nLab Feynman category

Contents

Idea

Feynman categories (Kaufmann-Ward 13) are monoidal categories with extra structures and properties that provide categorical way to corepresent operad-like objects.

They are a biequivalent approach to coloured operads and patterns (Kaufmann-Ward 13,Caviglia 15, Batanin-Kock-Weber 15).

Definition

Let $C$ be a groupoid, $C^\otimes$ the free symmetric monoidal category on $C$ and $M$ a symmetric monoidal category.

(Kaufman-Ward 2013, Defn 2.1) A symmetric strong monoidal functor $\tau: C^\otimes\to M$ is a Feynman category if the following are satisfied

• isomorphisms condition: $\tau^\otimes$ induces an equivalence of symmetric monoidal categories $C^\otimes \cong M_{iso}$

• hereditary condition: $\tau$ and $\tau^\otimes$ induce an equivalence of symmetric monoidal categories $(C\downarrow M)^\otimes_{iso}\cong(M\downarrow M)_{iso}$

• size condition: For any $\ast \in C$, $(M \downarrow \ast)$ is essentially small.

(Getzler 2009) A symmetric strong monoidal functor $\tau: C^\otimes\to M$ is a regular pattern if the following are satisfied

• $\tau$ is essentially surjective
• the induced functor of presheaves $\tau^{\hat{}} : M^{\hat{}}\to (C^\otimes)^{\hat{}}$ is strong monoidal for the Day convolution product

The latter condition on comma categories ensures the existence of certain (pointed) Kan extensions.

The condition of a pattern or that of being hereditary guarantees that there is a left adjoint free functor for the forgetful functor of restricting a monoidal functor from $M$ to $C$. This leads to a monadacity theorem for strong monoidal functors from $M$ to some cocomplete $D$, namely that they are equivalent to algebras over the monad of the adjunction.

Furthermore any morphisms $f$ in the 2-category of Feynman categories defines a pull–back $f^*$ on strong monoidal functors and a left adjoint push-forward $f_!$ computed by a left Kan extension. The theorem is that the Kan extension is strong monoidal.

Properties

Equivalence with coloured operads

The relationship to patterns and groupoid colored operads is given in Kaufmann-Ward 13 and was further upgraded to an equivalence of 2-categories between the 2-category of Feynman categories and that of coloured operads (Caviglia 15, Batanin-Kock-Weber 15). This allows to import results from Feynman categories such as W-constructions to the other theories.

References

Related items include operad, Feynman transform.

The axiomatics is proposed in

• Ralph M. Kaufmann, Benjamin C. Ward, Feynman categories, Astérisque 387 (2017), vii+161pp (arxiv:1312.1269)

A more recent survey is in

• Ralph M. Kaufmann, Lectures on Feynman categories (arxiv:1702.06843)

The biequivalence of Feynman catgeories with coloured operads is proven in

• Giovanni Caviglia, Section A.1 of: The Dwyer-Kan model structure for enriched coloured PROPs (arXiv:1510.01289)

• Michael Batanin, Joachim Kock, Mark Weber, Regular patterns, substitudes, Feynman categories and operads, Theory Appl. Categ. 33 (2018), 148–192 (arXiv:1510.08934)

A representation-theoretical viewpoint is given in

• Ralph M. Kaufmann, Feynman categories and Representation Theory, (arXiv:1911.10169)

A useful generalization is exhibited in

• Ralph M. Kaufmann, Jason Lucas, Decorated Feynman categories, arxiv/1602.00823

Certain bialgebras and Hopf algebras appear by universal constructions in the setting of Feynman categories:

• Imma Gálvez-Carrillo, Ralph M. Kaufmann, Andrew Tonks, Three Hopf algebras and their common simplicial and categorical background, arxiv:1607.00196

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes–Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework.

Role of left Kan extensions of specific kind in operadic theory, including in the setup of Feynman categories is investigated in

• Mark Weber, Algebraic Kan extensions along morphisms of internal algebra classifiers, arxiv/1511.04911

Getzler’s axiomatics of regular patterns is similar in spirit to Feynman categories.

• Ezra Getzler, Operads revisited, in: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, vol. 269 of Progr. Math., pp. 675–698 (2009) math/0701767

A connection of Feynman categories to (a generalization of) profunctors and to rewriting systems within a proposal to categorification of the cyclic operads are exhibited in

• Pierre-Louis Curien, Jovana Obradović, Categorified cyclic operads, arxiv/1706.06788

Interesting pair of functors (not an adjoint pair!) between operadic categories and Feynman categories is among the topics studied in

• Michael Batanin, Martin Markl, Koszul duality for operadic categories, arxiv.org/abs/1812.02935