{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T05:18:45Z","timestamp":1775798325150,"version":"3.50.1"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,11,18]],"date-time":"2025-11-18T00:00:00Z","timestamp":1763424000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,11,18]],"date-time":"2025-11-18T00:00:00Z","timestamp":1763424000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["686616"],"award-info":[{"award-number":["686616"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In Igusa, Todorov and Weyman (Picture groups of finite type and cohomology in type\n                    <jats:italic>A<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>n<\/jats:italic>\n                    <\/jats:sub>\n                    <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"http:\/\/arxiv.org\/abs\/1609.02636\" ext-link-type=\"uri\">arXiv:1609.02636<\/jats:ext-link>\n                    ), we introduced \u201cpicture groups\u201d and computed the cohomology of the picture group of type\n                    <jats:inline-formula>\n                      <jats:tex-math>$$A_n$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . This is the same group what was introduced by Loday (Contemp Math 265: 99\u2013127, 2000) where he called it the \u201cStasheff group\u201d. In this paper, we give an elementary combinatorial interpretation of the \u201ccluster morphism category\u201d constructed in as reported by Igusa and Todorov, (in: Signed exceptional sequences and the cluster morphism category,\n                    <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"http:\/\/arxiv.org\/abs\/1706.02041\" ext-link-type=\"uri\">arXiv:1706.02041<\/jats:ext-link>\n                    ) in the special case of the linearly oriented quiver of type\n                    <jats:inline-formula>\n                      <jats:tex-math>$$A_n$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . We prove that the classifying space of this category is locally\n                    <jats:italic>CAT<\/jats:italic>\n                    (0) and thus a\n                    <jats:inline-formula>\n                      <jats:tex-math>$$K(\\pi ,1)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . We prove a more general statement that classifying spaces of certain \u201ccubical categories\u201d are locally\n                    <jats:italic>CAT<\/jats:italic>\n                    (0). The objects of our category are the classical noncrossing partitions introduced by Kreweras (Discrete Math 1: 333\u2013350, 1972) . The morphisms are binary forests. This paper is independent of as reported by Igusa and Todorov (in: Signed exceptional sequences and the cluster morphism category,\n                    <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"http:\/\/arxiv.org\/abs\/1706.02041\" ext-link-type=\"uri\">arXiv:1706.02041<\/jats:ext-link>\n                    )and as reported by Igusa, Todorov and Weyman (in: Picture groups of finite type and cohomology in type\n                    <jats:italic>A<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>n<\/jats:italic>\n                    <\/jats:sub>\n                    <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"http:\/\/arxiv.org\/abs\/1609.02636\" ext-link-type=\"uri\">arXiv:1609.02636<\/jats:ext-link>\n                    )except in the last section where we use as reported by Igusa and Todorov (in: Signed exceptional sequences and the cluster morphism category,\n                    <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"http:\/\/arxiv.org\/abs\/1706.02041\" ext-link-type=\"uri\">arXiv:1706.02041<\/jats:ext-link>\n                    ) to compare our category with the category with the same name given by Hubery and Krause (J Eur Math Soc 18: 2273\u20132313, 2016).\n                  <\/jats:p>","DOI":"10.1007\/s10485-025-09838-8","type":"journal-article","created":{"date-parts":[[2025,11,18]],"date-time":"2025-11-18T16:03:46Z","timestamp":1763481826000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Category of Noncrossing Partitions"],"prefix":"10.1007","volume":"33","author":[{"given":"Kiyoshi","family":"Igusa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,11,18]]},"reference":[{"key":"9838_CR1","doi-asserted-by":"crossref","unstructured":"Bessis, D.: The dual braid monoid, Ann. Sci. \u00c9cole Norm. Sup. (4) 36, 5: 647\u2013683 (2003)","DOI":"10.1016\/j.ansens.2003.01.001"},{"issue":"4","key":"9838_CR2","doi-asserted-by":"publisher","first-page":"1983","DOI":"10.1090\/S0002-9947-07-04282-1","volume":"360","author":"T Brady","year":"2008","unstructured":"Brady, T., Watt, C.: Non-crossing partition lattices in finite real reflection groups. Trans. Amer. Math. Soc. 360(4), 1983\u20132005 (2008)","journal-title":"Trans. Amer. Math. Soc."},{"issue":"2","key":"9838_CR3","doi-asserted-by":"publisher","first-page":"572","DOI":"10.1016\/j.aim.2005.06.003","volume":"204","author":"AB Buan","year":"2006","unstructured":"Buan, A.B., Marsh, R.J., Reiten, I., Reineke, M., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572\u2013618 (2006)","journal-title":"Adv. Math."},{"key":"9838_CR4","unstructured":"Crawley-Boevey, E.: sequences of representations of quivers, Representations of algebras (Ottawa, ON,: CMS Conf. Proc., vol. 14, Amer. Math. Soc. Providence, RI 1993, 117\u2013124 (1992)"},{"issue":"3","key":"9838_CR5","doi-asserted-by":"publisher","first-page":"1347","DOI":"10.1090\/S0002-9947-05-03753-0","volume":"358","author":"P Caldero","year":"2006","unstructured":"Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters ($$A_n$$ case). Trans. Amer. Math. Soc. 358(3), 1347\u20131364 (2006)","journal-title":"Trans. Amer. Math. Soc."},{"key":"9838_CR6","unstructured":"Fr\u00e9d\u00e9ric, C., Nadeau, P.: Combinatorics of the categories of noncrossing partitions. S\u00e9m. Lothar. Combin. B 78 (2017)"},{"issue":"3","key":"9838_CR7","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1017\/S0305004100066275","volume":"100","author":"SM Gersten","year":"1986","unstructured":"Gersten, S.M., Howie, J.: Counterexamples to a conjecture about spherical diagrams. Math. Proc. Cambridge Philos. Soc. 100(3), 539\u2013543 (1986)","journal-title":"Math. Proc. Cambridge Philos. Soc."},{"key":"9838_CR8","doi-asserted-by":"crossref","unstructured":"Gromov, M.: Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol.\u00a08, Springer, New York, pp.\u00a075\u2013263 (1987)","DOI":"10.1007\/978-1-4613-9586-7_3"},{"issue":"10","key":"9838_CR9","doi-asserted-by":"publisher","first-page":"2273","DOI":"10.4171\/JEMS\/641","volume":"18","author":"A Hubery","year":"2016","unstructured":"Hubery, A., Krause, H.: A categorification of non-crossing partitions. J. Eur. Math. Soc. 18(10), 2273\u20132313 (2016). https:\/\/doi.org\/10.4171\/JEMS\/641","journal-title":"J. Eur. Math. Soc."},{"key":"9838_CR10","unstructured":"Igusa, K.: The $${W}h_3(\\pi )$$-obstruction for pseudoisotopy, Ph.D. thesis, Princeton University, (1979)"},{"key":"9838_CR11","unstructured":"Igusa, K., Kim, M., Todorov, G., Weyman, J.: Periodic trees and propictures, Unpublished preprint available at http:\/\/people.brandeis.edu\/~igusa\/Papers\/PeriodicProPics.pdf"},{"issue":"6","key":"9838_CR12","doi-asserted-by":"publisher","first-page":"1125","DOI":"10.1016\/S0040-9383(00)00002-1","volume":"40","author":"K Igusa","year":"2001","unstructured":"Igusa, K., Orr, K.E.: Links, pictures and the homology of nilpotent groups. Topology 40(6), 1125\u20131166 (2001)","journal-title":"Topology"},{"key":"9838_CR13","unstructured":"Igusa, K., Todorov, G.: Signed exceptional sequences and the cluster morphism category, arXiv:1706.02041"},{"key":"9838_CR14","unstructured":"Igusa, K., Todorov, G.: Which cluster morphism categories are $$CAT(0)$$, arXiv:2203.16679"},{"key":"9838_CR15","unstructured":"Igusa, K., Todorov, G., Weyman, J.: Periodic trees and semi-invariants, arXiv:1407.0619"},{"key":"9838_CR16","unstructured":"Igusa, K., Todorov G., Weyman, J.: Picture groups of finite type and cohomology in type $$A_n$$, arXiv:1609.02636"},{"issue":"6","key":"9838_CR17","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1112\/S0010437X09004023","volume":"145","author":"C Ingalls","year":"2009","unstructured":"Ingalls, C., Thomas, H.: Noncrossing partitions and representations of quivers. Compos. Math. 145(6), 1533\u20131562 (2009)","journal-title":"Compos. Math."},{"issue":"4","key":"9838_CR18","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1016\/0012-365X(72)90041-6","volume":"1","author":"G Kreweras","year":"1972","unstructured":"Kreweras, G.: Sur les partitions non crois\u00e9es d\u2019un cycle. Discrete Math. 1(4), 333\u2013350 (1972)","journal-title":"Discrete Math."},{"key":"9838_CR19","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1090\/conm\/265\/04245","volume":"265","author":"J-L Loday","year":"2000","unstructured":"Loday, J.-L.: Homotopical syzygies. Contemp. Math. 265, 99\u2013127 (2000)","journal-title":"Contemp. Math."},{"key":"9838_CR20","unstructured":"Lane S.M.: Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol.\u00a05, Springer-Verlag, New York, (1998)"},{"key":"9838_CR21","doi-asserted-by":"crossref","unstructured":"Peiffer, R.: \u00dcber Identit\u00e4ten zwischen relationen, Math. Ann., 121 (1949\/1950), 67\u201399","DOI":"10.1007\/BF01329617"},{"key":"9838_CR22","first-page":"275","volume":"2013","author":"D Speyer","year":"2011","unstructured":"Speyer, D., Thomas, H.: Acyclic cluster algebras revisited, \u201cAlgebras, quivers and representations. Proc. Abel Symp. 2013, 275\u2013298 (2011)","journal-title":"Proc. Abel Symp."},{"key":"9838_CR23","doi-asserted-by":"crossref","unstructured":"Stallings, J.R.: A graph-theoretic lemma and group-embeddings, Combinatorial group theory and topology (Alta, Utah,: Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ 1987, 145\u2013155 (1984)","DOI":"10.1515\/9781400882083-008"}],"container-title":["Applied Categorical Structures"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-025-09838-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10485-025-09838-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10485-025-09838-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T15:38:08Z","timestamp":1765294688000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10485-025-09838-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,18]]},"references-count":23,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["9838"],"URL":"https:\/\/doi.org\/10.1007\/s10485-025-09838-8","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-2609391\/v1","asserted-by":"object"}]},"ISSN":["0927-2852","1572-9095"],"issn-type":[{"value":"0927-2852","type":"print"},{"value":"1572-9095","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,18]]},"assertion":[{"value":"20 February 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 November 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 November 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"40"}}