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Explore the combinatorial foundations of opetopic shapes in this mathematical colloquium lecture dedicated to the memory of Marek Zawadowski. Delve into the framework of opetopes and opetopic sets, originally introduced by Baez and Dolan for handling higher structures and higher coherences in mathematics. Examine the many-to-one shapes that exist in all dimensions and discover their various representational forms, from abstract constructions using polynomial monads to concrete implementations as zoom complexes and oriented posets of faces. Learn about the groundbreaking work establishing precise isomorphisms between different combinatorial presentations, including the innovative concept of opetopes as epiphytes - recursively defined tree structures with nodes and edges decorated by lower-dimensional epiphytes. Navigate through the intricate combinatorial descriptions including epiphytes, zoom complexes, and oriented face structures, gaining insight into how these mathematical objects bridge abstract categorical theory with concrete combinatorial representations. Understand the collaborative research efforts that have advanced this field and appreciate the mathematical elegance of these higher-dimensional structures that are fundamental to modern category theory and algebraic topology.
