Alphabeta MathArticle items · 02 — Eilenberg--Mac Lane space
Presents 27 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Whiskering and horizontal composition of natural transformations
- Group extensions, sections, complements, and split extensions
- Group isomorphisms, automorphisms and the set Aut(G)
- Relation, dom R, ran R, fld R, and the specialisations "relation from A to B" and "relation on A"
- The Cartesian product A × B := { z ∈ P(P(A ∪ B)) : ∃ a ∈ A ∃ b ∈ B z = (a,b) }
- Reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive, and connex relations on a set
- The power set P(x) = { z : z ⊆ x }
- A function is a relation f with (a,b) ∈ f and (a,c) ∈ f implying b = c; f : A → B, the value f(a), domain and codomain
- The Axiom Schema of Separation: for each formula φ, ∀ p̄ ∀ x ∃ y ∀ z (z ∈ y ↔ (z ∈ x ∧ φ(z,p̄)))
- Injection, surjection, bijection
- Category, object, morphism, domain, codomain, identity, composition, and hom-collection
- Opposite category Cᵒᵖ
- Loop classes form the group π₁(X,x₀) under concatenation
- Subgroup
- The intersection of a nonempty family of subgroups of G is a subgroup of G
- Group and abelian group
- A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
- Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree
- Semigroup and monoid
- Left identity, right identity, and two-sided identity for a binary operation
- Binary operation on a set; associativity, commutativity, and a subset closed under the operation
- Free group on a set of generators
- The normal closure of a subset of a group
- The quotient group G/N and coset product (gN)(hN)=ghN
- Words in an alphabet with formal inverses, elementary cancellation, and reduced words
- Reduced words form the free group on an alphabet
- Relators and relations; finitely generated, finitely related, and finite presentations
- The normal closure of R is the set of finite products of conjugates of elements of R and their inverses
- Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group
- In ⟨ X∣ R⟩, the words u and v represent the same element if and only if u⁻¹v∈⟨⟨ R⟩⟩
- A homomorphism that kills a normal subgroup factors uniquely through the quotient group
- Powers gⁿ: natural exponents in a monoid and integer exponents in a group, with g⁰ = e
- The natural numbers ℕ (von Neumann)
- The well-ordering principle
- Homotopy relative to a fixed subspace, and path homotopy relative to endpoints, are equivalence relations
- Based loops and the fundamental group
- Continuity of a map of topological spaces at a point and globally
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Paths, path-connected spaces and path components
- Simple graphs on an arbitrary vertex set
- Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph
- Graph isomorphisms, automorphisms and graph complements
- Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
- The quotient vector space V/W and its canonical projection
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Vector space over a field
- In any vector space 0_F v = 0_V, λ 0_V = 0_V, (-λ)v = -(λ v), (-1_F)v = -v, and λ v = 0_V forces λ = 0_F or v = 0_V
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open
- Connected components, quasicomponents, and totally disconnected spaces
- Natural transformation and its components
- Identity natural transformation and vertical composition
- Functor category [C,D]
- Isomorphism, groupoid, and connected category
2 · Logical flowchart
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3 · Items
DefinitionLibrary prerequisite
Eilenberg--Mac Lane space
DefinitionLibrary prerequisite
Equivalence, quasi-inverse, and adjoint equivalence of categories
DefinitionLibrary prerequisite
Equivalence of group extensions with fixed kernel and fixed quotient
DefinitionLibrary prerequisite
Equivalence relation, equivalence class, and the quotient set A/∼
DefinitionLibrary prerequisite
A free group action has no nonidentity element fixing a point
DefinitionLibrary prerequisite
Covariant functor, identity functor, composite functor, and contravariant functor
DefinitionLibrary prerequisite
Fundamental groupoid of a space
DefinitionLibrary prerequisite
The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups
DefinitionLibrary prerequisite
The geometric simplex spanned by affinely independent vertices
DefinitionLibrary prerequisite
Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges
DefinitionLibrary prerequisite
Left group actions, transitive actions, and faithful actions
DefinitionLibrary prerequisite
Monoid homomorphism and group homomorphism
DefinitionLibrary prerequisite
Group presentation by generators and relations
DefinitionLibrary prerequisite
Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
DefinitionLibrary prerequisite
Higher homotopy group by based cubes
DefinitionLibrary prerequisite
Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
DefinitionLibrary prerequisite
Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
DefinitionLibrary prerequisite
Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints
DefinitionLibrary prerequisite
The homomorphism on fundamental groups induced by a pointed continuous map
DefinitionLibrary prerequisite
The kernel and image of a group homomorphism
DefinitionLibrary prerequisite
Labelled directed graphs, their underlying simple graphs, and label-preserving isomorphisms
DefinitionLibrary prerequisite
Lifts of maps, paths, and homotopies through a covering map
DefinitionLibrary prerequisite
Linear hyperplane
DefinitionLibrary prerequisite
Linear subspace of a vector space
DefinitionLibrary prerequisite
Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point
DefinitionLibrary prerequisite
Morphisms of group extensions
DefinitionLibrary prerequisite