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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| The integers as a subgroup of the rationals | 12:40 PM - September 10, 2025 |
| Order of a power of an element | 12:40 PM - September 10, 2025 |
| Problem bank | 12:39 PM - September 10, 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 | 7:21 AM - March 31, 2025 |
| Exterior algebras | 7:22 AM - March 31, 2025 |
| Symmetric algebras | 7:23 AM - March 31, 2025 |
| Symmetric and alternating tensors | 7:23 AM - March 31, 2025 |
| Tensor algebras | 7:23 AM - March 31, 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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| Abelian categories | 12:33 PM - March 31, 2025 |
| Additive categories | 12:34 PM - March 31, 2025 |
| Adjoints | 12:38 PM - March 31, 2025 |
| Categories | 8:51 AM - June 13, 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 | 12:35 PM - March 31, 2025 |
| Exact sequences and chain homology | 12:35 PM - March 31, 2025 |
| Examples of adjoints | 12:39 PM - March 31, 2025 |
| Functor categories | 12:46 PM - March 31, 2025 |
| Functors | 12:46 PM - March 31, 2025 |
| Natural transformations | 12:47 PM - March 31, 2025 |
| Preadditive categories | 12:36 PM - March 31, 2025 |
| Properties of adjoints | 12:39 PM - March 31, 2025 |
| Special morphisms | 12:47 PM - March 31, 2025 |
| Spectral sequences | 12:37 PM - March 31, 2025 |
| The Salamander Lemma | 12:37 PM - March 31, 2025 |
| Universal arrows and elements | 12:48 PM - March 31, 2025 |
| Universal Properties I - Inspiring Examples | 2:10 PM - July 01, 2025 |
| Universal Properties II - Commutative diagrams, cones and limits | 3:08 PM - August 06, 2025 |
| Universal Properties III - Yoneda's Lemma | 3:08 PM - August 06, 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
| Note | Last Updated |
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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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| Note | Last Updated |
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| Systems of equations | 2:56 PM - August 06, 2024 |