rmatthes / largecatmodules

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

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