Polynomial and Analytic Functors and Monads, revisited

Szawiel, Stanisław; Zawadowski, Marek

doi:10.48550/arxiv.1506.04317

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“…The +-construction was further generalized to Cartesian monads by Leinster [23], and interpreted combinatorially by Kock, Joyal, Batanin, and Mascari [22], using the calculus of polynomial functors of Gambino and Hyland [16]. An even more conceptual interpretation in this direction was given by Szawiel and Zawadowski [34,35,36]. For another approach using syntactic methods, see Curien, Thanh, and Mimram [12].…”

Section: Introductionmentioning

confidence: 99%

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Kriz

2021

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In this paper, I introduce a new generalization of the concept of an operad, further generalizing the concept of an opetope introduced by Baez and Dolan, who used this for the definition of their version of non-strict n-categories. Opetopes arise from iterating a certain construction on operads called the +-construction, starting with monoids. The first step gives rise to plain operads, i.e. operads without symmetries. The permutation axiom in a symmetric operad, however, is an additional structure resulting from permutation of variables, independent of the structure of a monoid. Even though we can apply the +-construction to symmetric operads, there is the possibility of introducing a completely different kind of permutations on the higher levels by again permuting variables without regard to the structure on the previous levels. Defining and investigating these structures is the main purpose of this paper. The structures obtained in this way is what I call n-actads. In n-actads with n > 1, the permutations on the different levels give rise to a certain special kind of n-fold category. I also explore the concept of iterated algebras over an n-actad (generalizing an algebra and module over an operad), and various types of iterated units. I give some examples of algebras over 2-actads, and show how they can be used to construct certain new interesting homotopy types of operads. I also discuss a connection between actads and ordinal notation.

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Section: Introductionmentioning

confidence: 99%

“…The concept essentially allows one to label the entries of a tree with the elements of I (for example, see (36)). In the case of I = {0} and I = {0, n}, the concept gives the notions of algebras and modules, respectively.…”

Section: Introductionmentioning

confidence: 99%

Actads

Kriz

2021

Preprint

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Polynomial and Analytic Functors and Monads, revisited

Abstract: We describe an abstract 2-categorical setting to study various notions of polynomial and analytic functors and monads.

Cited by 1 publication

References 5 publications

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