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 $\mathrm{pt}\stackrel{\sigma }{\to }I\stackrel{\tau }{\leftarrow }\mathrm{pt}$, and suppose that every object $X$ of $C$ is $I$-undirected (i.e. $[\mathrm{pt},X]\simeq [I,X]$).

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

Then we call the pair $(X,\mathrm{dX})$ an object with directed path space $\mathrm{dX}$ (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 $\gamma \in d_I$, $\varphi \in \mathrm{dX}$, then $\gamma \circ \varphi \in \mathrm{dX}$. If e.g $d_I=[I,I]$, then this condition means that $dX$ 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 $\varphi :I^{v2}\to X$. 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 $a•b:=\varphi \circ \psi$. Then $\mathrm{dX}$ 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 $p\in \mathrm{dX}$, $\varphi :X\to Y$, then $\varphi \circ p\in d_Y$). 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 $d_I=\{\mathrm{monotonic}\mathrm{maps}I\to I\}$.

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)