Monads of sets

Publisher Summary This chapter focuses on monads of sets—monads in the category S of sets and (total) functions. Monads were first surfaced to codify resolutions for sheaf cohomology. They are established as a standard concept of category theory. Monad research is shaped by applications to functional programming languages. The larger study of monads, in a general category, puts the theory of monads of sets in proper perspective. A balanced account of monads of sets in its current incipient stage, with the hope of stimulating further research is presented. A category-theoretic foundation based on monads for the cohomology of an object in a category is focused. There is a general theory of monadic completion which provides a best monadic approximation for a large class of set-valued functors. There is also an existence theorem to the effect that, under reasonable conditions, every forgetful functor between monadic has a left adjoint.

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