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We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential image can be classified by a Segal condition. This theorem can be used to recover a related statement, due to Andre Joyal and Joachim Kock, concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full.
In this work we provide a definition of a coloured operad as a monoid in some monoidal category, and develop the machinery of Grobner bases for coloured operads. Among the examples for which we show the existance of a quadratic Grobner basis we con
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 $inf
We prove a bicategorical analogue of Quillens Theorem A. As an application, we deduce the well-known result that a pseudofunctor is a biequivalence if and only if it is essentially surjective on objects, essentially full on 1-cells, and fully faithful on 2-cells.
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers.
We combine Homotopy Type Theory with axiomatic cohesion, expressing the latter internally with a version of adjoint logic in which the discretization and codiscretization modalities are characterized using a judgmental formalism of crisp variables. T