nLab cyclic category

Redirected from "cycle category".

The cyclic category

1. Idea
2. Definitions
3. Structure of the cyclic category
4. Properties

General
Generalized Reedy model structure

5. Related concepts
6. References

1. Idea

The cyclic category (Connes 83, see Cartier 85) typically denoted $\Lambda$ (or sometimes $\mathcal{C}$ ) 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.

2. Definitions

Multiple descriptions of the cyclic 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 ordinary function composition. The category $L$ is (equivalent to) the category $\Lambda_\infty$ called the paracyclic category by Nikolaus and Scholze.

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 2.1. Notice that $f \in L(m, n)$ is completely determined by the values $f(0), \ldots, f(m)$ . There is a faithful embedding $i \colon \Delta \to L$ 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 2.2. Of course the successor function $\tau \colon \mathbb{Z} \to \mathbb{Z}$ 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 $A$ .

3. Structure of the cyclic category

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

Proposition 3.1. 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 3.2. $L$ 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 3.3. $\Lambda$ 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 3.4. 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 2.1, 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 3.5. 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 3.6. 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 3.7. 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 3.5. 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 3.6, 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 }

4. Properties

General

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

Theorem 4.1.

$\Aut_\Lambda([n]) = \mathbf{Z}/(n+1)\mathbf{Z}$
$\Lambda([n],[m]) = \Delta([n],[m])\times \mathbf{Z}/(n+1)\mathbf{Z}$ (as a set)
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 Krasauskas 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 (see Berger-Moerdijk 08, example 2.7). Hence “cyclic spaces” carry a generalized Reedy model structure.

5. Related concepts

6. References

The original definition:

Alain Connes, Cohomologie cyclique et foncteurs $Ext^n$ , C.R.A.S. 296 (1983), Série I, 953-958 (pdf, pdf).
Pierre Cartier, Section 1.6 of: Homologie cyclique : rapport sur des travaux récents de Connes, Karoubi, Loday, Quillen…, Séminaire Bourbaki: volume 1983/84, exposés 615-632, Astérisque, no. 121-122 (1985), Exposé no. 621 (numdam:SB_1983-1984__26__123_0)

Exposition:

Cary Malkiewich, A visual introduction to cyclic sets and cyclotomic spectra, 2015 (pdf)

Textbook account:

Jean-Louis Loday, The Cyclic Category, Tor and Ext Interpretation (doi:10.1007/978-3-662-21739-9_6) and Cyclic Spaces and $S^1$ -Equivariant Homology (doi:10.1007/978-3-662-21739-9_7)

Chapters 6 and 7 in: Cyclic Homology, Grundlehren 301, Springer 1992 (doi:10.1007/978-3-662-21739-9)