Working in an arbitrary category endowed with a fixed $$({\mathcal {E}}, {\mathcal {M}})$$ -factorization system such that $${\mathcal {M}}$$ is a fixed class of monomorphisms, we first define and study a concept of codense morphisms with respect to a given categorical interior operator i. Some basic properties of these morphisms are discussed. In particular, it is shown that i-codenseness is preserved under both images and dual images under morphisms in $${\mathcal {M}}$$ and $${\mathcal {E}}$$ , respectively. We then introduce and investigate a notion of quasi-open morphisms with respect to i. Notably, we obtain a characterization of quasi i-open morphisms in terms of i-codense subobjects. Furthermore, we prove that these morphisms are a generalization of the i-open morphisms that are introduced by Castellini. We show that every morphism which is both i-codense and quasi i-open is actually i-open. Examples in topology and algebra are also provided.

中文翻译：

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对内部操作员的凝聚性和开放性

在具有固定 $$({\mathcal {E}}, {\mathcal {M}})$$ - 分解系统的任意类别中工作，使得 $${\mathcal {M}}$$ 是一个固定类关于单态性，我们首先定义和研究关于给定分类内部算子 i 的同态态射的概念。讨论了这些态射的一些基本性质。特别是，表明在 $${\mathcal {M}}$$ 和 $${\mathcal {E}}$$ 中的态射下，图像和对偶图像下的 i 码密性都得到了保留。然后我们介绍并研究关于 i 的准开态射的概念。值得注意的是，我们根据 i-codes 子对象获得了准 i-open 态射的特征。此外，我们证明了这些态射是 Castellini 引入的 i-open 态射的推广。我们证明了每个既是 i-codes 又是拟 i-open 的态射实际上是 i-open。还提供了拓扑和代数方面的示例。