Constructors for Action V G
for some concrete categories #
We construct Action (Type u) G
from a [MulAction G X]
instance and give some applications.
Bundles a type H
with a multiplicative action of G
as an Action
.
Instances For
Given a family F
of types with G
-actions, this is the limit cone demonstrating that the
product of F
as types is a product in the category of G
-sets.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The G
-set G
, acting on itself by left multiplication.
Instances For
The G
-set Gⁿ
, acting on itself by left multiplication.
Instances For
We have Fin 1 → G ≅ G
as G
-sets, with G
acting by left multiplication.
Instances For
If X
is a type with [Fintype X]
and G
acts on X
, then G
also acts on
FintypeCat.of X
.
Bundles a finite type H
with a multiplicative action of G
as an Action
.
Instances For
Shorthand notation for the quotient of G
by H
as a finite G
-set.
Equations
- One or more equations did not get rendered due to their size.
Instances For
If N
is a normal subgroup of G
, then this is the group homomorphism
sending an element g
of G
to the G
-endomorphism of G ⧸ₐ N
given by
multiplication with g⁻¹
on the right.
Equations
- Action.FintypeCat.toEndHom N = { toFun := fun (v : G) => { hom := Quotient.lift (fun (σ : G) => ⟦σ * v⁻¹⟧) ⋯, comm := ⋯ }, map_one' := ⋯, map_mul' := ⋯ }