## What is Representable used for in Haskell?

I am looking to understand what does Representable stand for in Haskell. The definitionRepresentable endofunctors over the category of Haskell types are isomorphic to the reader monad and so inherit a very large number of properties for free.
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## What is the main difference between Free Monoid and Monoid?

Looks like I have a pretty clear understanding what a Monoid is in Haskell, but last time I heard about something called a free monoid.
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## `(a -> b) -> (c -> d)` in Haskell?

This is yet another Haskell-through-category-theory question.Let's take something simple and well-known as an example. fmap? So fmap :: (a -> b) -> f a -> f b, omitting the fact that f is actually a Functor. As far as I understand, (a -> b) -> f a -> f b...
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## Why is there a distinction between co and contravariant functors in Haskell but not Category Theory?

This answer from a Category Theory perspective includes the following statement:...the truth is that there's no real distinction between co and contravariant functor, because every functor is just a covariant functor.
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## How does compiler figure out fixed point of a functor and how cata work at leaf level?

I feel like understanding the abstract concept of fixed point of a functor, however, I am still struggling to figure out the exact implementation of it and its catamorphism in Haskell.
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## Do the monadic liftM and the functorial fmap have to be equivalent?

(Note: I'm phrasing the question using Haskell terminology; answers are welcome to use the same terminology and/or the mathematical language of category theory, including proper mathematical definitions and axioms where I speak of functor and monad laws.)
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## What does “lax” mean in “lax monoidal functor”?

I know that the Applicative class is described in category theory as a "lax monoidal functor" but I've never heard the term "lax" before, and the nlab page on lax functor a bunch of stuff I don't recognize at all, re: bicategories and things that I didn't know we cared...
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## What is the purpose of `pure` in Applicative Functor

Meet the Applicative typeclass. It lies in the Control.Applicative module and it defines two methods, pure and <*>. It doesn't provide a default implementation for any of them, so we have to define them both if we want something to be an applicative functor.
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## Where is the bifunctor for functions in haskell?

I can't find bifunctor analog of fmap.Explanation:Functor for objects - datatype constructor. Type - a -> f a
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## Categorical structure in Haskell

Hask is usually thought to be the category whose objects are types and morphisms are functions. However, I've seen Conor McBride (@pigworker) warn against the use of Hask multiple times (1, 2, 3):
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