nLab
writer 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

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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) 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 𝒞\mathcal{C}, given any object/type WW there is a comonad

W×():𝒞𝒞 W \times (-) \colon \mathcal{C} \to \mathcal{C}

given by forming the Cartesian product with WW

If here WW is equipped with the structure of a monoid, then W×()W \times (-) also canonically inherits the structure of a monad.

In the context of monads in computer science this is called the writer monad. This is the case of interest in functional programming languages such as Haskell, where a program for the form XW×YX \longrightarrow W \times Y behaves like a program taking input of type XX to output of type YY while in addition producing output of type WW, which is aggregated using the the monoid when the functions are composed, creating a sort of side channel.

Properties

Relation to reader monad and state monad

In a cartesian closed category/type theory 𝒞\mathcal{C}, the writer comonad W×()W\times (-) is left adjoint to the reader monad [W,][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

𝒞 /W WW * W𝒞 \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

WW *=W×():𝒞𝒞 \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

*W* \ast \leftarrow W \rightarrow \ast

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

*WidW*. \ast \leftarrow W \stackrel{id}{\rightarrow} W \rightarrow \ast \,.

References

Last revised on August 21, 2019 at 13:10:59. See the history of this page for a list of all contributions to it.