## Catalan categories

### Jonathan Cohen – 16 January 2008

I'll introduce Catalan categories, which form a coherent categorification of the theory of higher-order associativity. These structures are interesting from several perspectives but, within the context of combinatorial algebra, the main interest stems from the fact that they can be leveraged in order to obtain new presentations for the higher Thompson groups $F_{n,1}$. The coherence axioms for catalan categories directly generalise Mac Lane's axiomatisation of coherent associativity, with an extra dynamic coming into play in the higher-order case. Subsequently, I'll move onto symmetric catalan categories, which encode an action of the symmetric group. These in turn yield a presentation of the Higman-Thompson groups $G_{n,1}$ and their axiomatisation generalises Mac Lane's axioms for coherent associativity and commutativity. I'll briefly review the necessary results from my previous talk, making this talk essentially self-contained.

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