directed object' (Rev #3, changes)

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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 $\mathrm{CV}$~~ C~~ V be a~~ closed~~ monoidal~~ (V-enriched~~ category,~~ for~~ let~~ some~~$\mathrm{VC}$~~ V~~ C~~ )~~ ~~ homotopical~~ be~~ category,~~~~ let~~$*V$~~ *~~ V~~ ~~ -enriched,~~ denote~~ closed~~ the~~ monoidal~~ tensor~~ homotopical~~ unit~~ category,~~ of~~ let$C*$~~ C~~ * ~~ which~~ denote~~ we~~~~ assume~~~~ to~~~~ be~~ the~~ terminal~~ tensor~~ object,~~ unit~~ let~~ of$\mathrm{IC}$~~ I~~ C ~~ denote~~ which we assume to be the~~ interval~~ terminal~~ object~~ object,~~ of~~ let$\mathrm{CI}$~~ C~~ I~~ ,~~ ~~ let~~ denote the interval object of$\mathrm{XC}$~~ X~~ C~~ ~~ ,~~ be~~ let~~ a~~~~ pointed~~~~ object~~~~ of~~$\mathrm{CX}$~~ C~~ X~~ .~~ ~~ Let~~ be a pointed object of$\mathrm{DC}$~~ D~~ C~~ ~~ .~~ be~~ Let~~ the~~~~ functor~~$D:C\to V$~~ D:C\to~~ D~~ V~~ 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 *$d*-\mathrm{directed}\mathrm{paths}\mathrm{of}$ d*-directed d paths of X$), -directed satisfies paths of$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~~$\mathrm{dddp}(X)$~~ d~~ ddp(X)~~ X~~ is closed under~~ pushout.~~ the tensor product. For$\alpha ,\beta \in \mathrm{ddp}(X)$~~ \alpha~~ \alpha,\beta\in ddp(X) , their pushout$\alpha \otimes \beta $ \alpha\otimes \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

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

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

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