There are many interesting situations in which algebraic structure can be described by operads [1, 12, 13, 14, 17, 20, 27, 32, 33, 34, 35]. Let (C,⊗, k) be a symmetric monoidal closed category (Section 2) with all small limits and colimits. It is possible to define two types of operads (Definition 6.1) in this setting, as well as algebras and modules over these operads. One type, called Σ-opera...

Guided by the microcosm principle of Baez-Dolan and by the algebraic definitions of operads of Kelly and Fiore, we introduce two “monoid-like” definitions of cyclic operads, one for the original, “exchangable-output” characterisation of GetzlerKapranov, and the other for the alternative “entries-only” characterisation, both within the category of Joyal’s species of structures. Relying on a resu...

The works of Poincaré, Birkhoff, Witt and Cartier, Milnor, Moore on the connected cocommutative Hopf algebras translated in the language of operads means that the triple of operads (Com,As, Lie) endowed with the Hopf compatiblity relation is good. In this paper, we focus on left dipterous (resp. right dipterous) algebras which are associative algebras with an extra left (resp. right) module on ...

The framed n-discs operad f Dn is studied as semidirect product of SO(n) and the little n-discs operad. Our equivariant recognition principle says that a grouplike space acted on by f Dn is equivalent to the n-fold loop space on an SO(n)-space. Examples of f D2-spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of f Dn , which produces higher Batalin–Vilko...

We present an alternative approach to induced higher homotopy structures constructed by the ’basic perturbation lemma’. This approach is motivated by physical considerations and makes use of operads and their representations. As an application we prove that the BFV-complex controls the formal deformations of coisotropic submanifolds. This is established by identifying the P∞-algebra structure o...

In this work, we develop the machinery of Gröbner bases for coloured operads, which allows us to establish a useful criterion Koszulness operad. Among examples show existence quadratic basis, consider seminal Lie-Rinehart operad whose algebras include pairs (functions, vector fields).