Opetopic algebras III: Presheaf models of homotopy-coherent opetopic algebras

单纯群胚的预层同伦理论

Homotopy theory of presheaves of simplicial groupoids

我们证明了单格群胚在格点$\mathcal{C}$上的预层范畴是右真单模型范畴。定义了2-群胚的预层$G$ -torson，单纯群的预层$G$-torson和单纯群的预层$G$-torson，并利用同伦类$[\ast, \bar{W}G]$对$G$ -torson进行了分类.

余代数的单预层

Simplicial presheaves of coalgebras

在小Grothendieck点上的交换单位环的预层上的单纯R-余代数范畴被赋予了左真的、单纯的、余纤维生成的模型范畴结构，其中弱等价是基础单纯预层的局部弱等价。这个模型范畴自然地与单纯预层的R-局部同伦理论和Quillen算子的单纯R-模的同伦理论联系在一起。在R是代数闭（或完全）域的预层的特殊情况下，我们研究了与单纯预层的R-局部同伦范畴的比较。若R是代数闭域的一个预层，证明了单纯预层的R-局部同伦范畴完全忠实地嵌入单纯R-余代数的同伦范畴.

内部和局部同伦理论

Internal and local homotopy theory

在Grothendieck topos中有一个关于单纯对象的完善的同伦理论，民间传说说弱等价在$L_{ω_1, ω}$的几何片段中是可公理化的。我们表明，它实际上是一个理论的预层型，即分类的预层拓扑。作为推论，我们得到了一个新的证明，证明了作为局部弱同伦等价的单纯预层的局部Kan纤维化正是具有期望的局部提升性质的态射.

同伦加权余极限

Homotopy weighted colimits

设V是共纤生成的闭对称monoidal模型范畴，M是模型V-范畴。我们说一个图D的加权上极限W*D是一个同伦加权上极限，如果图D是逐点相干的，并且权W在[C^op，V]上的投影模型结构中是相干的。然后，我们继续通过同伦张量和普通（锥形）同伦上极限来描述这样的同伦加权上极限。这是著名同构W*D= \int^C（W \tensor D）的同伦版本。在证明了这种同伦分解一般，我们研究了一些特殊情况下的细节。对于单纯集，张量可以用锥同伦上极限代替直到弱等价，因此加权同伦上极限没有附加值。对于模型dg-范畴，情况完全不同，其中解悬不能由圆锥同伦余极限构造。在最后一节中，我们刻画了在丰富预层范畴上诱导Quillen等价的V-函子。

A general method to construct cube-like categories and applications to homotopy theory

In this paper, we introduce a method to construct new categories which look like "cubes", and discuss model structures on the presheaf categories over them. First, we introduce a notion of thin-powered structure on small categories, which provides a generalized notion of "power-sets" on categories. Next, we see that if a small category $\mathcal{R}$ admits a good thin-powered structure, we can construct a new category $\square(\mathcal{R})$ called the cubicalization of the category. We also see that $\square(\mathcal{R})$ is equipped with enough structures so that many arguments made for the c…

Cocycles in Local Higher Category Theory

We develop a model structure on presheaves of small simplicially enriched categories on a site $\mathscr{C}$, for which the weak equivalences are 'stalkwise' weak equivalences for the Bergner model structure. This model structure is right proper, and can be connected via a zig-zag of Quillen equivalences to local analogues of the Joyal and Rezk model structures. Because the local Bergner model structure is right proper, we can apply Jardine's cocycle categories to study its homotopy category. As an application of the cocycle theory, we describe the maps $[*, X]$ in the homotopy category of the…