directed object (Rev #3)

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).

The interval object in any of these categories is Δ[1]\Delta[1]

Definition (object with directed pathspace)

Let CC be a category with an interval object II, and suppose that every object XX of CC is II-undirected?.

To be explicit, fix a subset d I pthom(I,I) pt d_I \subset {}_{pt}hom(I,I)_{pt} of the endomorphisms of the given interval object II regarded as a cospan ptIptpt \to I \leftarrow pt to be called the directed endomorphisms of the interval object. Let d X[I,X]d_X\subset [I,X] be a subset of the hom-set? [IX][I \to X].

Then we call the pair (X,d X)(X, d_X) an object with directed path space d Xd_X if the following conditions (attributed to Marco Grandis) are satisfied:

  1. (Constant paths) Every map IptXI \to \pt \to X is directed;

  2. (Reparametrisation) For γd Xhom(I,X)\gamma \in d_X \subset hom(I,X) and every ϕd Ihom(I,I)\phi \in d_I \subset hom(I,I), also γϕ\gamma \circ \phi is in d Xd_X;

  3. (Concatenation) Let a,b:IXa,b:I\to X be consecutive wrt. II (i.e. pt τI aX\pt \to^{\tau} I \to^{a} X equals pt σI bX\pt \to^{\sigma} I \to^{b} X), let I v2I^{v2} denote the pushout of σ\sigma and τ\tau, then by the universal property of the pushout there is a map ϕ:I v2X\phi:I^{v2}\to X. By definition of the interval object (described there in the section “Intervals for Trimble ω\omega-categories”) there is a unique morphism ψ:II v2\psi:I\to I^{v2}. Then the composition of aa and bb is defined by ab:=ϕψa\bullet b:=\phi\circ \psi. Then d Xd_X shall be closed under composition of consecutive paths.

We define a morphism of objects with directed path space to be a morphism of their underlying objects that preserves directed paths. Objects with directed path space and morphisms thereof define a category denoted by d ICd_I{C}.

CC is a subcategory of d ICd_I{C}.

Revision on November 7, 2012 at 22:39:36 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.