directed object' (Rev #5, changes)

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Added | ~~Removed~~ | ~~Chan~~ged

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 $*\stackrel{0}{\to}I\stackrel{1}{\leftarrow}$~~ I~~ *\stackrel{0}{\to} I\stackrel{1}{\leftarrow} 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~~$\mathrm{Dd}$~~ D~~ d~~ .~~ ~~ In~~ of~~ this~~~~ case~~$\mathrm{dD}X$~~ d~~ D~~ X~~ ~~ is~~ preserving~~ called~~ the~~ a~~ terminal object and satisfying the properties below. In this case$d X$*direction for $X$*~~ .~~ ~~ A~~ is~~ global~~ called~~ element~~ a~~ of~~~~$dX$~~*direction for $X$*~~ ~~ .~~ is~~ A~~ called~~ global~~ a~~ element of$dX$*$d$-directed path in $X$*~~ .~~ is called a*$d$-directed path in $X$* and their collection we denote by $ddp(X)$ the properties are:

(1) The~~ collection~~~~ of~~$\mathrm{dD}$~~ d~~ D~~ -directed~~ ~~ path~~ image~~ in~~ of every map$I\to *\to X$ I\to *\to X ~~ satisfies~~ factoring over the~~ following~~ point~~ properties:~~ is in$ddp(X)$.

~~ A~~ (2)$ddp(X)$ is*direction for $X$*~~ defined~~ closed~~ to~~ under~~ be~~ the~~ a~~ tensor~~ subobject~~ product. For$\mathrm{dX\alpha},\beta \in \mathrm{ddp}(X)$~~ dX~~ \alpha,\beta\in ddp(X)~~ ~~ ,~~ of~~ their pushout$[\alpha I\otimes ,\beta X]$~~ [I,X]~~ \alpha\otimes \beta ~~ whose~~ is~~ collection~~ called their~~$ddp(X)$~~*composition*~~ ~~ .~~ of~~~~ global~~~~ elements,~~~~ called~~*directed paths*~~ (or more precisely ~~*$d$-directed paths of $X$*~~), satisfies~~

~~ (1)~~ A~~ The~~~~$D$~~*directed object* ~~ image~~ is~~ of~~ defined~~ every~~ to~~ map~~ be a pair$I{}_{d}\to *\to X:=(X,\mathrm{dX})$~~ I\to~~ {}_d~~ *\to~~ X:=(X,dX)~~ X~~ ~~ factoring~~ consisting~~ over~~ of~~ the~~ an~~ point~~ object~~ is~~~~ in~~$\mathrm{ddp}(X)$~~ ddp(X)~~ X of $C$ and a direction $dX$ for $X$.

~~ (2)~~ A~~$ddp(X)$~~*morphism of directed objects*~~ ~~~~ is~~~~ closed~~~~ under~~~~ the~~~~ tensor~~~~ product.~~~~ For~~$\mathrm{\alpha f},:\beta \in \mathrm{ddp}(X,\mathrm{dX})\to (Y,\mathrm{dy})$~~ \alpha,\beta\in~~ f:(X,dX)\to~~ ddp(X)~~ (Y,dy)~~ ,~~ ~~ their~~ is~~ pushout~~ defined to be a pair$\alpha (\otimes f\beta ,\mathrm{df})$~~ \alpha\otimes~~ (f,df)~~ \beta~~ ~~ is~~ making~~ 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 $X$.

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

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.