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We use Luries symmetric monoidal envelope functor to give two new descriptions of $infty$-operads: as certain symmetric monoidal $infty$-categories whose underlying symmetric monoidal $infty$-groupoids are free, and as certain symmetric monoidal $infty$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $infty$-operads, as a localization of a presheaf $infty$-category, and we use this to give a simple proof of the equivalence between Luries and Barwicks models for $infty$-operads.
Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $infty$-categories. One of our main results is an $infty$-categorical ge
In this short note we prove that two definitions of (co)ends in $infty$-categories, via twisted arrow $infty$-categories and via $infty$-categories of simplices, are equivalent. We also show that weighted (co)limits, which can be defined as certain (
We study lax families of adjoints from a fibrational viewpoint, obtaining a version of the mate correspondence for (op)lax natural transformations of functors from an $infty$-category to the $(infty,2)$-category of $infty$-categories. We apply this t
Indexed symmetric monoidal categories are an important refinement of bicategories -- this structure underlies several familiar bicategories, including the homotopy bicategory of parametrized spectra, and its equivariant and fiberwise generalizations.
We use the basic expected properties of the Gray tensor product of $(infty,2)$-categories to study (co)lax natural transformations. Using results of Riehl-Verity and Zaganidis we identify lax transformations between adjunctions and monads with commut