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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Fiber bundle

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Summary

A locally trivial projection $p\colon E\to B$ with typical fiber $F$ .

Record metadata

Canonical name: Fiber bundle
Concept ID: fiber_bundle
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Topology
Domains: Topology, Differential geometry
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

total space $E$
base $B$
fiber $F$

Data

projection $p:E\to B$
local trivializations

Axioms / constraints

local product form

Notes

A bundle specifies total space, base space, projection, typical fiber, and local trivializations; it is not structure on a single topological space alone.

Concept sources

Wikipedia — Fiber bundle — Fiber bundle · expository · source ID wp-fiber-bundle
nLab — Fiber bundle — Fiber bundle · reference · source ID nlab-fiber-bundle
Morris — Topology Without Tears — Topology Without Tears · textbook · source ID topology-without-tears

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

topological spaces + locally trivial projection(construction junction)Fiber bundle

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: j_fiber_bundle__out
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: impose local triviality

Authored explanation

Give $E$ , $B$ , and $F$ topologies and a continuous projection $p\colon E\to B$ ; require local trivializations $p^{-1}(U)\cong U\times F$ that commute with projection to $U$ .

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Wikipedia — Fiber bundle — Fiber bundle · expository · source ID wp-fiber-bundle
nLab — Fiber bundle — Fiber bundle · reference · source ID nlab-fiber-bundle
Morris — Topology Without Tears — Topology Without Tears · textbook · source ID topology-without-tears

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Fiber bundle bundle + group action(construction junction)

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: j_principal_bundle__fiber_bundle
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: locally trivial projection

Authored explanation

Supply the Fiber bundle structure as one jointly required input.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Wikipedia — Principal bundle — Principal bundle · expository · source ID wp-principal-bundle
nLab — Principal bundle — Principal bundle · reference · source ID nlab-principal-bundle
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

Fiber bundle Vector bundle

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_fiber_vector
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + linear fibers

Authored explanation

Give each fiber a vector-space structure with linear transition functions.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Wikipedia — Vector bundle — Vector bundle · expository · source ID wp-vector-bundle
nLab — Vector bundle — Vector bundle · reference · source ID nlab-vector-bundle
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

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Fiber bundle

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Notes

Concept sources

Incoming relations (arrows to this concept)

topological spaces + locally trivial projection(construction junction)Fiber bundle

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Fiber bundle bundle + group action(construction junction)

Authored explanation

How to interpret this relation type

Relation sources

Fiber bundle Vector bundle

Authored explanation

How to interpret this relation type

Relation sources