Category Theory

This page is a sub-page of the page on our Learning Object Repository

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The sub-pages of this page are:

A Categorial Manifesto
Categorical Informatics
Functors
Limits and Colimits
Functor Categories
• Category of Bundles (over a Base Space)
• Naturally Related Functors and Processes
Adjoint Functors
Institution Theory
• The Human Category
Infinity Categories

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Related KMR-pages:

Representation and Reconstruction of Numbers
Numbers and their Digits in different Bases
Shift of Base for Numbers.
Mathematical Cogwheels
BioEntropy

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Books:

Basic Category Theory, by Tom Leinster, Cambridge University Press, 2014.
Conceptual Mathematics – A first introduction to categories,
by F. William Lawvere and Stephen H. Schanuel, Cambridge University Press, 1997.
Categories for the Working Mathematician, by Saunders Mac Lane, Springer Verlag, 1971.
Category Theory for the Sciences, by David Spivak, 2014.
Seven Sketches in Compositionality: An Invitation to Applied Category Theory,
by Brendan Fong and David Spivak, arXiv.org, 2018.
Category Theory in Context by Emily Riehl

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Other related sources of information:

Category theory at Wikipedia.
Abelian category at Wikipedia.
Comma category and slice category at Wikipedia.
Groupoid in category theory at Wikipedia.
Magma at Wikipedia.
Alexander Grothendieck at Wikipedia.
Lambda, the Ultimate
From Design Patterns to Category Theory by Mark Seemann.
From Design Patterns to Category Theory at Hacker News.
Workshop on Applied Category Theory in Leiden/The Netherlands, April 30 – May 4, 2018.
Workshops: From Design Patterns to Universal Abstractions,
Workshops 2019, at NDC, Sydney, Australia.
• Ambjörn’s archive on category theory at my.confolio.org
A Networked World by David Spivak, 2019.
Applied Category Theory Seminar, 2019.
With Category Theory, Mathematics Escapes From Equality,
by Kevin Hartnett, October 10, 2019.
Conducting the Mathematical Orchestra from the Middle at an interview with Emily Riehl at Quanta Magazine.
Emmy Noether – the grandmother of Category Theory.
Samuel Eilenberg and Saunders Mac Lane – the fathers of Category Theory.
The Last Mathematician from Hilbert’s Göttingen: Saunders Mac Lane as Philosopher of Mathematics.
Coherence condition at Wikipedia.
Categorification and de-categorification at Wikipedia.
Homology (mathematics) at Wikipedia.
David Spivak on Category Theory, posted by Simon Willerton at n-category café.
Category Theory for Programmers – the Preface by Bartosz Milewski, 2014.

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Key Articles:

What is a Concept? Joseph Goguen, in Dau, Mugnier & Stumme (Eds), Conceptual Structures: Common Semantics for Sharing Knowledge, Proceedings of the 13th International Conference on Conceptual Structures, Kassel, Germany, July 2005, Springer Verlag, Lecture Notes on Artificial Intelligence (LNAI 3596), ISBN-13 978-3-540-27783-5.
An Elementary Theory of the Category of Sets (long version) with commentary,
by F. William Lawvere.
An informal introduction to Topos theory, by Tom Leinster, arXiv, 2011.
Ologs – A categorical framework for knowledge representation, by David Spivak and Robert Kent, 2016.

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A category is a pattern, and a functor is an analogy between two patterns:

CategoryPattern-FunctorAnalogy

Five dogmas of category theory
(Joseph Goguen: A Categorical Manifesto, 1989):

Five dogmas of category theory

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Negozilla:

Negozilla

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The Bohr-Copernicus functor:

The Bohr-Copernicus functor

Updating theory through experiment:
(expressed in Knowledge Algebra):

Updating Theory through Experiment

Updating experiment through theory
(expressed in Knowledge Algebra):

Updating Experiment through Theory

Functorial transformation:

Functorial Transformation

A pattern is exemplified by another pattern:

A Pattern is Exemplified by another Pattern

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A functorial transformation \, f \, between two knowlecules \, K^1_{now} \, and \, K^2_{now} \, :

Functorial mapping between two knowlecules

Since each path equivalence in the syntactic graph expresses a fact, we can write:

\, [{K^1_{now}}]^{S_{yn}\times S_{yn}}_{S_{em}\times S_{em}} \, \, \xrightarrow[\; that \; respects \; facts \;]{\; graph \; morphism \;} \, [{K^2_{now}}]^{S_{yn}\times S_{yn}}_{S_{em}\times S_{em}} \,

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Functorial transformation:

Functorial Transformation

A functorial transformation \, f \, between two knowlecules \, K^1_{now} \, and \, K^2_{now} \, :

Functorial mapping between two knowlecules

Since each path equivalence in the syntactic graph expresses a fact, we can write:

\, [{K^1_{now}}]^{S_{yn}\times S_{yn}}_{S_{em}\times S_{em}} \, \, \xrightarrow[\; that \; respects \; facts \;]{\; graph \; morphism \;} \, [{K^2_{now}}]^{S_{yn}\times S_{yn}}_{S_{em}\times S_{em}} \,

A difference that makes a difference:

A difference that makes a difference

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How a difference makes a difference:

PrePatterns ---> NowPatterns ---> PostPatterns

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Interpreting objects through concepts:

Interpreting objects through concepts

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What we agree on and what we obey:

What we agree on and what we obey

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We as the limit and colimit of Me and You:

We as limit and colimit of Me and You

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Interpreting an Object through a Model:

Interpreting an Object through a Model

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Physically and mentally augmented senses:

Physically and Mentally Augmented Senses

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Overview of Human Communication:

Overview of Human Communication

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Functorial and non-functorial patterns:

Functorial and non-functorial patterns

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Communicating through thought-graphs:

Communicating through thought-graphs

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Mapping a thought-square 1:

Mapping a thought-square 1

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Mapping a thought-square 2:

Mapping a thought-square 2

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Mapping a thought-square 3:

Mapping a thought-square 3

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Mapping a thought-square 4:

Mapping a thought-square 4

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Mapping a thought-square 5:

Mapping a thought-square 5

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Mapping a thought-square 6:

Mapping a thought-square 6

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Thoughts and behavior patterns

Thoughts- and behavior patterns

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No drivers to learn when all triangles commute

No drivers to learn  when all triangles commute

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Learning kicks in when some triangles don’t commute

Learning "kicks in" when some triangles don't commute

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Better knowledge and skills restore commutativity

Better knowledge and skills restore commutativity

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Mental models – Ingrained behavior

Mental models - ingrained behavior

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Inter-reflection and inter-action

Inter-reflection and inter-action

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Shared mental models – shared ingrained behavior

Shared mental models - shared ingrained behavior

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Union and intersection of mental models and ingrained behavior

Union and intersection of mental models and ingrained behavior

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Unrepresented inter-relation mode

Unrepresented interrelation mode

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