Spahn
directed object' (Rev #2, 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 , which we assume to be the terminal object, letII denote the interval object of CC, let XX be a pointed object of CC . LetDD be the functor D:CVD:C\to V, X\mapsto [I,X]$.

A direction for in X C X C is defined to be a subobject subfunctor of dX D dX D . of In this case[dI,X] [I,X] d X satisfying is called adirection for XX. A global element of dXdX is called a dd-directed path in XX.

(1) The every collection map of I d*X I\to d *\to X -directed factoring path over in the point is a (generalized) element ofdX d X . satisfies the following properties:

(2) AdXd Xdirection for XX is closed defined under to pushout. be a subobjectdXdX of [I,X][I,X] whose collection ddp(X)ddp(X) of global elements, called directed paths (or more precisely d*directedpathsofd*-directed paths of X$), satisfies

A (1) global The (or just generalized) element of d DX d D X is image called of every mapdirected path in XXI*XI\to *\to X factoring over the point is in ddp(X)ddp(X).

A (2) The collection of global elements ofdirected objectdXd X is defined closed to under be pushout. a For pair dαX:=(X,dX) {}_d \alpha X:=(X,dX) , consisting of an object X β X \beta global elements of C dX C d X , and their a pushout direction dX αβ dX \alpha\otimes \beta for is called theirXXcomposition.

A morphism of directed objects object is defined to be a pairf dX:=(X,dX)(Y,dy) f:(X,dX)\to {}_d (Y,dy) X:=(X,dX) is consisting defined of to an be object a pair(Xf,df) (f,df) X making ofCC and a direction dXdX for XX.

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

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 18:40:37 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.