Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userModular operads and the nerve theorem
166
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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 consider the seminal Lie-Rinehart operad whose algebras include pairs (functions, vector fields).
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.
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. This is a version of Quillens Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.
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. This yields type theories that we call spatial and cohesive, in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the fundamental $infty$-groupoid or shape), disentangling the identifications of Homotopy Type Theory from the continuous paths of topology. In a further refinement called real-cohesion, the shape is determined by continuous maps from the real numbers, as in classical algebraic topology. This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwers fixed-point theorem.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
