# Why applicatives are monoidal

27 Oct 2019 ⇐ Notes archive(This is an entry in my technical notebook. There will likely be typos, mistakes, or wider logical leaps — the intent here is to “let others look over my shoulder while I figure things out.”)

“Applicative functors are […] lax monoidal functors with tensorial strength.”

I still don’t quite have the foundation to grok the “lax” and “tensorial strength” bits (I will, some day). Yet, seeing applicatives as monoidal always felt out of reach.

They’re often introduced with the `pure`

and `apply`

duo (sketched out below for `Optional`

).

(An aside, it finally clicked that the choice of “applicative” is, because, well, it supports function application inside of the functor’s context.)

And then, coincidentally, I recently asked:

is there any special terminology around types that support a

`zip`

in the same way functors have a`map`

and monads have a`flatMap`

?

To which, Matthew Johnson let me in on the secret.

`zip`

is an alternate way to formulate applicative

!!!.

That makes way more sense and sheds light on why Point-Free treated `map`

, `flatMap`

, and `zip`

as their big three instead of introducing `pure`

and `apply`

.

I can only *sort of* see that `apply`

implies monoidal-ness (pardon the formality) in that it reduces an `Optional<(A) -> B>`

and `Optional<A>`

into a single `Optional<B>`

. However, the fact that they contained different shapes always made me wonder.

`zip`

relays the ability to combine more readily. “Hand me an `Optional<A>`

and an `Optional<B>`

and I’ll give you an `Optional<(A, B)>`

.”

Yesterday, I finally got around to defining `apply`

in terms of `zip`

to see the equivalence.

Funnily enough, Brandon pointed me to exercise 25.2 which asks exactly this.

In short,

- Functors allow for a
`map`

. - Applicatives, a
`zip`

. - Monads, a
`flatMap`

.