Spahn directed object' (Rev #2, 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 $C$ 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$\mathrm{<span><del\; class="diffmod">\; X</del><ins\; class="diffmod">\; C</ins></span>}$ X C is defined to be a subobject subfunctor of$\mathrm{<span><del\; class="diffmod">\; dX</del><ins\; class="diffmod">\; D</ins></span>}$ dX D . of In this case$[dI,X]$ [I,X] d X satisfying is called adirection for $X$. A global element of $dX$ is called a $d$-directed path in $X$.

(1) The every collection map of$\mathrm{<span><del\; class="diffmod">\; I</del><ins\; class="diffmod">\; d</ins></span>}\to *\to X$ 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$\mathrm{<span><del\; class="diffmod">\; d</del><ins\; class="diffmod">\; D</ins></span>}X$ d D X is image called of every mapdirected path in $X$$I\to *\to X$ factoring over the point is in $ddp(X)$.

A (2) The collection of global elements ofdirected object$d X$ is defined closed to under be pushout. a For pair${}_d\alpha X:=(X,\mathrm{dX})$ {}_d \alpha X:=(X,dX) , consisting of an object$\mathrm{<span><del\; class="diffmod">\; X</del><ins\; class="diffmod">\; \beta </ins></span>}$ X \beta global elements of$\mathrm{<span><del\; class="diffmod">\; C</del><ins\; class="diffmod">\; d</ins></span>}X$ C d X , and their a pushout direction$\mathrm{<span><del\; class="diffmod">\; dX</del><ins\; class="diffmod">\; \alpha </ins></span>}\otimes \beta$ dX \alpha\otimes \beta for is called their$X$composition.

A morphism of directed objects object is defined to be a pair$f{}_{d}X:=(X,\mathrm{dX})\to (Y,\mathrm{dy})$ f:(X,dX)\to {}_d (Y,dy) X:=(X,dX) is consisting defined of to an be object a pair$(Xf,\mathrm{df})$ (f,df) X making of$C$ and a direction $dX$ for $X$.

Definition

A directed-path-space objects is defined.