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\begin{document}

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\section*{factorization algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{prefactorization_algebra}{Prefactorization algebra}\dotfill \pageref*{prefactorization_algebra} \linebreak
\noindent\hyperlink{factorization_algebras}{Factorization algebras}\dotfill \pageref*{factorization_algebras} \linebreak
\noindent\hyperlink{homotopy_factorization_algebras}{Homotopy factorization algebras}\dotfill \pageref*{homotopy_factorization_algebras} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{factorization algebra} is an [[algebra over an operad]] where the [[operad]] in question is like the [[little disk operad]], but with each disk embedded into a given [[manifold]] $X$.

This way a factorization algebra is an assignment of a [[chain complex]] $V_D$ to each [[ball]] $D \subset X$ embedded in $X$, and for each collection of non-intersecting embedded balls $D_1 , \cdots, D_n \subset D \subset X$ sitting inside a bigger embedded ball $D$ in $X$ a morphism

\begin{displaymath}
V_{D_1} \otimes V_{D_2} \otimes \cdots \otimes V_{D_n} \to V_{D}
\end{displaymath}
such that composition of such operations is suitably respected.

In Euclidean ([[Wick rotation|Wick rotated]]) [[field theory]] factorization algebras serve to axiomatize the [[operator product expansion]].

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

\hypertarget{prefactorization_algebra}{}\subsubsection*{{Prefactorization algebra}}\label{prefactorization_algebra}

\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[topological space]] write $Fact_X$ be the colored [[operad]] in [[Set]] whose

\begin{itemize}%
\item [[object]]s are the [[connected]] [[open subset]]s of $X$;


\item the [[hom-set]] $Fact_X(\{U_i\}_i, V)$ is the singleton precisely if the $U_i$ are all in $V$ and are pairwise disjoint and is the [[empty set]] otherwise.



\end{itemize}
This specifies composition uniquely.

\end{defn}
\begin{defn}
\label{PrefactorizationAlgebra}\hypertarget{PrefactorizationAlgebra}{}
For $(C, \otimes)$ a [[symmetric monoidal category|symmetric monoidal]] [[abelian category]] let $End(C)$ be its [[endomorphism operad]]. A \textbf{prefactorization algebra} on $X$ with values in $C$ is an [[algebra over an operad]] over $Fact_X$ in $C$, hence a morphism of operads

\begin{displaymath}
\mathcal{F} : Fact_X \to End(C)
  \,.
\end{displaymath}
\end{defn}
These definitions appear in ``\href{https://web.archive.org/web/20150909210342/http://math.northwestern.edu/~costello/factorization.pdf}{Factorization algebras in quantum field theory}'' by Costello and Gwilliam.

\hypertarget{factorization_algebras}{}\subsubsection*{{Factorization algebras}}\label{factorization_algebras}

\begin{defn}
\label{FactorizingCover}\hypertarget{FactorizingCover}{}
For $X$ a [[topological space]] and $U \subset X$ an [[open subset]], a [[open cover]] $\{U_i \hookrightarrow U\}_{i \in I}$ is called a \textbf{factorizing cover} if for every [[finite set]] of points $\{x_1, \cdots, x_k\} \subset U$ there is a finite subset $\{U_{i_j}\}_{j \in J \subset I}$ of pairwise disjoint open subsets such that each point is contained in their union.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Every [[Hausdorff space]] admits a factorizing cover.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
For a factorizing cover $\{U_i \to U\}_{i \in I}$ write $P I$ for the set of finite subsets $\alpha \subset I$ such that for $j,j' \in \alpha$ we have $U_j \cap U_{j'} = \emptyset$.

Given a \hyperlink{PrefactorizationAlgebra}{prefactorization algebra} $\mathcal{F}$ and $\alpha \in P I$ write

\begin{displaymath}
\mathcal{F}(\alpha) := \otimes_{j \in \alpha} F(U_j)
\end{displaymath}
and for $\alpha_1, \cdots, \alpha_k \in P I$ write

\begin{displaymath}
\mathcal{F}(\alpha_1, \cdots, \alpha_k) = 
    \bigotimes_{(j_1, \cdots, j_k) \in \alpha_1 \times \cdots \times \alpha_k}
    \mathcal{F}(U_{j_1} \cap \cdots \cap U_{j_k})
  \,.
\end{displaymath}
For each $1 \leq i \leq k$ there is a canonical morphism

\begin{displaymath}
p_i : \mathcal{F}(\alpha_1,\cdots, \alpha_k)
  \to 
   \mathcal{F}(\alpha_1, \cdots, \alpha_{i-1}, \alpha_{i+1}, \cdots, \alpha_k)
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A prefactorization algebra $\mathcal{F} : Fact_X \to End(C)$ is called a \textbf{factorization algebra} if for every open subset $U \subset X$ and every \hyperlink{FactorizingCover}{factorizing cover} $\{U_i \to U\}_{i \in I}$ the sequence

\begin{displaymath}
\bigoplus_{\alpha_1, \alpha_2 \in P I}
   \mathcal{F}(\alpha_1, \alpha_2)
  \stackrel{p_1 - p_2}{\to}
    \bigoplus_{\beta \in P I}
   \mathcal{F}(\beta)
   \to 
   \mathcal{F}(U)
   \to 0
\end{displaymath}
is an [[exact sequence]].

\end{defn}
These definitions appear \href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.8322&rep=rep1&type=pdf}{here}.

See also at \emph{[[cosheaf]]}.

\hypertarget{homotopy_factorization_algebras}{}\subsubsection*{{Homotopy factorization algebras}}\label{homotopy_factorization_algebras}

Let now $(C,\otimes)$ specifically be a [[category of chain complexes]].

\begin{defn}
\label{}\hypertarget{}{}
A prefactorization algebra $\mathcal{F} : Fact_X \to End(X)$ is a \textbf{homotopy factorization algebra} if for all factorizing covers $\{U_i \to U \subset X\}_{i \in I}$ the canonical morpshim

\begin{displaymath}
\bigoplus_{k \geq 0} \bigoplus_{\alpha_1, \cdots, \alpha_k \in P I}
   \mathcal{F}(\alpha_1, \cdots, \alpha_k)[k-1]
   \to 
  \mathcal{F}(U)
\end{displaymath}
is a [[quasi-isomorphism]], where the [[differential]] on the left is defined by (\ldots{}).

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This is the analogue of a [[descent]] condition for [[simplicial presheaves]].

\end{remark}
These definitions appear \href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.8322&rep=rep1&type=pdf}{here}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[beta-gamma system]]

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

Factorization algebras have some similarity with

\begin{itemize}%
\item [[Hochschild homology]];


\item [[blob homology]], [[factorization homology]], [[topological chiral homology]];


\item [[local net]]s in [[AQFT]].


\item [[chiral algebra]], [[vertex operator algebra]]


\item [[nonabelian Poincaré duality]]



\end{itemize}
[[!include Isbell duality - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion of factorization algebra may be regarded as a slight variation on the concept \emph{[[chiral algebra]]} originally introduced in

\begin{itemize}%
\item [[Alexander Beilinson]], [[Vladimir Drinfeld]], \emph{[[Chiral Algebras]]}.

\end{itemize}
A definition formulated genuinely in [[Higher Algebra]] appears in section 4.1 \emph{Topological Chiral Homology} of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]}

\end{itemize}
This discusses how locally constant factorization algebras obtained from [[En-algebras]] induce [[extended quantum field theory|extended]] [[FQFTs]].

A fairly comprehensive account of factorization algebras as a formalization of [[perturbation theory|perturbative]] [[quantum field theory]] (see at \emph{[[factorization algebra of observables]]}) is in

\begin{itemize}%
\item [[Kevin Costello]], [[Owen Gwilliam]], \emph{[[Factorization algebras in perturbative quantum field theory]]}


\item [[Owen Gwilliam]], \emph{Factorization algebras and free field theories} PhD thesis ([[GwilliamThesis.pdf:file]])


\item [[Owen Gwilliam]], [[Kasia Rejzner]], \emph{Comparing nets and factorization algebras of observables: the free scalar field}, \href{https://arxiv.org/abs/1711.06674}{arxiv:1711.06674}



\end{itemize}
and the beginnings of

\begin{itemize}%
\item [[Kevin Costello]], \emph{Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4}, \href{http://arxiv.org/abs/1111.4234}{arXiv:1111.4234}
\item K. Costello, C. Scheimbauer, \emph{Lectures on mathematical aspects of (twisted) supersymmetric gauge theories}, pp. 57-88 in: Mathematical aspects of QFTs, D. Calaque, T. Strobl editors, Springer 2015

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Kevin Costello]] (with [[Owen Gwilliam]]), \emph{Factorization algebras in perturbative quantum field theory} in [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Strings, Field, Topology]], Oberwolfach report No 28, 2009 (\href{http://www.mfo.de/programme/schedule/2009/24/OWR_2009_28.pdf#page=12}{pdf})

This can also be found mentioned in the talk notes of the [[Northwestern TFT Conference 2009]], see in particular

\begin{itemize}%
\item notes by [[Christoph Wockel]], [[costello.pdf:file]]


\item notes by Evan Jenkins on the same talk: \href{http://www.math.uchicago.edu/~ejenkins/notes/nwtft/25may-kc.pdf}{Factorization algebras in perturbative quantum gravity}



\end{itemize}


\end{itemize}
There seems to be a close relation between the description of [[quantum field theory]] by factorization algebras and the proposal presented in

\begin{itemize}%
\item [[Stefan Hollands]], (\href{http://arxiv.org/abs/0802.2198}{arXiv:0802.2198})

\end{itemize}
The relation of locally constant factorization algebras to higher order [[Hochschild homology]] is in

\begin{itemize}%
\item [[Gregory Ginot]], Thomas Tradler, Mahmoud Zeinalian, \emph{Derived higher Hochschild homology, topological chiral homology and factorization algebras}, \href{http://arxiv.org/abs/1011.6483}{arxiv/1011.6483}
\item [[Gregory Ginot]], \emph{Notes on factorization algebras, factorization homology and applications}, \href{https://arxiv.org/abs/1307.5213}{arxiv/1307.5213}

\end{itemize}
A comparison with FQFT for TFTs is presented in

\begin{itemize}%
\item C. Scheimbauer, \emph{A factorization view on states/observables in topological field theories} \href{https://www.youtube.com/watch?v=nN-qtnbtpW4}{youtube} 19 min, string-math 2017, Hamburg

\end{itemize}
An [[(infinity,1)-category theory|(infinity,1)-category theoretic ]] treatment of higher factorization algebras is in

\begin{itemize}%
\item [[Dennis Gaitsgory]], [[John Francis]] \emph{Chiral Koszul duality} (\href{http://arxiv.org/abs/1103.5803}{arXiv:1103.5803})

\end{itemize}
A construction of [[chiral differential operator]]s via quantization of $\beta\gamma$ system in [[BV formalism]] with an intermediate step using factorization algebras:

\begin{itemize}%
\item Vassily Gorbounov, Owen Gwilliami, Brian Williams, \emph{Chiral differential operators via Batalin-Vilkovisky quantization}, \href{http://people.mpim-bonn.mpg.de/gwilliam/cdo.pdf}{pdf}
\item Brian Williams, \emph{The Virasoro vertex algebra and factorization algebras on Riemann surfaces}, Lett. Math. Phys. \textbf{107}:12, 2189–2237 (2017) \href{https://dx.doi.org/10.1007/s11005-017-0982-7}{doi}

\end{itemize}
A version of bosonic string theory related to factorization algebras is presented in

\begin{itemize}%
\item Owen Gwilliam, Brian Williams, \emph{The holomorphic bosonic string?, \href{https://arxiv.org/abs/1711.05823}{arxiv/1711.05823}}

\end{itemize}
A higher-dimensional generalization of vertex algebras is suggested in the framework of factorization algebras in

\begin{itemize}%
\item Emily Cliff, \emph{Universal factorization spaces and algebras}, \href{http://arxiv.org/abs/1608.08122}{arxiv/1608.08122}

\end{itemize}
\begin{quote}%
We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a factorization algebra or space by an 'etale morphism of schemes, and hence to define the notion of a universal factorization space or algebra. This provides a generalization to higher dimensions and to non-linear settings of the notion of a vertex algebra.


\end{quote}
Homotopy prefactorization algebras play also a major role in

\begin{itemize}%
\item Donald Yau, \emph{Homotopical quantum field theory}, \href{https://arxiv.org/abs/1802.08101}{arxiv/1802.08101}, 302 pages

\end{itemize}
[[!redirects factorization algebras]]



\end{document}