From operator categories to higher operads

From Operator Categories to Topological Operads

In this paper it is shown that an assortment of operads near to many topologists’ hearts enjoy (homotopy) universal properties of an expressly combinatorial nature. These include the operads An and En. The main idea that makes this possible is the new notion of an operator category, which controls the homotopy types of these operads in a strong sense.

Higher Operads, Higher Categories

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. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n + 1)-operad A, in such a way that the algebras for A ma...

Algebras of higher operads as enriched categories II

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to this task. The resulting theory includes the Gray tensor product of 2-categories and the Crans tensor product [12] of Gray categories. Moreover much of the pre...

AN INTRODUCTION TO HIGHER CLUSTER CATEGORIES

In this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. We focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of Fomin and Reading, and colored quiver mutation.