Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories
Reflexive homology and dihedral homology are the homology theories associated to the reflexive and dihedral crossed simplicial groups respectively. The former has recently been shown to capture interesting information about $C_2$-equivariant homotopy theory and its structure is related to the study...
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Académie des sciences
2025-03-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.698/ |
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| author | Graves, Daniel |
| author_facet | Graves, Daniel |
| author_sort | Graves, Daniel |
| collection | DOAJ |
| description | Reflexive homology and dihedral homology are the homology theories associated to the reflexive and dihedral crossed simplicial groups respectively. The former has recently been shown to capture interesting information about $C_2$-equivariant homotopy theory and its structure is related to the study of “real” objects in algebraic topology. The latter has long been of interest for its applications in $O(2)$-equivariant homotopy theory and connections to Hermitian algebraic $K$-theory. In this paper, we show that the reflexive and dihedral homology theories can be interpreted as functor homology over categories of non-commutative sets, after the fashion of Pirashvili and Richter’s result for the Hochschild and cyclic homology theories. |
| format | Article |
| id | doaj-art-2719c2940b2b4feaa6fe4848ffb4949c |
| institution | Matheson Library |
| issn | 1778-3569 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-2719c2940b2b4feaa6fe4848ffb4949c2025-08-01T07:20:04ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692025-03-01363G1435510.5802/crmath.69810.5802/crmath.698Pirashvili–Richter-type theorems for the reflexive and dihedral homology theoriesGraves, Daniel0https://orcid.org/0000-0003-1136-9295Lifelong Learning Centre, University of Leeds, Woodhouse, Leeds, LS2 9JT, UKReflexive homology and dihedral homology are the homology theories associated to the reflexive and dihedral crossed simplicial groups respectively. The former has recently been shown to capture interesting information about $C_2$-equivariant homotopy theory and its structure is related to the study of “real” objects in algebraic topology. The latter has long been of interest for its applications in $O(2)$-equivariant homotopy theory and connections to Hermitian algebraic $K$-theory. In this paper, we show that the reflexive and dihedral homology theories can be interpreted as functor homology over categories of non-commutative sets, after the fashion of Pirashvili and Richter’s result for the Hochschild and cyclic homology theories.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.698/Reflexive homologydihedral homologyfunctor homologycrossed simplicial groupinvolutive non-commutative setsinvolutive algebra |
| spellingShingle | Graves, Daniel Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories Comptes Rendus. Mathématique Reflexive homology dihedral homology functor homology crossed simplicial group involutive non-commutative sets involutive algebra |
| title | Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories |
| title_full | Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories |
| title_fullStr | Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories |
| title_full_unstemmed | Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories |
| title_short | Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories |
| title_sort | pirashvili richter type theorems for the reflexive and dihedral homology theories |
| topic | Reflexive homology dihedral homology functor homology crossed simplicial group involutive non-commutative sets involutive algebra |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.698/ |
| work_keys_str_mv | AT gravesdaniel pirashvilirichtertypetheoremsforthereflexiveanddihedralhomologytheories |