Simplicial Approach to Frobenius Algebras in the Category of Relations

Abstract Frobenius algebras in the category of sets and relations ( $${\textbf {Rel}}$$ 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 $${\textbf {Rel}}$$ 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 $$\varepsilon $$ 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 $${\textbf {Rel}}$$ Rel within the framework of $$\varepsilon $$ ε -simplicial sets. These findings lay the groundwork for the development of a convenient model structure in future research.