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Mathlib.CategoryTheory.Action

Actions as functors and as categories #

From a multiplicative action M ↻ X, we can construct a functor from M to the category of types, mapping the single object of M to X and an element m : M to map X → X given by multiplication by m. This functor induces a category structure on X -- a special case of the category of elements. A morphism x ⟶ y in this category is simply a scalar m : M such that m • x = y. In the case where M is a group, this category is a groupoid -- the action groupoid.

@[simp]
theorem CategoryTheory.actionAsFunctor_map (M : Type u_1) [Monoid M] (X : Type u) [MulAction M X] :
∀ {X_1 Y : CategoryTheory.SingleObj M} (x : X_1 Y) (x_1 : X), (CategoryTheory.actionAsFunctor M X).map x x_1 = x x_1
@[simp]

A multiplicative action M ↻ X viewed as a functor mapping the single object of M to X and an element m : M to the map X → X given by multiplication by m.

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    def CategoryTheory.ActionCategory (M : Type u_1) [Monoid M] (X : Type u) [MulAction M X] :

    A multiplicative action M ↻ X induces a category structure on X, where a morphism from x to y is a scalar taking x to y. Due to implementation details, the object type of this category is not equal to X, but is in bijection with X.

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      The projection from the action category to the monoid, mapping a morphism to its label.

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        The canonical map ActionCategory M X → X. It is given by fun x => x.snd, but has a more explicit type.

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        • x.back = x.snd
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          • CategoryTheory.ActionCategory.instCoeTC = { coe := fun (x : X) => (), x }
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          theorem CategoryTheory.ActionCategory.back_coe {M : Type u_1} [Monoid M] {X : Type u} [MulAction M X] (x : CategoryTheory.ActionCategory M X) :
          (), x.back = x

          An object of the action category given by M ↻ X corresponds to an element of X.

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            theorem CategoryTheory.ActionCategory.hom_as_subtype (M : Type u_1) [Monoid M] (X : Type u) [MulAction M X] (p : CategoryTheory.ActionCategory M X) (q : CategoryTheory.ActionCategory M X) :
            (p q) = { m : M // m p.back = q.back }
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            The stabilizer of a point is isomorphic to the endomorphism monoid at the corresponding point. In fact they are definitionally equivalent.

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              def CategoryTheory.ActionCategory.homOfPair {X : Type u} {G : Type u_2} [Group G] [MulAction G X] (t : X) (g : G) :
              (), g⁻¹ t (), t

              A target vertex t and a scalar g determine a morphism in the action groupoid.

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                theorem CategoryTheory.ActionCategory.homOfPair.val {X : Type u} {G : Type u_2} [Group G] [MulAction G X] (t : X) (g : G) :
                def CategoryTheory.ActionCategory.cases {X : Type u} {G : Type u_2} [Group G] [MulAction G X] {P : a b : CategoryTheory.ActionCategory G X⦄ → (a b)Sort u_3} (hyp : (t : X) → (g : G) → P (CategoryTheory.ActionCategory.homOfPair t g)) ⦃a : CategoryTheory.ActionCategory G X ⦃b : CategoryTheory.ActionCategory G X (f : a b) :
                P f

                Any morphism in the action groupoid is given by some pair.

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                  theorem CategoryTheory.ActionCategory.cases' {X : Type u} {G : Type u_2} [Group G] [MulAction G X] ⦃a' : CategoryTheory.ActionCategory G X ⦃b' : CategoryTheory.ActionCategory G X (f : a' b') :
                  ∃ (a : X) (b : X) (g : G) (ha : a' = (), a) (hb : b' = (), b) (hg : a = g⁻¹ b), f = CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ) (CategoryTheory.CategoryStruct.comp (CategoryTheory.ActionCategory.homOfPair b g) (CategoryTheory.eqToHom ))

                  Given G acting on X, a functor from the corresponding action groupoid to a group H can be curried to a group homomorphism G →* (X → H) ⋊ G.

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                    theorem CategoryTheory.ActionCategory.uncurry_obj {X : Type u} {G : Type u_2} [Group G] [MulAction G X] {H : Type u_3} [Group H] (F : G →* (XH) ⋊[mulAutArrow] G) (sane : ∀ (g : G), (F g).right = g) :
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                    theorem CategoryTheory.ActionCategory.uncurry_map {X : Type u} {G : Type u_2} [Group G] [MulAction G X] {H : Type u_3} [Group H] (F : G →* (XH) ⋊[mulAutArrow] G) (sane : ∀ (g : G), (F g).right = g) :
                    ∀ {x b : CategoryTheory.ActionCategory G X} (f : x b), (CategoryTheory.ActionCategory.uncurry F sane).map f = (F f).left b.back
                    def CategoryTheory.ActionCategory.uncurry {X : Type u} {G : Type u_2} [Group G] [MulAction G X] {H : Type u_3} [Group H] (F : G →* (XH) ⋊[mulAutArrow] G) (sane : ∀ (g : G), (F g).right = g) :

                    Given G acting on X, a group homomorphism φ : G →* (X → H) ⋊ G can be uncurried to a functor from the action groupoid to H, provided that φ g = (_, g) for all g.

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