Category Theory

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


The sub-pages of this page are:

A Categorial Manifesto
Categorical Informatics
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


Related KMR-pages:

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



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,, 2018.
Category Theory in Context by Emily Riehl


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
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.


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.


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


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

Five dogmas of category theory





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


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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