Marcelo Fiore (University of Cambridge, UK), An Algebraic Combinatorial Approach to the Abstract Syntax of Opetopic Structures



The starting point of the talk will be the identification of structure common to tree-like combinatorial objects, exemplifying the situation with abstract syntax trees (as used in formal languages) and with opetopes (as used in higher-dimensional algebra). The emerging mathematical structure will be then formalized in a categorical setting, unifying the algebraic aspects of the theory of abstract syntax and the theory of opetopes. This realization conceptually allows one to transport viewpoints between these, now bridged, mathematical theories and I will explore it here in the direction of higher-dimensional algebra, giving an algebraic combinatorial framework for a generalisation of the slice construction for generating opetopes. The technical work will involve setting up a microcosm principle for near-semirings and subsequently exploit it in the cartesian closed bicategory of generalised species of structures. Connections to (cartesian and symmetric monoidal) equational theories, type theory, lambda calculus, and algebraic combinatorics will be mentioned in passing.