+-- {: .num_defn} ###### 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 [[nLab:co-span]] for this notation). =-- +-- {: .num_defn} ###### Definition (directed object) Let $C$ be a closed monoidal (V-enriched for some $V$) homotopical category, let $*$ denote the tensor unit of $C$ which we assume to be the terminal object, let $I$ denote the interval object of $C$, let $X$ be a pointed object of $C$. Let $D$ be the functor $D:C\to V$, X\mapsto [I,X]$. A *direction in $C$* is defined to be a subfunctor of $D$. In this case $d X$ is called a *direction for $X$*. A global element of $dX$ is called a *$d$-directed path in $X$*. The collection of $d$-directed path in $X$ satisfies the following properties: A *direction for $X$* is defined to be a subobject $dX$ of $[I,X]$ whose collection $ddp(X)$ of global elements, called *directed paths* (or more precisely *$d*-directed paths of $X$), satisfies (1) The $D$ image of every map $I\to *\to X$ factoring over the point is in $ddp(X)$. (2) The collection of global elements of $d X$ is closed under pushout. For $\alpha$, $\beta$ global elements of $d X$, their pushout $\alpha\otimes \beta$ is called their *composition*. 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 $$\array{ X& \stackrel{f}{\to}& Y \\ \downarrow^d&&\downarrow^d \\ dX& \stackrel{df}{\to}& d Y }$$ =-- +-- {: .num_remark} ###### Remark Let $C$ be $V$-enriched, closed monoidal homotopical category, let $I$ denote the tensor unit of $C$. =-- +-- {: .num_defn} ###### Definition A *directed-path-space objects* is defined. =--