Related papers: Algebras of higher operads as enriched categories …
One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2-categories. In this paper we continue the developments of [3] and [2] by understanding the…
In this article we extend the theory of lax monoidal structures, also known as multitensors, and the monads on categories of enriched graphs that they give rise to. Our first principal result -- the lifting theorem for multitensors --…
In this paper we unify previous developments on higher operads and multitensors into a single framework in which the interplay between multitensors on a category V, and monads on the category of graphs enriched in V, is taken as…
The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the…
We give a framework for comparing on the one hand theories of n-categories that are weakly enriched operadically, and on the other hand n-categories given as algebras for a contractible globular operad. Examples of the former are the…
We continue to develop the theory of monads and multitensors. The central result of this paper - the lifting theorem for multitensors - enables us to see the Gray tensor product of 2-categories and the Crans tensor product of Gray…
We construct an $(\infty,2)$-version of the (lax) Gray tensor product. On the 1-categorical level, this is a binary (or more generally an $n$-ary) functor on the category of $\Theta_2$-sets, and it is shown to be left Quillen with respect…
Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key mathematical disciplines, notably homotopy…
We construct a (lax) Gray tensor product of $(\infty,2)$-categories and characterize it via a model-independent universal property. Namely, it is the unique monoidal biclosed structure on the $\infty$-category of $(\infty,2)$-categories…
It is well-known that the tensor product of two bialgebras constitutes the binary product in the category of cocommutative bialgebras and morphisms of bialgebras between them. In this paper, we extend this result to triangular bialgebras…
We discuss the folklore construction of the Gray tensor product of 2-categories as obtained by factoring the map from the funny tensor product to the cartesian product. We show that this factorisation can be obtained without using a…
In this paper we introduce a strict monoidal subcategory of the category of matrices, suitable to address a higher representation theoretic analogue of radicals (non-semisimplicity) in ordinary representation theory. We show the extent to…
Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or…
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics,…
It was argued by Crans that it is too much to ask that the category of Gray-categories admit a well behaved monoidal biclosed structure. We make this precise by establishing undesirable properties that any such monoidal biclosed structure…
In this paper, we present an infinity-categorical version of the theory of monoidal categories. We show that the infinity category of spectra admits an essentially unique monoidal structure (such that the tensor product preserves colimits…
In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs,…
We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this…
We construct in a unifying way skew-multicategories and multicategories of double and Gray-categories that we call Gray (skew) multicategories. We study their different versions depending on the types of functors and higher transforms. We…
The purpose of this dissertation is to set up a theory of generalized operads and multicategories, and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed…