Spahn
directed object' (Rev #5, 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 *0I1 I *\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 functor D:CVD:C\to V, X[I,X]X\mapsto [I,X].

A direction in CC is defined to be a subfunctor of D d D d . In of this case d DX d D X is preserving called the a terminal object and satisfying the properties below. In this casedirection for XXdXd X . A is global called element a ofdXdXdirection for XX . is A called global a element ofdd-directed path in XXdXdX . is called add-directed path in XX and their collection we denote by ddp(X)ddp(X) the properties are:

(1) The collection of d D d D -directed path image in of every mapI*X I\to *\to X satisfies factoring over the following point properties: is inddp(X)ddp(X).

A (2)direction for XXddp(X)ddp(X) is defined closed to under be the a tensor subobject product. For dX α,βddp(X) dX \alpha,\beta\in ddp(X) , of their pushout[αI,βX] [I,X] \alpha\otimes \beta whose is collection called theirddp(X)ddp(X)composition . of global elements, calleddirected paths (or more precisely dd-directed paths of XX), satisfies

(1) A TheDDdirected object image is of defined every to map be a pairI d*X:=(X,dX) I\to {}_d *\to X:=(X,dX) X factoring consisting over of the an point object is inddp(X) ddp(X) X of CC and a direction dXdX for XX.

(2) Addp(X)ddp(X)morphism of directed objects is closed under the tensor product. For α f , :βddp(X,dX)(Y,dy) \alpha,\beta\in f:(X,dX)\to ddp(X) (Y,dy) , their is pushout defined to be a pairα(fβ,df) \alpha\otimes (f,df) \beta is making called theircomposition.

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.

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 23:15:39 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.