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Kaipel, Maximilian ORCID: 0009-0000-3034-2421 (2025). The category of a partitioned fan. Journal of the London Mathematical Society, 111 (2). pp. 1-41. Wiley. ISSN 0024-6107

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Abstract

In this paper the notion of an admissible partition of a simplicial polyhedral fan is introduced and the category of a partitioned fan is defined as a generalisation of the ‐cluster morphism category of a finite‐dimensional algebra. This establishes a complete lattice of categories around the ‐cluster morphism category, which is closely tied to the fan structure. We prove that the classifying spaces of these categories are cube complexes, which reduces the process of determining if they are spaces to three sufficient conditions. We characterise when these conditions are satisfied for fans in and prove that the first one, the existence of a certain faithful functor, is satisfied for hyperplane arrangements whose normal vectors lie in the positive orthant. As a consequence, we obtain a new infinite class of algebras for which the ‐cluster morphism category admits a faithful functor and for which the cube complexes are spaces. In the final section, we also offer a new algebraic proof of the relationship between an algebra and its ‐vector fan.

Item Type:	Article
Creators:	Creators Email ORCID ORCID Put Code Kaipel, Maximilian UNSPECIFIED https://orcid.org/0009-0000-3034-2421 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-796480
Identification Number:	10.1112/jlms.70071
Journal or Publication Title:	Journal of the London Mathematical Society
Volume:	111
Number:	2
Page Range:	pp. 1-41
Date:	27 February 2025
Publisher:	Wiley
ISSN:	0024-6107
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	Mathematics
['eprint_fieldname_oa_funders' not defined]:	Publikationsfonds UzK
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/79648