directed object'

Definition (interval object)

Let CC be a closed monoidal (V-enriched for some VV) homotopical category, let ** denote the tensor unit of CC. Then the cospan category (same objects and cospans as morphisms) is VV-enriched, too.

An interval object is defined to be a cospan *aIb**\stackrel{a}{\rightarrow}I\stackrel{b}{\leftarrow}*.

The pushout I 2I^{\coprod_2} of this diagram satisfies *[I,I 2] *{}_*[ I,I^{\coprod_2}]_*\simeq is contractible (see co-span for this notation).

Definition (directed object by subobject)

Let VV be a monoidal category, let CC be VV-enriched, closed monoidal homotopical category, let ** denote the tensor unit of CC which we assume to be the terminal object, let *0I1*\stackrel{0}{\to} I\stackrel{1}{\leftarrow} denote the interval object of CC, let XX be a pointed object of CC. Let DD be the bifunctor D:C×CVD:C\times C\to V, (X,I)[I,X](X,I)\mapsto [I,X].

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

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

(2) f,gdXf,g\in dX are called to be a composable pair, if f¯0=g¯1\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 fg:=f¯g¯̲f\circ g:=\underline{\overline{f}\bullet\overline{g}}. Then the condition is that ddp(X)ddp(X) is closed under composition.

(3) If τhom(I,I)\tau\in hom (I,I) and fddp(X)f\in ddp(X) then f¯τ̲dX\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)ddp(X) is a functor

A directed object is defined to be a pair dX:=(X,dX){}_d X:=(X,dX) consisting of an object XX of CC and a direction dXdX for XX.

A morphism of directed objects f:(X,dX)(Y,dY)f:(X,dX)\to (Y,dY) is defined to be a pair (f,df)(f,df) making

Last revised on November 7, 2012 at 17:19:47. See the history of this page for a list of all contributions to it.