What is a Natural Transformation? Definition and Examples (Parts 1 and 2)
by Tai-Danae Bradley · Math3ma
Two illustrated blog posts (February 2017) defining natural transformations via commuting squares, with examples: cones, natural isomorphisms, equivariant maps of group actions, the vector space double dual, and representable functors. Part 2 is linked from Part 1. Best free gentle-but-correct written introduction dedicated specifically to this concept; precise definitions plus concrete math examples, not a summary-of-summaries.
More resources on Natural Transformations
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Natural Transformation
The nLab wiki entry defining natural transformations between functors, with component and naturality-square formulations, vertical and horizontal composition, and links to 2-categorical generalizations. Best used as a precise reference once the basic definitions of category and functor are familiar.
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Learn natural transformations in Category Theory for Programmers with Bartosz Milewski's insightful video course. Master functional programming concepts!
The Catsters: Natural Transformations 1-3A
Four short blackboard lectures (Natural transformations 1, 2, 3, 3A) by Eugenia Cheng of the Sheffield Catsters covering the definition, examples, functor categories and vertical/horizontal composition of natural transformations.
nLab
Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.