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GradientBasedLearningForCAP

Gradient Based Learning via Category Theory

2026.02-01

12 February 2026

Kamal Saleh
Email: kamal.saleh@uni-siegen.de
Homepage: https://github.com/kamalsaleh
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1 Introduction
 1.1 Overview
2 Examples
 2.1 Binary-Class Neural Network with Binary Cross-Entropy Loss Function
 2.2 Multi-Class Neural Network with Cross-Entropy Loss Function
 2.3 Neural Network with Quadratic Loss Function
 2.4 Next Local Minima
3 Expressions
 3.1 Constructors

  3.1-1 Expression
  3.1-2 ConstantExpression
 3.2 Attributes

  3.2-1 Variables
  3.2-2 String
  3.2-3 AsFunction
 3.3 Operations

  3.3-1 DummyInput
 3.4 Global Functions

  3.4-1 ConvertToConstantExpressions
  3.4-2 CreateContextualVariables
  3.4-3 AssignExpressions
 3.5 Examples
 3.6 GAP Categories

  3.6-1 IsExpression
  3.6-2 IsConstantExpression
4 Skeletal Category of Smooth Maps
 4.1 Constructors

  4.1-1 SkeletalCategoryOfSmoothMaps
  4.1-2 ObjectConstructor
  4.1-3 MorphismConstructor
  4.1-4 SmoothMap
 4.2 Attributes

  4.2-1 RankOfObject
  4.2-2 Map
  4.2-3 JacobianMatrix
 4.3 Operations

  4.3-1 Eval
  4.3-2 EvalJacobianMatrix
 4.4 Available Smooth Maps

  4.4-1 AvailableSkeletalSmoothMaps
  4.4-2 DummyInputForAffineTransformation
 4.5 Supported CAP Operations

  4.5-1 SkeletalSmoothMaps
  4.5-2 Generate Documentation
 4.6 Examples
 4.7 GAP Categories

  4.7-1 IsSkeletalCategoryOfSmoothMaps
  4.7-2 IsObjectInSkeletalCategoryOfSmoothMaps
  4.7-3 IsMorphismInSkeletalCategoryOfSmoothMaps
5 Category of Parametrised Morphisms
 5.1 Definition
 5.2 Constructors

  5.2-1 CategoryOfParametrisedMorphisms
  5.2-2 ObjectConstrutor
  5.2-3 MorphismConstructor
 5.3 Attributes

  5.3-1 UnderlyingCategory
  5.3-2 UnderlyingObject
  5.3-3 UnderlyingObject
  5.3-4 UnderlyingMorphism
 5.4 Operations

  5.4-1 ReparametriseMorphism
  5.4-2 FlipParameterAndSource
  5.4-3 Batchify
  5.4-4 NaturalEmbedding
  5.4-5 EmbeddingIntoCategoryOfParametrisedMorphisms
 5.5 Available Parametrised Morphisms
 5.6 Supported CAP Operations

  5.6-1 Category of Parametrised Smooth Maps
  5.6-2 Generate Documentation
 5.7 Examples
 5.8 GAP Categories

  5.8-1 IsCategoryOfParametrisedMorphisms
  5.8-2 IsObjectInCategoryOfParametrisedMorphisms
  5.8-3 IsMorphismInCategoryOfParametrisedMorphisms
6 Neural Networks
 6.1 Definition
 6.2 Operations

  6.2-1 NeuralNetworkLogitsMorphism
  6.2-2 NeuralNetworkPredictionMorphism
  6.2-3 NeuralNetworkLossMorphism
 6.3 Examples
7 Category of Lenses
 7.1 Definition
 7.2 Constructors

  7.2-1 CategoryOfLenses
  7.2-2 ObjectConstructor
  7.2-3 MorphismConstructor
 7.3 Attributes

  7.3-1 UnderlyingPairOfObjects
  7.3-2 UnderlyingPairOfMorphisms
  7.3-3 GetMorphism
  7.3-4 PutMorphism
 7.4 Operations

  7.4-1 ReverseDifferentialLensFunctor
 7.5 Optimizers
 7.6 Supported CAP Operations

  7.6-1 Category of Lenses of Smooth Maps
 7.7 Examples
 7.8 GAP Categories

  7.8-1 IsCategoryOfLenses
  7.8-2 IsObjectInCategoryOfLenses
  7.8-3 IsMorphismInCategoryOfLenses
8 Fitting Parameters
 8.1 Introduction
 8.2 Notes on Batching
 8.3 Operations

  8.3-1 OneEpochUpdateLens
  8.3-2 OneEpochUpdateLens
  8.3-3 Fit
 8.4 Examples
9 CAP Operations for GradientBasedLearningForCAP
 9.1 Basic Operations

  9.1-1 MultiplicationForMorphisms
  9.1-2 ReverseDifferentialWithGivenObjects
  9.1-3 ReverseDifferential
 9.2 Add-Methods

  9.2-1 AddMultiplicationForMorphisms
  9.2-2 AddReverseDifferential
  9.2-3 AddReverseDifferentialWithGivenObjects
10 Tools
 10.1 Mathematical Operations

  10.1-1 Relu
  10.1-2 KroneckerDelta
  10.1-3 MultiplyMatrices
 10.2 List Operations

  10.2-1 Enumerate
  10.2-2 SplitDenseList
  10.2-3 SplitDenseList
 10.3 Helper Functions

  10.3-1 SelectBasedOnCondition
  10.3-2 CallFuncListBasedOnCondition
 10.4 Python Integration

  10.4-1 Diff
  10.4-2 LazyDiff
  10.4-3 JacobianMatrixUsingPython
  10.4-4 LazyJacobianMatrix
  10.4-5 ScatterPlotUsingPython
  10.4-6 SimplifyExpressionUsingPython
  10.4-7 LaTeXOutputUsingPython
  10.4-8 AsCythonFunction
Index

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