Spahn directed object' (Rev #6)

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 $*\stackrel{0}{\to} I\stackrel{1}{\leftarrow}$ denote the interval object of $C$, let $X$ be a pointed object of $C$. Let $D$ be the bifunctor $D:C\times C\to V$, $(X,I)\mapsto [I,X]$.

A direction in $C$ is defined to be a subfunctor $d$ of $D(I,-)$ preserving the terminal object and satisfying the properties below. In this case $d X$ is called a direction for $X$. A global element of $dX$ is called a $d$-directed path in $X$ and their collection we denote by $ddp(X)$. The properties are:

(1) The $D$ image of every map $I\to *\to X$ factoring over the point is in $ddp(X)$.

(2) $f,g\in dX$ are called to be a composable pair, if $\overline f\circ 0=\overline g\circ 1$ where the overlined letters denote the adjoints under the hom adjunction. The composition of a composable pair is defined by $f\circ g:=\underline{\overline{f}\bullet\overline{g}}$. Then the condition is that $ddp(X)$ is closed under composition.

(3) If $\tau\in hom (I,I)$ and $f\in ddp(X)$ then $\underline{\overline{f}\circ \tau}\in dX$ where the underlined term denotes the adjoint (in the other direction) under the hom adjunction.

(4) $ddp(X)$ is a functor

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