Model categories of quiver representations

2019/02/06 by Holm, Henrik, Jorgensen, Peter · 1 citation
#18E30 #18E35 #18G55 #Category Theory (math.CT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1902.02387

Abstract

Gillespie's Theorem gives a systematic way to construct model category structures on \mathscrC( \mathscrM ), the category of chain complexes over an abelian category \mathscrM. We can view \mathscrC( \mathscrM ) as the category of representations of the quiver ⋯ → 2 → 1 → 0 → -1 → -2 → ⋯ with the relations that two consecutive arrows compose to 0. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, and following Iyama and Minamoto, their representations can be viewed as generalised chain complexes. Our result gives a systematic way to construct model category structures on many categories. This includes the category of N-periodic chain complexes, the category of N-complexes where ∂^N = 0, and the category of representations of the repetitive quiver ℤ A_n with mesh relations.

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