We show that for a commutative quantale V every functor Set --> V-cat has an enriched left- Kan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V-cat. Moreover, one can build functors on V-cat by equipping Set-functors with a metric.
A. Balan, A. Kurz, and J. Velebil, “Extensions of Functors From Set to V-cat,” Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany, 2015. doi: 10.4230/LIPIcs.CALCO.2015.17
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