted by: Todd Trimble on June 24, 2007 1:35 AM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Regarding Vaughan Jones’ planar algebras, introduced here, Bar-Natan’s canopoly construction should be of interest here. In what must be the only ArXiv to contain a picture of tins of Jolly Green Giant corn, a canopoly is defined (p. 31) as a collection of categories over a planar algebra, satisfying various conditions.

As it was supposed to mean a ‘city of cans’, not ‘power of cans’, Morrison and Walker corrected it to ‘canopolis’ here (p. 61).

Noah Snyder wrote:

…n-category theory cafe readers should note that “canopolis” means roughly “monoidal 2-category where the 0- and 1-morphisms have duals.”

So all this should be right up our street.

Posted by: David Corfield on June 10, 2007 9:59 AM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Why are they called planar since they involve tangles?

Posted by: jim stasheff on June 10, 2007 4:51 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Well, technically they don’t involve tangles. They involve tangle diagrams, which live in the plane.

Posted by: John Armstrong on June 10, 2007 6:21 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

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

Posted by: David Corfield on June 10, 2007 5:46 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

The cyclic category now has a new description – as a generalized Reedy category.

Posted by: David Corfield on September 22, 2008 2:08 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

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.

Posted by: John Baez on September 23, 2008 4:50 AM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

For crossed simplicial groups you want this paper.

Posted by: David Corfield on September 23, 2008 8:38 AM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Ah, if only I had time to read that. It looks fun! But our fall quarter starts tomorrow… I gotta prepare to teach 120 kids how to do integrals.

It’s more hassle than usual, since we’re switching to a new computer-based homework system where problems are automatically graded by the company that makes the textbook, and I need to learn how that system works.

Why are we switching to this system? Simple: it’s cheaper than paying teaching assistants to do it by hand! Some of the costs are being offloaded to the students: each student needs to pay \$30 to the textbook company for the privilege of having their homework graded. But the math department saves money. Thus the expensive calculus textbook companies have found a way to stave off the threat of free online textbooks.

Lest you think the textbook publishers are the villains here, I should add that our department had given up grading homework for calculus several years ago — we didn’t have enough money to pay people to grade it. So, things are actually getting better now.

On a brighter note, tomorrow I go to the first meeting of Vasiliy Dolgushev’s seminar on Higher Operads. The higher-dimensional algebra revolution is catching fire here at UCR! He’ll lead us through Batanin’s work on the Deligne conjecture. I’ll try to get him to make some notes available online.

Posted by: John Baez on September 24, 2008 4:26 PM | Permalink | Reply to this

### Re: The Curious Incident of the Dog in the Night-time

Now, I remember I was trying to relate the cardinalities of $\Delta$ and $\Lambda$ via the crossed simplicial group definition.

Posted by: David Corfield on September 23, 2008 8:48 AM | Permalink | Reply to this

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