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Monads in double categories. (English) Zbl 1225.18003

A double category is an internal category in the category of all categories. By formulating the formal theory of monads in the context of double categories, some naturally occuring functors of the theory become monad maps; something which is not true when the formal theory of monads is formulated relative to the 2-category of categories. In this paper the formal theory of monads relative to a double category is developed and a sufficient condition is given for when free monads can be constructed.
The paper is slightly hampered by the need to deploy quite a bit of categorical machinery to make its points, but it is nonetheless very clearly written and provides an accessible and interesting account of the topics under discussion. The abstract and introduction are well enough written to allow the reader to understand quickly, and in more detail, what the topics are that the paper is trying to cover.

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
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