论文标题
类别$θ_2$，派生的修改和单体类别的变形理论
The category $Θ_2$, derived modifications, and deformation theory of monoidal categories
论文作者
论文摘要
构建了一个复杂的$ c^\ bullet（c，d）（c，d）（f，g）（η，θ）$，构建了单型类别的davydov-yetter复合体。这里$ c，d $是$ \ bbbk $ -linear（dg）单类别类别，$ f，g \ colon c \ to d $是$ \ bbbk $ -linear（dg）严格的单型函数，$η，$η，θ\ colon f \ colon f \ colon f \ rightarrow g $是单型自然自然变速器。在道德上，它是``派生的修改''$η\rrightArrowθ$的复杂，同样对于dg类别的情况，一个人具有``派生自然变换''$ f \ rightarrow g $，由$ c $ -bimodule $ bimodule $ c $ c $ c $ c $ d（g =）$ c $ c $ c $ c $ c $ c。同样，提供了$ c^\ bullet（c，d）（f，g）（η，θ）$的固有同源代数解释为$ rhom $在$ c $的2个型号的Abelian类别中。 The complex $C^\bullet(C,D)(F,G)(η,θ)$ naturally arises from a 2-cocellular dg vector space $A(C,D)(F,G)(η,θ)\colon Θ_2\to C^\bullet(\Bbbk)$, as its $Θ_2$-totalization (here $Θ_2$ is the category dual to the category Joyal 2盘）。 结果表明，$ h^3（c^\ bullet（c，c）（\ mathrm {id}，\ mathrm {id}）（\ mathrm {id}，\ mathrm {id}}））$是$ \ bbbk $ \ bbkkkkkk $ \ bbkkkkk的矢量的同构））完整}变形。这意味着要变形以下数据：（a）基本的DG类别结构，（b）形态上的单体产物（对象上的单体产物是一个设定的理论基准，并且在变形下维持），（c）协会者。结果表明，$ c^\ bullet（c，d）（f，f）（\ mathrm {id}，\ mathrm {id}）$是同型$ e_2 $ -algebra。构想，$ c^\ bullet（c，c）（\ mathrm {id}，\ mathrm {id}）（\ mathrm {id}，\ mathrm {id}）$是同型$ e_3 $ -algebra;但是，证明需要更复杂的方法，我们希望在下一篇论文中完成它。
A complex $C^\bullet(C,D)(F,G)(η, θ)$, generalising the Davydov-Yetter complex of a monoidal category, is constructed. Here $C,D$ are $\Bbbk$-linear (dg) monoidal categories, $F,G\colon C\to D$ are $\Bbbk$-linear (dg) strict monoidal functors, $η,θ\colon F\Rightarrow G$ are monoidal natural transformations. Morally, it is a complex of ``derived modifications'' $η\Rrightarrow θ$, likewise for the case of dg categories one has the complex of ``derived natural transformations'' $F\Rightarrow G$, given by the Hochschild cochain complex of $C$ with coefficients in $C$-bimodule $D(F-,G=)$. As well, an intrinsic homological algebra interpretation of $C^\bullet(C,D)(F,G)(η,θ)$ as $RHom$ in an abelian category of 2-bimodules over $C$, is provided. The complex $C^\bullet(C,D)(F,G)(η,θ)$ naturally arises from a 2-cocellular dg vector space $A(C,D)(F,G)(η,θ)\colon Θ_2\to C^\bullet(\Bbbk)$, as its $Θ_2$-totalization (here $Θ_2$ is the category dual to the category of Joyal 2-disks). It is shown that $H^3(C^\bullet(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id})))$ is isomorphic to the vector space of the outer infinitesimal deformations of the $\Bbbk$-linear monoidal category which we call {\it full} deformations. It means that the following data is to be deformed: (a) the underlying dg category structure, (b) the monoidal product on morphisms (the monoidal product on objects is a set-theoretical datum and is maintained under the deformation), (c) the associator. It is shown that $C^\bullet(C,D)(F,F)(\mathrm{id},\mathrm{id})$ is a homotopy $e_2$-algebra. Conjecturally, $C^\bullet(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id})$ is a homotopy $e_3$-algebra; however the proof requires more sophisticated methods and we hope to complete it in our next paper.
