Michael Batanin

The universal property of the multitude of trees

Abstract A vital ingredient in the first authors definition of weak ω -category is his description, in terms of trees, of the free (strict) ω -category on a globular set. The induced monad on the category of globular sets shares many of the properties of the monoid monad (describable in terms of words) on the category of sets. Benabou has shown how the simplicial category arises from the monoid monad. The present paper studies the object arising similarly from the ω -category monad.

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 (Batanin, Adv Math 136:39–103, 1998) 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 may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)-category is something like a category enriched in weak n-categories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.

On the Penon method of weakening algebraic structures

Abstract We consider a generalization of the Penon approach to the definition of weak n -category and compare his definition with that of the author.

CROSSED INTERVAL GROUPS AND OPERATIONS ON THE HOCHSCHILD COHOMOLOGY

We prove that the operad B of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that B is a certain crossed interval extension of an operad T whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.

Locally constant n-operads as higher braided operads

We introduce a category of locally constant n-operads which can be considered as the category of higher braided operads. For n = 1, 2,1 the homotopy category of locally constant n-operads is equivalent to the homotopy category of classical nonsymmetric, braided and symmetric operads correspondingly. 1991 Math. Subj. Class. 18D20 , 18D50, 55P48

Baez-Dolan stabilization via (semi-)model categories of operads

We describe a proof of the Baez–Dolan Stabilization Hypothesis for Rezk’s model of weak n-categories. This proof proceeds via abstract homotopy theory, and en route we discuss a version of left Bousfield localization which does not require left properness. We also discuss conditions under which various categories of operads can be made left proper, but these conditions are difficult to be satisfied, as a counterexample in the context of simplicial sets demonstrates.