Related papers: Monoidal envelopes and Grothendieck construction f…

We provide a categorical framework for mathematical objects for which there is both a sort of "independent" and "dependent" composition. Namely we model them as duoidal categories in which both monoidal structures share a unit and the first…

We compare two models for $\infty$-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all…

Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids. In particular, we show that the usual model category structure on the category of simplicial…

In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism…

We define a model category structure on a slice category of simplicial spaces, called the "Segal group action" structure whose fibrant-cofibrant objects may be viewed as representing spaces $X$ with a coherent action of a given Segal group…

We compare two approaches to the homotopy theory of infinity-operads. One of them, the theory of dendroidal sets, is based on an extension of the theory of simplicial sets and infinity-categories which replaces simplices by trees. The other…

We describe a Grothendieck construction for non-symmetric operads with values in categories, and hence in groupoids and posets. The construction produces a 2-category which is operadically fibered over the category D of finite non-empty…

Let $\mathcal{B}$ be a subcategory of a given category $\mathcal{D}$. Let $\mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $\mathcal{B}$ to $\mathcal{D}$ such that $\mathcal{B}$…

We exhibit the simplex category $\Delta$ and Segal's category $\Gamma$ as $\infty$-categorical localizations of the dendroidal categories $\Omega_\pi$ and $\Omega$ introduced by Moerdijk and Weiss. As an application we obtain an equivalence…

We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use…

This paper studies the homotopy theory of the Grothendieck construction using model categories and semi-model categories, provides a unifying framework for the homotopy theory of operads and their algebras and modules, and uses this…

We construct so called Hall monoidal categories (and Hall modules thereover) and exhibit them as a categorification of classical Hall and Hecke algebras (and certain modules thereover). The input of the (functorial!) construction are…

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…

The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state…

The goal of the paper is to establish and to investigate a fully faithful embedding of the category of group operads into that of crossed interval groups. For this, we introduce a monoidal structure on the slice of the category of operads…

We define and study opfibrations of $V$-enriched categories when $V$ is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with…

This paper gives an explicit description of the categorical operad whose algebras are precisely symmetric monoidal categories. This allows us to place the operad in a sequence of four, and therefore a sequence of four successively stricter…

We use fibrations of complete Segal spaces to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes with a universal fibration that…

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…

In this monograph, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A.…