Power maps and epicyclic spaces

Burghelea, Dan; Fiedorowicz, Zbigniew; Gajda, Wojciech

doi:10.1016/0022-4049(94)90081-7

Cited by 15 publications

(29 citation statements)

References 5 publications

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“…3 shows that the duality of B-semimodules E with rkE = 1 and rkE < ∞ behaves similarly to the duality holding for finite dimensional vector spaces over fields and it produces in particular the transposition of linear maps defined as follows. Let E * = Hom B (E, B), F * = Hom B(F, B)…”

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confidence: 82%

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Projective geometry in characteristic one and the epicyclic category

Connes¹,

Consani²

2015

Nagoya Math. J.

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We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield of "max-plus integers" Z max . Finite dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of Z max . The associated projective spaces are finite and provide a mathematically consistent interpretation of J. Tits' original idea of a geometry over the absolute point. The selfduality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

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confidence: 82%

“…By definition (cf. Definition 1.1 of [3]), the epicyclic categoryΛ is obtained by adjoining to the cyclic category Λ, new morphisms π k n ∶ [k(n + 1) − 1] → [n] for n ≥ 0, k ≥ 1, which fulfill the following relations:…”

Section: Proposition 28mentioning

confidence: 99%

Projective geometry in characteristic one and the epicyclic category

Connes¹,

Consani²

2015

Nagoya Math. J.

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“…The two composites along the top edges and along the left and bottom edges of the larger diagram are the values on (f 0 , f 1 ) of the two composites of diagram (6). The whole process is naturally cosimplicial in [n] and so gives us the necessary homotopy:…”

Section: Homotopy Coherence and Crossed Complexesmentioning

confidence: 99%

Spaces of Maps into Classifying Spaces for Equivariant Crossed Complexes, II: The General Topological Group Case

Brown¹,

Golasiński²,

Porter³

et al. 2001

K-Theory

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The results of a previous paper [3] on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg-Zilber theory from [19], and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces.

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“…[2], Definition 1.1) by adjoining to Λ new morphisms Id k n ∶ Ψ k (n) → n for n ≥ 0, k ≥ 1, which fulfill the following relations (i) Id B.1 Cyclic homology and cyclic modules Definition B.1 A cyclic module E is a covariant functor Λ → Ab from the cyclic category to the category of abelian groups.…”

Section: Appendix B Epicyclic Modules and The λ-Operationsmentioning

confidence: 99%

“…Let (G, G + ) be an oriented groupoïd fulfilling the three conditions of Proposition 2.6. Let x ∈ G (0) , consider the set G x = {γ ∈ G s(γ) = x} with the total order defined by (2) and with the action of Z given, for γ x ∈ G + the positive generator of G…”

Section: Proposition 26mentioning

confidence: 99%

The cyclic and epicyclic sites

Connes¹,

Consani²

2015

Rend. Sem. Mat. Univ. Padova

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We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of "max-plus integers" Z max . An object of this category is a pair (E, K) of a semimodule E over an algebraic extension K of Z max . The morphisms are projective classes of semilinear maps between semimodules. The epicyclic topos sits over the arithmetic topos N × of [6] and the fibers of the associated geometric morphism correspond to the cyclic site. In two appendices we review the role of the cyclic and epicyclic toposes as the geometric structures supporting cyclic homology and the lambda operations.

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Power maps and epicyclic spaces

Cited by 15 publications

References 5 publications

Projective geometry in characteristic one and the epicyclic category

Projective geometry in characteristic one and the epicyclic category

Spaces of Maps into Classifying Spaces for Equivariant Crossed Complexes, II: The General Topological Group Case

The cyclic and epicyclic sites

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