directed object (Rev #1)

Let Δ\Delta denote the simplex category. This is the category having finite ordinals as objects and as morphisms monotone maps thereof.

Let X:Δ opSetX:\Delta^{op}\to Set be a simplicial set?. The category of simplicial sets we denote 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:{1d 1d 00} opSetq:\{1 \stackrel{\overset{d_0}{\leftarrow}}{\underset{d_1}{\leftarrow}} 0\}^{op}\to Set is called a quiver?. This is just a directed graph perhaps with multiple edges and loops.

Denote the category of quivers with natural transformations thereof as morphisms by Quiver:=Psh(Q)Quiver:=Psh(Q).

Revision on November 7, 2012 at 18:40:07 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.