The web monoid and opetopic sets

Abstract

We develop a new definition of opetopic sets. There are two main technical ingredients. The first is the systematic use of fibrations, which are implicit in most of the approaches in the literature. Their explicit use leads to certain clarifications in the construction of opetopic sets and other constructions. The second is the "web monoid", which plays a role analogous to the "operad for operads" of Baez and Dolan, the "multicategory of function replacement" of Hermida, Makkai and Power. We demonstrate that the web monoid is closely related to the "Baez-Dolan slice construction" as defined by Kock, Joyal, Batanin and Mascari.

Theories of analytic monads

“…Some categories of algebras can be also described as algebras for a symmetric operad, and some can be described as algebras for a rigid † operad (Hermida et al 2000;Hermida et al 2001;Hermida et al 2002;Zawadowski 2011). This additional complication was necessary for the Hermida-Makkai-Power construction of opetopic sets, but in Szawiel and Zawadowski (2013b) we gave a more conceptual and simpler construction of opetopic sets based on multicategories with objects of one kind only. † What we call a 'rigid operad' was earlier called an 'operad with non-standard amalgamation', that is, a one-object version of the multicategories considered by C. Hermida, M. Makkai and J.…”

“…In fact, the multicategories considered in Hermida et al (2000;2001;2002) have yet another feature in that they have two kinds of objects (upper and lower). This additional complication was necessary for the Hermida-Makkai-Power construction of opetopic sets, but in Szawiel and Zawadowski (2013b) we gave a more conceptual and simpler construction of opetopic sets based on multicategories with objects of one kind only. A more detailed discussion of the comparison of rigid operads/multicategories with the multicategories considered in Hermida et al (2000;2001;2002) is presented in Zawadowski (2011, Sections 6.5 and 6.5).…”

Actads

“…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].…”

Polynomial and Analytic Functors and Monads, revisited

“…This was done in [Sz] to compare various algebraic definitions of opetopic sets, cf. [BD], [HMP], [Z2], [KJBM], [SZ2]. The polynomial functors are related to Kleisli algebras, whereas analytic functors are related to Eilenberg-Moore algebras.…”