Alphabeta MathGroupoid covers, arrangement topology and ribbons
Connect path groupoids and covering quotients with arrangement complements, Salvetti and Deligne models, spherical Garside structures and ribbons, preserving each finite or spherical hypothesis.
1 · Prerequisites
- Category, object, morphism, domain, codomain, identity, composition, and hom-collection
- Covariant functor, identity functor, composite functor, and contravariant functor
- Isomorphism, groupoid, and connected category
- Group and abelian group
- Group isomorphisms, automorphisms and the set Aut(G)
- Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
- Lifting criterion for maps from path-connected locally path-connected spaces
- Existence and uniqueness of homotopy lifts through a covering map
- Higher homotopy group by based cubes
- Higher homotopy groups are functorial and based homotopy invariant
- Eilenberg--Mac Lane space
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- CW complex with closure finiteness and weak topology
- Higher homotopy basepoint transport and moving homotopies
- Partial order and partially ordered set
- Lattices, distributive lattices, and order ideals
- Artin monoid and Artin group presentations, and the canonical monoid-to-group map
- Finite directed paths form the free category on a quiver
- Equivalence relation, equivalence class, and the quotient set A/∼
- Covering-space actions by disjoint translates of neighbourhoods
- The orbit map of a covering-space action is a covering, with the acting group equal to the deck group when the total space is path-connected
- Fundamental groupoid of a space
- Existence and uniqueness of path lifts through a covering map
- A covering map induces an injective homomorphism on fundamental groups
- Semilocally simply connected spaces with explicit basepoint convention
- The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
- Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point
- Deck transformations and the deck-transformation group of a covering
- Based loops and the fundamental group
- Universal covering spaces
- Paths, path-connected spaces and path components
- Left group actions, transitive actions, and faithful actions
- The homomorphism on fundamental groups induced by a pointed continuous map
- The kernel and image of a group homomorphism
- Locally finite arrangements and relative face terminology
- The tensor and direct-sum models of complexification are canonically complex-linearly isomorphic
- 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
- 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
- Complexification as ℂ⊗_ℝV with its canonical real-linear embedding
- The direct-sum model V⊕ iV of a complexification
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0
- For a linear operator, boundedness, continuity at 0, continuity, and Lipschitz continuity are equivalent
- Natural transformation and its components
- Semigroup and monoid
- Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why CAT is not formed
- Garside categories, periodic loops and cyclic sets · §2, basic triple and Definitions 2.2–2.5, footnote 3; printed pp. 9–10
- Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups
- Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups
- Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares
- Reduced-word lifts and the prefix-order convention
- Subgroup
- The normalizer N_G(H)={g∈ G:gHg⁻¹=H} of a subgroup
- Normal subgroup: invariance under conjugation
- An internal semidirect product and a complement to a normal subgroup
- An action of a group H on a group N by automorphisms
- The external semidirect product N⋊_α H
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Definition
Set-sized groupoids, functors and composition conversion
Proposition
Coverings preserve the higher-homotopy obstruction to K(pi,1)
Definition
Divisibility and atomicity in a category
Lemma
Positive degree and atoms of the presented Artin monoid
Definition
Path presentations, congruences and formal inverses
Proposition
Covering-space actions and fundamental-groupoid quotients
Definition
Complexified complements in an open convex cone
Definition
Categorical Garside structures and simple arrows
Lemma
Unique lifts, endpoint fibres and characteristic subgroups
Lemma
Reduced Coxeter lifts in the presented positive monoid
Lemma