Geodesic
The shape of a category up to directed homotopy
Theory and applications of categories, CT2004, Tome 15 (2005), pp. 95-146.

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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.
Classification : 55Pxx, 18A40, 68Q85
Keywords: omotopy theory, adjunctions, reflective subcategories, directed algebraic topology, fundamental category, concurrent processes 
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Marco Grandis. The shape of a category up to directed homotopy. Theory and applications of categories, CT2004, Tome 15 (2005), pp. 95-146. http://geodesic.mathdoc.fr/item/TAC_2005_15_a3/