Eilenberg-Moore comparison functor ($K$)

For $F⊣G:\mathcal{D}\to \mathcal{C}$ with induced monad $T=GF$, the Eilenberg-Moore comparison functor is

If each standard counit presentation is a coequalizer, the Eilenberg-Moore comparison functor is full and faithful. An monad algebra morphism $\alpha :GB\to GC$ makes ${\epsilon }_{C}F\alpha$ equalize $FG{\epsilon }_B$ and ${\epsilon }_{FGB}$; its unique descent is the required morphism $B\to C$. The triangle identities for an adjunction make $G{\epsilon }_B$ a split epimorphism, so descent has underlying arrow exactly $\alpha$. Epimorphic cancellation at ${\epsilon }_B$ proves faithfulness.

For an adjunction $F⊣G$, applying $G$ to its standard counit presentation gives $T^{2}GD\rightrightarrows TGD\to GD$, split by the monad units. Reflection therefore makes every counit presentation a coequalizer and gives full faithfulness by descent. Conversely, monad algebra forgetful functor creates split coequalizers makes an existing G-split quotient a coequalizer in the Eilenberg-Moore category. Fullness and faithfulness of the comparison lift its universal factorization back to the original category.

The free-forgetful adjunction for the Eilenberg-Moore category is terminal in the category of adjunctions inducing a fixed monad. The unique morphism into it is the Eilenberg-Moore comparison functor $B\mapsto (GB,G{\epsilon }_B)$. Its underlying objects and arrows are forced by the forgetful functor, and counit compatibility forces its algebra actions.

If $F⊣G:\mathcal{D}\to \mathcal{C}$ induces $T=GF$ and $\mathcal{D}$ has coequalizers of reflexive pairs, then the comparison $K:\mathcal{D}\to {\mathcal{C}}^T$ has a left adjoint. On a $T$-algebra $(A,a)$ it is the coequalizerThe pair is reflexive through $F{\eta }_A$, and its universal property gives $L⊣K$.

For $F⊣G$ inducing $T=GF$, the Eilenberg-Moore comparison functor has a left adjoint exactly when every algebra $(A,a)$ has a coequalizer $q_a:FA\to L(A,a)$ of $Fa,{\epsilon }_{FA}:FTA\rightrightarrows FA$. Transposition identifies maps equalizing this pair with monad algebra morphisms $(A,a)\to K(D)$, proving both directions of the criterion. The unit is $\lambda =G{q}_a{\eta }_A$ and satisfies $G{q}_a=\lambda a$. In the base category, $a:TA\to A$ is a split coequalizer of $Ta,{\mu }_A:T^{2}A\rightrightarrows TA$, split by ${\eta }_A$ and ${\eta }_{TA}$. Consequently $\lambda$ is invertible exactly when $G{q}_a$ is also a coequalizer. The unit is invertible everywhere precisely when $G$ preserves all these specified coequalizers; preservation of arbitrary coequalizers is not required.

When each comparison functor has a left adjoint, iterating the Eilenberg-Moore comparison functor produces the monadic tower. Its monadic length is the least number of comparison steps needed to reach an equivalence; an equivalence has length zero and a non-equivalence that is already monadic has length one.

For the nested partial unary operation category tower, each one-step forgetful functor is a monadic adjunction right adjoint: it reflects isomorphisms and creates its split coequalizers. The Beck monadicity theorem identifies the first Eilenberg-Moore category with the next category in the tower. A long composite has that same first monad because free extension of nested partial unary operations leaves no new top fixed points. Repeating the comparison forgets one fewer operation at each step. The remaining forgetful functor is not full until no operation remains to forget, giving monadic length $m-n$.