Large category of modules over monads on top of UniMaths and Display category
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README.md

largecatmodules

Large category of modules over monads on top of UniMath Signatures and signatures for higher order syntax.

Preliminaries are in the subfolder Modules/Prelims Signature and Signature related proofs are in the subfolder Modules/Signatures

To compile (coq 8.8): $ make

List of formalized propositions and definitions

  • Definition of signatures and their actions : Signatures/Signature

  • Representability of presentable signatures : Signatures/PresentableSignature

  • Representability of the codomain epimorphic morphism of signature : Signatures/EpiSigRepresentability

  • Adjunction in the category of modules over a specific monad R on Set Hom(M x R', N) ~ Hom(M , N') : Prelims/derivadj.v

  • A coproduct of presentable signatures is presentable : Signatures/PresentableSignatureCoproducts

  • The binproduct of a presentable signature with the tautological signature is presentable : Signatures/PresentableSignatureBinProdR

  • pointwise limits and colimits of modules : Prelims/LModuleColims

  • pointwise limits and colimits of signatures : Signatures/SignaturesColims

  • quotient monad : Prelims/quotientmonad

  • Epimorphisms of signatures are pointwise epimorphisms : Signatures/EpiArePointwise

  • Modularity in the context of a fibration : Prelims/FibrationInitialPushout

  • Modularity in the specific context of signatures and their models : Signatures/Modularity

The fact that algebraic signatures are representable is already proved in a different setting in the Heterogeneous Substitution System package of UniMaths. The adaptation to our setting is carried out in the files : Signatures/HssToSignature and Signatures/BindingSig

Summary of files

By folder

Prelims

  • modules : complements about modules

  • LModPbCommute : commutation of the pullback functor (change of base) with limits/colimits

  • DerivationIsFunctorial : Proof that derivation of modules is functorial

  • derivadj : Adjunction in the category of modules over a specific monad R on Set Hom(M x R', N) ~ Hom(M , N')

  • quotientmonad : the quotient monad construction

  • LModuleColims : limits and colimits of modules

  • LModuleBinProduct : direct definition of binary product of modules

  • LModuleCoproducts : direct definition of coproduct of modules There are also direct definitions for specific

  • CoproductsComplements, BinProductComplements,lib, SetCatComplements : self-explanatory

  • FibrationInitialPushout : modularity in the context of a fibration

Signatures

  • Signature : definition of signatures and their actions
  • EpiSigRepresentability : proof of the technical lemma : epimorphisms of signatures preserves representability
  • PresentableSignature : presentable signatures are representable.
  • quotientrep : quotient action construction
  • PresentableSignatureCoproducts : a coproduct of presentable signatures is presentable.
  • PresentableSignatureBinProdR : if a is presentable, then so is the product of a with the tautological signature
  • HssToSignature : Functor between signatures in the sense of heterogeneous substitution systems and our signatures.
  • BindingSig : adaptation of the proof in UniMath that algebraic signatures are representable
  • HssSignatureCommutation : Somme commutation rules between colimits/limits and the functor between signature in the sense of heterogeneous substitution systems and our signatures
  • SignaturesColims : colimits of signatures
  • SignatureBinproducts : direct definition of bin products of signatures
  • SignatureCoproduct : direct definition of coproducts of signatures
  • SignatureDerivation : derivation of signatures
  • Signatures/EpiArePointwise : epimorphisms of signatures are pointwise epimorphisms
  • Modularity : Modularity in the specific context of signatures and their models