nLab coreader comonad

Contents

Context

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
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit 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 setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
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

semantics

Mapping space

Contents

Idea

In a cartesian closed category/type theory π’ž\mathcal{C}, given any object/type WW there is a comonad

WΓ—(βˆ’):π’žβ†’π’ž W \times (-) \colon \mathcal{C} \to \mathcal{C}

given by forming the Cartesian product with WW, with the coproduct induced by the diagonal map on WW and the counit induced by the terminal map W→*W \to \ast.

Notice then if WW is furthermore equipped with the structure of a monoid, then WΓ—(βˆ’)W \times (-) also canonically inherits the structure of a monad, allowing aggregation of a program’s WW 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, the canonical comonad structure on WΓ—(βˆ’)W \times (-) is left adjoint to the reader monad, so that it is known as the reader comonad (eg. in the Haskell documentation for Control.Comonad.Reader) or coreader comonad (eg. Ahman & Uustalu (2019)).

Properties

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 \,.

References

Eg.

In Haskell:

See also:

Last revised on November 1, 2022 at 18:02:22. See the history of this page for a list of all contributions to it.