The cyclic category (Connes 83, see Cartier 85) typically denoted $\Lambda$ (or sometimes $\mathcal{C}$) is a small category whose presheaves β called cyclic sets or more generally cyclic objects β 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.
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 βspiraling propertyβ, 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.
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
is again faithful, so that the simplex category sits inside $\Lambda$.
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 βthe cyclic categoryβ to refer to $\Lambda$ is misleading, just like using βthe simplicial categoryβ 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 $A$.
To analyze the structure of $\Lambda$ further, we make a series of easy observations. These are largely based on Elmendorf 93.
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 $L$.
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
and
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 $L$.
$L$ is a self-dual category.
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, $f \mapsto f_\ast$.
$\Lambda$ is a self-dual category.
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 $\Lambda$.
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 $f(m) \leq n$).
The first assertion is immediate from the adjunction $f^\ast \dashv f$. The second follows from the deduction
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.
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 $n+1$.
The cyclic group $\mathbb{Z}/(m+1)$ acts on $\Delta^{op}(m, n)$ via the following formula for $f \in L(m, n), f(0) = 0$:
or in other words, via $(k \cdot f)(i) \coloneqq f(k+i) - f(k)$.
Clearly $k \cdot f \in \{g \in L(m, n): g(0) = 0\}$. We calculate
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.
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 $k \cdot f = \tau^{-k} \circ f \circ \tau^{f^\ast(k)}$.
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
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:
We reiterate the development in the section on structure in summary form:
$\Aut_\Lambda([n]) = \mathbf{Z}/(n+1)\mathbf{Z}$
$\Lambda([n],[m]) = \Delta([n],[m])\times \mathbf{Z}/(n+1)\mathbf{Z}$ (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 Krasausukas or equivalently crossed simplicial groups of Loday and Fiedorowicz.
The cyclic category is a generalized Reedy category, as explained here.
The cycle category is a generalized Reedy category (see Berger-Moerdijk 08, example 2.7). Hence βcyclic spacesβ carry a generalized Reedy model structure.
The original definition:
Alain Connes, Cohomologie cyclique et foncteurs $Ext^n$, C.R.A.S. 296 (1983), SΓ©rie I, 953-958 (pdf, pdf).
Pierre Cartier, Section 1.6 of: Homologie cyclique : rapport sur des travaux rΓ©cents de Connes, Karoubi, Loday, Quillenβ¦, SΓ©minaire Bourbaki: volume 1983/84, exposΓ©s 615-632, AstΓ©risque, no. 121-122 (1985), ExposΓ© no. 621 (numdam:SB_1983-1984__26__123_0)
Exposition:
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)
See also:
V. Drinfeld, On the notion of geometric realization, arXiv:math.CT/0304064
Alain Connes, Noncommutative geometry, Academic Press 1994
(also at http://www.alainconnes.org)
R. Krasauskas, Skew-simplicial groups, (Russian) Litovsk. Mat. Sb. 27 (1987), no. 1, 89β99, MR88m:18022 (English transl. R. Krasauskas, Skew-simplicial groups, Lith. Math. J. )
William Dwyer, Daniel Kan, Normalizing the cyclic modules of Connes, Comment. Math. Helv. 60 (1985), no. 4, 582β600.
William Dwyer, Mike Hopkins, Daniel Kan, The homotopy theory of cyclic sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 281β289.
Z. Fiedorowicz, Jean-Louis Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991), no. 1, 57β87, MR91j:18018, doi
Anthony Elmendorf, A simple formula for cyclic duality, Proc. Amer. Math. Soc. Volume 118, Number 3 (July 1993), 709-711. (pdf)
As a generalized Reedy category:
Relation to the paracyclic category:
See also
Wikipedia, Cyclic category
Blog discussion
Last revised on July 12, 2021 at 18:10:52. See the history of this page for a list of all contributions to it.