nLab cyclic category

The cyclic category

The cyclic category


The cyclic category (Connes 83, see Cartier 85) typically denoted Ξ›\Lambda (or sometimes π’ž\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.


Multiple descriptions of the cyclic category Ξ›\Lambda are possible, but a convenient starting point is to consider first a category LL whose objects are natural numbers nβ‰₯0n \geq 0, and where the hom-set L(m,n)L(m, n) consists of increasing functions f:β„€β†’β„€f: \mathbb{Z} \to \mathbb{Z} satisfying the β€œspiraling property”, that f(i+m+1)=f(i)+n+1f(i + m + 1) = f(i) + n + 1, with composition given by ordinary function composition. The category LL is (equivalent to) the category Ξ› ∞\Lambda_\infty called the paracyclic category by Nikolaus and Scholze.

Then, define Ξ›\Lambda to be a quotient category of LL having the same objects, with Ξ›(m,n)=L(m,n)/∼\Lambda(m, n) = L(m, n)/\sim where ∼\sim is the equivalence relation for which f∼gf \sim g means fβˆ’gf - g is a constant multiple of n+1n+1. Let q:Lβ†’Ξ›q: L \to \Lambda be the quotient.


Notice that f∈L(m,n)f \in L(m, n) is completely determined by the values f(0),…,f(m)f(0), \ldots, f(m). There is a faithful embedding i:Ξ”β†’Li \colon \Delta \to L which on objects is the identity, where f∈L(m,n)f \in L(m, n) belongs to the image of ii iff 0≀f(0)0 \leq f(0) and f(m)≀nf(m) \leq n. The composite

Ξ”β†ͺiLβ†’qΞ›\Delta \stackrel{i}{\hookrightarrow} L \stackrel{q}{\to} \Lambda

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


Of course the successor function Ο„:β„€β†’β„€\tau \colon \mathbb{Z} \to \mathbb{Z} gives a function Ο„ n∈L(n,n)\tau_n \in L(n, n) defined by Ο„ n(i)=i+1\tau_n(i) = i+1, which in turn induces a function q(Ο„)βˆˆΞ›(n,n)q(\tau) \in \Lambda(n, n) such that q(Ο„) n+1=1 nq(\tau)^{n+1} = 1_n. In this way, we have inclusions β„€/(n+1)β†ͺΞ›(n,n)\mathbb{Z}/(n+1) \hookrightarrow \Lambda(n, n) of cyclic groups inside Ξ›\Lambda.

Cyclic objects in a category CC are the contravariant functors Ξ› opβ†’C\Lambda^{\mathrm{op}}\to C, cocyclic objects are the covariant functors Ξ›β†’C\Lambda\to C. Note that Ξ›\Lambda itself is, via its inclusion into CatCat, an example of a cocyclic object in CatCat. (Thus, the common term β€œthe cyclic category” to refer to Ξ›\Lambda is misleading, just like using β€œthe simplicial category” to refer to the simplex category Ξ”\Delta.)

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

Structure of the cyclic category

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


Every morphism ff of LL, regarded as a functor β„€β†’β„€\mathbb{Z} \to \mathbb{Z}, has a left adjoint f *:β„€β†’β„€f^\ast: \mathbb{Z} \to \mathbb{Z} that is also a morphism of LL. Similarly, every morphism ff of LL has a right adjoint f *f_\ast belonging to LL.


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

f *(j+n+1)≀f *(j)+m+1 iff j+n+1≀f(f *(j)+m+1) iff j+n+1≀f(f *(j))+n+1 iff j≀f(f *(j)) iff f *(j)≀f *(j)\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) }


f *(j)+m+1≀f *(j+n+1) iff f *(j)≀f *(j+n+1)βˆ’mβˆ’1 iff j≀f(f *(j+n+1)βˆ’mβˆ’1) iff j≀f(f *(j+n+1))βˆ’nβˆ’1 iff j+n+1≀f(f *(j)+n+1) iff f *(j+n+1)≀f *(j+n+1).\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 (β„€,≀)(\mathbb{Z}, \leq) as a category is self-dual, every morphism ff of LL dually has a right adjoint that is a morphism of LL.


LL is a self-dual category.


The duality functor L opβ†’LL^{op} \to L is the identity on objects and takes f:mβ†’nf: m \to n to f *:nβ†’mf^\ast: n \to m. It is contravariant since the left adjoint of a composite fgf g is g *f *=(fg) *g^\ast f^\ast = (f g)^\ast. It is an equivalence because its inverse is the right-adjoint mapping, f↦f *f \mapsto f_\ast.


Ξ›\Lambda is a self-dual category.


If f∼gf \sim g in L(m,n)L(m, n), then f=Ο„ k(n+1)∘gf = \tau^{k (n+1)} \circ g for some kβˆˆβ„€k \in \mathbb{Z}. Observe that Ο„ *=Ο„ βˆ’1\tau^\ast = \tau^{-1}, so f *=g *βˆ˜Ο„ βˆ’k(n+1)=Ο„ βˆ’k(m+1)∘g *f^\ast = g^\ast \circ \tau^{-k(n+1)} = \tau^{-k(m+1)} \circ g^\ast where the last equation holds because g *:nβ†’mg^\ast: n \to m is spiraling. This shows f *∼g *f^\ast \sim g^\ast, i.e., the self-duality of LL descends to Ξ›\Lambda.


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


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

0≀f *(0) iff βˆ’1<f *(0) iff Β¬(f *(0)β‰€βˆ’1) iff Β¬(0≀f(βˆ’1)) iff f(βˆ’1)<0 iff f(m)<n+1 iff f(m)≀n\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 LL and Remark , shows that Ξ” op\Delta^{op} faithfully maps to LL by Ξ” op(m,n)β‰…{f∈L(m,n):f(0)=0}\Delta^{op}(m, n) \cong \{f \in L(m, n): f(0) = 0\}. Passing to the quotient q:Lβ†’Ξ›q: L \to \Lambda, this description also realizes Ξ” op\Delta^{op} as sitting inside Ξ›\Lambda, and the next result is immediate.


Every morphism f:mβ†’nf: m \to n in Ξ›\Lambda may be uniquely decomposed as f=Ο„ n f(0)gf = \tau_n^{f(0)} g where gg belongs to Ξ” op(m,n)βŠ‚L(m,n)\Delta^{op}(m, n) \subset L(m, n), and the exponent f(0)f(0) is considered modulo n+1n+1.


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

kβ‹…f=Ο„ βˆ’f(k)∘fβˆ˜Ο„ kk \cdot f = \tau^{-f(k)} \circ f \circ \tau^k

or in other words, via (kβ‹…f)(i)≔f(k+i)βˆ’f(k)(k \cdot f)(i) \coloneqq f(k+i) - f(k).


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

jβ‹…(kβ‹…f) = Ο„ βˆ’(kβ‹…f)(j)∘(kβ‹…f)βˆ˜Ο„ j = Ο„ βˆ’(f(j+k)βˆ’f(k))βˆ˜Ο„ βˆ’f(k)∘fβˆ˜Ο„ kβˆ˜Ο„ j = Ο„ βˆ’f(j+k)∘fβˆ˜Ο„ j+k = (j+k)β‹…f.\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)β‹…f)(i)=f(i+m+1)βˆ’f(0+m+1)=f(i)+n+1βˆ’(f(0)+n+1)=f(i)βˆ’f(0)=f(i),((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 β„€\mathbb{Z}-action (k,f)↦kβ‹…f(k, f) \mapsto k \cdot f factors through a β„€/(m+1)\mathbb{Z}/(m+1)-action.


Every morphism f:mβ†’nf: m \to n in Ξ›\Lambda may be uniquely decomposed as f=hΟ„ m βˆ’kf = h \tau_m^{-k} where hh belongs to Ξ”\Delta and kk is unique modulo m+1m+1. The cyclic group β„€/(n+1)\mathbb{Z}/(n+1) acts on Ξ”(m,n)β‰…{f∈L(m,n):0f(0)andf(m)≀n\Delta(m, n) \cong \{f \in L(m, n): 0 \f(0)\; and\; f(m) \leq n by the formula kβ‹…f=Ο„ βˆ’k∘fβˆ˜Ο„ f *(k)k \cdot f = \tau^{-k} \circ f \circ \tau^{f^\ast(k)}.


This follows from previous propositions by dualizing. For f∈L(m,n)f \in L(m, n) we write f *:nβ†’mf^\ast: n \to m uniquely in the form Ο„ m kg\tau_m^k g with gβˆˆΞ” op(n,m)g \in \Delta^{op}(n, m), by Proposition . Taking right adjoints, f=g *Ο„ m βˆ’kf = g_\ast \tau_m^{-k} where g *βˆˆΞ”(m,n)g_\ast \in \Delta(m, n). We define the action on Ξ”(m,n)\Delta(m, n) by conjugating the action on Ξ” op(n,m)\Delta^{op}(n, m) provided by Proposition , i.e., for fβˆˆΞ”(m,n)f \in \Delta(m, n) we define

kβ‹…f=(kβ‹…f *) *=[Ο„ βˆ’f *(k)∘f *βˆ˜Ο„ k] *=(Ο„ k) *∘f * *∘(Ο„ βˆ’f *(k)) *=Ο„ βˆ’k∘fβˆ˜Ο„ f *(k)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(Ο„ n)q(\tau_n) of Aut Ξ›([n])\Aut_\Lambda([n]) also by Ο„ n\tau_n, we saw Ο„ n n+1=id [n]\tau_n^{n+1} = \mathrm{id}_{[n]}. One may read off from the development above a (perhaps more standard, and equivalent) presentation of Ξ›\Lambda by generators and relations. In addition to the cosimplicial identities between the coboundaries Ξ΄ i\delta_i and codegeneracies Οƒ j\sigma_j and Ο„ n n+1=id\tau^{n+1}_n = \mathrm{id} there are the following identities:

Ο„ nΞ΄ i=Ξ΄ iβˆ’1Ο„ nβˆ’1,1≀i≀n Ο„ nΞ΄ 0=Ξ΄ n Ο„ nΟƒ i=Οƒ iβˆ’1Ο„ nβˆ’1,1≀i≀n Ο„ mΟƒ 0=Οƒ nΟ„ n+1 2\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 }



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

  1. Aut Ξ›([n])=Z/(n+1)Z\Aut_\Lambda([n]) = \mathbf{Z}/(n+1)\mathbf{Z}

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

  3. Any morphism ff in Ξ›([n],[m])\Lambda([n],[m]) can be uniquely written as a composition f=Ο•βˆ˜gf = \phi\circ g where Ο•βˆˆΞ”([n],[m])\phi\in\Delta([n],[m]) and g∈Aut Ξ›([n])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 β€œcyclic spaces” carry a generalized Reedy model structure.


The original definition:

  • Alain Connes, Cohomologie cyclique et foncteurs Ext nExt^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, Quillen…, 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)


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

Textbook account:

See also:

As a generalized Reedy category:

Relation to the paracyclic category:

See also

category: category

Last revised on April 25, 2023 at 15:36:56. See the history of this page for a list of all contributions to it.