This work is a contribution to a recent field, Directed
Algebraic Topology. Categories which appear as fundamental
categories of `directed structures', e.g. ordered topological
spaces, have to be studied up to appropriate notions of *directed
homotopy equivalence*, which are more general than ordinary equivalence
of
categories. Here we introduce * past* and * future equivalences
* of
categories - sort of symmetric versions of an adjunction - and use them
and
their combinations to get `directed models' of a category; in the simplest
case, these are the join of the *least full reflective* and the
*least full coreflective* subcategory.

Keywords: omotopy theory, adjunctions, reflective subcategories, directed algebraic topology, fundamental category, concurrent processes

2000 MSC: 55Pxx, 18A40, 68Q85

*Theory and Applications of Categories,*
Vol. 15, CT2004,
No. 4, pp 95-146.

http://www.tac.mta.ca/tac/volumes/15/4/15-04.dvi

http://www.tac.mta.ca/tac/volumes/15/4/15-04.ps

http://www.tac.mta.ca/tac/volumes/15/4/15-04.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/15/4/15-04.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/15/4/15-04.ps