Data.Category.RepresentableFunctor

Documentation

data Representable f repObj

Constructors

Representable

Fields

• representedFunctor :: f
• representingObject :: Obj (Dom f) repObj
• represent :: forall k z. (Dom f ~ k, Cod f ~ (->)) => Obj k z -> (f :% z) -> k repObj z
• universalElement :: forall k. (Dom f ~ k, Cod f ~ (->)) => f :% repObj

unrepresent :: (Functor f, Dom f ~ k, Cod f ~ (->)) => Representable f repObj -> k repObj z -> f :% z

covariantHomRepr :: Category k => Obj k x -> Representable (x :*-: k) x

contravariantHomRepr :: Category k => Obj k x -> Representable (k :-*: x) x

type InitialUniversal x u a = Representable ((x :*-: Cod u) :.: u) a

initialUniversal :: Functor u => u -> Obj (Dom u) a -> Cod u x (u :% a) -> (forall y. Obj (Dom u) y -> Cod u x (u :% y) -> Dom u a y) -> InitialUniversal x u a

An initial universal property, a universal morphism from x to u.

type TerminalUniversal x u a = Representable ((Cod u :-*: x) :.: Opposite u) a

terminalUniversal :: Functor u => u -> Obj (Dom u) a -> Cod u (u :% a) x -> (forall y. Obj (Dom u) y -> Cod u (u :% y) x -> Dom u y a) -> TerminalUniversal x u a

A terminal universal property, a universal morphism from u to x.

adjunctionInitialProp :: Adjunction c d f g -> Obj d y -> InitialUniversal y g (f :% y)

For an adjunction F -| G, each pair (FY, unit_Y) is an initial morphism from Y to G.

adjunctionTerminalProp :: Adjunction c d f g -> Obj c x -> TerminalUniversal x f (g :% x)

For an adjunction F -| G, each pair (GX, counit_X) is a terminal morphism from F to X.

initialPropAdjunction :: forall f g c d. (Functor f, Functor g, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) => f -> g -> (forall y. InitialUniversal y g (f :% y)) -> Adjunction c d f g

terminalPropAdjunction :: forall f g c d. (Functor f, Functor g, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) => f -> g -> (forall x. TerminalUniversal x f (g :% x)) -> Adjunction c d f g