Portability | non-portable (class-associated types) |
---|---|
Stability | experimental |
Maintainer | Edward Kmett <ekmett@gmail.com> |
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 type annotations.
- module Control.Category.Braided
- class Bifunctor p k k k => HasIdentity k p i | k p -> i
- class (Associative k p, HasIdentity k p i) => Monoidal k p i | k p -> i where
- class (Coassociative k p, HasIdentity k p i) => Comonoidal k p i | k p -> i where
Documentation
module Control.Category.Braided
class Bifunctor p k k k => HasIdentity k p i | k p -> iSource
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
HasIdentity Hask (,) Void | |
HasIdentity Hask p i => HasIdentity Hask (Flip p) i |
class (Associative k p, HasIdentity k p i) => Monoidal k p i | k p -> i whereSource
class (Coassociative k p, HasIdentity k p i) => Comonoidal k p i | k p -> i whereSource
A comonoidal category satisfies the dual form of the triangle identities
bimap idr id = coassociate . bimap id idl bimap id idl = coassociate . bimap idr id
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
(Associative Hask p, Comonoidal Hask p i) => Comonoidal Hask (Flip p) i |