| Portability | non-portable (class-associated types) |
|---|---|
| Stability | experimental |
| Maintainer | Edward Kmett <ekmett@gmail.com> |
Control.Category.Monoidal
Description
A Monoidal category is a category with an associated biendofunctor that has an identity,
which satisfies Mac Lane''s pentagonal and triangular coherence conditions
Technically we usually say that category is Monoidal, but since
most interesting categories in our world have multiple candidate bifunctors that you can
use to enrich their structure, we choose here to think of the bifunctor as being
monoidal. This lets us reuse the same Bifunctor over different categories without
painful newtype wrapping.
- type family Id k p :: *
- class Associative k p => Monoidal k p where
- class Disassociative k p => Comonoidal k p where
Documentation
Denotes that we have some reasonable notion of Identity for a particular Bifunctor in this Category. This
notion is currently used by both Monoidal and Comonoidal
class Associative k p => Monoidal k p whereSource
class Disassociative k p => Comonoidal k p whereSource
A comonoidal category satisfies the dual form of the triangle identities
first idr = disassociate . second idl second idl = disassociate . first idr
This type class is also (ab)used for the inverse operations needed for a strict (co)monoidal category.
A strict (co)monoidal category is one that is both Monoidal and Comonoidal and satisfies the following laws:
idr . coidr = id idl . coidl = id coidl . idl = id coidr . idr = id
Instances
| Comonoidal (->) Either | |
| Comonoidal (->) (,) |