Public:Comonads

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UNDER CONSTRUCTION

Different representations

Recall that a comonad on a category \(\mathcal{A}\) is a triple \((C,\epsilon,\delta)\) such that

1. \(C\colon \mathcal{A} \to \mathcal{A}\) is a functor,
2. \(\epsilon\colon C \to \mathrm{Id}_\mathcal{A}\) is a natural transformation,
3. \(\delta\colon C \to C^2\) is a natural transformation, and
4. the following diagrams commute

\[ \begin{array}{rcl} C & \overset{\delta}\rightarrow & C^2 \\ {}_\delta\downarrow & & \downarrow {}_{\delta_C} \\ C^2 & \overset{C\delta}\rightarrow & C^3 \end{array} \qquad \qquad \begin{array}{rrl} C & \overset{\delta}\rightarrow & C^2 \\ & {}_{\id}\searrow & \downarrow {}_{\epsilon_C} \\ & & C \end{array} \qquad \qquad \begin{array}{rrl} C & \overset{\delta}\rightarrow & C^2 \\ & {}_{\mathrm{id}}\searrow & \downarrow {}_{C\epsilon} \\ & & C \end{array} \]

Equivalently, it can be represented as a Kleisli-Manes triple \((C, \epsilon, \overline{(-)})\) such that

1. \(C\) is a mapping \(\mathrm{obj}(\mathcal{A}) \to \mathrm{obj}(\mathcal{A})\)
2. \(\epsilon\) is a collection of morphisms \(\epsilon_A\colon C(A) \to A\), for every object \(A\in \mathcal A\)
3. for every morphism \(f\colon C(A)\to B\) in \(\mathcal A\) we have a morphism \(\overline f\colon C(A)\to C(B)\) such that
   • (C1) \(\ \overline{\epsilon_A} = \id_{C(A)}\) for all objects \(A\in \mathcal{A}\)
   • (C2) \(\ \overline f \fcmp \epsilon_B = f\) for all morphisms \(f\colon C(A) \to B\)
   • (C3) \(\ \overline{\overline f \fcmp g} = \overline f \fcmp \overline g\) for all morphisms \(f\colon C(A) \to B\) and \(g\colon C(B) \to E\)

Here \(f \fcmp g\) is the diagramatic composition of arrows, equivalent to the more standard \(g \circ f\).

To transfer from the standard representation to the Kleisli-Manes we assign:

• \((C,\epsilon,\delta) \mapsto (C, \epsilon, \overline{(-)}) \) where \(\overline f = \delta_A \fcmp C(f)\) for every \(f\colon C(A) \to B\)

And conversely:

• \((C, \epsilon, \overline{(-)}) \mapsto (C,\epsilon,\delta)\) where, for a morphism \(f\colon A \to B\), the morphism \(C(f)\colon C(A) \to C(B)\) is \(\overline{\epsilon_A \fcmp f}\) and \(\delta_A\) is equal to \(\overline{\id_{C(A)}}\).

In fact, this is a bijective correspondence, cf. Proposition 1.6 in [Moggi91 ]

Relative comonads

TODO

Comonad morphism

Given comonads \(C\) and \(D\) on categories \(\mathcal A\) and \(\mathcal B\), respectively, a comonad morphism \((F,\lambda)\colon D \to C\) is given by a functor \(F\colon \mathcal B\to \mathcal A\) and a natural transformation \(\lambda\colon CF \to FD\) such that

\[ \begin{array}{rcl} CF & \stackrel{\lambda}{\to} & FD \\ & {}_{\epsilon^C_F}\searrow & \downarrow_{F\epsilon^D} \\ & & F \end{array} \qquad \qquad \begin{array}{rcl} CF & --- & -\stackrel{\lambda}{--} & \to & FD \\ {}_{\delta^C_F}\downarrow & & & & \downarrow {}_{F\delta^D} \\ C^2F & \stackrel{C(\lambda)}{\to} & CFD & \stackrel{\lambda_D}{\to} & FD^2 \end{array} \]

In particular, if the two comonads \(C,D\) are on the same category \(\mathcal A\) a comonad morphism \(\lambda\colon C\to D\) is required to make the two following diagrams commute \[ \begin{array}{rcl} C & \stackrel{\lambda}{\to} & D \\ & {}_{\epsilon^C}\searrow & \downarrow_{\epsilon^D} \\ & & \Id \end{array} \qquad \qquad \begin{array}{rcl} C & --- & -\stackrel{\lambda}{--} & \to & D \\ {}_{\delta^C}\downarrow & & & & \downarrow {}_{\delta^D} \\ C^2 & \stackrel{C(\lambda)}{\to} & C\circ D & \stackrel{\lambda_D}{\to} & D^2 \end{array} \]

Given two morphisms \((F,\lambda),(F',\lambda')\colon D\to C\) of comonads. A morphism of comonad morphisms \(\gamma\colon (F,\lambda) \Rightarrow (F',\lambda')\) is given by a natural transformation \(\gamma\colon F\to F'\) such that \[ \begin{array}{rcl} CF & \stackrel{C\gamma}{\to} & CF'\\ {}_{\lambda}\downarrow & & {}_{\lambda'}\downarrow \\ FD & \stackrel{\gamma_D}{\to} & F'D \end{array} \]

Let's assume we have a functor \(F\colon \mathcal B\to \mathcal A\) and comonads \(C\) and \(D\) on \(\mathcal A\) and \(\mathcal B\), respectively. Then,

1. comonad morphisms \((F,\lambda)\colon D\to C\) are in a bijective correspondence with functors \(\widehat F\colon EM(D) \to EM(C)\) such that \[ \array{ EM(D) & \stackrel{\widehat{F}}{\to} & EM(C) \\ \downarrow & & \downarrow \\ \mathcal B & \stackrel{F}{\to} & \mathcal A } \]
2. If \(EM(D)\) has equalisers of reflexive pairs and \(F\) has a right adjoint, then any lifting \(\widehat F\) has a right adjoint.
3. If \(T\) has a left adjoint and \(\lambda\colon D\to C\) is invertible, then the lifting \(\widehat T\) associated with \(\lambda\) has a left adjoint.

For details see Wisbauer's "Algebras versus Coalgebras" listed below.

References

• Robert Wisbauer. Algebras versus coalgebras, Applied Categorical Structures 16.1 (2008): 255-295.
• Gabriella Böhm, Tomasz Brzeziński, Robert Wisbauer. Monads and comonads on module categories. Journal of Algebra 322.5 (2009): 1719-1747.
• Aura Bârdeş. Bimonads in a 2-category.

TODO add more

Mentioned above: