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In particular, for these operadic categories we will study concrete examples of binary quadratic operads, describe their Koszul duals and prove that they are Koszul. This includes operads (for operadic categories) whose algebras are the most important operad- and PROP-like structures such as the classical operads, their variants such as cyclic or modular operads, and also diverse versions of PROPs such as wheeled properads, dioperads, <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mfrac><mml:mn>1<\/mml:mn><mml:mn>2<\/mml:mn><\/mml:mfrac><\/mml:math>PROPs, and still more exotic objects such as permutads and pre-permutads.<\/jats:p>","DOI":"10.32408\/compositionality-5-4","type":"journal-article","created":{"date-parts":[[2023,6,16]],"date-time":"2023-06-16T10:05:51Z","timestamp":1686909951000},"page":"4","source":"Crossref","is-referenced-by-count":5,"title":["Koszul duality for operadic categories"],"prefix":"10.46298","volume":"5","author":[{"given":"Michael","family":"Batanin","sequence":"first","affiliation":[{"name":"Institute of Mathematics of the Czech Academy of Sciences, \u017ditn\u00e1 25, 115 67 Prague 1, The Czech Republic"},{"name":"MFF UK, Sokolovsk\u00e1 83, 186 75 Prague 8, The Czech Republic"}]},{"given":"Martin","family":"Markl","sequence":"additional","affiliation":[{"name":"Institute of Mathematics of the Czech Academy of Sciences, \u017ditn\u00e1 25, 115 67 Prague 1, The Czech Republic"},{"name":"MFF UK, Sokolovsk\u00e1 83, 186 75 Prague 8, The Czech Republic"}]}],"member":"25203","published-online":{"date-parts":[[2023,6,16]]},"reference":[{"key":"0","doi-asserted-by":"crossref","unstructured":"D.W. 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