双置换索引范畴的格罗滕迪克构造与伪对称逆K-理论
The Grothendieck Construction of Bipermutative-Indexed Categories and Pseudo Symmetric Inverse K-Theory
AI总结：
本文研究小紧双置换范畴上的格罗滕迪克构造，证明其是伪对称Cat-多函子并提升为非对称Cat-多等价，进而证明逆K-理论保持伪对称E∞-代数结构。
摘要：
格罗滕迪克构造是指标化范畴与opfibration之间的基本联系。本文在Cat-富集多范畴的背景下，详细研究了小紧双置换范畴上的格罗滕迪克构造，并应用于逆K-理论和伪对称E∞-代数。 小范畴C上的普通格罗滕迪克构造是一个2-等价，它将C-指标化范畴映到C上的opfibration。我们证明，小紧双置换范畴D上的格罗滕迪克构造是一个伪对称Cat-多函子，但一般不是对称意义下的Cat-多函子。当考虑向D的投影时，我们证明D上的格罗滕迪克构造提升为一个非对称Cat-多等价，其陪域是一个非对称Cat-多范畴，其对象为D上的小置换opfibration。 作为应用，我们证明从Γ-范畴到小置换范畴的逆K-理论是一个伪对称Cat-多函子，但不是对称意义下的Cat-多函子。由此可知，逆K-理论保持由非对称和伪对称Cat-多函子参数化的代数结构，但一般不保持Cat-多函子。作为特例，我们观察到逆K-理论将Γ-范畴中的伪对称E∞-代数映到小置换范畴中的伪对称E∞-代数。
英文摘要：
The Grothendieck construction is a fundamental link between indexed categories and opfibrations. This work is a detailed study of the Grothendieck construction over a small tight bipermutative category in the context of Cat-enriched multicategories, with applications to inverse K-theory and pseudo symmetric E-infinity-algebras. The ordinary Grothendieck construction over a small category C is a 2-equivalence that sends a C-indexed category to an opfibration over C. We show that the Grothendieck construction over a small tight bipermutative category D is a pseudo symmetric Cat-multifunctor that is generally not a Cat-multifunctor in the symmetric sense. When the projection to D is taken into account, we prove that the Grothendieck construction over D lifts to a non-symmetric Cat-multiequivalence whose codomain is a non-symmetric Cat-multicategory with small permutative opfibrations over D as objects. As applications we show that inverse K-theory, from Gamma-categories to small permutative categories, is a pseudo symmetric Cat-multifunctor but not a Cat-multifunctor in the symmetric sense. It follows that inverse K-theory preserves algebraic structures parametrized by non-symmetric and pseudo symmetric Cat-multifunctors but not Cat-multifunctors in general. As a special case, we observe that inverse K-theory sends pseudo symmetric E-infinity-algebras in Gamma-categories to pseudo symmetric E-infinity-algebras in small permutative categories.
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