Additive Grothendieck pretopologies and presentations of tensor categories

The Grothendieck construction for model categories

The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As a main result, we establish an equivalence between suitable diagrams of model categories indexed by $\mathcal{M}$ and a new notion of \textbf{model fibrations} over $\mathcal{M}$. When $\mathcal{M}$ is a model category, our construction endows the Grothendieck construction with a model structure which gives a presentation of Lurie's $\infty$-categorical Grot…

Linear polygraphs applied to categorification

We introduce two applications of polygraphs to categorification problems. We compute first, from a coherent presentation of an $n$-category, a coherent presentation of its Karoubi envelope. For this, we extend the construction of Karoubi envelope to $n$-polygraphs and linear $(n,n-1)$-polygraphs. The second problem treated in this paper is the construction of Grothendieck decategorifications for $(n,n-1)$-polygraphs. This construction yields a rewriting system presenting for example algebras categorified by a linear monoidal category. We finally link quasi-convergence of such rewriting systems…

Lex 模态的投射表示

Projective Presentations of Lex Modalities

本文定义了同伦类型论中拓扑模态的投射表示，作为 Grothendieck 拓扑的内部化，通过局部选择原理和层条件提供计算工具，并应用于代数理论分类拓扑斯，证明拟凝聚模在多种拓扑斯间的上同调稳定性。

On notions of compactness, object classifiers and weak Tarski universes

We prove a correspondence between $κ$-small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively $κ$-compact maps in their underlying quasi-categories for suitably large regular cardinals $κ$. We thus obtain a transition result between weakly universal small fibrations in the (type theoretic) injective Dugger-Rezk-style standard presentations of model toposes and object classifiers in Grothendieck $\infty$-toposes in the sense of Lurie.

作为某类型论模型的球状弱ω-范畴

Globular weak $ω$-categories as models of a type theory

本研究针对表述弱ω-范畴理论的依赖类型论CaTT，通过证明其初始性猜想，形式化验证了该类型论的模型与Maltsiniotis定义的球状弱ω-范畴精确对应，为相关类型论的神经定理推广提供了可能。