Spahn
directed object (Rev #5, changes)

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Definition (some toposes of arrows)

Let Δ\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 sSet:=Psh(Δ)s Set:=Psh(\Delta).

Let Δ 0\Delta_0 be the terminal category (the category with one object ** and one morphism id *id_*. Then Psh(Δ 0)=Set discPsh(\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:={1d 1d 00} opQ:=\{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}\{s,t,id_0,id_1\}. QQ is called the walking quiver.

A functor q:Setq:\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)Quiver:=Psh(Q).

Idea

Are One there central for topic in higher category theory is the objects question to determine a realisation-and-nerve adjunctionX(||N):CDN 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 ofSet discSet_{disc}spaces . directed past space objectsdXdX?

The For example, the instanceinterval object(||N):GrpdNTop/ (||\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 itsd Δ[1][Δ[1],Δ[1]]=sSet([1],[1])d_{\Delta[1]}\subset [\Delta[1], \Delta[1]]=sSet([1],[1])fundamental \infty-groupoid. For N=SingN=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 (,1)(\infty,1)-categories.

Definitions

Definition (some toposes of arrows)

Let Δ\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 sSet:=Psh(Δ)s Set:=Psh(\Delta).

Let Δ 0\Delta_0 be the terminal category (the category with one object ** and one morphism id *id_*. Then Psh(Δ 0)=Set discPsh(\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:={1d 1d 00} opQ:=\{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}\{s,t,id_0,id_1\}. QQ is called the walking quiver.

A functor q:Setq:\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)Quiver:=Psh(Q).

Are there for the objects XX in sSetsSet, QuiverQuiver or Set discSet_{disc} directed past space objects dXdX?

The interval object in any of these categories is Δ[1]\Delta[1]. Let d Δ[1] pt[Δ[1],Δ[1]] ptd_{\Delta[1]}\subset {}_pt[\Delta[1], \Delta[1]]_{pt}, let dX[Δ[1],X]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.