https://github.com/unimath/largecatmodules
Large category of modules over monads on top of UniMaths and Display category
https://github.com/unimath/largecatmodules
Last synced: 5 months ago
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Large category of modules over monads on top of UniMaths and Display category
- Host: GitHub
- URL: https://github.com/unimath/largecatmodules
- Owner: UniMath
- Created: 2016-11-29T12:19:39.000Z (over 9 years ago)
- Default Branch: master
- Last Pushed: 2024-09-06T11:55:20.000Z (over 1 year ago)
- Last Synced: 2025-03-25T14:44:37.521Z (12 months ago)
- Language: Coq
- Size: 960 KB
- Stars: 12
- Watchers: 6
- Forks: 8
- Open Issues: 7
-
Metadata Files:
- Readme: README.md
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README
# 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