A category of noncrossing partitions
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866913750596976640 |
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| author | Igusa, Kiyoshi |
| author_facet | Igusa, Kiyoshi |
| contents | In [17], we introduced ``picture groups'' and computed the cohomology of the picture group of type $A_n$. This is the same group what was introduced by Loday [20] where he called it the ``Stasheff group''. In this paper, we give an elementary combinatorial interpretation of the {\color{blue}``cluster morphism category'' constructed in [13] in the special case of the linearly oriented quiver of type $A_n$.} We prove that the classifying space of this category is locally $CAT(0)$ and thus a $K(π,1)$. We prove a more general statement that classifying spaces of certain ``cubical categories'' are locally $CAT(0)$. The objects of our category are the classical noncrossing partitions introduced by Kreweras [19]. The morphisms are binary forests. This paper is independent of [13] and [17] except in the last section where we use [13] to compare our category with the category with the same name given by Hubery and Krause [9]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1411_0196 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | A category of noncrossing partitions Igusa, Kiyoshi Representation Theory 16G20 In [17], we introduced ``picture groups'' and computed the cohomology of the picture group of type $A_n$. This is the same group what was introduced by Loday [20] where he called it the ``Stasheff group''. In this paper, we give an elementary combinatorial interpretation of the {\color{blue}``cluster morphism category'' constructed in [13] in the special case of the linearly oriented quiver of type $A_n$.} We prove that the classifying space of this category is locally $CAT(0)$ and thus a $K(π,1)$. We prove a more general statement that classifying spaces of certain ``cubical categories'' are locally $CAT(0)$. The objects of our category are the classical noncrossing partitions introduced by Kreweras [19]. The morphisms are binary forests. This paper is independent of [13] and [17] except in the last section where we use [13] to compare our category with the category with the same name given by Hubery and Krause [9]. |
| title | A category of noncrossing partitions |
| topic | Representation Theory 16G20 |
| url | https://arxiv.org/abs/1411.0196 |