caa a22 4500 445378697 CHVBK 20180317143011.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s10485-009-9202-7 doi (NATIONALLICENCE)springer-10.1007/s10485-009-9202-7 Power John Department of Computer Science, University of Bath, BA2 7AY, Bath, UK aut Unicity of Enrichment over Cat or Gpd [Elektronische Daten] [John Power] Enrichment of ordinary monads over Cat or Gpd is fundamental to Max Kelly's unified theory of coherence for categories with structure. So here, we investigate existence and unicity of enrichments of ordinary functors, natural transformations, and hence also monads, over Cat and Gpd. We show that every ordinary natural transformation between 2-functors whose domain 2-category has either tensors or cotensors with the arrow category is 2-natural. We use that to prove that an ordinary monad, or endofunctor, on such a 2-category has at most one enrichment over Cat or Gpd. We also describe a monad on Cat that has no enrichment. So enrichment over Cat is a non-trivial property of a monad rather than a structure that is additional to it. Finally, we present an example, due to Kelly, of V other than Cat or Gpd and an ordinary monad for which more than one enrichment over V exists, showing that our main theorem is specific to Cat and Gpd. Springer Science+Business Media B.V., 2009 Enrichment nationallicence Monads nationallicence Tensors nationallicence Cotensors nationallicence Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 293-299 0927-2852 19:1<293 2011 19 10485 https://doi.org/10.1007/s10485-009-9202-7 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10485-009-9202-7 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Power John Department of Computer Science, University of Bath, BA2 7AY, Bath, UK aut NATIONALLICENCE 773 0- Applied Categorical Structures Springer Netherlands 19/1(2011-02-01), 293-299 0927-2852 19:1<293 2011 19 10485 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer