Dendroidal Segal spaces and infinity-operads

The category of dendroidal sets is an extension of that of simplicial sets, suitable for constructing nerves, not just of categories but also of (coloured) operads. It was introduced with this purpose, and with the aim of giving an inductive definition of weak higher categories, in [14,15]. This category dSet of dendroidal sets carries a symmetric monoidal closed structure which is closely related to the Boardman-Vogt tensor product of operads, and the inclusion of the category sSet of simplicial sets into dSet can in fact be identified with the forgetful functor, from the slice (or comma) category of dendroidal sets over the unit η of the monoidal structure back to dendroidal sets, via an explicit isomorphism of categories (0.0.1) dSet /η = sSet Dendroidal sets carry a very rich homotopical structure, which we began to explore in [7]. For example, there is a monoidal Quillen model structure on dSet , whose fibrant objects include all nerves of operads. In fact, these fibrant objects can be thought of as simple combinatorial models of the notion of operad-up-tohomotopy or "∞-operad". Like any Quillen model structure, this model structure on dendroidal sets induces another model structure on any slice category. Under the identification dSet /η = sSet this induced model structure can be shown to coincide with the Joyal model structure on simplicial sets, whose fibrant objects are most commonly known under the name "∞-categories" (and are also referred to as quasi-categories, weak Kan complexes, or inner Kan complexes [11,13,4]).

These ∞-categories model a notion of category-up-to-homotopy. Other ways of modelling such a notion have occurred in the literature, including the theory of Segal categories [17,2] and of complete Segal spaces [16]. The latter two concepts are both based on the much older observation that a simplicial set X is the nerve of a category if and only if the canonical map (0.0.2)

sending a simplex to its one-dimensional ribbons, is an isomorphism. Indeed, Simpson and Rezk both base their theories on bisimplicial sets X for which the map (0.0.2) is a weak equivalence of simplicial sets (and replacing the fibred product on the right hand side by its homotopy version). Building on the work of Simpson and Rezk, the relation between these different ways of modelling categories-upto-homotopy was recently made precise through the work of Bergner, Joyal and Tierney, and Lurie. Indeed, Simpson’s Segal categories, Rezk’s complete Segal spaces, and Joyal’s ∞-categories all arise as the fibrant objects in a specific Quillen model category structure, and these different model category structures have now been related to each other by explicit Quillen equivalences [2,13]. Moreover, they are all Quillen equivalent to the model category of simplicial categories discovered by Bergner [2], thus providing a strictification or rigidification result for each of these notions of category-up-to-homotopy. The goal of this paper and its sequel [8] is to develop analogous theories of Segal operads (rather than categories) and complete dendroidal (rather than simplicial) Segal spaces, to relate these to each other and to dendroidal sets via Quillen equivalences, and to prove a strictification result for each of them by relating them to simplicial operads. By a simple slicing procedure like in (0.0.1), the earlier results just mentioned for categories-up-to-homotopy can all be recovered from our results, which can in this sense be said to be more general.

In more detail, then, we will consider the category sdSet of simplicial objects in dendroidal sets, or what is the same, dendroidal spaces. We will define a Segal type condition on the objects of this category, based on an extension to trees of “the union of 1-dimensional ribbons in an n-simplex” to which we will refer as the Segal core of a tree. In Section 5, we will establish a closed model category structure on sdSet whose fibrant objects satisfy a tree-like Segal condition involving these Segal cores, and a completeness condition like the one of Rezk, and prove (Corollary 6.7) that this model category is Quillen equivalent to our earlier model category structure on dendroidal sets [7]. The definitions and proofs of these results are based on some elementary observations about these Segal cores presented in Section 2, and on a characterization of weak equivalences between ∞-operads as maps which are “essentially surjective and fully faithful” in a suitable sense (Theorem 3.5). The proof also exploits the hybrid nature of the objects of sdSet , which can be viewed alternatively as simplicial objects in one category or as dendroidal objects in another. In fact, the first view point is taken in Section 4, while the second viewpoint underlies the notion of complete dendroidal Segal space and the formulation of the main equivalence 6.7. The relation between these two view points is most clearly expressed by Theorem 6.6 which equates two seemingly different model ca

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