A Comparison of the functors of points of Supermanifolds

The local functors of points of Supermanifolds

We study the local functor of points (which we call the Weil-Berezin functor) for smooth supermanifolds, providing a characterization, representability theorems and applications to differential calculus.

The Schwarz-Voronov Embedding of ${\mathbb Z}_{2}^{n}$-Manifolds

Informally, ${\mathbb Z}_2^n$-manifolds are 'manifolds' with ${\mathbb Z}_2^n$-graded coordinates and a sign rule determined by the standard scalar product of their ${\mathbb Z}_2^n$-degrees. Such manifolds can be understood in a sheaf-theoretic framework, as supermanifolds can, but with significant differences, in particular in integration theory. In this paper, we reformulate the notion of a ${\mathbb Z}_2^n$-manifold within a categorical framework via the functor of points. We show that it is sufficient to consider ${\mathbb Z}_2^n$-points, i.e., trivial ${\mathbb Z}_2^n$-manifolds for whic…

A note on separable functors and monads

For an adjoint pair $(F, G)$ of functors, we prove that $G$ is a separable functor if and only if the defined monad is separable and the associated comparison functor is an equivalence up to retracts. In this case, under an idempotent completeness condition, the adjoint pair $(F, G)$ is monadic. This applies to the comparison between the derived category of the category of equivariant objects in an abelian category and the category of equivariant objects in the derived category of the abelian category.

半可分函子与收缩等价条件

Semiseparable functors and conditions up to retracts

本文结合幂等完备化研究半可分函子，引入收缩等价意义下的（余）反射与双反射概念，刻画了伴随函子的半可分性，推广了Chen的可分情形结果，并给出Balmer预三角范畴结果的半可分版本。

非交换n元Γ-半环的K-理论中的基本比较、基变换和下降定理

Fundamental comparison, base-change, and descent theorems in the $K$-theory of non-commutative n-ary Gamma-semirings

本文在非交换n元Γ-半环的代数K-理论中，通过双有限、槽敏感的Γ-模和完美复形，建立了由标量扩张和限制诱导的函子性映射，证明了导出的Morita不变性、Beck-Chevalley型基变换和投影公式，并推导了Zariski超下降、切除和局部化序列以及fpqc下降，最后给出了K_Γ(X)的几何解释及其在Γ-线性非交换动机中的泛性质。