Dendroidal Segal spaces and infinity-operads

We introduce the dendroidal analogs of the notions of complete Segal space and of Segal category, and construct two appropriate model categories for which each of these notions corresponds to the property of being fibrant. We prove that these two model categories are Quillen equivalent to each other, and to the monoidal model category for infinity-operads which we constructed in an earlier paper. By slicing over the monoidal unit objects in these model categories, we derive as immediate corollaries the known comparison results between Joyal’s quasi-categories, Rezk’s complete Segal spaces, and Segal categories.

💡 Research Summary

The paper develops a homotopy‑theoretic framework for ∞‑operads by introducing dendroidal analogues of Segal spaces and complete Segal spaces. Starting from the category of dendroidal sets—presheaves on the category of trees—the authors recall the operadic model structure, in which cofibrations are monomorphisms, weak equivalences are operadic weak equivalences, and fibrant objects model ∞‑operads.

A dendroidal Segal space is defined as a dendroidal presheaf X satisfying two homotopical conditions. First, X must have fillers for all inner horns, mirroring the inner Kan condition for simplicial sets. Second, for any tree T with input subtrees T₁,…,Tₙ, the canonical Segal map
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