Research

Unicity of Enrichment over Cat or Gpd


Reference:

Power, J., 2011. Unicity of Enrichment over Cat or Gpd. Applied Categorical Structures, 19 (1), pp. 293-299.

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    Official URL:

    http://dx.doi.org/10.1007/s10485-009-9202-7

    Abstract

    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.

    Details

    Item Type Articles
    CreatorsPower, J.
    DOI10.1007/s10485-009-9202-7
    Uncontrolled Keywordscotensors, enrichment, tensors, monads
    DepartmentsFaculty of Science > Computer Science
    Publisher StatementPower_ACS_2011_19_1_293.pdf: The original publication is available at www.springerlink.com
    RefereedYes
    StatusPublished
    ID Code23104

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