directed object (Rev #8, changes)

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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”.

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 $q:Q\to \mathrm{Set}$~~ q:\to~~ q: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 $[\Delta[1], X]$.