We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation and explicit computation. To relate our definition to the classical definition, we recast the Baez-Dolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the Baez-Dolan construction, starting with the trivial monad. We show that our notion of opetope agrees with Leinster's. Next we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the Baez-Dolan construction. A final section is devoted to example computations, and indicates also how the calculus of opetopes is well-suited for machine implementation.
Deep Dive into Polynomial functors and opetopes.
We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation and explicit computation. To relate our definition to the classical definition, we recast the Baez-Dolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the Baez-Dolan construction, starting with the trivial monad. We show that our notion of opetope agrees with Leinster’s. Next we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the Baez-Dolan construction. A final section is devoted to example computations, and indicates also how the calculus of opetopes is well-suited for machine implementation.
Among a dozen or so existing definitions of weak higher categories, the opetopic approach is one of the most intriguing, since it is based on a collection of 'shapes' that had not previously been studied: the opetopes. Opetopes are combinatorial structures parametrising higher-dimensional many-in/one-out operations, and can be seen as higher-dimensional generalisations of trees. They are important combinatorial structures on their own, 'as pervasive in higher-dimensional algebra as simplices are in geometry', according to Leinster [14,p.216]. Opetopes and opetopic higher categories were introduced by Baez and Dolan in the seminal paper [1], and the theory has been developed further by Hermida-Makkai-Power [9], Leinster [14], Cheng [2], [3], [4], [5], and others. It is in a sense a theory from scratch, compared to several other theories of higher categories which build on large bodies of preexisting machinery and experience, e.g. simplicial methods. The full potential of the opetopic approach may depend on a deeper understanding of the combinatorics of opetopes.
At the conference on n-categories: Foundations and applications at the IMA in Minneapolis, June 2004, much time was dedicated to opetopes, but it became clear that a concise and direct definition of opetopes was lacking, and that there was no practical way to represent higher-dimensional opetopes on the blackboard. In fact, there did not seem to exist a general method to represent concrete opetopes in any way, algebraic, graphical, or by machine. 1 The best definitions are very abstract and not very handson: e.g. Leinster’s definition in terms of iterated free cartesian monads [14], or the Hermida-Makkai-Power [9] definition of opetopic sets (there called multitopic sets), followed by a theorem that this category is a presheaf category, hence characterising a category of opetopes (there called multitopes).
As to graphical representations of opetopes in low dimensions, the current method is based on a polytope interpretation of opetopes (which is at the origin of the terminology: the word ‘opetope’ comes from ‘operation’ and ‘polytope’). Leinster [14, § 7.4] has constructed a geometric realisation functor which provides support for this interpretation, although the polytopes in general cannot be piece-wise linear objects in Euclidean space. Moreover, geometrical objects in dimension higher than 3 are inherently difficult to represent graphically, and currently one resorts to Lego-like drawings in which the individual faces of the polytopes are drawn separately, with small arrows as a recipe to indicate how they are supposed to fit together.
The goal of this paper is to come closer to the combinatorics. Our initial idea was to represent an opetope as a tree with some circles, which we now call constellations. This works in dimension 4 (cf. 1.11 below), but it does not seem to be sufficient to capture the possible opetopes in dimension 5 and higher. Pursuing the idea, what we eventually found was a representation in terms of a sequence of trees with circles, and in fact it is basically the notion of metatree originally proposed by Baez and Dolan. That notion was never really developed, though: in the original paper [1] the claim that metatrees could express opetopes was not really substantiated, and in the subsequent literature there seems to be no mention of the metatree notion. The presence of circles makes a conceptual difference, and it also reveals a certain shortcoming in the original notion of metatree, related to units (cf. 1.21).
We hasten to point out that our notion of opetope coincides with the notion due to Leinster [14] (cf. the explicit comparison culminating in Theorem 3. 16), not with the original Baez-Dolan definition: we work consistently with non-planar trees, which means our opetopes are ‘un-ordered’ like abstract geometric objects, whereas the original Baez-Dolan opetopes come equipped with an ordering of their faces. In our version, the planar aspect is only a particular feature of low dimensional opetopes.
While our opetopes agree with Leinster’s, the description we provide is completely elementary and does not even make reference to category theory. We think that our description can serve as the famous ‘5-minute definition’ that was previously missing, and that it can provide a convenient tool for communicating opetopical ideas. We also indicate how our approach is well-suited for machine manipulation.
Opetopes were introduced to parametrise higher-dimensional substitution operations. Surprisingly, opetopes arise also in another way, namely from computads and higher-dimensional pasting theory, and we wish to mention that a very different combinatorial approach has been developed in this setting by Palm [15]. A computad is a strict ω-category which is dimension-wise free. This notion was devised by Street [18] as a tool for describing higher-dimensional compositions in strict ncategories. In the works of Johnson [10] and Po
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