{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T10:14:41Z","timestamp":1780049681752,"version":"3.53.1"},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,4,19]],"date-time":"2026-04-19T00:00:00Z","timestamp":1776556800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,4,19]],"date-time":"2026-04-19T00:00:00Z","timestamp":1776556800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Palacky University Olomouc"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2026,6]]},"abstract":"Abstract<\/jats:title>\n \n Frobenius algebras in the category of sets and relations (\n \n \n $${\\textbf {Rel}}$$<\/jats:tex-math>\n \n Rel<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ) 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\n \n \n $${\\textbf {Rel}}$$<\/jats:tex-math>\n \n Rel<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n 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\n \n \n $$\\varepsilon $$<\/jats:tex-math>\n \n \u03b5<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -simplicial sets\u2014simplicial sets with marked edges\u2014that 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\n \n \n $${\\textbf {Rel}}$$<\/jats:tex-math>\n \n Rel<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n within the framework of\n \n \n $$\\varepsilon $$<\/jats:tex-math>\n \n \u03b5<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -simplicial sets. These findings lay the groundwork for the development of a convenient model structure in future research.\n <\/jats:p>","DOI":"10.1007\/s10485-026-09855-1","type":"journal-article","created":{"date-parts":[[2026,4,19]],"date-time":"2026-04-19T10:32:25Z","timestamp":1776594745000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Simplicial Approach to Frobenius Algebras in the Category of Relations"],"prefix":"10.1007","volume":"34","author":[{"given":"Dominik","family":"Lachman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,19]]},"reference":[{"key":"9855_CR1","volume-title":"Higher Categories and Homotopical Algebra. Cambridge Studies in Advanced Mathematics","author":"D-C Cisinski","year":"2019","unstructured":"Cisinski, D.-C.: Higher Categories and Homotopical Algebra. 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