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\begin{document}

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\section*{factorization homology}

\begin{quote}%
Currently this entry consists mainly of notes taken live in a talk by [[John Francis]] at \href{http://maths-old.anu.edu.au/esi/2012/}{ESI Program on K-Theory and Quantum Fields} (2012), without as yet, any double-checking or polishing. So handle with care for the moment.


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

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

[[!include higher algebra - contents]]

\hypertarget{aqft}{}\paragraph*{{AQFT}}\label{aqft}

[[!include AQFT and operator algebra 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_cobordism_hypothsis}{Relation to cobordism hypothsis}\dotfill \pageref*{relation_to_cobordism_hypothsis} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{dimension_1}{Dimension 1}\dotfill \pageref*{dimension_1} \linebreak
\noindent\hyperlink{from_fold_loop_spaces}{From $n$-fold loop spaces}\dotfill \pageref*{from_fold_loop_spaces} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{videos}{Videos:}\dotfill \pageref*{videos} \linebreak


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

\emph{Factorization homology} is a notion of [[homology theory]] for [[framed manifold|framed]] $n$-[[dimension|dimensional]] [[manifolds]] with coefficients in [[En-algebras]], due to (\hyperlink{Francis}{Francis}). It is similar in spirit to \emph{[[factorization algebras]]}, \emph{[[blob homology]]} and \emph{[[topological chiral homology]]}. In fact the definition of factorization homology turns out to be equivalent to that of topological chiral topology (\hyperlink{Francisb}{Francis b}).

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

Write $Mfd_n^{\coprod}$ for the category of [[manifolds]] with [[embeddings]] as morphisms. This is naturally a [[topological category]], hence regard it as an [[(infinity,1)-category]]. Regard it furthermore as a [[symmetric monoidal (∞,1)-category]] with [[tensor product]] given by [[disjoint union]].

For $k$ a [[field]], write $Mod_k$ for the [[symmetric monoidal (∞,1)-category]] of $k$-[[chain complexes]].

Let $H(Mfd_n^{\coprod}, Mod_k)$ be the [[sub-(∞,1)-category]] of those [[monoidal (∞,1)-functors]] $F : Mfd_n^{op} \to Mod_k$ which are ``[[cosheaves]]'' in that for any decomposition of a manifold $X$ into submanifolds $X'$ and $X''$ with overlap $O$, we have an [[equivalence in an (∞,1)-category|equivalence]]

\begin{displaymath}
F(X) \simeq F(X') \otimes_{F(O)}F(X'')
  \,.
\end{displaymath}
Next, let $Disk_n \subset Mfd_n$ be the full [[sub-(∞,1)-category]] on those [[manifolds]] which are finite [[disjoint unions]] of the [[Cartesian space]] $\mathbb{R}^n$.

Restriction along this inclusion gives an [[(∞,1)-functor]]

\begin{displaymath}
H(Mfd_n, Mod_k)
  \to
  Disk_n-Alg (Mod_k)
\end{displaymath}
This turns out to be an [[equivalence of (∞,1)-categories]]. The inverse is defined to be \emph{factorization homology}

\begin{displaymath}
FactorizationHomology
   :
  Disk_n-Alg (Mod_k)
  \to
  H(Mfd_n, Mod_k)
  \,.
\end{displaymath}
which sends an $n$-disk algebra $A : Disk_n \to Mod_k$ to the functor that sends a manifold $X$ to the [[derived functor|derived]] [[coend]]

\begin{displaymath}
\int^X A = \mathbb{E}_X \otimes_{Disk_n} A
\end{displaymath}
of $A$ with

\begin{displaymath}
\mathbb{E}_X 
    : 
  Disk_n \stackrel{Emb(-,X)}{\to} Top \stackrel{C_\bullet(-)}{\to} Mod_k
  \,.
\end{displaymath}
This is equivalent to [[topological chiral homology]], to be thought of as a topological version of [[chiral algebras]]. A version with values in [[homotopy types]] instead of chain complexes was given by Salvatore and [[Graeme Segal]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_cobordism_hypothsis}{}\subsubsection*{{Relation to cobordism hypothsis}}\label{relation_to_cobordism_hypothsis}

From a functor $F \in H(Mfd_n, Mod_k)$ we get an [[extended TQFT]] with values in $k$-linear $(\infty,n)$-categories

$Z_F : Bord_n \to Cat_n(k)$ which sends a $k$-manifold $X$ to $F(X \times \mathbb{R}^{n-k})$, regarded as a [[bimodule]] between the analogous boundary restriction, and hence as a [[n-morphism|k-morphism]] in $Cat_n(k)$.

From a $Disk_n$-algebra $A$ we obtain the corresponding [[delooping]] $\mathbf{B}A \in (Cat_n(k)_{dualizable})^{O(n)}$ which is a $k$-linear [[(infinity,n)-category]] that is a [[fully dualizable object]]. The [[cobordism hypothesis]] identifies this with cobordism representations, and the claim is that this identification is compatible factorization homology.

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

\hypertarget{dimension_1}{}\subsubsection*{{Dimension 1}}\label{dimension_1}

A $Disk_1$-algebra $A$ in $Mod_k$ is equivalently a [[differential graded algebra]].

The value of the corresponding $F_A \in H(Mfd_1, Mod_k)$ on the circle is the [[Hochschild homology]] of $A$

\begin{displaymath}
\int_{S^1} A \simeq \int_{\mathbb{R}^1} A \otimes_{\int_{S^0 \times \mathbb{R}}A} \int_{\mathbb{R}^1} A \simeq HH_\bullet(A)
  \,.
\end{displaymath}
\hypertarget{from_fold_loop_spaces}{}\subsubsection*{{From $n$-fold loop spaces}}\label{from_fold_loop_spaces}

Given a [[topological space]] $Z$ we get a $Disk_n$-algebra

\begin{displaymath}
Disk_n^\coprod 
  \stackrel{Maps_{compact}(-,Z)}{\to}
  Top
  \stackrel{C_\ast(-)}{\to} Mod_k
\end{displaymath}
Where $Maps_{compact}(\mathbb{R}^n, Z) \simeq \Omega^n Z$ is the [[n-fold loop space]] of $Z$.

\textbf{Theorem} (Salvatore and [[Jacob Lurie|Lurie]])

If $Z$ is $(n-1)$-[[n-connected object of an (infinity,1)-category]]

\begin{displaymath}
\int_X C_\ast(\Omega^n Z) \simeq C_\ast Maps_{compact}(X,Z)
  \,.
\end{displaymath}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[topological chiral homology]], [[factorization algebra]], [[blob homology]]


\item [[local net of observables]]



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

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

The definition appears in section 3 of

\begin{itemize}%
\item [[John Francis]], \emph{The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings} (\href{http://arxiv.org/abs/1104.0181}{arXiv:1104.0181})

\end{itemize}
A detailed account is in

\begin{itemize}%
\item [[John Francis]], \emph{Factorization homology of topological manifolds} (\href{http://de.arxiv.org/abs/1206.5522}{arXiv:1206.5522})

\end{itemize}
A survey that also covers [[factorization algebras]] is

\begin{itemize}%
\item [[Grégory Ginot]], \emph{Notes on factorization algebras, factorization homology and applications}, \href{http://arxiv.org/abs/1307.5213}{arXiv:1307.5213}.

\end{itemize}
See also

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Algebra]]}, section 5.3.


\item [[Hiro Lee Tanaka]], \emph{Manifold calculus is dual to factorization homology}, at \href{http://math.mit.edu/conferences/talbot/2012/_2012.php}{Talbot 2012:\_ Calculus of functors} (\href{http://math.mit.edu/conferences/talbot/2012/notes/14_Tanaka_FactorizationHomology%28hiro%29.pdf}{pdf})



\end{itemize}
Some applications are

\begin{itemize}%
\item [[Ben Knudsen]], \emph{Betti numbers and stability for configuration spaces via factorization homology}, \href{http://arxiv.org/abs/1405.6696}{arXiv:1405.6696}.


\item [[Quoc Ho]], \emph{Densities and stability via factorization homology}, \href{https://arxiv.org/abs/1802.07948}{arXiv:1802.07948}.



\end{itemize}
Application to higher [[Hochschild cohomology]] is discussed in

\begin{itemize}%
\item [[Grégory Ginot]], Thomas Tradler, [[Mahmoud Zeinalian]], \emph{Higher Hochschild cohomology, Brane topology and centralizers of $E_n$-algebra maps}, (\href{http://arxiv.org/abs/1205.7056}{arXiv:1205.7056})


\item [[Geoffroy Horel]], \emph{Higher Hochschild homology of the Lubin-Tate ring spectrum}, \href{http://geoffroy.horel.org/HHC%20of%20the%20LT%20ring%20spectrum.pdf}{pdf}.



\end{itemize}
Application to stratified spaces with tangential structures is discussed in

\begin{itemize}%
\item [[David Ayala]], [[John Francis]], [[Hiro Lee Tanaka]], \emph{Factorization homology of stratified spaces}, (\href{http://arxiv.org/abs/1409.0848}{arXiv:1409.0848})

\end{itemize}
A [[duality]] theorem for factorization homology, generalizing [[Poincare duality]] for [[manifolds]] and [[Koszul duality]] for [[E-n algebras]].

\begin{itemize}%
\item [[David Ayala]], [[John Francis]], \emph{Poincar\'e{}/Koszul duality}, \href{http://arxiv.org/abs/1409.2478}{arXiv:1409.2478}.

\end{itemize}
Discussion in the context of [[extended TQFT]] appears in

\begin{itemize}%
\item [[Claudia Scheimbauer]], \emph{Factorization homology as a fully extended topological field theory} (\href{https://people.maths.ox.ac.uk/scheimbauer/ScheimbauerThesis.pdf}{pdf})

\end{itemize}
\hypertarget{videos}{}\subsection*{{Videos:}}\label{videos}

\begin{itemize}%
\item \href{http://www.birs.ca/events/2015/5-day-workshops/15w5125/videos}{Videos from from the BIRS Workshop 15w5125} on \emph{Factorizable Structures in Topology and Algebraic Geometry}.

\end{itemize}


\end{document}