CAP-based packages

Categorical doctrines

CAP defines and allows defining (hierarchies of) categorical doctrines.

There are several CAP-based packages that define various categorical doctrines.
Defining a doctrine means several things:
1. Type specification of all the categorical operations which are required by the doctrine: In particular this means specifying those operations involved in the definition of a certain (co)limit (e.g., a kernel) which exists in the doctrine (e.g., Abelian category). This includes the specification of the input and output^1,² of each such operation. Specifying a certain limit means specifying all operations involved in the definition of that limit.
  For example, specifying “kernel” means specifying all operations entering the definition of the “kernel”:
  - KernelObject³,
  - KernelEmbedding⁴,
  - KernelLift⁵.
2. Doctrine-specific derivations of categorical operations from other ones: For example in an Abelian category the LiftAlongMonomorphism can be derived from⁶
  - CokernelProjection,
  - KernelLift,
  - Compose,
  - Inverse.
  This is wrong in other doctrines like pre-Abelian, and does not even make sense in doctrines without kernels and cokernels.
3. Doctrine-specific logical implications: For example, in an Abelian category the algorithm IsIsomorphism( ImageEmbedding( α ) ) is equivalent to the algorithm IsEpimorphism( α ). Again, this is wrong in other doctrines like pre-Abelian.
Even if GAP (automatically) organizes the doctrines in hierarchies, the relations among the category (instances) fulfilling these doctrines are much richer than these hierarchies.⁶ These relations are given by several functors, natural transformations between (parallel) functors, and adjunctions (also described by natural transformations) between (anti-parallel) functors.
For example, in set-theoretic modelling of mathematics one would say that a group is a monoid (in which each element is invertible). In category-theoretic modelling of mathematics one would say that there is a fully faithful forgetful functor from the category of groups to that of monoids which has a left adjoint and a right adjoint:
- The left adjoint adds formal inverses to the monoid (Grothendieck construction in the commutative case).
- The right adjoint maps a monoid to its group of units.

Category constructors and building towers of categories

The CAP-based packages provide various category constructors which take a combinatorial or algebraic data structure, or even a category or several categories (of a certain doctrine) as input and provide another category (of a possibly different doctrine) as output. The true strength of CAP is that it allows concatenating these category constructor to build arbitrary high towers.

Functors and natural transformations

The CAP-based packages provide various functors between these categories as well as natural transformations between (parallel) functors.

CompilerForCAP

CompilerForCAP should be able to use the logical rules valid in the various doctrines to compile these towers of categories —built by applying multiple category constructors— and produce highly efficient (parallel) code (ultimately in HPC-GAP/Julia).

Packages part of or based on CAP_project:

Name	Description	Documentation	Build Status	Code Coverage	Status
CAP	Categories, Algorithms, Programming				deposited
Algebroids	Algebroids and bialgebroids as preadditive categories generated by enhanced quivers				dev
ActionsForCAP	Actions and Coactions for CAP				dev
AttributeCategoryForCAP	Automatic enhancement with attributes of a CAP category				dev
Bicomplexes	Bicomplexes for Abelian categories				dev
CartesianCategories	Cartesian and cocartesian categories and various subdoctrines				dev
CategoriesWithAmbientObjects	Categories with objects having ambient objects				dev
CatReps	Representations and cohomology of finite categories				dev
CompilerForCAP	Speed up and verify categorical algorithms				dev
ComplexesAndFilteredObjectsForCAP	Implementation of complexes, cocomplexes and filtered objects for CAP				dev
ComplexesCategories	Category of (co)chain complexes of an additive category				dev
DerivedCategories	Derived categories of Abelian categories				dev
ExteriorPowersCategories	The graded category of exterior powers of objects of a given Ab-category				dev
FinSetsForCAP	The elementary topos of (skeletal) finite sets				dev
FinGSetsForCAP	The elementary topos of (skeletal) finite G-sets				dev
FiniteCocompletions	Finite (co)product/(co)limit (co)completions				dev
FpCategories	Finitely presented categories by generating quivers and relations				dev
FreydCategoriesForCAP	Freyd categories - Formal (co)kernels for additive categories				dev
FunctorCategories	Categories of functors				dev
GeneralizedMorphismsForCAP	Implementations of generalized morphisms for the CAP project				deposited
GradedCategories	Graded closures of categories				dev
GradedModulePresentationsForCAP	Presentations for graded modules				dev
GroupRepresentationsForCAP	Skeletal category of group representations for CAP				dev
HeckeCategories	Categorification of Hecke algebras				dev
HomologicalAlgebraForCAP	Homological algebra algorithms for CAP				dev
HomotopyCategories	Homotopy categories of additive categories				dev
InternalExteriorAlgebraForCAP	Constructions for Modules over the Internal Exterior Algebra for CAP				dev
InternalModules	Modules over internal algebras				dev
IntrinsicCategories	Intrinsic categories for CAP				dev
IntrinsicGradedModules	Finitely presented graded modules over computable graded rings allowing multiple presentations and the notion of elements				dev
IntrinsicModules	Finitely presented modules over computable rings allowing multiple presentations and the notion of elements				dev
LazyCategories	Construct an equivalent lazy category out of a CAP category				dev
LinearAlgebraForCAP	Category of Matrices over a Field for CAP				deposited
Locales	Locales, frames, coframes, meet semi-lattices of locally closed subsets, and Boolean algebras of constructible sets				dev
MachineLearningForCAP	Exploring categorical machine learning in CAP				dev
ModulePresentationsForCAP	Category R-pres for CAP				deposited
ModulesOverLocalRingsForCAP	Category of modules over a local ring modeled by Serre quotients for CAP				dev
MonoidalCategories	Monoidal and monoidal (co)closed categories				deposited
PreSheaves	Categories of (co)presheaves				dev
QPA2	Quivers and Path Algebras	N/A	N/A	N/A	dev
QuotientCategories	Quotient categories				dev
StableCategories	Stable categories of additive categories				dev
SubcategoriesForCAP	Subcategory and other related constructors for CAP categories				dev
ToolsForCategoricalTowers	Tools for CategoricalTowers				dev
Toposes	Elementary toposes				dev
ToricSheaves	Toric sheaves as Serre quotients				dev
TriangulatedCategories	Framework for triangulated categories				dev
ZariskiFrames	(Co)frames/Locales of Zariski closed/open subsets of affine, projective, or toric varieties				dev
ZXCalculusForCAP	The category of ZX-diagrams				dev