Visual Category Theory

The current full set in either PDF or paperback or both is available for purchase from Software Diagnostics Services

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Concepts from category theory were used as metaphors for some trace and log analysis patterns (see Mathematical Concepts in Software Diagnostics and Software Data Analysis) and also as a foundation of software diagnostics (see Categorical Foundations of Software Diagnostics) as a part of Theoretical Software Diagnostics. However, category theory abstractions are very challenging to apprehend correctly, require a steep learning curve for non-mathematicians, and, for people with traditional naïve set theory education, a paradigm shift in thinking. The book utilizes a novel approach to teach category theory and abstract mathematics in general by using LEGO® bricks. This method was discovered when applying the same technique to teach machine learning, its data structures and algorithms, particularly directed graphs.

Part 0 (ISBN-13: 978-1912636396) covers universe and sets, set-builder notation, set membership, set inclusion, subsets as members, membership vs. subset, powerset, relations, functions, domain, codomain, range, injection, surjection, bijection, product, union, intersection, set difference, symmetric set difference, sets of functions, function composition, inverse functions.


Download Free Part 0

In order to facilitate earlier adoption and feedback, the book was split into small manageable parts. Part 1 (ISBN-13: 978-1912636402) is currently available on Leanpub and Amazon Kindle Store. It covers the definition of categories, arrows, the composition and associativity of arrows, retracts, equivalence, covariant and contravariant functors, natural transformations, and 2-categories.


Book Sample

CoPart 1 (ISBN-13: 978-1912636815) is a dual complement to Visual Category Theory Brick by Brick, Part 1. It is currently available on Leanpub and Amazon Kindle Store. The original series translated abstract categorical concepts into the language of LEGO® bricks, and the CoPart series implement the opposite way of translating brick constructions to the standard diagram language of category theory that should benefit comprehension of definitions. Since usual categorical diagrams are black and white and occupy less space on paper, CoParts include additional color-enhanced diagrams in the spirit of brick constructions when arrow source and target parts use different colors. These CoParts from CoSeries (named after opposite categories with reversed arrows) keep the same 1-to-1 page correspondence between Parts and CoParts. Page layout is also similar: location of explanatory notes (written using standard mathematical notation) is the same — only bricks are replaced by letters, dots, and arrows. Therefore, this CoSeries can be used independently from the original series or together.


Book Sample

Part 2 (ISBN-13: 978-1912636419) is currently available on Leanpub and Amazon Kindle Store. It covers duality, products, coproducts, biproducts, initial and terminal objects, pointed categories, matrix representation of morphisms, and monoids.


Book Sample

CoPart 2 (ISBN-13: 978-1912636822) is a dual complement to Visual Category Theory Brick by Brick, Part 2. It is currently available on Leanpub and Amazon Kindle Store.


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Part 3 (ISBN-13: 978-1912636426) is currently available on Leanpub and Amazon Kindle Store. It covers adjoint functors, diagram shapes and categories, cones and cocones, limits and colimits, pullbacks and pushouts.


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Part 4 (ISBN-13: 978-1912636433) is currently available on Leanpub and Amazon Kindle Store. It covers non-concrete categories, group objects, monoid, group, opposite, arrow, slice, and coslice categories, forgetful functors, monomorphisms, epimorphisms, and isomorphisms.


Book Sample

Part 5 (ISBN-13: 978-1912636440) is currently available on Leanpub and Amazon Kindle Store. It covers exponentials and evaluation in sets and categories, subobjects, equalizers, equivalence classes and quotients, coequalizers, congruence categories, morphism functors, and presheaves.


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Part 6 (ISBN-13: 978-1912636457) is currently available on Leanpub and Amazon Kindle Store. It covers ideas that require a leap of abstraction: vertical and whisker compositions of natural transformations, identity and isomorphism of functors, equivalence, isomorphism, and adjoint equivalence of categories, functor and morphism categories, natural transformations as functors, representable functors, category of presheaves, Yoneda embedding and lemma. It also includes an index for parts 1 - 6.


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Part 7 (ISBN-13: 978-1912636464) is currently available on Leanpub and Amazon Kindle Store. It covers ideas related to functional programming: exponentials, disjoint unions, endofunctors and natural transformations, partial and total functions, monads.


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The first 5 parts are available as Visual Category Theory bundle on Leanpub.

Amazon 9-book Kindle bundle

All 8 parts together are now available in paperback format:

  • Title: Visual Category Theory Brick by Brick: Diagrammatic LEGO® Reference
  • Authors: Dmitry Vostokov
  • Publisher: OpenTask (October 2021)
  • Language: English
  • Product Dimensions: 16.5 x 16.5 cm (6.5 x 6.5 in)
  • Paperback: 172 pages
  • ISBN-13: 978-1912636389

Also available on Amazon and Barnes&Noble.

Applications of category theory to software diagnostics also include Software Codiagnostics and Diagnostic Operads.

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