Spahn directed object (Rev #5, changes)

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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 $Quiver:=Psh(Q)$ .

Idea

~~Are~~ One ~~there~~ central ~~for~~ topic in higher category theory is the ~~objects~~ question to determine a realisation-and-nerve adjunction $X (| | ⊣ N) : C \overset{N}{D}$ X (||\dashv N):C\stackrel{N} D in between some higher category of higher categories $sSet C$ ~~sSet~~ C , and some category $Quiver D$ ~~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 $Δ N [1]$ ~~\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.

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