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Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\pm }$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mo>\u00b1<\/mml:mo>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results.\n                  <\/jats:p>","DOI":"10.1007\/s10485-026-09868-w","type":"journal-article","created":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T13:39:09Z","timestamp":1778765949000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Invertibility and Parity in Symmetric Monoidal Categories"],"prefix":"10.1007","volume":"34","author":[{"given":"Nick","family":"Gurski","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Niles","family":"Johnson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,14]]},"reference":[{"key":"9868_CR1","doi-asserted-by":"publisher","unstructured":"Abramsky, S.: No-cloning in categorical quantum mechanics. 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