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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -equivariant homotopy theory and connections to Hermitian algebraic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -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.