directed object' (Rev #4)

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).

Let $V$ be a monoidal category, let $C$ be $V$-enriched, closed monoidal 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) $ddp(X)$ is closed under the tensor product. For $\alpha,\beta\in ddp(X)$, 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

$\array{
X&
\stackrel{f}{\to}&
Y
\\
\downarrow^d&&\downarrow^d
\\
dX&
\stackrel{df}{\to}&
d Y
}$

Let $C$ be $V$-enriched, closed monoidal homotopical category, let $I$ denote the tensor unit of $C$.

A *directed-path-space objects* is defined.