论文标题
张量类别的cosimimplicial单体和变形理论
Cosimplicial monoids and deformation theory of tensor categories
论文作者
论文摘要
我们在对称的单体类别$ {\ bf v} $中引入了$ n $ commutativity（$ 0 \ le n \ le n \ le \ le \ infty $），其中$ n = 0 $对应于$ {\ bf v cosim sosim cosim cosem cosem cos cos yn = \ n = \ n = \ n = \ n = \ n = \ n = \ n = \ n = \。如果$ {\ bf v} $具有单型模型结构（在某些温和的技术条件下），则$ n $ cosimplicial monoid的总对象具有天然$ e_ {n+1} $ - 代数结构。 我们的主要应用是张量类别和张量函子的变形理论。我们表明，张量函子的变形复合物是$ 1 $的共同体cosimimplicial monoid，因此具有与DeLigne猜想相关的Assoiative Algebra hochschild Complect上的$ e_2 $ -algebra结构相似。我们进一步证明，张量类别的变形复合物是$ 2 $ commutative cosimimplicial monoid的总复合体，因此自然是$ e_3 $ -algebra。我们通过Delannoy路径的语言及其非交换升降机使这些结构非常明确。我们研究了这些结构如何在具体的例子中表现出来。
We introduce a notion of $n$-commutativity ($0\le n\le \infty$) for cosimplicial monoids in a symmetric monoidal category ${\bf V}$, where $n=0$ corresponds to just cosimplicial monoids in ${\bf V,}$ while $n=\infty$ corresponds to commutative cosimplicial monoids. If ${\bf V}$ has a monoidal model structure we show (under some mild technical conditions) that the total object of an $n$-cosimplicial monoid has a natural $E_{n+1}$-algebra structure. Our main applications are to the deformation theory of tensor categories and tensor functors. We show that the deformation complex of a tensor functor is a total complex of a $1$-commutative cosimplicial monoid and, hence, has an $E_2$-algebra structure similar to the $E_2$-structure on Hochschild complex of an associative algebra provided by Deligne's conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a $2$-commutative cosimplicial monoid and, therefore, is naturally an $E_3$-algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples.
