不同范畴中的半投射与半内射
Semiprojectivity and semiinjectivity in different categories
摘要：
投射性和内射性是范畴论中的基本概念。我们认为自然弱化称为半投射性和半内射性，并研究这些概念在不同的类别。例如，在度量空间的范畴中，（半）内射对象恰好是绝对（邻域）收缩。证明了平凡群是唯一的半内射群，而平凡群与自由群的每个自由积都是半内射的.对于紧致度量空间$X$，我们将交换C*-代数$C(X)$联系起来。这种关联是逆变的，因此$X$的半内射性与$C(X)$的半投射性有关。我们与Adam Sørensen一起证明了$C(X)$在所有C*-代数范畴中是半投射的当且仅当$X$是维数至多为1的绝对邻域收缩核。
英文摘要：
Projectivity and injectivity are fundamental notions in category theory. We consider natural weakenings termed semiprojectivity and semiinjectivity, and study these concepts in different categories. For example, in the category of metric spaces, (semi)injective objects are precisely the absolute (neighborhood) retracts. We show that the trivial group is the only semiinjective group, while every free product of a finitely presented group and a free group is semiprojective. To a compact, metric space $X$ we associate the commutative C*-algebra $C(X)$. This association is contravariant, whence semiinjectivity of $X$ is related to semiprojectivity of $C(X)$. Together with Adam Sørensen, we showed that $C(X)$ is semiprojective in the category of all C*-algebras if and only if $X$ is an absolute neighborhood retract with dimension at most one.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
