Related papers: On rectification and enrichment of infinity proper…
We prove a rectification theorem for enriched infinity-categories: If V is a nice monoidal model category, we show that the homotopy theory of infinity-categories enriched in V is equivalent to the familiar homotopy theory of categories…
A theory of $\infty$-properads is developed, extending both the Joyal-Lurie $\infty$-categories and the Cisinski-Moerdijk-Weiss $\infty$-operads. Every connected wheel-free graph generates a properad, giving rise to the graphical category…
In this paper we initiate the study of enriched $\infty$-operads. We introduce several models for these objects, including enriched versions of Barwick's Segal operads and the dendroidal Segal spaces of Cisinski and Moerdijk, and show these…
This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via…
We continue the study of enriched infinity categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched infinity categories are associative monoids in an especially designed monoidal category of…
We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching…
A categorical formalism for directed graphs is introduced, featuring natural notions of morphisms and subgraphs, and leading to two elementary descriptions of the free-properad monad, first in terms of presheaves on elementary graphs,…
In this paper, we settle the homotopy properties of the infinity-morphisms of homotopy (bial)-gebras over properads, i.e. algebraic structures made up of operations with several inputs and outputs. We start by providing the literature with…
We recall several categories of graphs which are useful for describing homotopy-coherent versions of generalized operads (e.g. cyclic operads, modular operads, properads, and so on), and give new, uniform definitions for their morphisms.…
This note is an introduction to several generalizations of the dendroidal sets of Moerdijk--Weiss. Dendroidal sets are presheaves on a category of rooted trees, and here we consider indexing categories whose objects are other kinds of…
We prove a number of results involving categories enriched over \textsc{CMet}, the category of complete metric spaces with possibly infinite distances. The category \textsc{CPMet} of intrinsic complete metric spaces is locally…
We show how the model structure on the category of simplicially-enriched (colored) props induces a model structure on the category of simplicially-enriched (colored) properads. A similar result holds for dioperads.
We study the categories governing infinity (wheeled) properads. The graphical category, which was already known to be generalized Reedy, is in fact an Eilenberg-Zilber category. A minor alteration to the definition of the wheeled graphical…
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…
In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete…
We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of…
We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of…
We construct a new model category presenting the homotopy theory of presheaves on "inverse EI $(\infty,1)$-categories", which contains universe objects that satisfy Voevodsky's univalence axiom. In addition to diagrams on ordinary inverse…
This paper introduces a skew variant of the notion of enriched category, suitable for enrichment over a skew-monoidal category, the main novelty of which is that the elements of the enriched hom-objects need not be in bijection with the…
We introduce a general notion of enrichment for homotopy-coherent algebraic structures described by Segal conditions, using the framework of "algebraic patterns" developed in our previous work. This recovers several known examples of…