The cyclic category (Connes 83, see Cartier 85) typically denoted (or sometimes ) 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 are possible, but a convenient starting point is to consider first a category whose objects are natural numbers , and where the hom-set consists of increasing functions satisfying the βspiraling propertyβ, that , with composition given by ordinary function composition. The category is (equivalent to) the category called the paracyclic category by Nikolaus and Scholze.
Then, define to be a quotient category of having the same objects, with where is the equivalence relation for which means is a constant multiple of . Let be the quotient.
Notice that is completely determined by the values . There is a faithful embedding which on objects is the identity, where belongs to the image of iff and . The composite
is again faithful, so that the simplex category sits inside .
Of course the successor function gives a function defined by , which in turn induces a function such that . In this way, we have inclusions of cyclic groups inside .
Cyclic objects in a category are the contravariant functors , cocyclic objects are the covariant functors . Note that itself is, via its inclusion into , an example of a cocyclic object in . (Thus, the common term βthe cyclic categoryβ to refer to is misleading, just like using βthe simplicial categoryβ to refer to the simplex category .)
If is an abelian category then the category of -presheaves on is usually called (Connes's) category of cyclic modules in .
To analyze the structure of further, we make a series of easy observations. These are largely based on Elmendorf 93.
Every morphism of , regarded as a functor , has a left adjoint that is also a morphism of . Similarly, every morphism of has a right adjoint belonging to .
By the spiraling property of , for any the comma category as a subset of has a lower bound in and hence is well-ordered. It is also nonempty, and we define to be the least element of . In other words is the least such that . It is easy to check that obeys the spiraling property , since
and
Also, since as a category is self-dual, every morphism of dually has a right adjoint that is a morphism of .
is a self-dual category.
The duality functor is the identity on objects and takes to . It is contravariant since the left adjoint of a composite is . It is an equivalence because its inverse is the right-adjoint mapping, .
is a self-dual category.
If in , then for some . Observe that , so where the last equation holds because is spiraling. This shows , i.e., the self-duality of descends to .
For a morphism , we have iff , and iff . Hence iff ( and ).
The first assertion is immediate from the adjunction . 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 and Remark , shows that faithfully maps to by . Passing to the quotient , this description also realizes as sitting inside , and the next result is immediate.
Every morphism in may be uniquely decomposed as where belongs to , and the exponent is considered modulo .
Clearly . We calculate
Moreover, so that the -action factors through a -action.
Every morphism in may be uniquely decomposed as where belongs to and is unique modulo . The cyclic group acts on by the formula .
This follows from previous propositions by dualizing. For we write uniquely in the form with , by Proposition . Taking right adjoints, where . We define the action on by conjugating the action on provided by Proposition , i.e., for we define
and this conjugation preserves the action axioms.
Denoting the generator of also by , we saw . One may read off from the development above a (perhaps more standard, and equivalent) presentation of by generators and relations. In addition to the cosimplicial identities between the coboundaries and codegeneracies and there are the following identities:
We reiterate the development in the section on structure in summary form:
(as a set)
Any morphism in can be uniquely written as a composition where and .
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.
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 , 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 -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:
From a topological perspective
See also
Wikipedia, Cyclic category
Blog discussion
Last revised on September 20, 2024 at 14:15:37. See the history of this page for a list of all contributions to it.