Home

Welcome to Covariance

Eilenberg and Mac Lane | Natural transformations

..."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."

Note	Last Updated
Summer REU 2024	1:06 PM - March 31, 2025
Tensor Products IV - The Adjoint Property	1:05 PM - March 31, 2025
Tensor Products III - Balanced Maps and a Universal Property of the Tensor Product	1:04 PM - March 31, 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
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
Examples of classical conic duality	7:24 AM - March 31, 2025

Category theory

Topic
Abelian categories
Additive categories
Adjoints
Categories
Chain complexes
Diagram chases without elements
Diagram lemmas
Double complexes and mural maps
Exact sequences and chain homology
Examples of adjoints
Functor categories
Functors
Natural transformations
Preadditive categories
Properties of adjoints
Special morphisms
Spectral sequences
The Salamander Lemma
Universal arrows and elements
Universal Properties I - Inspiring Examples
Universal Properties II - Commutative diagrams, cones and limits
Universal Properties III - Yoneda's Lemma

Field theory

Topic
A question about finite fields

Group theory

Topic
A condition to be non-cyclic
A condition under which a group must be abelian
A condition under which a group must be abelian (2)
A dihedral group that is not an internal direct product
A group isomorphic to a subgroup of a direct product of quotient groups
A group isomorphic to an internal direct product
A group of upper-triangular matrices
A group with a trivial automorphism group
A group with a trivial automorphism group (2)
A property of the order of an element
An automorphism of a group of odd order
Another condition for a group to be abelian
Automorphisms of a finite cyclic group
Closely related subgroups of a finite group
Comparing cosets
Computations in symmetric groups
Computations with inner automorphisms
Computing an automorphism group
Counting morphisms between specified groups
Directly proving the existence of an element of a desired order
Elements of finite order
Elements of order 2
Existence of a normal subgroup of finite index
Existence of an identity element in a group
Existence of automorphisms
Finding all morphisms between two groups
Image of a normal subgroup and induced morphisms
Image of the identity is the identity
Indices and intersections
Inner and outer automorphisms
Inner automorphisms and the center of a group
Inner automorphisms of an alternating group
Nonexistence of a simple group of a given order
Nonexistence of morphisms between two groups
Nonexistence of small nonabelian groups
Normal subgroups
Normal subgroups with trivial intersection
Normality and the operation on cosets
Normality and the operation on cosets (defunct)
Normalizer of a subgroup
Normalizers and centralizers
Order of a power of an element
Order of an element in a finite group
Order of elements in a symmetric group
Orders of elements in a quotient group
Preimage of a subgroup
Product of two subgroups
Products of quotient groups
Projection onto a quotient
Properties of the center of a group
Property of the order of an element
Proving Lagrange's Theorem
Stabilizer of a coset
Subgroups of a group of even order
The cyclic group of order 2020
The integers as a subgroup of the rationals
The Third Isomorphism Theorem
Verifying axioms of a group
Working modulo 11

Module theory

Topic
Bimodule morphisms
Bimodules
Direct products of modules
Direct products vs. direct sums vs. sums
Direct sums of modules
Exact Sequences I - Illustrative Examples
Exact Sequences II - Exact Sequences
Exact Sequences III - Morphisms of Exact Sequences
Exact Sequences IV - Exact Sequences and Functors
Examples of free modules
Free modules
Generators for modules and submodules
Jordan Canonical Form I - Definition
Jordan Canonical Form II - Computation
Linear independence, rank and the structure of free modules
Module morphisms
Module morphisms and submodules
Modules
Modules over a PID - The Fundamental Theorem
Noetherian modules
Quotient modules
Rational Canonical Form I - Definition
Rational Canonical Form II - Additional Properties
Rational Canonical Form III - Computation
Submodules
Sums of submodules
Tensor Products I - Extending scalars
Tensor Products II - Tensor products of bimodules
Tensor Products III - Balanced Maps and a Universal Property of the Tensor Product
Tensor Products IV - The Adjoint Property
The 2-category of bimodules
The Isomorphism Theorems for Modules

Ring theory

Topic
A maximal ideal in a function ring
A non-PID
A ring in which all prime ideals are maximal
An evaluation morphism
An isomorphism of rings
Automorphisms of a ring
Boolean algebras
Boolean rings are commutative
Characteristic of a ring
Characteristic of a ring (2)
Chinese Remainder Theorem
Constructing a field extension
Constructing the field with eight elements
Dimension of a PID
Euclidean domains are PIDs
Evaluation at i
Existence of an identity element in a finite ring
Existence of certain ring morphisms
Generator for a field extension
Graded rings
Group of units of a product
Ideals in a polynomial ring
Ideals in a polynomial ring (2)
Idempotent elements in a ring
Idempotent elements in a ring (2)
Image of an evaluation morphism
Image of the identity element under a ring morphism
Maximal ideals in a PID
Morphism from the Gaussian integers
Morphism from the Gaussian integers (2)
Nilpotent elements
Nilpotent elements in a ring
Nonzero prime ideals are maximal in a PID
Polynomials with even constant term
Prime and irreducible elements in a commutative ring
Prime ideals and quotient rings
Principal ideal domains (PIDs)
Properties of Boolean rings
Properties of the annihilator
Proving an ideal is prime
Quotienting out nilpotent elements
Quotients and direct products
The field with eight elements
The field with nine elements
The kernel of an evaluation morphism
The nilradical of a ring
The structure of the integers as both a group and a ring
Using the Chinese Remainder Theorem

Tropical algebraic geometry

Topic
Investigation into tropical tangency loci
Linear tropical varieties
Quadratic tropical varieties
Tropical varieties

Software

Blender

Topic

Houdini

Topic

L A T E X

Topic
Systems of equations