nLab writer 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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
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
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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



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.

In the context of monads in computer science this is called the writer comonad. This is of interest in functional programming languages such as Haskell, where a program for the form X⟢WΓ—YX \longrightarrow W \times Y behaves like taking input of type XX to output of type YY while in addition producing output (β€œside effect”) of type WW.

If here WW is 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. (See writer monad.)


Relation to reader monad and state monad

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

The composite of writer 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 writer 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 \,.


Last revised on March 13, 2021 at 03:53:07. See the history of this page for a list of all contributions to it.