We introduce the concept of a dendroidal set. This is a generalization of the notion of a simplicial set, specially suited to the study of (coloured) operads in the context of homotopy theory. We define a category of trees, which extends the category used in simplicial sets, whose presheaf category is the category of dendroidal sets. We show that there is a closed monoidal structure on dendroidal sets which is closely related to the Boardman-Vogt tensor product of (coloured) operads. Furthermore, we show that each (coloured) operad in a suitable model category has a coherent homotopy nerve which is a dendroidal set, extending another construction of Boardman and Vogt. We also define a notion of an inner Kan dendroidal set, which is closely related to simplicial Kan complexes. Finally, we briefly indicate the theory of dendroidal objects in more general monoidal categories, and outline several of the applications and further theory of dendroidal sets.
Abstract. Dendroidal sets offer a formalism for the study of ∞-operads akin to the formalism of ∞-categories by means of simplicial sets. We present here an account of the current state of the theory while placing it in the context of the ideas that led to the conception of dendroidal sets. We briefly illustrate how the added flexibility embodied in ∞-operads can be used in the study of A∞-spaces and weak n-categories in a way that cannot be realized using strict operads.
With the blessing of hind sight we consider the problem of metrizability and
show that the classical Bing-Nagata-Smirnov Theorem and a more recent result of
Flagg give complementary answers to the metrization problem, that are in a
sense dual to each other.Comment: 3 pages54E3
a b s t r a c tWe prove a Dold-Kan type correspondence between the category of planar dendroidal abelian groups and a suitably constructed category of planar dendroidal chain complexes. Our result naturally extends the classical Dold-Kan correspondence between the category of simplicial abelian groups and the category of non-negatively graded chain complexes.
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