AI 中文总结
研究预交叉模和交叉模的卡卢日宁 - 克拉斯纳嵌入，通过范畴框架，为预交叉模引入嵌入，在交叉模情形遇阻碍，部分结果积极，完整嵌入构建困难。
AI 中文摘要
卡卢日宁 - 克拉斯纳嵌入表明，对于群扩张，中间群可嵌入其核与余核的 wreath 积。近期此构造在范畴框架中被推广。本文将该范畴框架应用于预交叉模和交叉模（群上的）。对于预交叉模，严格依范畴步骤引入此前未阐述的卡卢日宁 - 克拉斯纳嵌入。对于交叉模，虽有部分积极结果，但构建完整嵌入困难得多。
英文摘要
The Kaluzhnin--Krasner embedding establishes that, for a group extension, the middle group can be embedded into the wreath product of its kernel and cokernel. Recently, this construction has been generalised in a categorical framework, recovering both the classical group-theoretic result and its analogue for Lie In this article, we apply this categorical framework to two specific cases: precrossed modules and crossed modules (over groups). For precrossed modules, we introduce a Kaluzhnin--Krasner embedding -- previously unformulated -- by rigorously following the steps of the categorical procedure. In contrast, for crossed modules, we encounter substantial obstacles: while we present partial positive results, we also explain why constructing a full-fledged Kaluzhnin--Krasner embedding in this context is far more difficult.
