Spahn directed object' (Rev #1, changes)

Definition (interval object)

Let $C$ be a closed monoidal (V-enriched for some $V$) homotopical category, let $*$ denote the tensor unit of $C$. Then the cospan category (same objects and cospans as morphisms) is $V$-enriched, too.

An interval object is defined to be a cospan $*\stackrel{a}{\rightarrow}I\stackrel{b}{\leftarrow}*$.

The pushout $I^{\coprod_2}$ of this diagram satisfies ${}_*[ I,I^{\coprod_2}]_*\simeq$ is contractible (see co-span for this notation).

Definition (directed object)

Let $C$ be a closed monoidal (V-enriched for some $V$) homotopical category, let $*$ denote the tensor unit of $C$, let $I$ denote the interval object of $C$, let $X$ be a pointed object of $C$.

A direction for $X$ is defined to be a subobject $dX$ of $[I,X]$ satisfying

(1) every map $I\to *\to X$ factoring over the point is a (generalized) element of $d X$.

(2) $d X$ is closed under pushout.

A global (or just generalized) element of $d X$ is called directed path in $X$

A directed object is defined to be a pair ${}_d X:=(X,dX)$ consisting of an object $X$ of $C$ and a direction $dX$ for $X$.

A morphism of directed objects $f:(X,dX)\to (Y,dy)$ is defined to be a pair $(f,df)$ making

Definition

A directed-path-space objects is defined.