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\begin{document}

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\section*{Feynman category}

[[!redirects Feynman categories]]

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

(Kaufmann, Ward 2013) A symmetric strong monoidal functor $\tau: C^\otimes\to M$ is a \textbf{Feynman category} if the following are satisfied

\begin{itemize}%
\item $\tau$ is a groupoid
\item $\tau$ induces an equivalence of groupoids $C^\otimes \cong M_{iso}$
\item $\tau$ induces an equivalence of groupoids $(C\downarrow M)^\otimes_{iso}\cong(M\downarrow M)_{iso}$

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

\begin{itemize}%
\item $\tau$ is [[essentially surjective functor|essentially surjective]]
\item the induced functor of presheaves $\tau^{\hat{}} : M^{\hat{}}\to (C^\otimes)^{\hat{}}$ is strong monoidal for the [[Day convolution]] product

\end{itemize}
The latter condition on [[comma categories]] ensures the existence of certain (pointed) [[Kan extension]]s.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

Related items include [[operad]], [[Feynman transform]].

The axiomatics is proposed in

\begin{itemize}%
\item Ralph M. Kaufmann, Benjamin C. Ward, \emph{Feynman categories}, Astérisque 387 (2017), vii+161pp. \href{https://arxiv.org/abs/1312.1269}{arxiv/1312.1269}

\end{itemize}
A more recent survey is in

\begin{itemize}%
\item Ralph M. Kaufmann, \emph{Lectures on Feynman categories}, \href{https://arxiv.org/abs/1702.06843}{arxiv:1702.06843}

\end{itemize}
A useful generalization is exhibited in

\begin{itemize}%
\item Ralph M. Kaufmann, Jason Lucas, \emph{Decorated Feynman categories}, \href{https://arxiv.org/abs/1602.00823}{arxiv/1602.00823}

\end{itemize}
Certain bialgebras and Hopf algebras appear by universal constructions in the setting of Feynman categories:

\begin{itemize}%
\item Imma Gálvez-Carrillo, Ralph M. Kaufmann, Andrew Tonks, \emph{Three Hopf algebras and their common simplicial and categorical background}, \href{https://arxiv.org/abs/1607.00196}{arxiv:1607.00196}

\end{itemize}
\begin{quote}%
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.


\end{quote}
Role of left [[Kan extension]]s of specific kind in operadic theory, including in the setup of Feynman categories is investigated in

\begin{itemize}%
\item Mark Weber, \emph{Algebraic Kan extensions along morphisms of internal algebra classifiers}, \href{https://arxiv.org/abs/1511.04911}{arxiv/1511.04911}

\end{itemize}
Getzler's axiomatics of regular patterns is similar in spirit to Feynman categories.

\begin{itemize}%
\item [[Ezra Getzler]], \emph{Operads revisited}, in: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, vol. 269 of Progr. Math., pp. 675–698 (2009) \href{https://arxiv.org/abs/math/0701767}{math/0701767}

\end{itemize}
A comparison of both with related notions like Day-Street [[substitude]]s as well as [[colored operad]]s is in

\begin{itemize}%
\item Michael Batanin, Joachim Kock, Mark Weber, \emph{Regular patterns, substitudes, Feynman categories and operads}, \href{https://arxiv.org/abs/1510.08934}{arxiv/1510.08934}

\end{itemize}
A connection of Feynman categories to (a generalization of) profunctors and to rewriting systems within a proposal to categorification of the [[cyclic operad]]s are exhibited in

\begin{itemize}%
\item Pierre-Louis Curien, Jovana Obradović, \emph{Categorified cyclic operads}, \href{https://arxiv.org/abs/1706.06788}{arxiv/1706.06788}

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

\begin{itemize}%
\item M. Batanin, M. Markl, \emph{Koszul duality for operadic categories}, \href{https://arxiv.org/abs/1812.02935}{arxiv.org/abs/1812.02935}

\end{itemize}


\end{document}