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arXiv · 2203.16679

Which cluster morphism categories are CAT(0)

Kiyoshi Igusa·Gordana Todorov
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Abstract

The cluster morphism category of an hereditary algebra was introduced in [5] to show that the picture space of an hereditary algebra of finite representation type is a $K(\pi,1)$ for the associated picture group, thereby allowing for the computation of the homology of picture groups of finite type as carried out in [7] for the case of $A_n$. In this paper we show that the cluster morphism category is a $CAT(0)$-category for hereditary algebras of finite or tame type with only small tubes. As a consequence, we get that the classifying space of the cluster morphism category is a locally $CAT(0)$ space and, as a consequence of that, we get that this classifying space is a $K(\pi,1)$.

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Kiyoshi Igusa, Gordana Todorov. 2022-03-30. Which cluster morphism categories are CAT(0). https://arxiv.org/abs/2203.16679

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