Operads in iterated monoidal categories

This paper is the first of two articles which develop the notion of protoperads. In this one, we construct a new monoidal product on the category of reduced S-modules. We study the associated monoids, called protoperads, which are a generalization of operads. As operads encode algebraic operations with several inputs and one outputs, protoperads encode algebraic operations with the same number of inputs and outputs. We describe the underlying combinatorics of protoperads, and...

I exhibit a pair of non-symmetric operads that, although not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if `algebraic theory' is understood as meaning `monad', operads cannot be regarded as algebraic theories of a special kind.

In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. In a second part we develop the thoery of S-modules of [EKMM]., which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Fin...

Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure containing the prescribed internal structure is studied. Following the work of Lack, these universal objects must be constructed from simplicial objects arising from our monad-theoretic framework, as certain 2-categorical colimits called c...

In this article, we describe how coalgebraic structures on operads induce algebraic structures on their categories of algebras and coalgebras.

Dendroidal sets offer a formalism for the study of $\infty$-operads akin to the formalism of $\infty$-categories by means of simplicial sets. We present here an account of the current state of the theory while placing it in the context of the ideas that lead to the conception of dendroidal sets. We briefly illustrate how the added flexibility embodied in $\infty$-operads can be used in the study of $A_{\infty}$-spaces and weak $n$-categories in a way that cannot be realized u...

We use Lurie's symmetric monoidal envelope functor to give two new descriptions of $\infty$-operads: as certain symmetric monoidal $\infty$-categories whose underlying symmetric monoidal $\infty$-groupoids are free, and as certain symmetric monoidal $\infty$-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of $\infty$-operads, as a localization of a presheaf $\infty$-category,...

A new generalisation of the notion of space, called "vectoid", is suggested in this work. Basic definitions, examples and properties are presented, as well as a construction of direct product of vectoids. Proofs of more complicated properties not used later are just sketched. Classifying vectoids of simplest algebraic structures, such as objects, algebras and coalgebras, are studied in some detail afterwards. Apart from giving interesting examples of vectoids not coming from ...

In this paper we prove the equivalence of two symmetric monoidal $\infty$-categories of $\infty$-operads, the one defined in Lurie's book on Higher Algebra and the one based on dendroidal spaces. V.2 Some corrections made and exposition slightly altered.

In this paper we develop the theory of presentations for globular operads and construct presentations for the globular operads corresponding to several key theories of $n$-category for $n \leqslant 4$.