nLab cowriter comonad



Categorical algebra

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




Let 𝒞\mathcal{C} be a closed monoidal category with tensor product and internal hom denoted by

respectively, and consider a monoid internal to 𝒞\mathcal{C}:

AMon(𝒞). A \,\in\, Mon\big(\mathcal{C}\big) \,.

Recall that the “AA-action monad” induced by AA – often called the “AA-writer monad” when considered as a monad in computer science – is the endofunctor of forming the tensor product with AA

A(-):𝒞𝒞 A \otimes (\text{-}) \;\colon\; \mathcal{C} \to \mathcal{C}

equipped with unit and multiplication structure morphisms induced, under the tensor product, from the unit and multiplication on AA, respectively.

The analogous construction but with the tensor product replaced by the internal hom

[A,-]:𝒞𝒞 [A,\, \text{-}] \;\colon\; \mathcal{C} \to \mathcal{C}

which in computer science/type theory tends to be written

(A-):𝒞𝒞, (A \to \text{-}) \;\colon\; \mathcal{C} \to \mathcal{C} \,,

gives a comonad which hence deserves to be called the coaction comonad or cowriter comonad induced by AA.

Alternatively, one might think to dualize the notion of the writer monad by considering a comonoid-object

BCoMon(𝒞) B \,\in\, CoMon(\mathcal{C})

and the induced comonad structure on B(-)B \otimes (\text{-}).

But if 𝒞\mathcal{C} is in fact a compact closed category, then these two notions agree: In this case the internal hom out of the monoid object AA is the tensor product with the dual comonoid object

[A,-]A *(-). [A,\,\text{-}] \;\simeq\; A^\ast \otimes (\text{-}) \,.

Finally, if AA carries both monoid- and comonoid-structure in a compatible way to make a Frobenius monoid, then the induced (co)writer (co)monad structures are compatible to make a Frobenius monad.

(co)monad nameunderlying endofunctor(co)monad structure induced by
reader monadW(-)W \to (\text{-}) on cartesian typesunique comonoid structure on WW
coreader comonadW×(-)W \times (\text{-}) on cartesian typesunique comonoid structure on WW
writer monadA(-)A \otimes (\text{-}) on monoidal typeschosen monoid structure on AA
cowriter comonadA(-) A(-)\array{A \to (\text{-}) \\ \\ A \otimes (\text{-})} on monoidal typeschosen monoid structure on AA

chosen comonoid structure on AA
Frobenius (co)writerA(-) A(-)\array{A \to (\text{-}) \\ A \otimes (\text{-})} on monoidal typeschosen Frobenius monoid structure


The terminology “cowriter comonad” is used for instance in:

Last revised on August 11, 2023 at 20:11:11. See the history of this page for a list of all contributions to it.