Equivalence of Applicative Functors and Multifunctors
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909082936410112 |
|---|---|
| author | Abel, Andreas |
| author_facet | Abel, Andreas |
| contents | McBride and Paterson introduced Applicative functors to Haskell, which are equivalent to the lax monoidal functors (with strength) of category theory. Applicative functors F are presented via idiomatic application $\_\circledast\_ : F (A \to B) \to F A \to F B$ and laws that are a bit hard to remember. Capriotti and Kaposi observed that applicative functors can be conceived as multifunctors, i.e., by a family liftA$_n$ : $(A_1 \to ... \to A_n \to C) \to F A_1 \to ... \to F A_n \to F C$ of zipWith-like functions that generalize pure $(n=0)$, fmap $(n=1)$ and liftA2 $(n=2)$. This reduces the associated laws to just the first functor law and a uniform scheme of second (multi)functor laws, i.e., a composition law for liftA. In this note, we rigorously prove that applicative functors are in fact equivalent to multifunctors, by interderiving their laws. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivalence of Applicative Functors and Multifunctors Abel, Andreas Programming Languages Logic in Computer Science 68N18 D.1.1 McBride and Paterson introduced Applicative functors to Haskell, which are equivalent to the lax monoidal functors (with strength) of category theory. Applicative functors F are presented via idiomatic application $\_\circledast\_ : F (A \to B) \to F A \to F B$ and laws that are a bit hard to remember. Capriotti and Kaposi observed that applicative functors can be conceived as multifunctors, i.e., by a family liftA$_n$ : $(A_1 \to ... \to A_n \to C) \to F A_1 \to ... \to F A_n \to F C$ of zipWith-like functions that generalize pure $(n=0)$, fmap $(n=1)$ and liftA2 $(n=2)$. This reduces the associated laws to just the first functor law and a uniform scheme of second (multi)functor laws, i.e., a composition law for liftA. In this note, we rigorously prove that applicative functors are in fact equivalent to multifunctors, by interderiving their laws. |
| title | Equivalence of Applicative Functors and Multifunctors |
| topic | Programming Languages Logic in Computer Science 68N18 D.1.1 |
| url | https://arxiv.org/abs/2401.14286 |