Summary (address issue #65) for equations

Author: amblafont

Modules/SoftEquations/.package/files

@@ -5,4 +5,5 @@ quotientrepslice.v
 Equation.v
 quotientequation.v
 InitialEquationRep.v
-SignatureOverDerivation.v
+SignatureOverDerivation.v
+Summary.v

Modules/SoftEquations/InitialEquationRep.v

@@ -91,3 +91,35 @@ Section QuotientRep.
 
 End QuotientRep.
 
+(** Mere reformulation which hides the initial object in the main statment (isInitial becomes Initial) *)
+Section QuotientRepInit.
+
+  Local Notation MOD Mon := (category_LModule Mon SET).
+  Local Notation MONAD := (Monad SET).
+  Local Notation SIG := (signature SET).
+
+  Variable (choice : AxiomOfChoice.AxiomOfChoice_surj).
+  Context {Sig : SIG}.
+  Context (epiSig : ∏ (R S : Monad _)
+                      (f : Monad_Mor R S),
+                    isEpi (C := [ SET , SET]) ( f : nat_trans _ _) ->
+                    isEpi (C := [ SET , SET]) (# Sig f : nat_trans _ _)%ar).
+
+  Local Notation REP := (model Sig).
+
+  Local Notation REP_CAT := (rep_fiber_category Sig).
+
+  Context {O : UU} (eq : O -> soft_equation choice epiSig ) .
+
+  Local Notation REP_EQ := (model_equations eq ).
+  Local Notation REP_EQ_PRECAT := (precategory_model_equations eq).
+
+  Lemma soft_equations_preserve_initiality : Initial REP_CAT -> Initial REP_EQ_PRECAT.
+  Proof.
+    intro init.
+    eapply mk_Initial.
+    apply push_initiality.
+    exact (pr2 init).
+  Qed.
+End QuotientRepInit.
+

Modules/SoftEquations/Summary.v

@@ -0,0 +1,323 @@
+(** Summary of the formalization (kernel)
+
+Tip: The Coq command [About ident] prints where the ident was defined
+ *)
+
+
+Require Import UniMath.Foundations.PartD.
+
+Require Import UniMath.CategoryTheory.Monads.Monads.
+Require Import UniMath.CategoryTheory.Monads.LModules. 
+Require Import UniMath.CategoryTheory.SetValuedFunctors.
+Require Import UniMath.CategoryTheory.HorizontalComposition.
+Require Import UniMath.CategoryTheory.functor_categories.
+Require Import UniMath.CategoryTheory.categories.category_hset.
+Require Import UniMath.CategoryTheory.categories.category_hset_structures.
+
+Require Import UniMath.CategoryTheory.Categories.
+Require Import UniMath.Foundations.Sets.
+Require Import UniMath.CategoryTheory.Epis.
+Require Import UniMath.CategoryTheory.EpiFacts.
+
+Require Import Modules.Prelims.lib.
+Require Import Modules.Prelims.quotientmonad.
+Require Import Modules.Prelims.quotientmonadslice.
+Require Import Modules.Signatures.Signature.
+Require Import Modules.SoftEquations.ModelCat.
+Require Import Modules.Prelims.modules.
+
+Require Import Modules.SoftEquations.quotientrepslice.
+Require Import Modules.SoftEquations.SignatureOver.
+Require Import Modules.SoftEquations.Equation.
+Require Import Modules.SoftEquations.quotientrepslice.
+Require Import Modules.SoftEquations.quotientequation.
+
+Require Import UniMath.CategoryTheory.limits.initial.
+Require Import Modules.SoftEquations.InitialEquationRep.
+
+Require Import UniMath.CategoryTheory.DisplayedCats.Auxiliary.
+Require Import UniMath.CategoryTheory.DisplayedCats.Core.
+Require Import UniMath.CategoryTheory.DisplayedCats.Constructions.
+Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations.
+
+Require Import UniMath.CategoryTheory.Subcategory.Core.
+Require Import UniMath.CategoryTheory.Subcategory.Full.
+
+Local Notation MONAD := (Monad SET).
+Local Notation MODULE R := (LModule R SET).
+
+(**
+The command:
+
+Check (x ::= y) 
+
+succeeds if and only if [x] is convertible to [y]
+
+*)
+Notation  "x ::= y"  := ((idpath _ : x = y) = idpath _) (at level 70, no associativity).
+
+Fail Check (true ::= false).
+Check (true ::= true).
+
+(** *******************
+
+ Definition of a signature
+
+The more detailed definition can be found in Signature/Signature.v
+
+ ****** *)
+
+Check (
+    signature_data (C := SET) ::=
+      (** a signature assigns to each monad a module over it *)
+      ∑ F : (∏ R : MONAD, MODULE R),
+            (** a signature sends monad morphisms on module morphisms *)
+            ∏ (R S : MONAD) (f : Monad_Mor R S), LModule_Mor _ (F R) (pb_LModule f (F S))
+  ).
+
+(** Functoriality for signatures:
+- [signature_idax] means that # F id = id
+- [signature_compax] means that # F (f o g) = # F f o # F g
+ *)
+Check (∏ (F : signature_data (C:= SET)),
+       is_signature  F ::=
+         (signature_idax F × signature_compax F)).
+
+Check (signature SET ::= ∑ F : signature_data, is_signature F).
+
+
+Local Notation SIGNATURE := (signature SET).
+(** *******************
+
+ Definition of a model of a signature
+
+The more detailed definitions of models can be found in Signatures/Signature.v
+The more detailed definitions of model morphisms can be found in SoftSignatures/ModelCat.v
+
+ ****** *)
+
+(** The tautological module R over the monad R *)
+Local Notation Θ := tautological_LModule.
+
+Check (∏ (R : MONAD), (Θ R : functor _ _) ::= R).
+
+(**
+A model of a signature S is a monad R with a module morphism from S R to R, called an action.
+*)
+Check (∏ (S : SIGNATURE), model S ::=
+    ∑ R : MONAD, LModule_Mor R (S R) (Θ R)).
+
+(** the action of a model *)
+Check (∏ (S : SIGNATURE) (M : model S),
+       model_τ M ::= pr2 M).
+
+(**
+Given a signature S, a model morphism is a monad morphism commuting with the action
+*)
+Check (∏ (S : SIGNATURE) (M N : model S),
+       rep_fiber_mor M N  ::=
+         ∑ g:Monad_Mor M N,
+             ∏ c : SET, model_τ M c · g c = ((#S g)%ar:nat_trans _ _) c ·  model_τ N c ).
+
+
+Local Notation  "R →→ S" := (rep_fiber_mor  R S) (at level 6).
+
+
+(** The category of 1-models *)
+Check (∏ (S : SIGNATURE),
+       rep_fiber_precategory S ::=
+         (precategory_data_pair
+            (** the object and morphisms of the category *)
+            (precategory_ob_mor_pair (model S) rep_fiber_mor)
+            (** the identity morphism *)
+            (λ R, rep_fiber_id R)
+            (** Composition *)
+            (λ M N O , rep_fiber_comp)
+           ,,
+             (** This second component is a proof that the axioms of categories are satisfied *)
+             is_precategory_rep_fiber_precategory_data S)
+         ).
+
+
+(** *******************
+
+ Definition of an S-signature, or that of a signature over a 1-signature S
+
+The more detailed definitions can be found in SoftSignatures/SignatureOver.v
+
+ ****** *)
+
+(**
+a signature over a 1-sig S assigns functorially to any model of S a module over the underlying monad.
+*)
+Check (∏ (S : SIGNATURE),
+       signature_over S ::=
+         ∑ F :
+           (** model -> module over the monad *)
+           (∑ F : (∏ R : model S, MODULE R),
+                  (** model morphism -> module morphism *)
+                  ∏ (R S : model S) (f : R →→ S),
+                  LModule_Mor _ (F R) (pb_LModule f (F S))),
+
+               (** functoriality conditions (see SignatureOver.v) *)
+               is_signature_over S F).
+
+Local Notation "F ⟹ G" := (signature_over_Mor _ F G) (at level 39).
+
+(**
+a morphism of oversignature is a natural transformation
+*)
+Check (∏ (S : SIGNATURE)
+         (F F' : signature_over S),
+       signature_over_Mor S F F' ::=
+         (** a family of module morphism from F R to F' R for any model R *)
+         ∑ (f : (∏ R : model S, LModule_Mor R (F R) (F' R))),
+         (** subject to naturality conditions (see SignatureOver.v for the full definition) *)
+           is_signature_over_Mor S F F' f
+      ).
+
+(** Definition of an oversignature which preserve epimorphisms in the category of natural transformations
+(Cf SoftEquations/quotientequation.v *)
+Check (∏ (S : SIGNATURE)
+         (F : signature_over S),
+       isEpi_overSig F ::=
+        ∏ R S (f : R →→ S),
+                   isEpi (C := [SET, SET]) (f : nat_trans _ _) ->
+                   isEpi (C := [SET, SET]) (# F f : nat_trans _ _)%sigo
+       ).
+
+(** Definition of a soft-over signature (SoftEquations/quotientequation.v) 
+
+It is a signature Σ such that for any model R, and any family of model morphisms 
+(f_j : R --> d_j), the following diagram can be completed in the category
+of natural transformations:
+
+<<<
+           Σ(f_j)
+    Σ(R) ----------->  Σ(d_j)
+     |
+     |
+     |
+ Σ(π)|
+     |
+     V
+    Σ(S)
+
+>>>
+
+where π : R -> S is the canonical projection (S is R quotiented by the family (f_j)_j
+
+ *)
+Check (∏ (S : SIGNATURE)
+         (F : signature_over S)
+         (** The axiom of choice is necessary to make quotient monad *)
+         (ax_choice : AxiomOfChoice.AxiomOfChoice_surj)
+         (** S preserves epimorphisms of monads *)
+         (isEpi_sig : ∏ (R R' : MONAD)
+                        (f : Monad_Mor R R'),
+                        (isEpi (C:= [SET,SET]) (f : nat_trans _ _) ->
+                      isEpi (C:= [SET,SET]) ((#S f)%ar : nat_trans _ _))),
+       isSoft ax_choice isEpi_sig F
+       ::=
+         (∏ (R : model S)
+            (J : UU)(d : J -> (model S))(f : ∏ j, R →→ (d j))
+            X (x y : (F R X : hSet))
+            (pi := projR_rep S isEpi_sig ax_choice d f),
+          (∏ j, (# F (f j))%sigo X x  = (# F (f j))%sigo X y )
+          -> (# F pi X x)%sigo = 
+            (# F pi X y)%sigo  )
+       ).
+
+
+
+
+
+(* **********
+
+Definition of Equations. See SoftEquations/Equation.v
+
+************ **)
+
+(** An equation over a signature S is a pair of two signatures over S, and a signature over morphism between them *)
+Check (∏ (S : SIGNATURE),
+       equation (Sig := S) ::=
+    ∑ (S1 S2 : signature_over S), S1 ⟹ S2 × S1 ⟹ S2).
+
+(** Soft equation: the domain must be an epi over-signature, and the target
+    must be soft (SoftEquations/quotientequation)
+ *)
+Check (∏ (S : SIGNATURE)
+         (** The axiom of choice is necessary to make quotient monad *)
+         (ax_choice : AxiomOfChoice.AxiomOfChoice_surj)
+         (** S preserves epimorphisms of monads *)
+         (isEpi_sig : ∏ (R R' : MONAD)
+                        (f : Monad_Mor R R'),
+                        (isEpi (C:= [SET,SET]) (f : nat_trans _ _) ->
+                      isEpi (C:= [SET,SET]) ((#S f)%ar : nat_trans _ _))),
+
+       soft_equation ax_choice isEpi_sig ::=
+        ∑ (e : equation), isSoft ax_choice isEpi_sig (pr1 (pr2 e)) × isEpi_overSig (pr1 e)).
+(** 
+Definition of the category of 2-models of a 1-signature with a family of equation.
+
+It is the full subcategory of 1-models satisfying all the equations
+
+See SoftEquations/Equation.v for details
+
+*)
+
+(** a model R satifies the equations if the R-component of the two over-signature morphisms are equal for any
+    equation of the family *)
+Check (∏ (S : SIGNATURE)
+         (** a family of equations indexed by O *)
+         O (e : O -> equation (Sig := S))
+         (R : model S)
+       ,
+    satisfies_all_equations_hp e R ::=
+         (
+           (∏ (o : O),
+            (** the first half-equation of the equation [e o] *)
+            pr1 (pr2 (pr2 (e o))) R =
+            (** the second half-equation of the equation [e o] *)
+            pr2 (pr2 (pr2 (e o))) R)
+          ,,
+            (** this second component is a proof that this predicate is hProp *)
+            _)
+           ).
+
+(** The category of 2-models  is the full subcategory of the category [rep_fiber_category S]
+  satisfying all the equations *)
+Check (∏ (S : SIGNATURE)
+         (** a family of equations indexed by O *)
+         O (e : O -> equation (Sig := S)),
+
+   precategory_model_equations e ::=
+    full_sub_precategory (C := rep_fiber_precategory S)
+                         (satisfies_all_equations_hp e)).
+
+
+(** *********************** 
+
+Our main result : if a 1-signature Σ generates a syntax, then the 2-signature over Σ
+consisting of any family of soft equations over Σ also generates a syntax
+(SoftEquations/InitialEquationRep.v)
+
+*)
+Check (soft_equations_preserve_initiality :
+         (** The axiom of choice is needed for quotient monads/models *)
+         ∏ (choice : AxiomOfChoice.AxiomOfChoice_surj)
+           (** The 1-signature *)
+           (Sig : SIGNATURE)
+           (** The 1-signature must be an epi-signature *)
+           (epiSig : ∏ (R S : Monad SET) (f : Monad_Mor R S),
+                     isEpi (C := [SET, SET]) (f : nat_trans R S)
+                     → isEpi (C := [SET, SET])
+                             ((# Sig)%ar f : nat_trans (Sig R)  (pb_LModule f (Sig S))))
+           (** A family of equations *)
+           (O : UU) (eq : O → soft_equation choice epiSig),
+         (** If the category of 1-models has an initial object, .. *)
+         Initial (rep_fiber_category Sig)
+        (** .. then the category of 2-models has an initial object *)
+         → Initial (precategory_model_equations (λ x : O, eq x))).
+
+