Cyclic category
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In mathematics, the cyclic category or cycle category or category of cycles is a category of finite cyclically ordered sets and degree-1 maps between them. It was introduced by {{harvtxt|Connes|1983}}.
Definition
The cyclic category Λ has one object Λn for each natural number n = 0, 1, 2, ...
The morphisms from Λm to Λn are represented by increasing functions f from the integers to the integers, such that f(x+m+1) = f(x)+n+1, where two functions f and g represent the same morphism when their difference is divisible by n+1.
Informally, the morphisms from Λm to Λn can be thought of as maps of (oriented)
necklaces with m+1 and n+1 beads. More precisely, the morphisms can be identified with homotopy classes of degree 1 increasing maps from S1 to itself that map the subgroup Z/(m+1)Z to Z/(n+1)Z.
Properties
The number of morphisms from Λm to Λn is (m+n+1)!/m!n!.
The cyclic category is self dual.
The classifying space BΛ of the cyclic category is a classifying space BS1of the circle group S1.
Cyclic sets
A cyclic set is a contravariant functor from the cyclic category to sets. More generally a cyclic object in a category C is a contravariant functor from the cyclic category to C.
See also
References
- {{Citation |last=Connes |first=Alain |authorlink=Alain Connes |year=1983 |title=Cohomologie cyclique et foncteurs Extn |language=French |journal=Comptes Rendus de l'Académie des Sciences, Série I |volume=296 |issue=23 |pages=953–958 |url=http://www.alainconnes.org/docs/n83.pdf |accessdate=15 May 2011 |mr=777584 |archive-date=4 March 2016 |archive-url=https://web.archive.org/web/20160304212700/http://www.alainconnes.org/docs/n83.pdf |url-status=dead }}
- {{Citation |last=Connes |first=Alain |authorlink=Alain Connes |year=2002 |chapter=Noncommutative Geometry Year 2000 |editor-last=Fokas |editor-first=A. |title=Highlights of mathematical physics |isbn=0-8218-3223-9 |pages=49–110 |chapter-url=http://www.alainconnes.org/docs/2000.pdf |accessdate=15 May 2011 |arxiv=math/0011193|bibcode=2000math.....11193C }}
- {{Citation |last1=Kostrikin |first1=A. I. |authorlink=Alexei Kostrikin |last2=Shafarevich |first2=I. R.|authorlink2=Igor Shafarevich |year=1994 |title=Algebra V: Homological algebra |series=Encyclopaedia of Mathematical Sciences |volume=38 |publisher=Springer |isbn=3-540-53373-7 |pages=60–61}}
- {{Citation | last1=Loday | first1=Jean-Louis | title=Cyclic homology | url=https://books.google.com/books?id=KaLshoPoSlsC | publisher=Springer-Verlag | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-53339-9 | mr=1217970 | year=1992 | volume=301}}
External links
- [http://ncatlab.org/nlab/show/cycle+category Cycle category] in nLab