nLab adjoint monad

Definition

A monad $(T,\mu,\eta)$ is left (resp. right) adjoint to a comonad $(G,\delta,\epsilon)$, if its underlying endofunctor $T$ is left adjoint (resp. right adjoint) to the underlying 1-morphism $G$ of the comonad, and $\delta$ and $\epsilon$ are conjugate/adjoint/mate 2-cells to $\mu$ and $\eta$ in the sense explained below.

Construction

In fact given a monad $\mathbf{T} = (T,\mu^T,\eta^T)$ which has a right adjoint $G$, automatically $G$ is a part of a comonad $\mathbf{G} = (G,\delta^G,\epsilon^G)$ where $\delta^G$ and $\epsilon^G$ are in some sense dual to $\mu^T$ and $\eta^T$.

Thus there is a bijective correspondence between monads having a right adjoint and comonads having a left adjoint (and “2-dually” between comonads having a right adjoint and monads having a left adjoint, as explained below). This is little more than a consequence of two general facts:

1. If $T\dashv G$ then $T^k \dashv G^k$ for every natural number $k$.

2. Given two adjunctions $S\dashv T$ and $S'\dashv T'$ where $S,S': B\to A$, then there is a bijection between the natural transformations $\phi:S'\Rightarrow S$ and natural transformations $\psi:T\Rightarrow T'$ such that

$$\array{ A (S,-) &\to& B(-,T) \\ {}^{\mathllap{A(\phi,-)}}\downarrow &&\downarrow {}^{\mathrlap{B(-,\psi)}} \\ A(S',-)&\to & B(-,T') }$$

where the horizontal arrows are the natural bijections given by the adjunctions. Eilenberg & Moore 1965 would write $\phi\dashv\psi$ and talk about “adjointness for morphisms” (of functors), which is of course relative to the given adjunctions among functors. MacLane calls the correspondence conjugation (p 99-102 in Categories for Working Mathematician). It is a special case, of a general construction of mates.

If $\eta,\eta'$ and $\epsilon,\epsilon'$ are their unit and counit of course the upper arrow is $(S M\stackrel{f}\to N)\mapsto T f\circ \eta_M$ and the lower arrow $(S'M\stackrel{g}\to N)\mapsto T'g\circ\eta'_M$. Thus the condition renders as

$$T'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ T f\circ\eta_M$$

or $T'f\circ T'\phi_M\circ\eta'_M = T'f\circ \psi_{SM}\circ\eta_M$. Given $\phi$, the uniqueness of $B(-,\psi)$ is clear from the above diagram, as the horizontal arrows are invertible. $B(-,\psi)$ determines $\psi$, namely $\psi_N = B(-,\psi)(id_N)$. For the existence of $\psi$ (given $\phi$) satisfying the above equation, one proposes that $\psi$ is the composition $\psi = T'\epsilon \circ T'\phi T \circ \eta'T$, i.e.

$$T\stackrel{\eta' T}\longrightarrow T'S' T\stackrel{T'\phi T}\longrightarrow T' S T \stackrel{T'\epsilon}\longrightarrow T'$$

and checks that it works. The inverse is similarly given by the composition

$$S'\stackrel{S'\eta}\longrightarrow S' T S\stackrel{S'\psi S}\longrightarrow S'T'S\stackrel{\epsilon' S}\longrightarrow S$$

This correspondence now enables in our special case to dualize $\mu^T$ to $\delta^G$, and similarly unit to the counit.

Duality

The construction above easily dualises to a bijection between comonads on $\mathcal{C}$ having a right adjoint and monads having a left adjoint. However, 1-categorical duality is not enough to analyse this: in the opposite category $\mathcal{C}^{op}$, monads and comonads are swapped and so are left and right adjoints (see opposite adjunction); thus we still get a correspondence between comonads having a left adjoint and monads having a right adjoint.

Instead, as explained in math.SE:a/3564754, one can generalise the construction above to monads and adjunctions in any 2-category; the special case under consideration here being Cat. The dual construction then follows by instantiating the construction in $\mathbf{Cat}^{op}$, the 1-cell dual of $\mathbf{Cat}$: this swaps left and right adjoints but leaves monads and comonads unchanged.

Examples

Example

(adjoint monads induced from adjoint triples of adjoint functors)
Every adjoint triple (of adjoint functors) $F^*\dashv F_* \dashv F^!$ induces an adjoint pair $F_* F^*\dashv F_* F^!$. The endofunctor $F_* F^*$ is underlying a monad induced by the adjunction $F^*\dashv F_*$ and $F_* F^!$ is underlying a comonad induced by the adjuntion $F_*\dashv F^!$. This pair of a monad and a comonad are adjoint.

(See also at adjoint modality.)

Properties

General

Proposition

Given an adjoint pair of a monad and comonad

$$\lozenge \,\dashv\, \Box$$

on some category $\mathcal{C}$, then there is an equivalence between their Eilenberg-Moore categories of algebras over $\lozenge$ and coalgebras over $\Box$, compatible with their forgetful functors to $\mathcal{C}$:

This is due to Eilenberg & Moore 1965, where it is implied by the last part of Prop. 3.3. In the more explicit form above the statement may be found in MacLane & Moerdijk 1992, Thm. 2 on p. 249,

General adjoint (co)algebras $End(A)$

Given a small category $A$, the endofunctor category $End(A)$ (of endofunctors and natural transformations between them, with vertical composition as composition) is monoidal with respect to the composition as the tensor product of objects (endofunctor) and Godement product (horizontal composition) as the tensor product of morphisms (natural transformations). Hence we can consider operads and algebras over operads, as well as, dually, coalgebras over cooperads; or some other framework for general algebras and coalgebras (or even props).

In any case, given an adjunction $T\dashv G$, operations $T^n\to T$ dualize to cooperations $G\to G^n$, and more generally multioperations $T^k\to T^l$ dualize to the multioperations $G^l\to G^k$. We would like to sketch the proof that the identities for operations on $T$, correspond to the identities on cooperations on $G$ (and more generally there is a duality among the identities for multioperations). This is essentially the consequence of

Lemma. (Zoran) Given the adjunction $T\dashv G$ with unit $\eta$ and counit $\epsilon$, and the sequence

$$T^k \stackrel{\alpha}\longrightarrow T^l\stackrel\beta\longrightarrow T^s$$

the composition $\alpha^*\circ\beta^*$ of the dual (in the above sense) sequence

$$G^k \stackrel{\alpha^*}\longleftarrow G^l\stackrel{\beta^*}\longleftarrow G^s$$

equal to the dual $(\beta\circ\alpha)^*$ of $\beta\circ\alpha$,

Proof. Mike Shulman notices that this is a special case of known contravariant functoriality of mates, but here is a direct proof.

We need to prove that the composition

$$G^s\stackrel{\eta_l G^s}\to G^l T^l G^s\stackrel{G^l\beta G^s}\to G^l T^s G^s\stackrel{G^l \epsilon_s}\to G^l\stackrel{\eta_k G^l}\to G^k T^k G^l\stackrel{G^k\alpha G^l}\to G^k T^l G^l\stackrel{G^k\epsilon_l}\to G^k$$

equals the composition

$$G^s\stackrel{\eta_k G^l}\to G^k T^k G^s\stackrel{G^k\alpha G^l}\to G^k T^l G^s\stackrel{G^k\beta G^s}\to G^k T^s G^s\stackrel{G^k\epsilon_s}\to G^k.$$

Note that in the two compositions there is an opposite order between the expressions involving $\alpha$ and those involving $\beta$. But anyway, their equality reduces to a naturality calculation (which in particular exchanges the order of $\alpha$ and $\beta$ in effect):

$$\array{ G^s &\stackrel{\eta_l G^s}\to & G^l T^l G^s &\stackrel{G^l\beta G^s}\to & G^l T^s G^s&\stackrel{G^l \epsilon_s}\longrightarrow& G^l\\ \eta^k G^s\downarrow &&\downarrow \eta_k G^l T^l G^s&&\downarrow \eta_k G^l T^s G^s&& \downarrow \eta_k G^l \\ G^k T^k G^s &\stackrel{G^k T^k\eta_l G^s}\longrightarrow &G^k T^k G^l T^l G^s &\stackrel{G^k T^k G^l \beta G^s}\longrightarrow & G^k T^k G^l T^s G^s &\stackrel{G^k T^k G^l \epsilon_s}\longrightarrow & G^k T^k G^l\\ G^k \alpha G^s \downarrow && \downarrow G^k \alpha G^l T^l G^s&&\downarrow G^k \alpha G^k T^s G^s&&\downarrow G^k \alpha G^l\\ G^k T^l G^s &\stackrel{G^k T^l\eta_l G^s}\longrightarrow & G^k T^l G^l T^l G^s &\stackrel{G^k T^l G^l \beta G^s}\longrightarrow & G^k T^l G^l T^s G^s &\stackrel{G^k T^l G^l\epsilon_s}\longrightarrow & G^k T^l G^l\\ G^k\beta G^s \downarrow &&\downarrow{\rho} &&\downarrow G^k \epsilon_l T^s G^s &&\downarrow G^k\epsilon_l \\ G^k T^s G^s &=&G^k T^s G^l &=&G^k T^s G^s&\stackrel{G^k \epsilon_s}\longrightarrow& G^k }$$

where $\rho := G^k \beta G^s \circ G^k \epsilon_l G^s = G^k\epsilon_l T^s G^s\circ G^k T^l G^l \beta G^s$. The commutativity of all small squares in the diagram is evident, except the lower left corner. This one follows by one of the triangle identities for the adjunction $T^l\dashv G^l$. Namely,

$$G^k \beta G^s = G^k \beta G^s \circ (G^k \epsilon_l T^l G^s\circ G^k T^l \eta_l G^s) = \rho \circ G^k T^l \eta_l G^s$$

Examples

Adjoint modalities

An adjoint modality is an example of a pair of adjoint monads.

References

Discussion of adjoint monads originates with

• Samuel Eilenberg, John Moore, Section 3 in: Adjoint functors and triples, Illinois J. Math. 9 3 (1965) 381-398 [doi:10.1215/ijm/1256068141]

(there called “adjoint triples”, sticking with the old term “triple” for “monad”, a terminology that now clashes with the modern use of adjoint triples of adjoint functors).

Textbook account:

• Saunders Mac Lane, Ieke Moerdijk, pp. 248 in: Sheaves in Geometry and Logic — A First Introduction to Topos Theory, Springer (1992) [doi:10.1007/978-1-4612-0927-0]

Discussion in the context of ambidextrous adjunctions and Frobenius monads:

• Ross Street, Frobenius monads and pseudomonoids, J. Math. Phys. 45 3930 (2004) [doi:10.1063/1.1788852]

• Aaron Lauda, Frobenius algebras and ambidextrous adjunctions, Theory and Applications of Categories 16 4 (2006) 84-122 [arXiv:math/0502550, tac:16-04]