Spahn
directed object' (Rev #1, changes)

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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)

Let CC be a closed monoidal (V-enriched for some VV) homotopical category, let ** denote the tensor unit of CC, let II denote the interval object of CC, let XX be a pointed object of CC.

A direction for XX is defined to be a subobject dXdX of [I,X][I,X] satisfying

(1) every map I*XI\to *\to X factoring over the point is a (generalized) element of dXd X.

(2) dXd X is closed under pushout.

A global (or just generalized) element of dXd X is called directed path in XX

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

X f Y d d dX df dY\array{ X& \stackrel{f}{\to}& Y \\ \downarrow^d&&\downarrow^d \\ dX& \stackrel{df}{\to}& d Y }
Remark

Let CC be VV-enriched, closed monoidal homotopical category, let II denote the tensor unit of CC.

Definition

A directed-path-space objects is defined.

Revision on November 5, 2012 at 17:21:46 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.