AI 中文总结
该研究利用代数小对象引理构造格罗滕迪克∞群胚的代数协调子，定义受控理论的格罗滕迪克实现并构造球状模型，提出广义推出猜想以支撑半模型结构存在性及同伦假设。
AI 中文摘要
我们利用代数小对象引理构造格罗滕迪克∞群胚的代数协调子，以更直接地自由附加协调数据的方法，替代此前基于单子分配序列的相关方法。给定一个受控理论，我们定义非约化与约化格罗滕迪克实现，生成∞-Lawvere理论，并将该构造函子性地扩展到受控理论的连通图表。我们应用该框架构造幺半∞群胚、对称幺半∞群胚、协调∞群与皮卡∞群的球状模型，在∞-Lawvere理论的模型范畴上定义典范半模型结构，并提出广义推出猜想，该猜想蕴含这些半模型结构的存在性及格罗滕迪克∞群胚的同伦假设。
英文摘要
We introduce a construction of algebraic coherators for Grothendieck $\infty$-groupoids using the algebraic small object argument, replacing previous approaches we have used based on distributive series of monads with a more direct method for freely adjoining coherence data. Given a controlled theory, we define unreduced and reduced Grothendieck realizations, producing $\infty$-Lawvere theories and extending this construction functorially to connected diagrams of controlled theories. We apply this framework to construct globular models for monoidal $\infty$-groupoids, symmetric monoidal $\infty$-groupoids, coherent $\infty$-groups, and Picard $\infty$-groupoids. We define canonical semi-model structures on categories of models over $\infty$-Lawvere theories and formulate a generalized pushout conjecture that implies the existence of these semi-model structures and the Homotopy Hypothesis for Grothendieck $\infty$-groupoids.
