nLab directed object

Idea

The idea of the following text is to begin with a category of objects presumed undirected and construct from that a supercategory of directed objects, analogous to how Marco Grandis developed directed topological spaces out of the usual undirected ones.

(Rather different approaches to a notion of “directed object” will exist. See also at directed homotopy theory and directed homotopy type theory.)

Definition (tentative)

Let $C$ be a Trimble omega-category with interval object $pt\stackrel{\sigma}{\to}I\stackrel{\tau}{\leftarrow}pt$, and suppose that every object $X$ of $C$ is $I$-undirected (i.e. $[pt,X]\simeq [I,X]$).

Let $d_I \subset {}_{pt}hom(I,I)_{pt}$ be a subset of the set of co-span-endomorphisms of $pt\stackrel{\sigma}{\to}I\stackrel{\tau}{\leftarrow}pt$. Let $dX\subset [I,X]$ be a subset of the hom-set $[I , X]$.

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

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

2. (Reparametrisation) If $\gamma\in d_I$, $\phi \in dX$, then $\gamma \circ \phi\in dX$. If e.g $d_I=[I,I]$, then this condition means that $d X$ is a sieve in $I/C$.

3. (Concatenation) Let $a,b:I\to X$ be consecutive wrt. $I$ (i.e. $\pt \to^{\tau} I \to^{a} X$ equals $\pt \to^{\sigma} I \to^{b} X$), let $I^{v2}$ denote the pushout of $\sigma$ and $\tau$, then by the universal property of the pushout there is a map $\phi:I^{v2}\to X$. By definition of the interval object (described there in the section “Intervals for Trimble $\omega$-categories”) there is a unique morphism $\psi:I\to I^{v2}$. Then the composition of $a$ and $b$ is defined by $a\bullet b:=\phi\circ \psi$. Then $dX$ 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 (i.e. if $p\in dX$, $\phi:X\to Y$, then $\phi\circ p\in d_Y$). Objects with directed path space and morphisms thereof define a category denoted by $d_I{C}$.

$C$ is a subcategory of $d_I{C}$.

Examples

• The category of directed topological spaces according to Grandis is of the above form $d_I{C}$ for $C =$ Top, $I = [0,1]$ and $d_I = \{monotonic maps I \to I\}$.

References

The definition and study of directed topological spaces was undertaken in

• Marco Grandis, Directed homotopy theory, I. The fundamental category (arXiv)

Applications of categories regarded as models for directed spaces are discussed in

• Tim Porter: Enriched categories and models for spaces of evolving states, Theoretical Computer Science, 405, (2008), pp. 88 - 100.

• Tim Porter, Enriched categories and models for spaces of

  dipaths. A discussion document and overview of some techniques_ (pdf)