• The
forgetful functor from topological spaces to sets is not monadic as it does not reflect isomorphisms: continuous bijections between (non-compact or non-Hausdorff) topological spaces need not be homeomorphisms. • shows that the functor from commutative
C*-algebras to sets sending such an algebra
A to the
unit ball, i.e., the set \{a \in A, \|a\| \le 1 \}, is monadic. Negrepontis also deduces
Gelfand duality, i.e., the equivalence of categories between the opposite category of compact Hausdorff spaces and commutative C*-algebras can be deduced from this. • The powerset functor from Setop to Set is monadic, where Set is the
category of sets. More generally Beck's theorem can be used to show that the powerset functor from Top to T is monadic for any topos T, which in turn is used to show that the topos T has finite colimits. • The forgetful functor from
semigroups to sets is monadic. This functor does not preserve arbitrary coequalizers, showing that some restriction on the coequalizers in Beck's theorem is necessary if one wants to have conditions that are necessary and sufficient. • If
B is a faithfully flat
commutative ring over the commutative ring
A, then the functor
T from
A modules to
B modules taking
M to
B⊗
AM is comonadic. This follows from the dual of Beck's theorem, as the condition that
B is flat implies that
T preserves finite limits, while the condition that
B is faithfully flat implies that
T reflects isomorphisms. A coalgebra over
T turns out to be essentially a
B-module with descent data, so the fact that
T is comonadic is equivalent to the main theorem of faithfully flat descent, saying that
B-modules with descent are equivalent to
A-modules. ==External links==