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Reflexive homology and dihedral homology are the homology theories associatedto the reflexive and dihedral crossed simplicial groups respectively. Theformer 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 forits applications in $O(2)$-equivariant homotopy theory and connections toHermitian algebraic $K$-theory. In this paper, we show that the reflexive anddihedral homology theories can be interpreted as functor homology overcategories of non-commutative sets, after the fashion of Pirashvili andRichter's result for the Hochschild and cyclic homology theories.

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