Spahn directed object' (Rev #5, changes)

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 $*\stackrel{0}{\to }I\stackrel{1}{\leftarrow }$ 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$\mathrm{<span><del\; class="diffmod">\; D</del><ins\; class="diffmod">\; d</ins></span>}$ D d . In of this case$\mathrm{<span><del\; class="diffmod">\; d</del><ins\; class="diffmod">\; D</ins></span>}X$ 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$\mathrm{<span><del\; class="diffmod">\; d</del><ins\; class="diffmod">\; D</ins></span>}$ 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)direction for $X$$ddp(X)$ is defined closed to under be the a tensor subobject product. For$\mathrm{<span><del\; class="diffmod">\; dX</del><ins\; class="diffmod">\; \alpha </ins></span>},\beta \in \mathrm{ddp}(X)$ 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, 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$I{}_d\to *\to X:=(X,\mathrm{dX})$ 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$\mathrm{<span><del\; class="diffmod">\; \alpha </del><ins\; class="diffmod">\; f</ins></span>}<span><del\; class="diffmod">\; ,</del><ins\; class="diffmod">\; :</ins></span>\beta \in \mathrm{ddp}(X,\mathrm{dX})\to (Y,\mathrm{dy})$ \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 theircomposition.

Definition

A directed-path-space objects is defined.