Abstract

Part 1: Let Q be a valued quiver. We already defined the picture LQ and the picture group G0(LQ). I will point out a few more facts about semi-invariant pictures and picture groups for valued (and modulated) quivers. This was in- spired by Tomoki Nakanishi’s comment about picture groups for valued quivers. Part 2: The original motivation for introducing Cluster Morphism Categories was to obtain cubical categories (with an additional structure “locally CAT ”) which was used to prove that the picture space is a K(π, 1) for the picture group, which was an important step for computing homology of picture groups. The notion of Cluster Morphism Category was beautifully generalized by Aslak Buan and Bethany Marsh and we already heard their talks. Also, Eric Hanson showed that the Picture space for Nakayama algebras is a K(π,1). It would be good to know which of the spaces are K(π,1) for Picture groups of algebras which are not hereditary of finite type.