Quotient.lean`

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2 declarations, 0 contain sorry, 0 declared as axioms

Group actions on quotients and equivariant homs

Imports (1)

Mathlib.GroupTheory.GroupAction.Quotient

Imported by 2 files

def MulAction.homEquivProdFixedPoints Documented declaration: homEquivProdFixedPoints line 20 source
Source documentation
Given a representative for each orbit of X under G, and for each x : X a choice of σ that sends x to the representative, we obtain a bijection between G-equivariant homs from X and the product of Stab(x)-fixed points over each orbit representative x.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, MulAction, MulAction.orbitRel, Eq, HSMul.hSMul, Equiv, MulActionHom, id, Set.range, MulAction.fixedPoints, Subtype, Membership.mem, Subgroup, MulAction.stabilizer, Set
Proof uses: Equiv.mk, Subtype.mk, MulActionHom.mk, Inv.inv
42 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
```
def MulAction.homEquivProdFixedPoints {G X Y : Type*} [Group G] [MulAction G X] [MulAction G Y]
    (σ : X → G) (hσ : ∀ a b, orbitRel G X a b → σ a • a = σ b • b) :
    (X →[G] Y) ≃ Π i : Set.range (fun i ↦ σ i • i), fixedPoints (stabilizer G i.1) Y where
…
```
def MulAction.sigmaRangeQuotientStabilizer Documented declaration: sigmaRangeQuotientStabilizer line 43 source
Source documentation
Given a representative for each orbit of X under G, and for each x : X a choice of σ that sends x to the representative, we obtain a bijection between X and ∐ G/stab(x) where the disjoint union runs through the representatives.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, MulAction, MulAction.orbitRel, Eq, HSMul.hSMul, Equiv, Sigma, Set.range, HasQuotient.Quotient, Subgroup, MulAction.stabilizer, Membership.mem, Set
Proof uses: Equiv.mk, MulAction.ofQuotientStabilizer, Sigma.fst, Sigma.snd, Sigma.mk, Subtype.mk, Inv.inv
30 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
```
def MulAction.sigmaRangeQuotientStabilizer
    {G X : Type*} [Group G] [MulAction G X]
    (σ : X → G) (hσ : ∀ a b, orbitRel G X a b → σ a • a = σ b • b) :
    (Σ i : Set.range (fun i ↦ σ i • i), G ⧸ stabilizer G i.1) ≃ X :=
…
```