Acyclic models theorem
The acyclic models theorem for categories with models and its applications
Recent posts
Posts sorted by last_modified_at (most recently updated first).
Computation of Homology
Practical computation of homology via relative homology and Mayer–Vietoris
Covering Spaces
Equivalent conditions for simply connected, covering spaces, and the Seifert–van Kampen theorem
Cup Product
The exterior product in cohomology, the definition of cup product, and the resulting ring structure
Homotopy
Classifying spaces via topological invariants and the fundamental group
Poincaré Duality
Duality between homology and cohomology via orientation sheaves and fundamental classes
Algebraic Structures
Binary operations defined on sets
Change of Base Ring
Restriction and extension of scalars via a ring homomorphism
Direct Product of Groups
Direct product of groups
Free Products
Free product and universal property
Grothendieck Groups
The Grothendieck group and the definition of integers
Group Actions
Group action
Group Homomorphisms
Definition and properties of group homomorphisms, kernels and images
Isomorphism Theorems
Isomorphism theorems
Products, Coproducts, and Tensor Products of Rings
Definition of the product, coproduct, and tensor product of rings
Quotient Rings and Ring Homomorphisms
Quotient ring and ring isomorphism theorems
Directed Sets
Directed sets and lattices
Examples of Equivalence Relations
Examples of equivalence relations, saturation of equivalence relations, isomorphism theorems
Filters, Ideals, and Galois Connections
Filter and ideal
Limits
Inverse limit and direct limit
Monotone Functions
Operations on ordered sets and monotone functions
Natural Numbers and Infinite Sets
Definition of natural numbers and properties of infinite sets
Operations on Cardinals
Operations on cardinal numbers
Operations on Functions
Inverse and composition of functions, surjective and injective functions
Order Relations Between Ordinals
Order relations between ordinals and the rigorous definition of cardinals
Properties of Products
Partial products, associativity and distributivity
Cardinals
Definition of Cardinal number
Functions
Basic Definition of Functions
Ordinals and Well-Ordered Sets
Definition of well-ordered sets, motivation for ordinals
Product of Sets
Product of Sets
Retraction and Section
Properties of Surjective and Injective Functions
ZFC Axioms
The axioms of set theory
Cohomology
The definition of cohomology and the universal coefficient theorem
Topological Manifolds
The definition and properties of topological manifolds as locally Euclidean spaces
Graded Modules
The definition of graded modules over a graded ring
Graded Rings
Definition and basic properties of a graded ring indexed by a monoid
Abelian Groups
Free abelian group, tensor product
Direct Products, Direct Sums, and Tensor Products of Modules
Product, coproduct, and tensor product in the module category
Modules
Definition of modules
Field of Fractions
Localization, ring of fractions, prime ideal
Homology
Definitions and properties of simplices
Restricted Sums
Restricted sums of groups
Definition of a Ring
The definition of a ring and its basic properties
Quotient Groups
Normal subgroups and quotient groups
Semigroups, Monoids, and Groups
Definitions of semigroups, monoids, and groups
Operations on Binary Relations
Inverse and Composition of Binary Relations
Axiom of Choice
The Axiom of Choice and its equivalents
Properties of Well-Ordered Sets
Definition of ordinals and properties of well-ordered sets
Elements of Ordered Sets
Greatest, least, maximal, and minimal elements of ordered sets
Equivalence Relations
The definition and properties of equivalence relations