Categorifications of Non-Integer Quivers: Types ????₄, ????₃ and ????₂(2????+1)

We define the notion of a weighted unfolding of quivers with real weights, and use this to provide a categorification of mutations of quivers of finite types H 4 H_4 , H 3 H_3 and I 2 ( 2 n + 1 ) I_2(2n+1) . In particular, the (un)folding induces a semiring action on the categories associated to the unfolded quivers of types E 8 E_8 , D 6 D_6 and A 2 n A_{2n} respectively. We then define the tropical seed pattern on the folded quivers, which includes c c - and g g -vectors, and show its compatibility with the unfolding.