Kleisli bicategories are a natural environment in which the combinatorics involved in various notions of algebraic theory can be handled in a uniform way. The setting allows a clear account of comparisons between such notions. Algebraic theories, symmetric operads and nonsymmetric operads are treated as examples.

We introduce a functor As from the category of posets to the category of nonsymmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction As provides a generalization of the associative operad because all of its generating operations are associative. This construction has a very singular property: the operads obtaine...

Talk 1: Big Goal of Alg Top, operads and model categories, fix notation for model categories, remarks about how difficult it is to verify model category axioms. Motivation from equivariant spectra, and discussion of Kervaire. Monoidal model categories, define the inherited model structure on the category of algebras over an operad. Basic facts about Bousfield localization. Preservation theorem ...

We show how to “interleave” the monad for operads and the monad for contractions on the category Coll of collections, to construct the monad for the operads-with-contraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions each of which acts on only one dimension of the underlying globular sets at a time. We then exhibit ...

We endow the normalized cacti with structure of an ?-operad by showing that its existing composition laws are associative up to all higher homotopies. The homotopies encoded a new topological operad bracketed trees which we relate both enrichment dendroidal category ? and Boardman-Vogt W-construction on operads.

We give conditions on a monoidal model category M and on a set of maps C so that the Bousfield localization of M with respect to C preserves the structure of algebras over various operads. This problem was motivated by an example due to Mike Hill which demonstrates that for the model category of equivariant spectra, preservation does not come for free, even for cofibrant operads. We discuss thi...

We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘S-operads’, and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O+ whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I0+ = I, I(...