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12 Reinterpretations of categories
 12.1 Introduction
 12.2 Tutorial
 12.3 Implementation details
 12.4 Relation to CompilerForCAP
 12.5 Attributes
 12.6 Constructors

12 Reinterpretations of categories

12.1 Introduction

The support for building towers of category constructors is one of the main design features of CAP. Many categories that appear in the various applications can be modeled by towers of multiple category constructors. The category constructor ReinterpretationOfCategory (12.6-1) allows adding one last layer on top which allows expressing the desired (re)interpretation of such a modeling tower. In particular, this category constructor allows specifying the name of the category together with customized methods for the operations

in order to reflect the desired (re)interpretation with a user-interface that is independent of the modeling tower (see below for details). Note that the same tower might have multiple interpretations.

Table: A tower of categories modeling the category R
R := ReinterpretationOfCategory( cat_n )
cat_n := CategoryConstructor_n( cat_{n-1} )
...
cat_1 := CategoryConstructor_1( non_categorical_input )

The reinterpretation R is isomorphic to the top category cat_n in the tower. In practice, the word tower stands more generally for a finite poset with a greatest element.

12.2 Tutorial

We will show how one can reinterpret a category with the following guiding example: We reinterpret Opposite( CategoryOfRows( R ) ) as CategoryOfColumns( R ) using ReinterpretationOfCategory (12.6-1) with the options described in the following (see CategoryOfColumnsAsOppositeOfCategoryOfRows.gi in FreydCategoriesForCAP for a full implementation).

  1. Set the options category_filter, category_object_filter, and category_morphism_filter to the filters corresponding to the data structure of the desired reinterpretation, e.g. IsCategoryOfColumns, IsCategoryOfColumsObject, and IsCategoryOfColumsMorphism.

  2. Set object_constructor, object_datum, morphism_constructor, and morphism_datum to the functions one would write for ObjectConstructor and so on for a primitive implementation of the desired reinterpretation. In our example, object_constructor takes the reinterpretation R (which lies in IsCategoryOfColumns due to the filter set in the first step) and an integer, and returns a CAP object in the category with attribute RankOfObject set to the integer, just like a primitive implementation of CategoryOfColumns would do.

  3. Set modeling_tower_object_constructor, modeling_tower_object_datum, modeling_tower_morphism_constructor, and modeling_tower_morphism_datum: modeling_tower_object_constructor gets the same input as object_constructor but must return the corresponding object in the tower cat_n. modeling_tower_object_datum has the same output as object_datum but gets the reinterpretation R and an object in the tower cat_n as an input. In our example, modeling_tower_object_constructor gets the reinterpretation R and an integer as in step 2 and wraps the integer as a CategoryOfRowsObject and the result as an object in the opposite category. modeling_tower_object_datum gets the reinterpretation R and an object in Opposite( CategoryOfRows( R ) ) (that is, an integer boxed as a category of rows object boxed as an object in the opposite category) and returns the underlying integer. modeling_tower_morphism_constructor and modeling_tower_morphism_datum are given analogously.

By composing modeling_tower_object_datum with object_constructor and modeling_tower_morphism_datum with morphism_constructor (with suitable source and range), ReinterpretationOfCategory defines a functor "Reinterpretation" from cat_n to R. Similarly, it defines a functor "Model" from R to cat_n by composing object_datum with modeling_tower_object_constructor and morphism_datum with modeling_tower_morphism_constructor (with suitable source and range). "Reinterpretation" should be an isomorphism of categories with inverse "Model". More precisely, one has to take care of the following things:

12.3 Implementation details

Operations in ReinterpretationOfCategory are implemented as follows:

  1. Apply object_datum and morphism_datum to the input to get the underlying data.

  2. Apply modeling_tower_object_constructor and modeling_tower_morphism_constructor to the underlying data to get objects and morphisms in the tower cat_n.

  3. Apply the operation of the tower cat_n.

  4. Apply modeling_tower_object_datum or modeling_tower_morphism_datum to the result to get the underlying data.

  5. Apply object_constructor or morphism_constructor to the underlying data to get an object or a morphism in the reinterpretation R.

The first two steps define the functor "Model" and the last two steps define the functor "Reinterpretation". "Reinterpretation" on objects and morphisms is called ReinterpretationOfObject and ReinterpretationOfMorphism in the code. "Model" on objects and morphisms is called ModelingObject and ModelingMorphism in the code.

12.4 Relation to CompilerForCAP

The operation of the tower cat_n (step 3 above) usually unboxes objects and morphisms, operates on the underlying data, and boxes the result. The unboxing usually cancels with step 2 above, and boxing the result usually cancels with step 4 above. If one now compiles the operations of the reinterpretation R, only the following steps remain:

  1. Apply object_datum and morphism_datum to the input to get the underlying data.

  2. Operate on the underlying data. (previously part of step 3)

  3. Apply object_constructor or morphism_constructor to the underlying data to get an object or a morphism in the reinterpretation R. (previously step 5)

This is exactly what a primitive implementation would look like. Thus, in many cases compiling a reinterpretation immediately gives a primitive implementation with no remaining references to the tower cat_n.

12.5 Attributes

12.5-1 ModelingCategory
‣ ModelingCategory( R )( attribute )

Returns: a category

The tower cat_n modeling the reinterpretation R.

12.6 Constructors

12.6-1 ReinterpretationOfCategory
‣ ReinterpretationOfCategory( category, options )( operation )

Returns: a category

Reinterprets a category category (the "modeling category") to form a new category R (the "reinterpretation") subject to the options given via options, which is a record with the following keys:

12.6-2 ReinterpretationFunctor
‣ ReinterpretationFunctor( R )( attribute )

Returns: a functor

Returns the functor from the modeling category ModelingCategory(R) to the reinterpretation R which maps each object/morphism to its reinterpretation.

12.6-3 ModelingObject
‣ ModelingObject( R, obj )( operation )

Returns: a CAP category object

Returns the object in ModelingCategory(R) modeling the object obj in the reinterpretation R.

12.6-4 ReinterpretationOfObject
‣ ReinterpretationOfObject( R, obj )( operation )

Returns: a CAP category object

Returns the reinterpretation in R of an object obj in ModelingCategory(R).

12.6-5 ModelingMorphism
‣ ModelingMorphism( R, mor )( operation )

Returns: a CAP category morphism

Returns the morphism in ModelingCategory(R) modeling the morphism mor in the reinterpretation R.

12.6-6 ReinterpretationOfMorphism
‣ ReinterpretationOfMorphism( R, source, obj, range )( operation )

Returns: a CAP category morphism

Returns the reinterpretation in R with given source and range of a morphism mor in ModelingCategory(R).

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