Other models for ∞-operads

Remark 12.4.5 (Other models for \(\infty \)-operads).

The span-based model used in this book and Lurie’s model over \(\Fin _*\) from Section 17.4 are not the only available descriptions of \(\infty \)-operads. In the dendroidal approach, introduced by Moerdijk and Weiss (2007), the simplex category is replaced by a category of trees. The presentation of \(\infty \)-operads by complete dendroidal Segal animae is due to Cisinski and Moerdijk (2013). A direct comparison with Lurie’s model is given by Hinich and Moerdijk (2024); see [Heuts and Moerdijk (2022)] for a comprehensive account of dendroidal homotopy theory.

The classical description of one-colored operads as associative algebras in symmetric sequences also admits a homotopy-coherent generalization due to Haugseng (2022). More generally, this construction treats varying animae of colors by means of a double \(\infty \)-category of symmetric collections; see [Haugseng (2023)] for a general introduction to \(\infty \)-operads. A related viewpoint describes pinned \(\infty \)-operads by analytic monads on slices \(\An _{/S}\), with the anima \(S\) of colors allowed to vary [Gepner et al. (2022); Haugseng (2023)]. This description is compared directly with both Lurie’s model and the span-based model through their Lawvere theories by Haugseng (2023). More general Segal-type formalisms, notably the theory of algebraic patterns developed by Chu and Haugseng (2021), provide a common framework for these and many similar homotopy-coherent algebraic structures. The general theory of envelopes for algebraic patterns developed by Barkan et al. (2022) contains our construction of operadic envelopes in Section 17.3 as a special case. We will not otherwise use the alternative presentations discussed in this remark.