Spahn directed object (Rev #7)

Idea

One central topic in higher category theory is the question to determine a realisation-and-nerve adjunction $(||\dashv N):C\stackrel{N}{\to} D$ 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 $X$ its fundamental $\infty$ -groupoid, denoted by $\Pi(X)$ . For $C=\infty Grpd=Kan$ , $N=Sing$ 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.

In the previous case $(||\dashv N):\infty Grpd\stackrel{N}{\to} Top/_{\sim}$ , the fact that $X$ is a topological space and consequently all the paths and higher paths in it are invertible corresponds to the fact that all morphisms and higher morphisms in $\Pi(X)$ are invertible.

Now we wish to describe some other adjunction $(||\dashv N):C\stackrel{N} {\to}D$ where $C=(m, n)Cat$ , for natural numbers $n\le m$ and since here not all higher morphisms in $X\in C$ are invertible we think of $N(X)$ as an object in whose path space some paths are not invertible and “can be traversed in only one direction”.

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 sets by $s Set:=Psh(\Delta)$ .

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 $\Delta[1]$ . The path space of $X$ is the internal hom object $[\Delta[1], X]$ .

Revision on November 9, 2012 at 02:47:48 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.