Functor Categories

This page is a sub-page of our page on Category Theory.

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Related KMR pages:

Adjoint Functors

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• The n-Category Café: A group blog on Category Theory by John C. Baez.

• Brendan Fong, The Algebra of Open and Interconnected Systems,
PhD Thesis in Computer Science, University of Oxford, 2016.

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A \, P_{attern} \, is \, e_{xemplified} \, by \, another \, P_{attern} \, :

A Pattern is Exemplified by another Pattern

A schema mapping and its corresponding pullback functor:
A schema mapping and its corresponding pull-back functor

Left and Right Adjoints to Pullback of Schemas:
Left and Right Adjoints to Pullback of Schemas

Functor Category of Annotations:
Functor Category of Annotations

From Aggregators to Functor Categories:
From Aggregators to Functor Categories

Pullback of an Annotation Schema:
Pullback of an annotation schema

Each annotation is a functor : Schema —> Sets, and the set of annotations on a given schema forms a functor category. A functor category has objects that are themselves functors and arrows that are given by natural transformations between some of these functors.

Pullback of a Functor Category:
Pullback of functor category

Translating from annotation-i to annotation-k:
Translate between two annotations

Notation: If the above diagram commutes (as is indicated by the # sign), the \, a_{nnotation-k} \, is said to be naturally related to the \, a_{nnotation-i} \, . Natural relationships are indicated by “implication arrows”, as seen in the diagram below.

Within the metadata (= annotation) community, a naturally related annotation, that is an annotation which an implication arrow is pointing towards, is called a “dumb-down” of the annotation that it is naturally related to, which is the annotation from which the same implication/”dumbification” arrow is pointing away. A dumb-down annotation represents a “coarsification” (= a coarser version) of the original annotation.

Functor categories:
Functor categories

Functor n-categories:
Functor n-categories

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/////////////////// Infrastructures for cross-institutional reasoning, p. 86

10. Databases as functor categories

10.1. Database schemas present categories
10. Databases as functor categories
10.2. Instances on a schema
10.2 Instances on a schema
10.3. Data migration
10.3. Data migration
10.4. The pullback migration functor ∆
10.4. The pullback migration functor ∆
10.5. Functors in three different contexts
Functors in three different contexts
10.6. Adjoint functors
10.6. Adjoint functors
10.7. The migration functors are adjoints to the pullback functor
10.7. The migration functors are adjoints to the pullback functor
10.8. The ‘product-oriented’ (right) push-forward makes joins
10.8. The ‘product-oriented’ (right) push-forward makes joins
10.9. The ‘sum-oriented’ (left) push-forward makes unions
10.9 The ‘sum-oriented’ (left) push-forward makes unions
10.10 Views
10.10 Views

/////// Infrastructures for cross-institutional reasoning, p. 489

48.14 A simple “SELECT” query using views
48.14. A simple “SELECT” query using views
48.15 The Grothendieck construction
The Grothendieck construction
48.16 The Grothendieck construction applied to database instances
48.16 The Grothendieck construction applied to database instances
48.17 A different perspective on data
48.17  A different perspective on data
48.18 RDF schema and stores
48.18 RDF schema and stores
48.19 Allowing for semi-structured data
48.19 Allowing for semi-structured data
48.20 Summary
48.20 Summary

49. EXCERPTS FROM SPIVAK, 2014 (p. 396)

49.0 Comments on Category Theory for the Sciences
49.0 What is a category?

49.1. Graphs
49.1 Graphs

49.2. What are databases?
49.2. What are databases?

49.3. What is a database schema?
49.3. What is a database schema?

49.4. What is an instance of a database schema?
49.4. What is an instance of a database schema?

49.5. The Grothendieck construction
49.5. The Grothendieck construction

49.6. The Grothendieck construction (example)
49.6. The Grothendieck construction (example)

49.7. The Grothendieck construction (application)
49.7. The Grothendieck construction (application)
(cont.)
49.7 (cont.)

49.8. Slogan 6.2.2.5 for the Grothendieck construction
49.8. Slogan 6.2.2.5 for the Grothendieck construction

49.9. Full subcategory
49.9. Full subcategory
(cont.)
49.9. Full subcategory (cont.)

49.10. Comma categories
49.10 Comma categories
Example:
49.10 Comma categories (example)

49.11. Arithmetic of categories
49.11. Arithmetic of categories

49.12. The definition of categories
49.12 The definition of categories

49.13. The category of graphs
49.13. The category of graphs

49.14. The category of categories
49.14. The category of categories

49.15. Database schemas present categories
49.15. Database schemas present categories

49.16 Instances on a schema S
49.16. Instances on a schema

49.17. Groupoids
49.17. Groupoids

49.18. The category of propositions
48.18. The category of propositions

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