There are situations where standard completion constructions are insufficient to accurately describe the relationship between two categories. The motivating example in the paper "Relative completions" (JPAA 192, 2004) was the presentation of the Effective topos as an exact completion of the category of partitioned assemblies. This presentation relies on the axiom of choice in Sets, and therefore does not work when we work over an arbitrary base topos. The solution is to define a relative version of the exact completion which preserves quotients of equivalence relations coming from the base topos. More precisely, it is defined by first freely adding all quotients, but then formally inverting the canonical comparison morphisms between the new quotients and the old ones from the base. Best regards, Pieter -----Original Message----- From: categories@mta.ca on behalf of Andree Ehresmann Sent: Mon 7/6/2009 4:08 AM To: Categories Subject: categories: Non-free cocompletions Vaughan Pratt writes

Incidentally, of what use are non-free cocompletions? Is there any reason not to define "cocompletion" to make it free?

I can indicate two important uses of non-free cocompletions, and more precisely cocompletions for particular classes of diagrams preserving some given colimits: 1. The construction of what, with Charles, we called the "prototype" and the "type" associated to a sketch (in "Categories of sketchd structures", Cahiers Top. et Geom. Diff. III-2, 1972) 2. The "complexification process" which, with Jean-Paul Vanbremeersch, we use extensively in our model for hierarchical evolutionary systems ("Memory Evolutive Systems: Hierarchy, Emergence, Cognition", Elsevier 2007) Kindly Andree [For admin and other information see: http://www.mta.ca/~cat-dist/ ]