<p>Frobenius algebras in the category of sets and relations (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\textbf {Rel}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">Rel</mi> </math></EquationSource> </InlineEquation>) 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\textbf {Rel}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">Rel</mi> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({\textbf {Rel}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">Rel</mi> </math></EquationSource> </InlineEquation> within the framework of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-simplicial sets. These findings lay the groundwork for the development of a convenient model structure in future research.</p>