nLab adjoint monad




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


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

Thus there is a bijective correspondence between monads having a right adjoint and comonads having a left adjoint (what Alexander Rosenberg calls duality). This is a little more than a consequence of two general facts:

  1. If TGT\dashv G then T kG kT^k \dashv G^k for every natural number kk.

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

A(S,) B(,T) A(ϕ,) B(,ψ) A(S,) B(,T) \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 ϕψ\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 η,η\eta,\eta' and ϵ,ϵ\epsilon,\epsilon' are their unit and counit of course the upper arrow is (SMfN)Tfη M(S M\stackrel{f}\to N)\mapsto T f\circ \eta_M and the lower arrow (SMgN)Tgη M(S'M\stackrel{g}\to N)\mapsto T'g\circ\eta'_M. Thus the condition renders as

T(fϕ M)η M=ψ NTfη MT'(f\circ\phi_M)\circ\eta'_M = \psi_N\circ T f\circ\eta_M

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

TηTTSTTϕTTSTTϵT 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

SSηSTSSψSSTSϵSS 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 μ T\mu^T to δ G\delta^G, and similarly unit to the counit.



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

(See also at adjoint modality.)




Given an adjoint pair of a monad and comonad

\lozenge \,\dashv\, \Box

on some category 𝒞\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 𝒞\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)End(A)

Given a small category AA, the endofunctor category End(A)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 TGT\dashv G, operations T nTT^n\to T dualize to cooperations GG nG\to G^n, and more generally multioperations T kT lT^k\to T^l dualize to the multioperations G lG kG^l\to G^k. We would like to sketch the proof that the identities for operations on TT, correspond to the identities on cooperations on GG (and more generally there is a duality among the identities for multioperations). This is essentially the consequence of

Lemma. (Zoran) Given the adjunction TGT\dashv G with unit η\eta and counit ϵ\epsilon, and the sequence

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

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

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

equal to the dual (βα) *(\beta\circ\alpha)^* of βα\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 sη lG sG lT lG sG lβG sG lT sG sG lϵ sG lη kG lG kT kG lG kαG lG kT lG lG kϵ lG k 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 sη kG lG kT kG sG kαG lG kT lG sG kβG sG kT sG sG kϵ sG k. 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 α\alpha and those involving β\beta. But anyway, their equality reduces to a naturality calculation (which in particular exchanges the order of α\alpha and β\beta in effect):

G s η lG s G lT lG s G lβG s G lT sG s G lϵ s G l η kG s η kG lT lG s η kG lT sG s η kG l G kT kG s G kT kη lG s G kT kG lT lG s G kT kG lβG s G kT kG lT sG s G kT kG lϵ s G kT kG l G kαG s G kαG lT lG s G kαG kT sG s G kαG l G kT lG s G kT lη lG s G kT lG lT lG s G kT lG lβG s G kT lG lT sG s G kT lG lϵ s G kT lG l G kβG s ρ G kϵ lT sG s G kϵ l G kT sG s = G kT sG l = G kT sG s G kϵ s G k\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 ρ:=G kβG sG kϵ lG s=G kϵ lT sG sG kT lG lβG s\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 lG lT^l\dashv G^l. Namely,

G kβG s=G kβG s(G kϵ lT lG sG kT lη lG s)=ρG kT lη lG s 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


Adjoint modalities

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


Discussion of adjoint monads originates with

(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:

Discussion in the context of ambidextrous adjunctions and Frobenius monads:

Last revised on August 10, 2023 at 17:53:40. See the history of this page for a list of all contributions to it.