# Spahn directed object' (Rev #3, changes)

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###### 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 co-span for this notation).

###### Definition (directed object)

Let  C V be a closed monoidal (V-enriched category, for let some V C ) homotopical be category, let * V -enriched, denote closed the monoidal tensor homotopical unit category, of let C * which denote we assume to be the terminal tensor object, unit let of I C denote which we assume to be the interval terminal object object, of let C I , let denote the interval object of X C , be let a pointed object of C X . Let be a pointed object of D C . be Let the functor 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*-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 d ddp(X) X is closed under pushout. the tensor product. For \alpha \alpha,\beta\in ddp(X) , their pushout \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

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

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

###### Definition

A directed-path-space objects is defined.

Revision on November 5, 2012 at 19:44:06 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.