DEF0250canonicalv1Simply Connected in Category τ
Categorical reinterpretation of simple connectivity: π₁^τ trivial iff every loop of transition functions in the Hartogs bulk is contractible. S³ is the terminal object among closed, simply connected 3-dimensional τ-spaces. Poincaré = uniqueness of the terminal object.
Payload
Simply Connected in Category τ
Categorical reinterpretation of simple connectivity: π₁^τ trivial iff every loop of transition functions in the Hartogs bulk is contractible. S³ is the terminal object among closed, simply connected 3-dimensional τ-spaces. Poincaré = uniqueness of the terminal object.
Simply Connected in Category τ
Summary
Categorical reinterpretation of simple connectivity: π₁^τ trivial iff every loop of transition functions in the Hartogs bulk is contractible. S³ is the terminal object among closed, simply connected 3-dimensional τ-spaces. Poincaré = uniqueness of the terminal object.
Statement
\label{def:simply-connected-tau}
A $\tau$-space $X$ (i.e., a presheaf on the primorial site enriched over the spectral algebra $A_{\mathrm{spec}}(\mathbb{L})$) is \textbf{simply connected in Category~$\tau$} if the automorphism group of its universal covering presheaf is trivial.
Proof / Justification
This item is definitional. No manuscript proof is required.
Source Context
- Registry source:
book-03.jsonlline 84 - Manuscript source:
2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch32-simply-connected-in-category-tau.texlines 33-35
Lean / Formalization Notes
- Formalization:
formalized - Module:
TauLib.BookIII.Doors.Poincare - Name:
simply_connected_check
Dependencies
- Canonical: III.R15, III.D01, III.D03, III.D25
Related Results
Generated by later projection phases.
Related Publications
Generated by later projection phases.
Revision Notes
- 2026-04-24: Initial pilot migration.
Identifiers
Aliases & legacy IDs
III.D35simply-connected-in-categorydef:simply-connected-tauRelease lines
corpus_v3_workingcorpus_v2Relations
Formalized by (2)
Appears in (1)
Downstream uses (computed) (4)
Items in the corpus that reference this one via load-bearing relations. Computed from the full corpus-v3 graph at build time.
Sources
Version & History
Status disclaimer
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