Connes' Cyclic Category

• Connes' cyclic category is a universal construct defined via finite cyclically ordered sets and cyclic permutation operators that model cyclic symmetry.
• It underpins cyclic (co)homology by serving as the indexing category for cyclic modules and spaces, thereby influencing noncommutative and homotopical geometry.
• Extensions like the epicyclic category integrate additional operations, enriching its structure with connections to projective geometry, Reedy theory, and higher categorical frameworks.

Connes' cyclic category, denoted by $\Lambda$, is a universal combinatorial and categorical construct encoding cyclic symmetry in homological invariants of algebras and related objects. It serves as the foundational indexing category for cyclic modules and cocyclic objects, determines the algebraic underpinnings of cyclic (co)homology, and manifests deep structural connections to higher category theory, algebraic topology, and noncommutative geometry. Over the last four decades, $\Lambda$ has undergone multiple conceptual reinterpretations: as a discrete category of combinatorial objects, as a geometric category of marked circles, as a crossed simplicial group, and as a categorical shadow of projective geometry in characteristic one. Its versatility continues to inform major advances in model category theory, topological and equivariant homotopy theory, and the paper of extended field theories.

1. Foundational Definitions and Presentations of the Cyclic Category

The cyclic category $\Lambda$ is most fundamentally the category whose objects are finite cyclically ordered sets, commonly labeled by the finite integers $[n] = \{0, 1, \ldots, n\}$, equipped with an additional "cyclic" generator. Morphisms in $\Lambda$ are generated by the standard face and degeneracy maps of the simplex category $\Delta$, augmented with a cyclic permutation operator

$$\tau_n: [n] \to [n], \quad \tau_n(i) = i+1 \mod (n+1)$$

subject to relations formalizing cyclic symmetry: $$\tau_n^{n+1} = \mathrm{id}, \quad \tau_n \circ d_0 = d_n, \quad \tau_n \circ d_i = d_{i-1} \circ \tau_{n-1} \;\text{ for } 1 \leq i \leq n,$$

$$\tau_n \circ s_i = s_{i-1} \circ \tau_{n+1} \;\text{ for } 1 \leq i \leq n, \quad \tau_n \circ s_0 = s_n \circ \tau_{n+1}^2.$$

(Bergner et al., 18 Sep 2024)

An alternative topological-geometric description identifies the objects as pairs $(S^1, \underline{n})$, where $S^1$ is the circle with $n+1$ marked points, and morphisms are compositions of orientation-preserving degree one self-maps $f: S^1 \to S^1$ carrying marked points to marked points up to homotopy. The mapping spaces in this "topological cyclic category" are contractible, with $\Lambda$ appearing as the homotopy category (Bergner et al., 18 Sep 2024).

Another deeply influential reformulation is as a crossed simplicial group: $\Lambda$ is the universal extension of $\Delta$ by cyclic automorphism groups $C_{n+1}$ such that each morphism uniquely factors as a composable pair of a simplicial map and a group element, reflecting the groupoidal nature and symmetry of $\Lambda$ (Bergner et al., 18 Sep 2024).

2. Universal Role in Cyclic (Co)Homology and Higher Structures

Connes' cyclic category is the organizing principle underlying cyclic modules (covariant functors $\Lambda \to \mathrm{Ab}$) and cocyclic objects (contravariant functors $\Lambda^{\mathrm{op}} \to \mathcal{C}$), which supply the data for defining and computing cyclic (co)homology of algebras and other structures (Bergner et al., 18 Sep 2024). This encoding of cyclic symmetry allows cyclic homology to be phrased as a derived functor on the abelian category of cyclic modules, in direct analogy to the classical use of simplicial objects in homological algebra (Connes et al., 2022).

The passage from the cyclic category to associated (co)homology theories is not restricted to the algebraic field: $\Lambda$ is the correct indexing category for model structures on "cyclic sets" (i.e., presheaves on $\Lambda$), cyclic spaces, cyclic (higher) Segal spaces, and equivariant models in stable homotopy theory. All these model-theoretic advances depend critically on the interleaving of simplicial and cyclic symmetries formalized in the structure of $\Lambda$ (Bergner et al., 18 Sep 2024).

3. Self-Duality, Geometric Realization, and Characteristic One Geometry

A fundamental property of the cyclic category is its self-duality: there exists a canonical anti-isomorphism $\Lambda \cong \Lambda^{\mathrm{op}}$, induced by cyclic transposition of the combinatorial data (Connes et al., 2013, Connes et al., 2022). This self-duality underpins the duality between cyclic and cocyclic modules and the existence of dual homology and cohomology theories, including the characterization of duplicial structures in higher categories (Slevin, 2016).

Geometric models for $\Lambda$ as a category of projective spaces over the idempotent (max-plus) semifield $\mathbb{F}$ in characteristic one have deepened the understanding of the underlying symmetry: objects are modules (semimodules) defined via restriction of scalars along Frobenius endomorphisms, and morphisms are projective classes of semilinear maps. The subcategory of "linear" morphisms is canonically isomorphic to $\Lambda$, and its enlargement with all semilinear maps yields the epicyclic category (Connes et al., 2013, Connes et al., 2022). In this setting, features like the self-duality of $\Lambda$ and the cyclic descent number of permutations acquire a natural geometric interpretation.

4. Model Structures and Reedy Theory on Cyclic Sets and Spaces

The structure of $\Lambda$ enables the construction of rich model categories, notably cyclic sets (functors $\Lambda^{\mathrm{op}} \to \mathbf{Set}$) and cyclic spaces (functors $\Lambda^{\mathrm{op}} \to \mathcal{S}$, with $\mathcal{S}$ a category of spaces or spectra). Weak equivalences and fibrations are detected on the underlying simplicial structures via the inclusion $j: \Delta \hookrightarrow \Lambda$. The model category of cyclic sets is Quillen equivalent to topological spaces with an $S^1$-action, with the representable cyclic set $\Lambda[n]$ realizing to $|\Delta[n]| \times S^1$ (Bergner et al., 18 Sep 2024).

A generalized Reedy structure on $\Lambda$ provides the machinery to define latching and matching objects in cyclic spaces, with automorphism group actions $C_{n+1}$. The resulting Reedy model structures are crucial for constructing and analyzing cyclic Segal spaces (with Segal conditions imposed on cyclic representables) and cyclic 2-Segal spaces (which localize with respect to polygonal triangulation maps), thereby extending higher category theory to settings with explicit cyclic symmetry (Bergner et al., 18 Sep 2024).

5. Extensions: Epicyclic and Pericyclic Categories

To encode additional operations (notably the $\lambda$-operations on cyclic homology), $\Lambda$ is naturally extended to the epicyclic category $\widetilde{\Lambda}$ by including morphisms corresponding to Frobenius twists—this corresponds formally to taking the semidirect product of $\Lambda$ with the multiplicative monoid $\mathbb{N}^{\ast}$ of positive integers. The epicyclic, and more generally the pericyclic, category $\Pi$ (defined as the pullback of $\Delta$ and $\mathbb{N}^{\ast}$ via appropriate degree functors) further unify various presentations of "epicyclic spaces" and enable a harmonized foundation for higher cyclic theory, including applications in topological cyclic homology (Caramello et al., 2014, Connes et al., 2022).

These generalizations have corresponding topos-theoretic incarnations: the (epi)cyclic toposes are classifying topoi for geometric theories of oriented groupoids (with or without a distinguished elementary cycle). The cyclic topos, in particular, is associated to the subcategory with morphisms preserving both the orientation and the elementary cycle, while the epicyclic topos relaxes the latter condition, generating a richer logical and arithmetic framework (Caramello et al., 2014).

6. Applications in Noncommutative and Higher Geometry

The structure of $\Lambda$ is central in noncommutative geometry, where it organizes cyclic modules and thereby cyclic (co)homology of algebras, Hopf (algebroid) cyclic cohomology (with parameters such as anti-Yetter–Drinfeld modules), and the realization of bilinear pairings in noncommutative motives as composition in suitable derived categories (Tabuada, 2010, Kowalzig et al., 2010). It further underlies the construction of model structures and invariants in contexts as diverse as the Chern–Connes pairing in noncommutative index theory, equivariant cyclic homology under quantization, braidings and quantum symmetries in higher representation theory, and the theory of categorified invariants via generalized Fredholm modules and traces in $C^*$-tensor categories (Beliakova et al., 2015, Quddus, 2016, Bartulović, 2022, Banerjee et al., 19 Apr 2024).

The canonical periodicity operator $S$ (related to cup-product with the 2-dimensional fundamental cyclic cocycle) and the duality properties of cyclic modules reflect intrinsic features of the cyclic category and govern the periodicity and stabilization phenomena in cyclic (co)homology and their applications. In higher categorical and motivic settings, understanding the nerve of categories with duplicial or cyclic structure is intimately connected to descent and coreflection properties indexed by $\Lambda$ (Slevin, 2016, Balodi et al., 2019, Balodi et al., 2021).

---

Through these foundational, geometric, model-theoretic, and homological roles, Connes' cyclic category has become an indispensable tool in modern mathematics, crucial both for explicit computational constructions and for conceptual advances in noncommutative, homotopical, and higher categorical geometry.