Opetopic algebras I: Algebraic structures on opetopic sets

Cédric Ho Thanh (1, 2) , Chaitanya Leena Subramaniam (1)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 PI.R2 - Design, study and implementation of languages for proofs and programs

We define a family of structures called ``opetopic algebras'', which are algebraic structures with an underlying opetopic set. Examples of such are categories, planar operads, and Loday's combinads over planar trees. Opetopic algebras can be defined in two ways, either as the algebras of a ``free pasting diagram'' parametric right adjoint monad, or as models of a small projective sketch over the category of opetopes. We define an opetopic nerve functor that fully embeds each category of opetopic algebras into the category of opetopic sets. In particular, we obtain fully faithful opetopic nerve functors for categories and for planar coloured $\Set$-operads. This paper is the first in a series aimed at using opetopic spaces as models for higher algebraic structures.

Mots clés: Projective sketch; Polynomial monad; Opetope; Opetopic set; Operad

Domaines: Catégories et ensembles [math.CT]

Cédric Ho Thanh, Chaitanya Leena Subramaniam. Opetopic algebras I: Algebraic structures on opetopic sets. 2019. ⟨hal-02343861v2⟩