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| Require Import Basics.Overture Basics.Tactics. | ||
| Require Import WildCat.Core WildCat.Equiv WildCat.Monoidal WildCat.Bifunctor | ||
| WildCat.NatTrans WildCat.Opposite WildCat.Products. | ||
| Require Import abstract_algebra. | ||
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| (** * Monoids and Comonoids *) | ||
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| (** Here we define a monoid internal to a monoidal category. Various algebraic theories such as groups and rings may also be internalized, however these specifically require a cartesian monoidal structure. The theory of monoids however has no such requirement and can therefore be developed in much greater generality. This can be used to define a range of objects such as R-algebras, H-spaces, Hopf algebras and more. *) | ||
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| (** * Monoid objects *) | ||
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| Section MonoidObject. | ||
| Context {A : Type} {tensor : A -> A -> A} {unit : A} | ||
| `{HasEquivs A, !Is0Bifunctor tensor, !Is1Bifunctor tensor} | ||
| `{!Associator tensor, !LeftUnitor tensor unit, !RightUnitor tensor unit}. | ||
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| (** An object [x] of [A] is a monoid object if it comes with the following data: *) | ||
| Class IsMonoidObject (x : A) := { | ||
| (** A multiplication map from the tensor product of [x] with itself to [x]. *) | ||
| mo_mult : tensor x x $-> x; | ||
| (** A unit of the multplication. *) | ||
| mo_unit : unit $-> x; | ||
| (** The multiplication map is associative. *) | ||
| mo_assoc : mo_mult $o fmap10 tensor mo_mult x $o associator x x x | ||
| $== mo_mult $o fmap01 tensor x mo_mult; | ||
| (** The multiplication map is left unital. *) | ||
| mo_left_unit : mo_mult $o fmap10 tensor mo_unit x $== left_unitor x; | ||
| (** The multiplication map is right unital. *) | ||
| mo_right_unit : mo_mult $o fmap01 tensor x mo_unit $== right_unitor x; | ||
| }. | ||
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| Context `{!Braiding tensor}. | ||
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| (** An object [x] of [A] is a commutative monoid object if: *) | ||
| Class IsCommutativeMonoidObject (x : A) := { | ||
| (** It is a monoid object. *) | ||
| cmo_mo :: IsMonoidObject x; | ||
| (** The multiplication map is commutative. *) | ||
| cmo_comm : mo_mult $o braid x x $== mo_mult; | ||
| }. | ||
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| End MonoidObject. | ||
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| Arguments IsMonoidObject {A} tensor unit {_ _ _ _ _ _ _ _ _ _} x. | ||
| Arguments IsCommutativeMonoidObject {A} tensor unit {_ _ _ _ _ _ _ _ _ _ _} x. | ||
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| Section ComonoidObject. | ||
| Context {A : Type} (tensor : A -> A -> A) (unit : A) | ||
| `{HasEquivs A, !Is0Bifunctor tensor, !Is1Bifunctor tensor} | ||
| `{!Associator tensor, !LeftUnitor tensor unit, !RightUnitor tensor unit}. | ||
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| (** A comonoid object is a monoid object in the opposite category. *) | ||
| Class IsComonoidObject (x : A) | ||
| := ismonoid_comonoid_op :: IsMonoidObject (A:=A^op) tensor unit x. | ||
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| (** We can build comonoid objects from the following data: *) | ||
| Definition Build_IsComonoidObject (x : A) | ||
| (** A comultplication map. *) | ||
| (co_comult : x $-> tensor x x) | ||
| (** A counit. *) | ||
| (co_counit : x $-> unit) | ||
| (** The comultiplication is coassociative. *) | ||
| (co_coassoc : associator x x x $o fmap01 tensor x co_comult $o co_comult | ||
| $== fmap10 tensor co_comult x $o co_comult) | ||
| (** The comultiplication is left counital. *) | ||
| (co_left_counit : left_unitor x $o fmap10 tensor co_counit x $o co_comult $== Id x) | ||
| (** The comultiplication is right counital. *) | ||
| (co_right_counit : right_unitor x $o fmap01 tensor x co_counit $o co_comult $== Id x) | ||
| : IsComonoidObject x. | ||
| Proof. | ||
| snrapply Build_IsMonoidObject. | ||
| - exact co_comult. | ||
| - exact co_counit. | ||
| - nrapply cate_moveR_eV. | ||
| symmetry. | ||
| nrefine (cat_assoc _ _ _ $@ _). | ||
| rapply co_coassoc. | ||
| - simpl; nrefine (_ $@ cat_idr _). | ||
| nrapply cate_moveL_Ve. | ||
| nrefine (cat_assoc_opp _ _ _ $@ _). | ||
| exact co_left_counit. | ||
| - simpl; nrefine (_ $@ cat_idr _). | ||
| nrapply cate_moveL_Ve. | ||
| nrefine (cat_assoc_opp _ _ _ $@ _). | ||
| exact co_right_counit. | ||
| Defined. | ||
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| (** Comultiplication *) | ||
| Definition co_comult {x : A} `{!IsComonoidObject x} : x $-> tensor x x | ||
| := mo_mult (A:=A^op) (tensor:=tensor) (unit:=unit) (x:=x). | ||
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| (** Counit *) | ||
| Definition co_counit {x : A} `{!IsComonoidObject x} : x $-> unit | ||
| := mo_unit (A:=A^op) (tensor:=tensor) (unit:=unit) (x:=x). | ||
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| Context `{!Braiding tensor}. | ||
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| (** A cocommutative comonoid objects is a commutative monoid object in the opposite category. *) | ||
| Class IsCocommutativeComonoidObject (x : A) | ||
| := iscommuatativemonoid_cocomutativemonoid_op | ||
| :: IsCommutativeMonoidObject (A:=A^op) tensor unit x. | ||
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| (** We can build cocommutative comonoid objects from the following data: *) | ||
| Definition Build_IsCocommutativeComonoidObject (x : A) | ||
| (** A comonoid. *) | ||
| `{!IsComonoidObject x} | ||
| (** Together with a proof of cocommutativity. *) | ||
| (cco_cocomm : braid x x $o co_comult $== co_comult) | ||
| : IsCocommutativeComonoidObject x. | ||
| Proof. | ||
| snrapply Build_IsCommutativeMonoidObject. | ||
| - exact _. | ||
| - exact cco_cocomm. | ||
| Defined. | ||
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| End ComonoidObject. | ||
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| (** ** Monoid enrichment *) | ||
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| (** A hom [x $-> y] in a cartesian category where [y] is a monoid object has the structure of a monoid. Equivalently, a hom [x $-> y] in a cartesian category where [x] is a comonoid object has the structure of a monoid. *) | ||
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| Section MonoidEnriched. | ||
| Context {A : Type} `{HasEquivs A} `{!HasBinaryProducts A} | ||
| (unit : A) `{!IsTerminal unit} {x y : A} | ||
| `{!HasMorExt A} `{forall x y, IsHSet (x $-> y)}. | ||
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| Section Monoid. | ||
| Context `{!IsMonoidObject _ _ y}. | ||
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| Local Instance sgop_hom : SgOp (x $-> y) | ||
| := fun f g => mo_mult $o cat_binprod_corec f g. | ||
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| Local Instance monunit_hom : MonUnit (x $-> y) := mo_unit $o mor_terminal _ _. | ||
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| Local Instance associative_hom : Associative sgop_hom. | ||
| Proof. | ||
| intros f g h. | ||
| unfold sgop_hom. | ||
| rapply path_hom. | ||
| refine ((_ $@L cat_binprod_fmap01_corec _ _ _)^$ $@ _). | ||
| nrefine (cat_assoc_opp _ _ _ $@ _). | ||
| refine ((mo_assoc $@R _)^$ $@ _). | ||
| nrefine (_ $@ (_ $@L cat_binprod_fmap10_corec _ _ _)). | ||
| refine (cat_assoc _ _ _ $@ (_ $@L _) $@ cat_assoc _ _ _). | ||
| nrapply cat_binprod_associator_corec. | ||
| Defined. | ||
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| Local Instance leftidentity_hom : LeftIdentity sgop_hom mon_unit. | ||
| Proof. | ||
| intros f. | ||
| unfold sgop_hom, mon_unit. | ||
| rapply path_hom. | ||
| refine ((_ $@L (cat_binprod_fmap10_corec _ _ _)^$) $@ cat_assoc_opp _ _ _ $@ _). | ||
| nrefine (((mo_left_unit $@ _) $@R _) $@ _). | ||
| 1: nrapply cate_buildequiv_fun. | ||
| unfold trans_nattrans. | ||
| nrefine ((((_ $@R _) $@ _) $@R _) $@ _). | ||
| 1: nrapply cate_buildequiv_fun. | ||
| 1: nrapply cat_binprod_beta_pr1. | ||
| nrapply cat_binprod_beta_pr2. | ||
| Defined. | ||
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| Local Instance rightidentity_hom : RightIdentity sgop_hom mon_unit. | ||
| Proof. | ||
| intros f. | ||
| unfold sgop_hom, mon_unit. | ||
| rapply path_hom. | ||
| refine ((_ $@L (cat_binprod_fmap01_corec _ _ _)^$) $@ cat_assoc_opp _ _ _ $@ _). | ||
| nrefine (((mo_right_unit $@ _) $@R _) $@ _). | ||
| 1: nrapply cate_buildequiv_fun. | ||
| nrapply cat_binprod_beta_pr1. | ||
| Defined. | ||
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| Local Instance issemigroup_hom : IsSemiGroup (x $-> y) := {}. | ||
| Local Instance ismonoid_hom : IsMonoid (x $-> y) := {}. | ||
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| End Monoid. | ||
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| Context `{!IsCommutativeMonoidObject _ _ y}. | ||
| Local Existing Instances sgop_hom monunit_hom ismonoid_hom. | ||
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| Local Instance commutative_hom : Commutative sgop_hom. | ||
| Proof. | ||
| intros f g. | ||
| unfold sgop_hom. | ||
| rapply path_hom. | ||
| refine ((_ $@L _^$) $@ cat_assoc_opp _ _ _ $@ (cmo_comm $@R _)). | ||
| nrapply cat_binprod_swap_corec. | ||
| Defined. | ||
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| Local Instance iscommutativemonoid_hom : IsCommutativeMonoid (x $-> y) := {}. | ||
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| End MonoidEnriched. | ||
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