Welcome to Covariance
..."category" has been defined in order to be able to define "functor" and "functor" has been defined in order to be able to define "natural transformation." Eilenberg and Mac Lane in Categories for the Working Mathematician (p. 28) | Natural transformations
Recently updated notes
| Note | Last Updated |
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| Homework 10 | 8:46 AM - December 10, 2025 |
| Study Guide for Final Exam | 7:51 AM - December 10, 2025 |
| The Salamander Lemma | 3:38 PM - December 09, 2025 |
Knowledge
Mathematics
Algebra theory
Not to be confused with the general concept of algebra (abstract or otherwise), an algebra is an algebraic structure similar to a module, only with two internal binary operations.
| Note | Last Updated |
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| Algebras | 2:03 PM - October 30, 2025 |
| Exterior algebras | 2:27 PM - November 06, 2025 |
| Symmetric algebras | 7:39 PM - November 02, 2025 |
| Symmetric and alternating tensors | 1:45 PM - November 04, 2025 |
| Tensor algebras | 7:42 PM - November 02, 2025 |
Algebraic geometry
| Note | Last Updated |
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| Examples of classical conic duality | 7:24 AM - March 31, 2025 |
Category theory
| Note | Last Updated |
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| Ab-categories | 6:51 AM - December 01, 2025 |
| Abelian categories | 6:55 AM - December 01, 2025 |
| Additive categories | 6:53 AM - December 01, 2025 |
| Adjoints | 8:45 PM - October 07, 2025 |
| Categories | 7:55 PM - October 07, 2025 |
| Chain complexes | 12:34 PM - March 31, 2025 |
| Diagram chases without elements | 7:26 PM - November 23, 2024 |
| Diagram lemmas | 12:34 PM - March 31, 2025 |
| Double complexes and mural maps | 6:40 AM - December 04, 2025 |
| Exact sequences and chain homology | 12:35 PM - March 31, 2025 |
| Examples of adjoints | 9:00 PM - October 07, 2025 |
| Functor categories | 8:42 PM - October 07, 2025 |
| Functors | 8:12 PM - October 07, 2025 |
| Kan extensions | 11:12 AM - November 24, 2025 |
| Natural transformations | 8:38 PM - October 07, 2025 |
| Properties of adjoints | 12:39 PM - March 31, 2025 |
| Special morphisms | 2:21 PM - November 18, 2025 |
| Spectral sequences | 12:37 PM - March 31, 2025 |
| The Salamander Lemma | 3:38 PM - December 09, 2025 |
| Universal arrows and elements | 12:48 PM - March 31, 2025 |
| Universal Properties I - Inspiring Examples | 7:56 PM - October 07, 2025 |
| Universal Properties II - Commutative diagrams, cones and limits | 3:08 PM - August 06, 2025 |
| Universal Properties III - Yoneda's Lemma | 10:14 AM - October 17, 2025 |
Differential equations
Field theory
| Note | Last Updated |
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| A question about finite fields | 10:26 AM - May 30, 2025 |
Group theory
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| Normal subgroups | 7:25 AM - March 31, 2025 |
Module theory
Representation theory
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| Representations of groups | 9:26 AM - July 08, 2025 |
Ring theory
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| Chinese Remainder Theorem | 9:26 AM - July 15, 2025 |
| Division rings | 9:54 AM - July 15, 2025 |
| Graded rings | 7:25 AM - March 31, 2025 |
| Least common multiples | 7:34 PM - July 21, 2024 |
| Principal ideal domains (PIDs) | 6:29 AM - May 07, 2024 |
Tropical algebraic geometry
| Note | Last Updated |
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| Investigation into tropical tangency loci | 7:27 AM - March 31, 2025 |
| Linear tropical varieties | 7:27 AM - March 31, 2025 |
| Quadratic tropical varieties | 7:27 AM - March 31, 2025 |
| Tropical varieties | 7:27 AM - March 31, 2025 |
Software
Blender
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Houdini
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| Systems of equations | 2:56 PM - August 06, 2024 |