Simplicial Approach to Frobenius Algebras in the Category of Relations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914208737656832 |
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| author | Lachman, Dominik |
| author_facet | Lachman, Dominik |
| contents | Frobenius algebras in the category of sets and relations ($\mathbf{Rel}$) 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 $\mathbf{Rel}$ 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 $\mathbf{Rel}$ within the framework of $ε$-simplicial sets. These findings lay the groundwork for the development of a convenient model structure in future research. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simplicial Approach to Frobenius Algebras in the Category of Relations Lachman, Dominik Category Theory 06C15, 18G30, 18M05 Frobenius algebras in the category of sets and relations ($\mathbf{Rel}$) 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 $\mathbf{Rel}$ 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 $\mathbf{Rel}$ within the framework of $ε$-simplicial sets. These findings lay the groundwork for the development of a convenient model structure in future research. |
| title | Simplicial Approach to Frobenius Algebras in the Category of Relations |
| topic | Category Theory 06C15, 18G30, 18M05 |
| url | https://arxiv.org/abs/2509.06193 |