Welcome to Covariance
The general fact of the uniqueness of the universal arrows implies the uniqueness of the ... object, up to a unique isomorphism (who wants more?).
Recently updated notes
| Note | Last Updated |
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| Study Guide for Final Exam | 7:51 AM - December 10, 2025 |
| The Salamander Lemma | 3:38 PM - December 09, 2025 |
| 2025-12-03 | 1:56 PM - December 04, 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 |