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largecatmodules

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

Preliminaries are in the subfolder Modules/Prelims 1-Signature related proofs are in the subfolder Modules/Signatures 2-Signature related proofs are in the subfolder Modules/SoftEquations

Requirement: the UniMath library (installed with $ make install)

To compile (Coq 8.9.0): $ make

List of some important formalized propositions and definitions

The file SoftEquations/Summary gives a summary of main formalized propositions and definitions for 2-signatures and elementary equations.

For the rest:

  • 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

  • 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 effective is already proved in a different setting in the Heterogeneous Substitution System package of UniMath. The adaptation to our setting is carried out in the files : Signatures/SigWithStrengthToSignature, Signatures/HssInitialModel and Signatures/BindingSig.

Summary of files

By folder

Prelims

  • quotientmonad, quotientmonadslice : the quotient monad construction

  • FibrationInitialPushout : modularity in the context of a fibration

  • 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')

  • LModulesFibration : fibration of left modules over monads

  • LModulesColims : limits and colimits of modules

  • LModulesBinProducts, LModulesCoproducts : direct definition of some particular colimits/limits of modules

  • PushoutsFromCoeqBinCoproducts : Pushouts from coequalizers and binary coproducts

  • FaithfulFibrationEqualizer : Faithful fibrations lift coequalizers

  • Opfibration : definition of opfibrations (adapted from the definition of fibrations in UniMath)

  • BinCoproductComplements, BinProductComplements , CoproductsComplements, EpiComplements LModulesComplements, SetCatComplements, lib : various complements

Signatures

Everything here is about 1-signatures

  • Signature : definition of signatures and the displayed category of models

  • ModelCat : direct definition of the category of models of a signature

  • EpiSigRepresentability : proof of the technical lemma : epimorphisms of signatures preserves representability

  • PresentableSignatures : presentable signatures are effective.

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

  • quotientrep : quotient model construction

  • HssInitialModel, BindingSig : adaptation of the proof in UniMath of initiality for strengthened signatures (in particular, for binding or algebraic signatures)

  • PreservesEpi : Epi-signatures

  • EpiArePointwise : epimorphisms of signatures are pointwise epimorphisms

  • PresentableSignatureCoproducts : a coproduct of presentable signatures is presentable.

  • PresentableSignatureBinProdR : if a is presentable, then so is the product of a with the tautological signature

  • SignaturesColims : colimits of signatures

  • SignatureBinproducts : direct definition of bin products of signatures

  • SignatureCoproduct : direct definition of coproducts of signatures

  • SignatureDerivation : derivation of signatures

  • SigWithStrengthToSignature : Functor between signatures with strength and our signatures.

  • HssSignatureCommutation : Somme commutation rules between colimits/limits and the functor between signatures with strength and our signatures

SoftEquations

This folder is about 2-signatures and elementary equations

  • Summary : summary of main propositions and definitions

  • SignatureOver : category of Σ-modules

  • CatOfTwoSignatures : category of 2-signatures, fibration of 2-models over it

  • Equation : equations, and category of models satisfying those equations

  • quotientequation : quotient model satisfying the equations

  • quotientrepslice : more general quotient model construction

  • AdjunctionEquationRep : algebraic 2-signatures are effective and related proofs

  • Modularity : modularity in the specific context of 2-signatures and their models

  • Examples/LCBetaEta : example of the lambda calculus modulo beta eta

  • SignatureOverAsFiber : (unused) alternative definition of Σ-modules as a displayed category over the category of 1-signatures

  • SignatureOverBinproducts : binary products of Σ-modules

  • SignatureOverDerivation : derivative of a Σ-module

  • BindingSig : complements about algebraic 1-signatures

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Large category of modules over monads on top of UniMaths and Display category

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