directed object (Rev #6)


One central topic in higher category theory is the question to determine a realisation-and-nerve adjunction (||N):CND(||\dashv N):C\stackrel{N}{\to} D between some higher category of higher categories CC and some category DD of spaces.

For example, the instance (||N):GrpdNTop/ (||\dashv N):\infty Grpd\stackrel{N}{\to} Top/_{\sim} is called homotopy hypothesis. In this case NN is said to assign to a space-modulo-weak-homotopy-equivalence XX its fundamental \infty-groupoid, denoted by Π(X)\Pi(X). For C=Grpd=KanC=\infty Grpd=Kan, 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.

In the previous case (||N):GrpdNTop/ (||\dashv N):\infty Grpd\stackrel{N}{\to} Top/_{\sim}, the fact that XX 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 Π(X)\Pi(X) are invertible.

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


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 sets by 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 02:09:48 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.