Janelidze's Categorical Galois Theory as a step in the Joyal and Tierney result

dagger $(\infty,1)$-范畴的模型 II：dagger 完备 Segal 空间与 Joyal--Tierney 等价

Models for dagger $(\infty,1)$-categories II: Dagger complete Segal spaces and the Joyal--Tierney equivalence

本文为dagger (∞,1)-范畴发展单纯空间模型，构造酉完备dagger Segal空间模型结构，并证明Joyal--Tierney等价的dagger类比，建立dagger Bergner、Joyal与Rezk模型间的Quillen等价链。

On the fibrewise effective Burnside $\infty$-category

Effective Burnside $\infty$-categories are the centerpiece of the $\infty$-categorical approach to equivariant stable homotopy theory. In this étude, we recall the construction of the twisted arrow $\infty$-category, and we give a new proof that it is an $\infty$-category, using an extremely helpful modification of an argument due to Joyal--Tierney. The twisted arrow $\infty$-category is in turn used to construct the effective Burnside $\infty$-category. We employ a variation on this theme to construct a fibrewise effective Burnside $\infty$-category. To show that this constuctionworks fibrewi…

CW-complexes in the Category of Small Categories

We compute the collection of CW-complexes in the model category of small categories constructed by Joyal and Tierney. More generally, if $X$ is a connected topological space, we show that the homotopy category of CW-complexes in Joyal-Tierney's model category of sheaves of sets on $X$ is equivalent to the homotopy category of groupoids. As an application of the ideas, we show that the algebraic $K$-theory groups of the category of pointed small categories are trivial, and more generally, the algebraic $K$-theory groups of any sufficiently "nice" Waldhausen category $\mathcal{A}$ of pointed sma…

Tannaka Theory for Topos

We consider locales $B$ as algebras in the tensor category $s\ell$ of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections $sh(B) \stackrel{q}{\longrightarrow} \mathcal{E}$ in Galois theory [An extension of the Galois Theory of Grothendieck, AMS Memoirs 151] and a Tannakian recognition theorem over $s\ell$ for the $s\ell$-functor $Rel(E) \stackrel{Rel(q^*)}{\longrightarrow} Rel(sh(B)) \cong (B$-$Mod)_0$ into the $s\ell$-category of discrete $B$-modules. Thus, a new Tannaka recognition theorem is obtained, essentially different from those …

Galois theory in bicategories

We develop a Galois (descent) theory for comonads within the framework of bicategories. We give generalizations of Beck's theorem and the Joyal-Tierney theorem. Many examples are provided, including classical descent theory, Hopf-Galois theory over Hopf algebras and Hopf algebroids, Galois theory for corings and group-corings, and Morita-Takeuchi theory for corings. As an application we construct a new type of comatrix corings based on (dual) quasi bialgebras.

高阶范畴伽罗瓦理论中提升条件的不合理有效性 I：一个拟范畴伽罗瓦定理

The Unreasonable Efficacy of the Lifting Condition in Higher Categorical Galois Theory I: a Quasi-categorical Galois Theorem

本文通过借鉴 Lurie 2024 的基础，将范畴伽罗瓦定理精化为拟范畴伽罗瓦定理，并揭示分解系统、高阶范畴伽罗瓦定理与高阶拓扑斯内部伽罗瓦理论之间的深层联系。