directed object (Rev #5, changes)

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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~~ One~~ there~~ central~~ for~~ topic in higher category theory is the~~ objects~~ question to determine a realisation-and-nerve adjunction$X(||\u22a3N):C\stackrel{N}{D}$~~ X~~ (||\dashv N):C\stackrel{N} D ~~ in~~ between some higher category of higher categories$\mathrm{sSetC}$~~ sSet~~ C~~ ,~~ and some category$\mathrm{QuiverD}$~~ Quiver~~ D ~~ or~~ of~~$Set_{disc}$~~*spaces*~~ ~~ .~~ directed~~~~ past~~~~ space~~~~ objects~~~~$dX$~~~~?~~

~~ The~~ For example, the instance~~interval object~~$(||\dashv N):\infty Grpd\stackrel{N}{\to} Top/_{\sim}$ ~~ in~~~~ any~~~~ of~~~~ these~~~~ categories~~ is called homotopy hypothesis. In this case$\mathrm{\Delta N}[1]$~~ \Delta[1]~~ N~~ .~~ ~~ Let~~ is said to assign to a space-modulo-weak-homotopy-equivalence its~~$d_{\Delta[1]}\subset [\Delta[1], \Delta[1]]=sSet([1],[1])$~~*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.

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 $\Delta[1]$. Let $d_{\Delta[1]}\subset {}_pt[\Delta[1], \Delta[1]]_{pt}$, let $dX\subset [\Delta[1],X]$ be a subset