# Spahn directed object' (Rev #5, 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 $V$ 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 *\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 D d . In of this case d D X is preserving called the a terminal object and satisfying the properties below. In this casedirection for $X$$d X$ . A is global called element a of$dX$direction for $X$ . is A called global a element of$d$-directed path in $X$$dX$ . is called a$d$-directed path in $X$ and their collection we denote by $ddp(X)$ the properties are:

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

A (2)direction for $X$$ddp(X)$ is defined closed to under be the a tensor subobject product. For dX \alpha,\beta\in ddp(X) , of their pushout [I,X] \alpha\otimes \beta whose is collection called their$ddp(X)$composition . of global elements, calleddirected 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\to {}_d *\to X:=(X,dX) X factoring consisting over of the an point object is in 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 \alpha,\beta\in f:(X,dX)\to ddp(X) (Y,dy) , their is pushout defined to be a pair \alpha\otimes (f,df) \beta is making called theircomposition.

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

$\array{ X& \stackrel{f}{\to}& Y \\ \downarrow^d&&\downarrow^d \\ dX& \stackrel{df}{\to}& d Y }$
###### 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 23:15:39 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.