nLab coreader comonad



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Mapping space



Given an object WW in a category π’ž\mathcal{C}, if all binary cartesian products with WW exist, then taking the cartesian product with WW assembles into a comonad

WΓ—(βˆ’):π’žβŸΆπ’ž W \times (-) \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}

whose comultiplication is induced by the diagonal map on WW and whose counit is induced by the terminal map W→*W \to \ast.

When WW is furthermore equipped with the structure of a monoid, then WΓ—(βˆ’)W \times (-) also canonically inherits the structure of a monad. (Regarded as a monad in computer science this models the aggregation of a program’s WW-typed outputs, corresponding to a sort of side channel.) Equipped with this monad-structure, the operation WΓ—(βˆ’)W \times (-) is known as the writer monad, see there for more.

On the other hand, if WW is further exponentiable then WΓ—(βˆ’)W \times (-) is left adjoint to the reader monad Wβ†’βˆ’W \to -, so that it is known as the reader comonad (eg. in the Haskell documentation for Control.Comonad.Reader) or coreader comonad (e.g. Ahman & Uustalu (2019)). It is also known as the product comonad (e.g. Uustalu & Vene 2008, p. 270) and the environment comonad (eg. in the Haskell documentation for Control.Comonad.Env).

Coreader comonads in π’ž op\mathcal{C}^op are exception monads in π’ž\mathcal{C}, because products turn into coproducts.


As an Indexed Comonad

If all binary cartesian products in CC exist, then the coreader comonad extends to a CC-indexed comonad on CC, i.e. a functor Cβ†’ComonadC \to Comonad. If CC has a terminal object 11, then the environment comonad 1Γ—βˆ’1 \times - is isomorphic to the identity comonad.

Generalization to Tensor Products

If CC is a monoidal category rather than having all binary products, then tensoring with an object Ξ“βŠ—βˆ’\Gamma \otimes - can be given an analogous comonad structure when Ξ“\Gamma has a comonoid structure. In a cartesian monoidal category every object has a unique comonoid structure, which is what induces the coreader comonad. Even without assuming objects carry comonoid structures, the action sending Ξ“\Gamma to the functor Ξ“βŠ—βˆ’\Gamma \otimes - defines a strong monoidal functor from CC to the monoidal category of endofunctors [C,C][C,C], a degenerate kind of graded comonad.

Relation to reader monad and state monad

In a cartesian closed category/type theory π’ž\mathcal{C}, the coreader comonad WΓ—(βˆ’)W\times (-) is left adjoint to the reader monad [W,βˆ’][W,-].

The composite of coreader comonad followed by reader monad is the state monad.

In terms of dependent type theory

If the type system is even a locally Cartesian closed category/dependent type theory then for each type WW there is the base change adjoint triple

π’ž /W⟢∏ W⟡W *βŸΆβˆ‘ Wπ’ž \mathcal{C}_{/W} \stackrel{\overset{\sum_W}{\longrightarrow}}{\stackrel{\overset{W^\ast}{\longleftarrow}}{\underset{\prod_W}{\longrightarrow}}} \mathcal{C}

In terms of this then the coreader comonad is equivalently the composite

βˆ‘ WW *=WΓ—(βˆ’):π’žβŸΆπ’ž \sum_W W^\ast = W\times (-) \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}

of context extension followed by dependent sum.

One may also think of this as being the integral transform through the span

*←Wβ†’* \ast \leftarrow W \rightarrow \ast

(with trivial kernel) or as the polynomial functor associated with the span

*←Wβ†’idWβ†’*. \ast \leftarrow W \stackrel{id}{\rightarrow} W \rightarrow \ast \,.


A coalgebra of the coreader comonad WΓ—βˆ’W \times - is equivalent to simply a morphism Aβ†’WA \to W for AA the carrier of the co-algebra. Furthermore, the category of coalgebras is the isomorphic to the slice category over WW. This shows that if all cartesian products with WW exist, then the forgetful functor Ξ£ W:C/Wβ†’C\Sigma_W \colon C / W \to C is comonadic.

(co)monad nameunderlying endofunctor(co)monad structure induced by
reader monadW→(-)W \to (\text{-}) on cartesian typesunique comonoid structure on WW
coreader comonadWΓ—(-)W \times (\text{-}) on cartesian typesunique comonoid structure on WW
writer monadAβŠ—(-)A \otimes (\text{-}) on monoidal typeschosen monoid structure on AA
cowriter comonadAβ†’(-) AβŠ—(-)\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})} on monoidal typeschosen monoid structure on AA

chosen comonoid structure on AA
Frobenius (co)writerAβ†’(-) AβŠ—(-)\array{A \to (\text{-}) \\ A \otimes (\text{-})} on monoidal typeschosen Frobenius monoid structure



In Haskell:

See also:

Last revised on February 6, 2024 at 22:45:22. See the history of this page for a list of all contributions to it.