The General Theory of Diads |
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Authors: | Toby Kenney |
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Institution: | 1.Dalhousie University,Halifax,Canada |
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Abstract: | A diad is a generalisation of a monad and a comonad. The idea is that we ignore the unit or counit, and consider only the
natural transformations between T and T
2. It turns out that almost all the constructions that we form for a monad or comonad can also be constructed from a related
diad. Diads were introduced in Kenney (Appl. Categ. Structures, 2008), where they give a generalisation of the results that the category of coalgebras for a finite-limit preserving comonad on
a topos is another topos, and that the category of algebras for a finite-limit preserving idempotent monad on a topos is another
topos. In that paper, we were only interested in a special class of diads called codistributive diads, and we considered only
the part of the theory of diads necessary to prove the result about finite-limit preserving diads in topoi. Here, we will
study general diads in greater detail. We will develop the general theory with constructions that extend the standard constructions
for monads and comonads. |
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Keywords: | |
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