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Polynomial functors and opetopes
 In preparation
"... We give an elementary and direct combinatorial definition of opetopes in terms of trees, wellsuited for graphical manipulation (e.g. drawings of opetopes of any dimension and basic operations like sources, target, and composition); a substantial part of the paper is constituted by drawings and exam ..."
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We give an elementary and direct combinatorial definition of opetopes in terms of trees, wellsuited for graphical manipulation (e.g. drawings of opetopes of any dimension and basic operations like sources, target, and composition); a substantial part of the paper is constituted by drawings and example computations. To relate our definition to the classical definition, we recast the BaezDolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the BaezDolan construction, starting with the trivial monad. Finally we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the BaezDolan construction. The calculus of opetopes is also wellsuited for machine implementation: in an appendix we show how to represent opetopes in XML, and manipulate them with simple Tcl scripts.
A PREHISTORY OF nCATEGORICAL PHYSICS
, 2008
"... We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, me ..."
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We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, membranes and spin foams.
Contents
, 2002
"... We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (EQ, TQ) and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the ‘free operad ’ monad cons ..."
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We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (EQ, TQ) and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the ‘free operad ’ monad construction on suitable monads. We use this to give an explicit description of the relationship between BaezDolan and Leinster opetopes.
HIGHER DIMENSIONAL ALGEBRAS VIA COLORED PROPS
, 809
"... Abstract. Starting from any unital colored PROP P, we define a category P(P) of shapes called Ppropertopes. Presheaves on P(P) are called Ppropertopic sets. For 0 ≤ n ≤ ∞ we define and study ntime categorified Palgebras as Ppropertopic sets with some lifting properties. Taking appropriate PROPs ..."
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Abstract. Starting from any unital colored PROP P, we define a category P(P) of shapes called Ppropertopes. Presheaves on P(P) are called Ppropertopic sets. For 0 ≤ n ≤ ∞ we define and study ntime categorified Palgebras as Ppropertopic sets with some lifting properties. Taking appropriate PROPs P, we obtain higher categorical versions of polycategories, 2fold monoidal categories, topological quantum field theories, and so on.