Are operads algebraic theories?

In this article we present a detailed study of the existing constructions of colilimits in the category of symmetrical operations. In addition, some examples of operads obtained from colimits of other operads are presented.

The aim of this note is to give a detailed account of how symmetric operads can be constructed from planar (non-symmetric) operads, and to carefully spell out the algebraic interplay between these two notions. It is a companion note to the main paper "Homotopical operadic calculus in positive characteristic".

This is a cornucopia of types of algebras with some of their properties from the operadic point of view.

The notion of 2--monoidal category used here was introduced by B.~Vallette in 2007 for applications in the operadic context. The starting point for this article was a remark by Yu. Manin that in the category of quadratic algebras (that is, "quantum linear spaces") one can also define 2--monoidal structure(s) with rather unusual properties. Here we give a detailed exposition of these constructions, together with their generalisations to the case of quadratic operads. Their p...

Tangent categories provide a categorical axiomatization of the tangent bundle. There are many interesting examples and applications of tangent categories in a variety of areas such as differential geometry, algebraic geometry, algebra, and even computer science. The purpose of this paper is to expand the theory of tangent categories in a new direction: the theory of operads. The main result of this paper is that both the category of algebras of an operad and its opposite cate...

This work addresses the homotopical analysis of enveloping operads in a general cofibrantly generated symmetric monoidal model category. We show the potential of this analysis by obtaining, in a uniform way, several central results regarding the homotopy theory of operadic algebras.

Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. The theory of operads, together with the algebraic setting and the tools accompanying it, promises advances in these two areas. On the one hand, opera...

We review definitions and basic properties of operads, PROPs and algebras over these structures.

We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.

We study different algebraic structures associated to an operad and their relations: to any operad $\mathbf{P}$ is attached a bialgebra,the monoid of characters of this bialgebra, the underlying pre-Lie algebra and its enveloping algebra; all of them can be explicitely describedwith the help of the operadic composition. non-commutative versions are also given. We denote by $\mathbf{b\_\infty}$ the operad of $\mathbf{b\_\infty}$ algebras, describing all Hopf algebra structures...