Comonads and Their Importance
The categorical dual definition is a formal definition of a comonad (or cotriple); this can be said quickly in the terms that a comonad for a category is a monad for the opposite category . It is therefore a functor from to itself, with a set of axioms for counit and comultiplication that come from reversing the arrows everywhere in the definition just given.
Since a comonoid is not a basic structure in abstract algebra, this is less familiar at an immediate level.
The importance of the definition comes in a class of theorems from the categorical (and algebraic geometry) theory of descent. What was realised in the period 1960 to 1970 is that recognising the categories of coalgebras for a comonad was an important tool of category theory (particularly topos theory). The results involved are based on Beck's theorem. Roughly what goes on is this: while it is simple set theory that a surjective mapping of sets is as good as the imposition of the equivalence relation 'in the same fiber', for geometric morphisms what you should do is pass to such a coalgebra subcategory.
Read more about this topic: Monad (category Theory)
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