On the notion of geometric realization

A simple approach to geometric realization of simplicial and cyclic sets

This is the same version that was previously only on my home page. We give a description of geometric realization which makes it evident that it commutes with products. A similar approach is used to treat cyclic sets. Our approach is similar to those of Drinfeld and Grayson.

Geometric Realizations of Tricategories

Any tricategory characteristically has associated various simplicial or pseudo-simplicial objects. This paper explores the relationship amongst three of them: the pseudo-simplicial bicategory so-called Grothendieck nerve of the tricategory, the simplicial bicategory termed its Segal nerve, and the simplicial set called its Street geometric nerve, and it proves the fact that the geometric realizations of all of these possible candidate 'nerves of the tricategory' are homotopy equivalent. Our results provide coherence for all reasonable extensions to tricategories of Quillen's definition of the …

The geometric realization of a simplicial Hausdorff space is Hausdorff

It is shown that the thin geometric realization of a simplicial Hausdorff space is Hausdorff. This proves a famous claim by Graeme Segal that the thin geometric realisation of a simplicial k-space is a k-space.

分次gentle代数的Koszul对偶的几何实现

A geometric realization of Koszul duality for graded gentle algebras

本文证明了同调光滑分次gentle代数的Koszul函子可由几何模型中的半旋转实现，并以此证明了涉及该函子的交维数公式。

A geometric realization of the $m$-cluster categories of type $\tilde{D_n}$

We show that a subcategory of the $m$-cluster category of type $\tilde{D_n}$ is isomorphic to a category consisting of arcs in an $(n-2)m$-gon with two central $(m-1)$-gons inside of it. We show that the mutation of colored quivers and $m$-cluster-tilting objects is compatible with the flip of an $(m+2)$-angulation. In the final part of this paper, we detail an example of a quiver of type $\tilde{D_7}$.

Derived noncommutative schemes, geometric realizations, and finite dimensional algebras

The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined as differential graded categories of a special type. We review and discuss different properties of both noncommutative schemes and morphisms between them. In addition, the concept of geometric realization for derived noncommutative scheme is introduced and problems of existence and construction of such realizations are discussed. We also study the construction of gluing noncommutative schemes via …