# The cycle category

## Idea

Alain Connes's cycle category $\Lambda$ (sometimes denoted $\mathcal{C}$), often called his cyclic category or category of cycles, is a small category whose presheaves – called cyclic sets or more generally cyclic objects – are somewhere intermediate between simplicial sets and symmetric sets. It strictly contains the simplex category, and has cyclic groups for automorphism groups. Among its virtues, it is a self-dual category.

The cycle category is used for the description of the cyclic structure on Hochschild homology/Hochschild cohomology and accordingly for the description of cyclic homology/cyclic cohomology.

## Definitions

Multiple descriptions of the cycle category $\Lambda$ are possible, but a convenient starting point is to consider first a category $L$ whose objects are natural numbers $n \geq 0$, and where the hom-set $L(m, n)$ consists of increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ satisfying the “spiraling property”, that $f(i + m + 1) = f(i) + n + 1$, with composition given by functional composition. Then, define $\Lambda$ to be a quotient category of $L$ having the same objects, with $\Lambda(m, n) = L(m, n)/\sim$ where $\sim$ is the equivalence relation for which $f \sim g$ means $f - g$ is a constant multiple of $n+1$. Let $q: L \to \Lambda$ be the quotient.

###### Remark

Notice that $f \in L(m, n)$ is completely determined by the values $f(0), \ldots, f(m)$. There is a faithful embedding $i: \Delta \to L$ which on objects is the identity, where $f \in L(m, n)$ belongs to the image of $i$ iff $0 \leq f(0)$ and $f(m) \leq n$. The composite

$\Delta \stackrel{i}{\hookrightarrow} L \stackrel{q}{\to} \Lambda$

is again faithful, so that the simplex category sits inside $\Lambda$.

###### Remark

Of course the successor function $\tau: \mathbb{Z} \to \mathbb{Z}$ gives a function $\tau_n \in L(n, n)$ defined by $\tau_n(i) = i+1$, which in turn induces a function $q(\tau) \in \Lambda(n, n)$ such that $q(\tau)^{n+1} = 1_n$. In this way, we have inclusions $\mathbb{Z}/(n+1) \hookrightarrow \Lambda(n, n)$ of cyclic groups inside $\Lambda$.

Cyclic objects in a category $C$ are the contravariant functors $\Lambda^{\mathrm{op}}\to C$, cocyclic objects? are the covariant functors $\Lambda\to C$. Note that $\Lambda$ itself is, via its inclusion into $Cat$, an example of a cocyclic object in $Cat$. (Thus, the common term “the cyclic category” to refer to $\Lambda$ is misleading, just like using “the simplicial category” to refer to the simplex category $\Delta$.)

If $A$ is an abelian category then the category of $A$-presheaves on $\Lambda$ is usually called (Connes's) category of cyclic modules in $A$.

## Structure of the cycle category

To analyze the structure of $\Lambda$ further, we make a series of easy observations. These are largely based on Elmendorf 93.

###### Proposition

Every morphism $f$ of $L$, regarded as a functor $\mathbb{Z} \to \mathbb{Z}$, has a left adjoint $f^\ast: \mathbb{Z} \to \mathbb{Z}$ that is also a morphism of $L$. Similarly, every morphism $f$ of $L$ has a right adjoint $f_\ast$ belonging to $L$.

###### Proof

By the spiraling property of $f$, for any $j \in \mathbb{Z}$ the comma category $(j \downarrow f)$ as a subset of $\mathbb{Z}$ has a lower bound in $\mathbb{Z}$ and hence is well-ordered. It is also nonempty, and we define $f^\ast(j)$ to be the least element of $(j \downarrow f)$. In other words $f^\ast(j)$ is the least $i$ such that $j \leq f(i)$. It is easy to check that $f^\ast$ obeys the spiraling property $f^\ast(j+n+1) = f^\ast(j)+m+1$, since

$\array{ f^\ast(j+n+1) \leq f^\ast(j)+m+1 & iff & j+n+1 \leq f(f^\ast(j)+m+1) \\ & iff & j+n+1 \leq f(f^\ast(j))+n+1 \\ & iff & j \leq f(f^\ast(j)) \\ & iff & f^\ast(j) \leq f^\ast(j) }$

and

$\array{ f^\ast(j)+m+1 \leq f^\ast(j+n+1) & iff & f^\ast(j) \leq f^\ast(j+n+1)-m-1 \\ & iff & j \leq f(f^\ast(j+n+1)-m-1) \\ & iff & j \leq f(f^\ast(j+n+1))-n-1 \\ & iff & j + n + 1\leq f(f^\ast(j) + n + 1) \\ & iff & f^\ast(j+n+1) \leq f^\ast(j+n+1). }$

Also, since $(\mathbb{Z}, \leq)$ as a category is self-dual, every morphism $f$ of $L$ dually has a right adjoint that is a morphism of $L$.

###### Corollary

$L$ is a self-dual category.

###### Proof

The duality functor $L^{op} \to L$ is the identity on objects and takes $f: m \to n$ to $f^\ast: n \to m$. It is contravariant since the left adjoint of a composite $f g$ is $g^\ast f^\ast = (f g)^\ast$. It is an equivalence because its inverse is the right-adjoint mapping, $f \mapsto f_\ast$.

###### Proposition

$\Lambda$ is a self-dual category.

###### Proof

If $f \sim g$ in $L(m, n)$, then $f = \tau^{k (n+1)} \circ g$ for some $k \in \mathbb{Z}$. Observe that $\tau^\ast = \tau^{-1}$, so $f^\ast = g^\ast \circ \tau^{-k(n+1)} = \tau^{-k(m+1)} \circ g^\ast$ where the last equation holds because $g^\ast: n \to m$ is spiraling. This shows $f^\ast \sim g^\ast$, i.e., the self-duality of $L$ descends to $\Lambda$.

###### Proposition

For a morphism $f \in L(m, n)$, we have $f^\ast(0) \leq 0$ iff $0 \leq f(0)$, and $0 \leq f^\ast(0)$ iff $f(m) \leq f(n)$. Hence $f^\ast(0) = 0$ iff ($0 \leq f(0)$ and $f(m) \leq n$).

###### Proof

The first assertion is immediate from the adjunction $f^\ast \dashv f$. The second follows from the deduction

$\array{ 0 \leq f^\ast(0) & iff & -1 \lt f^\ast(0) \\ & iff & \neg (f^\ast(0) \leq -1) \\ & iff & \neg (0 \leq f(-1)) \\ & iff & f(-1) \lt 0 \\ & iff & f(m) \lt n+1 \\ & iff & f(m) \leq n }$

where the step to the penultimate line used the spiraling property.

The previous proposition, in conjunction with the self-duality of $L$ and Remark , shows that $\Delta^{op}$ faithfully maps to $L$ by $\Delta^{op}(m, n) \cong \{f \in L(m, n): f(0) = 0\}$. Passing to the quotient $q: L \to \Lambda$, this description also realizes $\Delta^{op}$ as sitting inside $\Lambda$, and the next result is immediate.

###### Proposition

Every morphism $f: m \to n$ in $\Lambda$ may be uniquely decomposed as $f = \tau_n^{f(0)} g$ where $g$ belongs to $\Delta^{op}(m, n) \subset L(m, n)$, and the exponent $f(0)$ is considered modulo $n+1$.

###### Proposition

The cyclic group $\mathbb{Z}/(m+1)$ acts on $\Delta^{op}(m, n)$ via the following formula for $f \in L(m, n), f(0) = 0$:

$k \cdot f = \tau^{-f(k)} \circ f \circ \tau^k$

or in other words, via $(k \cdot f)(i) \coloneqq f(k+i) - f(k)$.

###### Proof

Clearly $k \cdot f \in \{g \in L(m, n): g(0) = 0\}$. We calculate

$\array{ j \cdot (k \cdot f) & = & \tau^{-(k \cdot f)(j)} \circ (k \cdot f) \circ \tau^j \\ & = & \tau^{-(f(j+k) - f(k))} \circ \tau^{-f(k)} \circ f \circ \tau^k \circ \tau^j \\ & = & \tau^{-f(j+k)} \circ f \circ \tau^{j+k} \\ & = & (j + k) \cdot f. }$

Moreover, $((m+1)\cdot f)(i) = f(i+m+1)-f(0+m+1) = f(i)+n+1 - (f(0)+n+1) = f(i) - f(0) = f(i)$, so that the $\mathbb{Z}$-action $(k, f) \mapsto k \cdot f$ factors through a $\mathbb{Z}/(m+1)$-action.

###### Proposition

Every morphism $f: m \to n$ in $\Lambda$ may be uniquely decomposed as $f = h \tau_m^{-k}$ where $h$ belongs to $\Delta$ and $k$ is unique modulo $m+1$. The cyclic group $\mathbb{Z}/(n+1)$ acts on $\Delta(m, n) \cong \{f \in L(m, n): 0 \f(0)\; and\; f(m) \leq n$ by the formula $k \cdot f = \tau^{-k} \circ f \circ \tau^{f^\ast(k)}$.

###### Proof

This follows from previous propositions by dualizing. For $f \in L(m, n)$ we write $f^\ast: n \to m$ uniquely in the form $\tau_m^k g$ with $g \in \Delta^{op}(n, m)$, by Proposition . Taking right adjoints, $f = g_\ast \tau_m^{-k}$ where $g_\ast \in \Delta(m, n)$. We define the action on $\Delta(m, n)$ by conjugating the action on $\Delta^{op}(n, m)$ provided by Proposition , i.e., for $f \in \Delta(m, n)$ we define

$k \cdot f = (k \cdot f^\ast)_\ast = [\tau^{-f^\ast(k)} \circ f^\ast \circ \tau^k]_\ast = (\tau^k)_\ast \circ f^\ast_\ast \circ (\tau^{-f^\ast(k)})_\ast = \tau^{-k} \circ f \circ \tau^{f^\ast(k)}$

and this conjugation preserves the action axioms.

Denoting the generator $q(\tau_n)$ of $\Aut_\Lambda([n])$ also by $\tau_n$, we saw $\tau_n^{n+1} = \mathrm{id}_{[n]}$. One may read off from the development above a (perhaps more standard, and equivalent) presentation of $\Lambda$ by generators and relations. In addition to the cosimplicial identities between the coboundaries $\delta_i$ and codegeneracies $\sigma_j$ and $\tau^{n+1}_n = \mathrm{id}$ there are the following identities:

$\array{ \tau_n\delta_i = \delta_{i-1}\tau_{n-1},\,\, 1\leq i \leq n\\ \tau_n\delta_0 = \delta_n\\ \tau_n\sigma_i = \sigma_{i-1}\tau_{n-1},\,\, 1\leq i \leq n\\ \tau_m\sigma_0 = \sigma_n\tau_{n+1}^2 }$

## Properties

### General

We reiterate the development in the section on structure in summary form:

###### Theorem
1. $\Aut_\Lambda([n]) = \mathbf{Z}/(n+1)\mathbf{Z}$

2. $\Lambda([n],[m]) = \Delta([n],[m])\times \mathbf{Z}/(n+1)\mathbf{Z}$ (as a set)

3. Any morphism $f$ in $\Lambda([n],[m])$ can be uniquely written as a composition $f = \phi\circ g$ where $\phi\in\Delta([n],[m])$ and $g\in\Aut_\Lambda([n])$.

The generalizations of this theorem are the starting point of the theory of skew-simplicial sets of Krasausukas or equivalently crossed simplicial groups of Loday and Fiedorowicz.

The cyclic category is a generalized Reedy category, as explained here.

### Generalized Reedy model structure

The cycle category is a generalized Reedy category. Hence “cyclic spaces” carry a generalized Reedy model structure.

Blog discussion

Literature: