directed object' (Rev #2, 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 $C$ 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 for in$\mathrm{XC}$ X C* is defined to be a

~~ (1)~~ The~~ every~~ collection~~ map~~ of$\mathrm{Id}\to *\to X$~~ I\to~~ d~~ *\to~~~~ X~~~~ ~~ -directed~~ factoring~~ path~~ over~~ in~~ the~~~~ point~~~~ is~~~~ a~~~~ (generalized)~~~~ element~~~~ of~~$dX$~~ d~~ X~~ .~~ satisfies the following properties:

~~ (2)~~ A~~$d X$~~*direction for $X$* is~~ closed~~ defined~~ under~~ to~~ pushout.~~ 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*

~~ A~~ (1)~~ global~~ The~~ (or~~~~ just~~~~ generalized)~~~~ element~~~~ of~~$\mathrm{dD}X$~~ d~~ D~~ X~~ ~~ is~~ image~~ called~~ of every map$I\to *\to X$ factoring over the point is in $ddp(X)$.*directed path in $X$*

~~ A~~ (2) The collection of global elements of$d X$ is*directed object*~~ defined~~ closed~~ to~~ under~~ be~~ pushout.~~ a~~ For~~ pair~~${}_{d}\alpha X:=(X,\mathrm{dX})$~~ {}_d~~ \alpha~~ X:=(X,dX)~~~~ ~~ ,~~ consisting~~~~ of~~~~ an~~~~ object~~$\mathrm{X\beta}$~~ X~~ \beta global elements of$\mathrm{Cd}X$~~ C~~ d X~~ ~~ ,~~ and~~ their~~ a~~ pushout~~ direction~~$\mathrm{dX\alpha}\otimes \beta $~~ dX~~ \alpha\otimes \beta ~~ for~~ is called their~~$X$~~*composition*.

A * morphism of directed objects object* is defined to be a pair$f{}_{d}X:=(X,\mathrm{dX})\to (Y,\mathrm{dy})$

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

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.