mathlib documentation

algebra.hom.non_unital_alg

Morphisms of non-unital algebras #

This file defines morphisms between two types, each of which carries:

The multiplications are not assumed to be associative or unital, or even to be compatible with the scalar actions. In a typical application, the operations will satisfy compatibility conditions making them into algebras (albeit possibly non-associative and/or non-unital) but such conditions are not required to make this definition.

This notion of morphism should be useful for any category of non-unital algebras. The motivating application at the time it was introduced was to be able to state the adjunction property for magma algebras. These are non-unital, non-associative algebras obtained by applying the group-algebra construction except where we take a type carrying just has_mul instead of group.

For a plausible future application, one could take the non-unital algebra of compactly-supported functions on a non-compact topological space. A proper map between a pair of such spaces (contravariantly) induces a morphism between their algebras of compactly-supported functions which will be a non_unital_alg_hom.

TODO: add non_unital_alg_equiv when needed.

Main definitions #

Tags #

non-unital, algebra, morphism

A morphism respecting addition, multiplication, and scalar multiplication. When these arise from algebra structures, this is the same as a not-necessarily-unital morphism of algebras.

Instances for non_unital_alg_hom
@[class]
structure non_unital_alg_hom_class (F : Type u_1) (R : out_param (Type u_2)) (A : out_param (Type u_3)) (B : out_param (Type u_4)) [monoid R] [non_unital_non_assoc_semiring A] [non_unital_non_assoc_semiring B] [distrib_mul_action R A] [distrib_mul_action R B] :
Type (max u_1 u_3 u_4)

non_unital_alg_hom_class F R A B asserts F is a type of bundled algebra homomorphisms from A to B.

Instances of this typeclass
Instances of other typeclasses for non_unital_alg_hom_class
  • non_unital_alg_hom_class.has_sizeof_inst
@[protected, simp]
@[ext]
theorem non_unital_alg_hom.ext {R : Type u} {A : Type v} {B : Type w} [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] [non_unital_non_assoc_semiring B] [distrib_mul_action R B] {f g : A →ₙₐ[R] B} (h : (x : A), f x = g x) :
f = g
theorem non_unital_alg_hom.congr_fun {R : Type u} {A : Type v} {B : Type w} [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] [non_unital_non_assoc_semiring B] [distrib_mul_action R B] {f g : A →ₙₐ[R] B} (h : f = g) (x : A) :
f x = g x
@[simp]
theorem non_unital_alg_hom.coe_mk {R : Type u} {A : Type v} {B : Type w} [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] [non_unital_non_assoc_semiring B] [distrib_mul_action R B] (f : A B) (h₁ : (m : R) (x : A), f (m x) = m f x) (h₂ : f 0 = 0) (h₃ : (x y : A), f (x + y) = f x + f y) (h₄ : (x y : A), f (x * y) = f x * f y) :
{to_fun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄} = f
@[simp]
theorem non_unital_alg_hom.mk_coe {R : Type u} {A : Type v} {B : Type w} [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] [non_unital_non_assoc_semiring B] [distrib_mul_action R B] (f : A →ₙₐ[R] B) (h₁ : (m : R) (x : A), f (m x) = m f x) (h₂ : f 0 = 0) (h₃ : (x y : A), f (x + y) = f x + f y) (h₄ : (x y : A), f (x * y) = f x * f y) :
{to_fun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄} = f
@[norm_cast]
theorem non_unital_alg_hom.coe_distrib_mul_action_hom_mk {R : Type u} {A : Type v} {B : Type w} [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] [non_unital_non_assoc_semiring B] [distrib_mul_action R B] (f : A →ₙₐ[R] B) (h₁ : (m : R) (x : A), f (m x) = m f x) (h₂ : f 0 = 0) (h₃ : (x y : A), f (x + y) = f x + f y) (h₄ : (x y : A), f (x * y) = f x * f y) :
{to_fun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄} = {to_fun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃}
@[norm_cast]
theorem non_unital_alg_hom.coe_mul_hom_mk {R : Type u} {A : Type v} {B : Type w} [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] [non_unital_non_assoc_semiring B] [distrib_mul_action R B] (f : A →ₙₐ[R] B) (h₁ : (m : R) (x : A), f (m x) = m f x) (h₂ : f 0 = 0) (h₃ : (x y : A), f (x + y) = f x + f y) (h₄ : (x y : A), f (x * y) = f x * f y) :
{to_fun := f, map_smul' := h₁, map_zero' := h₂, map_add' := h₃, map_mul' := h₄} = {to_fun := f, map_mul' := h₄}
@[protected, simp]
theorem non_unital_alg_hom.map_smul {R : Type u} {A : Type v} {B : Type w} [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] [non_unital_non_assoc_semiring B] [distrib_mul_action R B] (f : A →ₙₐ[R] B) (c : R) (x : A) :
f (c x) = c f x
@[protected, simp]
theorem non_unital_alg_hom.map_add {R : Type u} {A : Type v} {B : Type w} [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] [non_unital_non_assoc_semiring B] [distrib_mul_action R B] (f : A →ₙₐ[R] B) (x y : A) :
f (x + y) = f x + f y
@[protected, simp]
theorem non_unital_alg_hom.map_mul {R : Type u} {A : Type v} {B : Type w} [monoid R] [non_unital_non_assoc_semiring A] [distrib_mul_action R A] [non_unital_non_assoc_semiring B] [distrib_mul_action R B] (f : A →ₙₐ[R] B) (x y : A) :
f (x * y) = f x * f y
@[protected, simp]
@[protected, instance]
Equations

The composition of morphisms is a morphism.

Equations

The inverse of a bijective morphism is a morphism.

Equations
@[simp]

Operations on the product type #

Note that much of this is copied from linear_algebra/prod.

The first projection of a product is a non-unital alg_hom.

Equations

The second projection of a product is a non-unital alg_hom.

Equations

The prod of two morphisms is a morphism.

Equations

Taking the product of two maps with the same domain is equivalent to taking the product of their codomains.

Equations

The left injection into a product is a non-unital algebra homomorphism.

Equations

The right injection into a product is a non-unital algebra homomorphism.

Equations

Interaction with alg_hom #

@[protected, instance]
def alg_hom.non_unital_alg_hom_class {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] {F : Type u_1} [alg_hom_class F R A B] :
Equations
def alg_hom.to_non_unital_alg_hom {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (f : A →ₐ[R] B) :

A unital morphism of algebras is a non_unital_alg_hom.

Equations
@[protected, instance]
def alg_hom.non_unital_alg_hom.has_coe {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] :
Equations
@[simp]
theorem alg_hom.to_non_unital_alg_hom_eq_coe {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (f : A →ₐ[R] B) :
@[simp, norm_cast]
theorem alg_hom.coe_to_non_unital_alg_hom {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (f : A →ₐ[R] B) :