nLab relation between adjunctions and monads -- section

Relation between adjunctions and monads

There is a close relation between adjunctions (adjoint functors) and monads:

• Monad induced by an adjunction

• Category of adjunction-resolutions of a monad

• Semantics-Structure adjunction

Monad induced by an adjunction

Every adjunction $(L \dashv R)$ induces a monad $R \circ L$ and a comonad $L \circ R$.

(Huber 1961, §4; see eg. MacLane 1971, §VI.1 (p 134); Borceux 1994, vol. 2, prop. 4.2.1).

In detail:

Proposition

Let $(\mathcal{C},\mathcal{D},F,U,\eta,\epsilon)$ be a pair of adjoint functors ie $F \dashv U$ are adjoint functors where $F \colon \mathcal{C} \rightarrow \mathcal{D}$, $U \colon\mathcal{D} \rightarrow \mathcal{C}$, $\eta_{A}:A \rightarrow U(F(A))$ is the unit and $\epsilon_{B} \colon F(U(B)) \rightarrow B$ is the counit. Then:

• $U \circ F$ is a monad on $\mathcal{C}$, with unit $\eta$ and multiplication $U(\epsilon_{F(A)}):U(F(U(F(A)))) \rightarrow U(F(A))$.

• $F \circ U$ is a comonad on $\mathcal{D}$, with counit $\epsilon$ and comultiplication $F(\eta_{U(B)}):F(U(B)) \rightarrow F(U(F(U(B))))$.

Proof

We verify that we obtain a monad, the argument for the comonad is formally dual.

(1) We know that this diagram commutes:

By applying $U$, we obtain the first part of the unitaly of the monad:

(2) We know that this diagram commutes: By putting $B=F(A)$, we obtain the second part of the unitality of the monad:

(3) The naturality of $\epsilon$ is written, for every $f:B \rightarrow B'$:

We apply it to $f=\epsilon_{F(A)}$ and it gives:

Finally apply $U$ to this diagram to obtain:

which is exactly the associativity of the multiplication of the monad.

(4) The naturality of the multiplication $U(\epsilon_{F(A)}):U(F(U(F(A)))) \rightarrow U(F(A))$ is obtained by two whiskerings of the counit $\epsilon_{B}:U(F(B)) \rightarrow B$.

Category of adjunction-resolutions of a monad

An adjunction inducing a monad $T$ (as above) is also called a resolution of $T$.

There is in general more than one such resolution, in fact there is a category of adjunctions for a given monad whose morphisms are “comparison functors” (eg. MacLane 1971, §VI.3).

In this category:

• the initial object is the adjunction over the Kleisli category of the monad;

• the terminal object is that over the Eilenberg-Moore category of algebras, also called the monadic adjunction (recognized by monadicity theorems).

(e.g. Borceux 1994, vol. 2, prop. 4.2.2)

Semantics-structure adjunction

The above passage from adjunctions to monads and back to their monadic adjunctions constitutes itself an adjunction, sometimes called the semantics-structure adjunction.