# Spahn directed object (Rev #5)

## Idea

One central topic in higher category theory is the question to determine a realisation-and-nerve adjunction $(||\dashv N):C\stackrel{N} D$ between some higher category of higher categories $C$ and some category $D$ of spaces.

For example, the instance $(||\dashv N):\infty Grpd\stackrel{N}{\to} Top/_{\sim}$ is called homotopy hypothesis. In this case $N$ is said to assign to a space-modulo-weak-homotopy-equivalence its fundamental $\infty$-groupoid. For $N=Sing$ and $||$ geometric realization of topological spaces this is an equivalence, and moreover a Quillen equivalence of appropriate model categories and hence an equivalence of $(\infty,1)$-categories.

## Definitions

###### Definition (some toposes of arrows)

Let $\Delta$ denote the simplex category. This is the category having finite ordinals as objects and as morphisms monotone maps thereof.

We define the category of simplicial setsby $s Set:=Psh(\Delta)$.

Let $\Delta_0$ be the terminal category (the category with one object $*$ and one morphism $id_*$. Then $Psh(\Delta_0)=Set_{disc}$ is the discrete category of sets; this is the class of sets and the class of morphisms consists only of the identities.

Let $Q:=\{1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0\}^{op}$ denote the category with two objects and morphism set $\{s,t,id_0,id_1\}$. $Q$ is called the walking quiver.

A functor $q:\to Set$ is called a quiver?. This is just a directed graph perhaps with multiple edges and loops.

We denote the category of quivers with natural transformations thereof as morphisms by $Quiver:=Psh(Q)$.

Are there for the objects $X$ in $sSet$, $Quiver$ or $Set_{disc}$ directed past space objects $dX$?

The interval object in any of these categories is $\Delta$. Let $d_{\Delta}\subset {}_pt[\Delta, \Delta]_{pt}$, let $dX\subset [\Delta,X]$ be a subset

Revision on November 9, 2012 at 01:19:15 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.