Related papers: All Segal objects are generalised monads in spans

It is shown how double categories provide a direct abstract approach to coloured operads; namely, product-preserving normal lax functors from (Pb C)^op (the opposite of the double category of pullback squares in C) to Cat (the double…

Let $\mathscr X$ be an $\infty$-topos, for example the $\infty$-category of simplicial sheaves on a Grothendieck site. Then $\infty$-group sheaves are group objects in $\mathscr X$. Let $A\in\mathrm{Grp}\mathscr X$ be such a group object.…

Idempotent analogues of convexity are introduced. It is proved that the category of algebras for the capacity monad in the category of compacta is isomorphic to the category of $(\max,\min)$-idempotent biconvex compacta and their biaffine…

Notions of `operad' and `multicategory' abound. This work provides a single framework in which many of these various notions can be expressed. Explicitly: given a monad * on a category S, we define the term `(S,*)-multicategory', subject to…

In this paper, we address the construction of homotopy bicategories of $(\infty,2)$-categories, which we take as being modeled by 2-fold Segal spaces. Our main result is the concrete construction of a functor $h_2$ from the category of…

An object $P$ in a monoidal category $\mathcal{C}$ is called pivotal if its left dual and right dual objects are isomorphic. Given such an object and a choice of dual $Q$, we construct the category $\mathcal{C}(P,Q)$, of objects which…

Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case,…

Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any…

We introduce \emph{flagged $(\infty,n)$-categories} and prove that they are equivalent to Segal sheaves on Joyal's category ${\mathbf\Theta}_n$. As such, flagged $(\infty,n)$-categories provide a model-independent formulation of Segal…

In this paper we define a sequence of monads $\mathbb{T}^(\infty;n)$ $(n\in\mathbb{N})$ on $\infty$-$\mathbb{G}\text{r}$, the category of the $\infty$-graphs. We conjecture that algebras for $\mathbb{T}^(0;n)$ which are defined in a purely…

In this survey article, we give an introduction to the notion of a 2-Segal set and prove that 2-Segal sets are equivalent to pseudomonoids in the bicategory of spans. The proof utilizes graphical techniques for 2-Segal sets and spans that…

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…

Many structured categories of interest are most naturally described as algebras for a relative monad, but turn out nonetheless to be algebras for an ordinary monad. We show that, under suitable hypotheses, the left oplax Kan extension of a…

Given a family of model categories $\cal E \to \cal C$, we associate to it a homotopical category of derived, or Segal, sections $DSect(\cal C,\cal E)$ that models the higher-categorical sections of the localisation $L\cal E \to \cal C$.…

In this paper we give a summary of the comparisons between different definitions of so-called (\infty,1)-categories, which are considered to be models for \infty-categories whose n-morphisms are all invertible for n>1. They are also, from…

By building on the notions of internal projective and injective objects in a module category introduced by Douglas, Schommer-Pries, and Snyder, we extend the reconstruction theory for module categories of Etingof and Ostrik. More…

We develop a theory of enriched categories over a (higher) category M equipped with a class W of morphisms called homotopy equivalences. We call them Segal M_W -categories. Our motivation was to generalize the notion of "up-to-homotopy…

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of…

The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every…

We describe various equivalent ways of associating to an orbifold, or more generally a higher \'etale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we…