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Factorization algebra
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<h2 id="definition">Definition</h2> <h3>Prefactorization algebras</h3> <p>A factorization algebra is a prefactorization algebra satisfying some properties, similar to <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheafs</a> being a <a href="/facts/Presheaf/MM3hcRw4">presheaf</a> with extra conditions. </p><p>If M {\displaystyle M} is a <a href="/facts/Topological_space/v2ut7CbK">topological space</a>, a prefactorization algebra F {\displaystyle {\mathcal {F}}} of <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> on M {\displaystyle M} is an assignment of vector spaces F ( U ) {\displaystyle {\mathcal {F}}(U)} to <a href="/facts/Open_set/QfTCIqMu">open sets</a> U {\displaystyle U} of M {\displaystyle M} , along with the following conditions on the assignment: </p> <ul><li>For each inclusion U ⊂ V {\displaystyle U\subset V} , there's a <a href="/facts/Linear_map/Q1PBKNVV">linear map </a> m V U : F ( U ) → F ( V ) {\displaystyle m_{V}^{U}:{\mathcal {F}}(U)\rightarrow {\mathcal {F}}(V)} </li> <li>There is a linear map m V U 1 , ⋯ , U n : F ( U 1 ) ⊗ ⋯ ⊗ F ( U n ) → F ( V ) {\displaystyle m_{V}^{U_{1},\cdots ,U_{n}}:{\mathcal {F}}(U_{1})\otimes \cdots \otimes {\mathcal {F}}(U_{n})\rightarrow {\mathcal {F}}(V)} for each finite collection of open sets with each U i ⊂ V {\displaystyle U_{i}\subset V} and the U i {\displaystyle U_{i}} pairwise disjoint.</li> <li>The maps compose in the obvious way: for collections of opens U i , j {\displaystyle U_{i,j}} , V i {\displaystyle V_{i}} and an open W {\displaystyle W} satisfying U i , 1 ⊔ ⋯ ⊔ U i , n i ⊂ V i {\displaystyle U_{i,1}\sqcup \cdots \sqcup U_{i,n_{i}}\subset V_{i}} and V 1 ⊔ ⋯ V n ⊂ W {\displaystyle V_{1}\sqcup \cdots V_{n}\subset W} , the following diagram commutes.</li></ul> <p> ⨂ i ⨂ j F ( U i , j ) → ⨂ i F ( V i ) ↓ ↙ F ( W ) {\displaystyle {\begin{array}{lcl}&\bigotimes _{i}\bigotimes _{j}{\mathcal {F}}(U_{i,j})&\rightarrow &\bigotimes _{i}{\mathcal {F}}(V_{i})&\\&\downarrow &\swarrow &\\&{\mathcal {F}}(W)&&&\\\end{array}}} </p><p>So F {\displaystyle {\mathcal {F}}} resembles a <a href="/facts/Cosheaf/2tJmwzu7">precosheaf</a>, except the vector spaces are <a href="/facts/Tensor_product/7K3pR1ap">tensored</a> rather than <a href="/facts/Direct_sum/x3ljVsP6">(direct-)summed</a>. </p><p>The category of vector spaces can be replaced with any <a href="/facts/Symmetric_monoidal_category/AxhumNny">symmetric monoidal category</a>. </p> <h3>Factorization algebras</h3> <p>To define factorization algebras, it is necessary to define a Weiss <a href="/facts/Open_cover/2VUdelA4">cover</a>. For U {\displaystyle U} an open set, a collection of opens U = { U i | i ∈ I } {\displaystyle {\mathfrak {U}}=\{U_{i}|i\in I\}} is a Weiss cover of U {\displaystyle U} if for any finite collection of points { x 1 , ⋯ , x k } {\displaystyle \{x_{1},\cdots ,x_{k}\}} in U {\displaystyle U} , there is an open set U i ∈ U {\displaystyle U_{i}\in {\mathfrak {U}}} such that { x 1 , ⋯ , x k } ⊂ U i {\displaystyle \{x_{1},\cdots ,x_{k}\}\subset U_{i}} . </p><p>Then a factorization algebra of vector spaces on M {\displaystyle M} is a prefactorization algebra of vector spaces on M {\displaystyle M} so that for every open U {\displaystyle U} and every Weiss cover { U i | i ∈ I } {\displaystyle \{U_{i}|i\in I\}} of U {\displaystyle U} , the sequence ⨁ i , j F ( U i ∩ U j ) → ⨁ k F ( U k ) → F ( U ) → 0 {\displaystyle \bigoplus _{i,j}{\mathcal {F}}(U_{i}\cap U_{j})\rightarrow \bigoplus _{k}{\mathcal {F}}(U_{k})\rightarrow {\mathcal {F}}(U)\rightarrow 0} is <a href="/facts/Exact_sequence/leUJnwJb">exact</a>. That is, F {\displaystyle {\mathcal {F}}} is a factorization algebra if it is a cosheaf with respect to the Weiss topology. </p><p>A factorization algebra is <i>multiplicative</i> if, in addition, for each pair of disjoint opens U , V ⊂ M {\displaystyle U,V\subset M} , the structure map m U ⊔ V U , V : F ( U ) ⊗ F ( V ) → F ( U ⊔ V ) {\displaystyle m_{U\sqcup V}^{U,V}:{\mathcal {F}}(U)\otimes {\mathcal {F}}(V)\rightarrow {\mathcal {F}}(U\sqcup V)} is an isomorphism. </p> <h3>Algebro-geometric formulation</h3> <p>While this formulation is related to the one given above, the relation is not immediate. </p><p>Let X {\displaystyle X} be a <a href="/facts/Smoothness/mffL4ch2">smooth</a> <a href="/facts/Algebraic_curve/LnWsPmDP">complex curve</a>. A factorization algebra on X {\displaystyle X} consists of </p> <ul><li>A <a href="/facts/Quasicoherent_sheaf/h2NoUN7e">quasicoherent sheaf</a> V X , I {\displaystyle {\mathcal {V}}_{X,I}} over X I {\displaystyle X^{I}} for any finite set I {\displaystyle I} , with no non-zero local <a href="/facts/Section_(fiber_bundle)/DLnpXS5a">section</a> supported at the union of all partial diagonals</li> <li><a href="/facts/Functorial/l2u3rIBE">Functorial</a> isomorphisms of quasicoherent sheaves Δ J / I ∗ V X , J → V X , I {\displaystyle \Delta _{J/I}^{*}{\mathcal {V}}_{X,J}\rightarrow {\mathcal {V}}_{X,I}} over X I {\displaystyle X^{I}} for surjections J → I {\displaystyle J\rightarrow I} .</li> <li>(<i>Factorization</i>) Functorial isomorphisms of quasicoherent sheaves</li></ul> <p> j J / I ∗ V X , J → j J / I ∗ ( ⊠ i ∈ I V X , p − 1 ( i ) ) {\displaystyle j_{J/I}^{*}{\mathcal {V}}_{X,J}\rightarrow j_{J/I}^{*}(\boxtimes _{i\in I}{\mathcal {V}}_{X,p^{-1}(i)})} over U J / I {\displaystyle U^{J/I}} . </p> <ul><li>(<i>Unit</i>) Let V = V X , { 1 } {\displaystyle {\mathcal {V}}={\mathcal {V}}_{X,\{1\}}} and V 2 = V X , { 1 , 2 } {\displaystyle {\mathcal {V}}_{2}={\mathcal {V}}_{X,\{1,2\}}} . A global section (the <i>unit</i>) 1 ∈ V ( X ) {\displaystyle 1\in {\mathcal {V}}(X)} with the property that for every local section f ∈ V ( U ) {\displaystyle f\in {\mathcal {V}}(U)} ( U ⊂ X {\displaystyle U\subset X} ), the section 1 ⊠ f {\displaystyle 1\boxtimes f} of V 2 | U 2 Δ {\displaystyle {\mathcal {V}}_{2}|_{U^{2}\Delta }} extends across the diagonal, and restricts to f ∈ V ≅ V 2 | Δ {\displaystyle f\in {\mathcal {V}}\cong {\mathcal {V}}_{2}|_{\Delta }} .</li></ul> <h2 id="example">Example</h2> <h3>Associative algebra</h3> <p class="note">See also: <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a></p> <p>Any associative algebra A {\displaystyle A} can be realized as a prefactorization algebra A f {\displaystyle A^{f}} on R {\displaystyle \mathbb {R} } . To each <a href="/facts/Open_interval/e9se3vTe">open interval</a> ( a , b ) {\displaystyle (a,b)} , assign A f ( ( a , b ) ) = A {\displaystyle A^{f}((a,b))=A} . An arbitrary open is a disjoint union of countably many open intervals, U = ⨆ i I i {\displaystyle U=\bigsqcup _{i}I_{i}} , and then set A f ( U ) = ⨂ i A {\displaystyle A^{f}(U)=\bigotimes _{i}A} . The structure maps simply come from the multiplication map on A {\displaystyle A} . Some care is needed for infinite tensor products, but for finitely many open intervals the picture is straightforward. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Vertex_algebra/npzGRlsD">Vertex algebra</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Beilinson, Alexander; Drinfeld, Vladimir (2004). Chiral algebras. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3528-9. Retrieved 21 February 2023. <a href="978-0-8218-3528-9" target="_blank">978-0-8218-3528-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Costello, Kevin; Gwilliam, Owen (2017). Factorization algebras in quantum field theory, Volume 1. Cambridge. ISBN 9781316678626.{{cite book}}: CS1 maint: location missing publisher (link) <a href="9781316678626" target="_blank">9781316678626</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>