Spahn
directed object' (Rev #4, 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 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 II denote the interval object of CC, let XX be a pointed object of CC. Let DD be the functor D:CVD:C\to V , X\mapsto [I,X]$.X[I,X]X\mapsto [I,X].

A direction in CC is defined to be a subfunctor of DD. In this case dXd X is called a direction for XX. A global element of dXdX is called a dd-directed path in XX.

The collection of dd-directed path in XX satisfies the following properties:

A direction for XX is defined to be a subobject dXdX of [I,X][I,X] whose collection ddp(X)ddp(X) of global elements, called directed paths (or more precisely dd-directed paths of XX), satisfies

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

(2) ddp(X)ddp(X) is closed under the tensor product. For α,βddp(X)\alpha,\beta\in ddp(X), their pushout αβ\alpha\otimes \beta is called their composition.

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 20:36:40 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.