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Pirashvili--Richter-type theorems for the reflexive and dihedral homology theories

Daniel Graves

Abstract

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.

Pirashvili--Richter-type theorems for the reflexive and dihedral homology theories

Abstract

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 -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 -equivariant homotopy theory and connections to Hermitian algebraic -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.
Paper Structure (20 sections, 12 theorems, 44 equations)

This paper contains 20 sections, 12 theorems, 44 equations.

Table of Contents

  1. Functor homology
  2. Crossed simplicial groups
  3. A gathering of groups
  4. Crossed simplicial groups
  5. Functor homology theories
  6. A gathering of groupoids
  7. Involutive non-commutative sets
  8. The category of non-commutative sets
  9. The category of involutive non-commutative sets
  10. Involutive algebras, Loday functors and bar constructions
  11. Involutive algebras
  12. Loday functors and homology theories for involutive algebras
  13. The hyperoctahedral bar construction
  14. Bar constructions on categories of non-commutative sets
  15. The results of Pirashvili and Richter
  16. ...and 5 more sections

Key Result

Theorem 1

Let $A$ be an involutive, associative $k$-algebra. There exist isomorphisms of graded $k$-modules where $\mathcal{L}(A)$ is the dihedral Loday functor (also known as the dihedral bar construction); $\mathsf{H}_A$ is the hyperoctahedral bar construction; and the functor $B$ can be found in Definition functor-B-defn.

Theorems & Definitions (51)

  • Theorem
  • Theorem
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Example 2.4
  • ...and 41 more