Gijs Heuts — Page 2

arXiv2011

Algebras over infinity-operads

We develop a notion of an algebra over an infinity-operad with values in infinity-categories which is completely intrinsic to the formalism of dendroidal sets. Its definition involves the notion of a coCartesian fibration of dendroidal sets and extends Lurie's definition of a coCartesian fibration of simplicial sets. We show how, for a dendroidal set X, the coCartesian fibrations over X fit together to form an infinity-category coCart(X). Using a generalization of the Grothendieck construction, we prove that coCart(X) is equivalent to the infinity-category of algebras in infinity-categories over the simplicial operad associated to X. This equivalence can be restricted to give an equivalence between algebras taking values in infinity-groupoids (or equivalently, spaces) and the infinity-category of so-called left fibrations over X.

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arXiv2011

An infinite loop space machine for infinity-operads

This paper describes a consequence of the more general results of a previous paper which is of independent interest. We construct a functor from the category of dendroidal sets, which models the theory of infinity-operads, into the category of E-infinity-spaces. Applying May's infinite loop space machine for E-infinity-spaces then gives an infinite loop space machine for infinity-operads. We show that our machine exhibits the homotopy theory of E-infinity-spaces as a localization of the homotopy theory of infinity-operads.

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