dedekind::category::LeftActionFunctor< M, S, AddM, AddS, MultS, Act > Struct Template Reference

A semimodule's left action as a functor out of the opposite of the delooping of its scalars' product: \((s t) m = s (t m)\), so s then t along the walk acts as \(t s\), the opposite product, and \(1 m = m\) keeps the identity. Those laws are the semimodule's, which is_linear_action declares, so the functor laws are derived. A left action of a monoid is a right action of its opposite: the right actions are ActionFunctor. More...

Public Types
using	Σ_cat = Delooping<S, Opposite<MultS>>
using	Τ_cat = LeftActingOn<M, S, AddM, AddS, MultS, Act>
using	Domain = Σ_cat
using	Codomain = Τ_cat
template<typename U>
using	Shape = M

Public Member Functions
constexpr Codomain	operator() (const Domain &) const
constexpr LeftActing< Act, M, S >	operator() (const Element< S, Opposite< MultS > > &s) const
constexpr IdentityIn< Τ_cat >	operator() (const IdentityIn< Σ_cat > &) const

Public Attributes
Act	act {}

Detailed Description

template<std::semiregular M, std::semiregular S, std::semiregular AddM = std::plus<M>, std::semiregular AddS = std::plus<S>, std::semiregular MultS = std::multiplies<S>, std::semiregular Act = std::multiplies<>>
requires IsSemimodule<M, S, AddM, AddS, MultS, Act> && IsMonoid<S, Opposite<MultS>> && std::invocable<const Act&, S, M>
struct dedekind::category::LeftActionFunctor< M, S, AddM, AddS, MultS, Act >

A semimodule's left action as a functor out of the opposite of the delooping of its scalars' product: \((s t) m = s (t m)\), so s then t along the walk acts as \(t s\), the opposite product, and \(1 m = m\) keeps the identity. Those laws are the semimodule's, which is_linear_action declares, so the functor laws are derived. A left action of a monoid is a right action of its opposite: the right actions are ActionFunctor.

Template Parameters

M	the module.
S	the scalars.
AddM
AddS
MultS	the operations.
Act	the action.

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The documentation for this struct was generated from the following file:

• src/main/modules/dedekind/category/functor.cppm