Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarsegeometry from the foundations, taking a categorical point of view. In thispaper, we concentrate on the discrete case in which topology plays no role. Ourtheory is particularly suited to the development of the_Roe (C*-)algebras_C*(X) and their K-theory on the analytic side; we also hope that it will be ofuse in the strictly geometric/algebraic setting of controlled topology andalgebra. We leave these topics to future papers. Crucial to our approach are nonunital coarse spaces, and what we call_locally proper_ maps (which are actually implicit in [MR1988817]). Our_coarsecategory_ Crs generalizes the usual one: its objects are nonunital coarsespaces and its morphisms (locally proper) coarse maps modulo_closeness_. Crs ismuch richer than the usual unital coarse category. As such, it has all nonzerolimits and all colimits. We examine various other categorical issues. E.g., Crsdoes not have a terminal object, so we substitute a_termination functor_ whichwill be important in the development of exponential objects (i.e., "functionspaces") and also leads to a notion of_quotient coarse spaces_. To connect ourmethods with the standard methods, we also examine the relationship between Crsand the usual coarse category of Roe. Finally we briefly discuss some basic examples and applications. Topicsinclude_metric coarse spaces_,_continuous control_ [MR1277522], metric andcontinuously controlled_coarse simplices_,_sigma-coarse spaces_ [MR2225040],and the relation between quotient coarse spaces and the K-theory of Roealgebras (of particular interest for continuously controlled coarse spaces).
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