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Eilenberg and Mac Lane | Universal Properties I - Inspiring Examples

...whenever new abstract objects are constructed in a specified way out of given ones, it is advisable to regard the construction of the corresponding induced mappings on these objects as an integral part of their definition.

Note	Last Updated
REU Meeting - 2026-07-02	10:18 AM - July 02, 2026
Summer REU 2026 - Categorical representation theory	9:56 AM - July 02, 2026
Summer REU 2026 - Categorical Lie theory	9:55 AM - July 02, 2026

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.

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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

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Examples of classical conic duality	7:24 AM - March 31, 2025

Category theory

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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

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Fourier series solutions I - A problem with power series	3:38 PM - August 02, 2025
Fourier series solutions II - Periodic functions	9:03 AM - August 06, 2025
Fourier series solutions III - Series representations of periodic functions	11:40 AM - August 05, 2025
Fourier series solutions IV - Computing Fourier coefficients	3:17 PM - August 06, 2025
Fourier series solutions V - Computing some Fourier series	8:24 AM - August 07, 2025
Fourier series solutions VI - Inner product spaces	12:49 PM - August 08, 2025
Fourier series solutions VII - Solving differential equations with Fourier series	9:25 AM - August 13, 2025
Fourier transform I - Pushing Fourier series to the limit	2:45 PM - August 18, 2025
Fourier transform II - The Fourier transform and inverse transform	3:21 PM - August 18, 2025
Fourier transform III - Properties of the Fourier transform	9:52 AM - August 19, 2025
Fourier transform IV - Derivatives	2:06 PM - August 19, 2025
Fourier transform V - Convolution	2:37 PM - August 19, 2025
Fourier transform VI - Solving differential equations	3:20 PM - August 19, 2025
Frobenius Series Solutions I - Slightly generalizing power series	3:30 PM - July 30, 2025
Frobenius Series Solutions II - Some illustrative examples	3:30 PM - July 30, 2025
Frobenius Series Solutions III - Frobenius theory	3:23 PM - August 02, 2025
Laplace transform I - Desperate times	1:41 PM - August 21, 2025
Laplace transform II - The inverse Laplace transform	2:10 PM - August 21, 2025
Laplace transform III - The Shifting Theorems	1:17 PM - August 22, 2025
Laplace transform IV - Convolution redux	8:42 AM - August 25, 2025
Laplace transform V - Solving differential equations	2:54 PM - August 23, 2025
Power Series Solutions I - From polynomials to power series	7:16 PM - July 28, 2025
Power Series Solutions II - First steps with power series	3:48 PM - July 28, 2025
Power Series Solutions III - A more representative example	8:21 AM - July 29, 2025
Power Series Solutions IV - Analytic functions and ordinary points	9:04 AM - July 31, 2025

Field theory

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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

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Bimodule morphisms	1:02 PM - October 13, 2025
Bimodules	8:52 AM - October 14, 2025
Direct products of modules	1:51 PM - October 08, 2025
Direct products vs. direct sums vs. sums	2:29 PM - October 08, 2025
Direct sums of modules	2:11 PM - October 08, 2025
Exact Sequences I - Illustrative Examples	6:46 AM - October 24, 2025
Exact Sequences II - Exact Sequences	2:38 PM - October 23, 2025
Exact Sequences III - Morphisms of Exact Sequences	7:26 AM - October 24, 2025
Exact Sequences IV - Exact Sequences and Functors	6:39 AM - November 10, 2025
Examples of free modules	2:25 PM - October 09, 2025
Free modules	2:17 PM - October 09, 2025
Generators for modules and submodules	2:31 PM - October 09, 2025
Jordan Canonical Form I - Definition	10:59 AM - November 14, 2025
Jordan Canonical Form II - Computation	10:12 AM - November 18, 2025
Linear independence, rank and the structure of free modules	4:50 PM - November 05, 2025
Module morphisms	10:28 AM - October 08, 2025
Module morphisms and submodules	10:43 AM - October 08, 2025
Modules	10:15 AM - October 08, 2025
Modules over a PID - The Fundamental Theorem	1:01 PM - March 31, 2025
Noetherian modules	1:01 PM - March 31, 2025
Quotient modules	10:47 AM - October 08, 2025
Rational Canonical Form I - Definition	1:02 PM - March 31, 2025
Rational Canonical Form II - Additional Properties	1:02 PM - March 31, 2025
Rational Canonical Form III - Computation	6:50 AM - November 18, 2025
Submodules	10:36 AM - October 08, 2025
Sums of submodules	2:14 PM - October 08, 2025
Tensor Products I - Extending scalars	6:48 AM - October 13, 2025
Tensor Products II - Tensor products of bimodules	2:19 PM - October 14, 2025
Tensor Products III - Balanced Maps and a Universal Property of the Tensor Product	6:51 AM - October 17, 2025
Tensor Products IV - The Adjoint Property	8:48 AM - October 24, 2025
The 2-category of bimodules	6:46 AM - October 10, 2024
The Isomorphism Theorems for Modules	10:13 AM - October 10, 2025

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

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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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L A T E X

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Systems of equations	2:56 PM - August 06, 2024