Decorated Feynman categories

Abstract

Abstract. In [KW17], the new concept of Feynman categories was introduced to simplify the discussion of operad-like objects. In this present paper, we demonstrate the usefulness of this approach, by introducing the concept of decorated Feynman categories. The procedure takes a Feynman category F and a functor O to a monoidal category to produce a new Feynman category F decO . This in one swat explains the existence of non-sigma operads, non-sigma cyclic operads, and the non-sigma-modular operads of Markl as well as all the usual candidates simply from the category G, which is a full subcategory of the category of graphs of [BM08]. Moreover, we explain the appearance of terminal objects noted in [Mar15]. We can then easily extend this for instance to the dihedral case. Furthermore, we obtain graph complexes and all other known operadic type notions from decorating and restricting the basic Feynman category G of aggregates of corollas. We additionally show that the construction is functorial. There are further geometric and number theoretic applications, which will follow in a separate preprint.

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

“…. The construction of simplicial strings is captured by the nc-construction applied to the Feynman category ∆ * , * together with a decoration, that is the construction of F decO , see [KL17] and [KW17, §3.3], see §3 in particular §3.8 and 3.3.2. Finally, universal operations 3.9 explain the amputation mechanism.…”

Comprehensive factorisation systems

“…( F-ops )) → F and this factorisation is unique up to isomorphism. The existence of such a factorisation has been proved in [23], but its uniqueness is new.…”

“…Since the projection el F (F ) → F is a discrete opfibration, the hereditary condition of F (as formulated in Remark 2.2) lifts to el F (F ), see [23] for a detailed proof. The latter is thus a Feynman category over F. Since extensions along Feynman functors are computed as pointwise left Kan extensions, the fact that the usual category of elements construction defines a consistent comprehension scheme on Cat (cf.…”

“…One of the leading motivations of this text has been the recent construction by the second author and Lucas [23] of decorated Feynman categories, which play the role of Feynman categories of elements. A Feynman category is a special kind of symmetric monoidal category, and there is a comprehension scheme assigning to a Feynman category F the category of strong symmetric monoidal set-valued functors on F. The resulting comprehensive factorisation of a Feynman functor sheds light on Markl's recent definition of non-Σ-modular operads [27].…”

“…We first exhibit a Feynman category of elements el F (F ) over F with the universal property of Lemma 1.1 for each (F, V, ι)-operad F , closely following [23]. Indeed, the usual category of elements el(F ) of the underlying diagram F : F → Set comes equipped with a Feynman category structure: for objects (X, x ∈ F (X)) and (Y, y ∈ F (Y )), the tensor is given by (X,…”