Frobenius algebras in the category of sets and relations (Rel) serve as a unifying framework for various algebraic and combinatorial structures, including groupoids, effect algebras, and abstract circles. Recently, a nerve construction of simplicial sets for Frobenius algebras in Rel has been introduced. In this work, we investigate the lifting properties of these simplicial sets, linking them to the algebraic properties of Frobenius algebras. We introduce ε-simplicial sets -- simplicial sets with marked edges -- that enable the representation of a broader class of structures, such as test spaces from quantum logic. Our main results focus on weakly saturated classes generated by cofibrations, corresponding to specific lifting problems. Furthermore, we provide a characterization of Frobenius algebras in Rel within the framework of ε-simplicial sets. These findings lay the groundwork for the development of a convenient model structure in future research.
Dominik Lachman (Sun,) studied this question.