directed object



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

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

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

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

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

  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 dXdX 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 pdXp\in dX, ϕ:XY\phi:X\to Y, then ϕpd Y\phi\circ p\in d_Y). 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}.


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


The definition and study of directed topological spaces was undertaken in

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)

Last revised on November 8, 2012 at 00:42:23. See the history of this page for a list of all contributions to it.