Grothendieck ∞-群胚的归纳余生成器
The Inductive Coherator For Grothendieck Infinity Groupoids
AI总结：
本文推广单子的分配级数理论，构造两个可完备分配级数，分别诱导出归纳余生成器和严格∞-群胚理论，为Grothendieck∞-群胚提供归纳构造。
摘要：
我们将\cite{EC1}中关于单子的分配级数理论进行推广，将定义扩展为包含一个由自然数指标集$\bN$索引的单子族。在特定条件下，单子的分配级数在带点自函子范畴中具有余极限。我们将单子的\emph{可完备}分配级数定义为其诱导的带点自函子（若存在）可提升为单子的分配级数。随后，我们构造了用于在$Θ_0^\op$上的理论范畴中生成单子的因子分解系统，以形成两个\emph{可完备}的单子分配级数。第一个可完备的单子分配级数诱导一个单子，该单子将$Θ_0^\op$上的恒等理论发送到一个$(\infty,0)$-余生成器，其归纳构造模拟了归纳弱充实。第二个可完备的单子分配级数诱导一个单子，该单子将$Θ_0^\op$上的恒等理论发送到一个严格$\infty$-群胚的理论。
英文摘要：
We extend the theory of distributive series of monads of \cite{EC1} by extending the definition to include an $\bN$-indexed collection of monads. Under certain conditions, distributive series of monads will have a colimit in the category of pointed endofunctors. We define a \emph{completable} distributive series of monads to be a distributive series of monads whose induced pointed endofunctor, if it exists, lifts to a monad. We then construct factorization systems used to generate monads on the category of theories over $Θ_0^\op$, in order to form two \emph{completable} distributive series of monads. The first completable distributive series of monads induces a monad that sends the identity theory over $Θ_0^\op$ to an $(\infty,0)$-coherator whose inductive construction mimics inductive weak enrichment. The second completable distributive series of monads induces a monad that sends the identity theory over $Θ_0^\op$ to a theory for strict $\infty$-groupoids.
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