Brown, Higgins, Sivera — Nonabelian Algebraic Topology (Fast Track 1.05a) — Audit + Gap Plan

Book: Ronald Brown, Philip J. Higgins, Rafael Sivera, Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids (EMS Tracts in Mathematics Vol. 15, European Mathematical Society 2011, xxxvi + 668 pp.). Hosted free by the author at https://groupoids.org.uk/pdffiles/NAT-book.pdf.

Fast Track entry: 1.05a (Brown's sequel to 1.05 Topology and Groupoids). Added to the catalog 2026-05-17 — previously omitted from docs/catalogs/FASTTRACK_BOOKLIST.md although linked from the Fast Track source page inside the §1.05 Brown toggle as "his sequel book."

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units to write so that Nonabelian Algebraic Topology (NAT hereafter) is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

This pass is intentionally not a full P1 audit. NAT is a 668-page research monograph; a complete P1 inventory at line-number granularity is a multi-week job and is deferred to a dedicated audit pass. This plan works from NAT's canonical topic list and the authors' distinctive editorial choices, produces the gap punch-list, and stops there.

§1 What NAT is for

NAT is the higher-dimensional sequel to Brown's Topology and Groupoids. Where Topology and Groupoids (1.05) extends classical algebraic topology from fundamental groups to fundamental groupoids — gaining the disconnectedness-tolerant Seifert-van Kampen theorem — NAT extends the programme one dimension further, replacing groupoids with strict cubical ω-groupoids and crossed complexes. The payoff: explicit, computable nonabelian higher analogues of relative homotopy groups, with a higher-dim Seifert-van Kampen theorem (HvKT) that actually computes second relative homotopy groups in cases standard algebraic topology cannot.

Distinctive contributions, in roughly the order NAT develops them:

Filtered spaces as the right ambient category for higher Seifert-van Kampen. Replaces pairs and triads.
Crossed complex $C (X_{*})$ of a filtered space — a non-abelian chain complex carrying the second and higher relative homotopy groups together with the boundary operators that link them.
Higher Homotopy Seifert-van Kampen Theorem (HvKT). Brown-Higgins 1981; the central theorem of the book. Lets one compute crossed complexes of pushouts of filtered spaces — the higher-dim analogue of the group-theoretic SvKT — and hence calculate higher relative homotopy groups in concrete examples (relative $π_{2}$ of mapping cones with nontrivial $π_{1}$ , etc.).
Cubical ω-groupoid $ρ (X_{*})$ of a filtered space, and the equivalence of categories with crossed complexes (Brown-Higgins). Cubical methods are essential to the proof of HvKT because cubes compose in all directions; simplices do not.
Tensor product and homotopies of crossed complexes — gives the monoidal closed structure that lets one form function objects and prove the cubical equivalence.
Free crossed resolutions of groups and groupoids; the identities-among-relations problem.
Higher Whitehead products computed via the crossed complex framework (where classical methods stall).
Connections to nonabelian cohomology, classifying spaces of crossed complexes, and pointer toward the further programme (∞-groupoid models, Grothendieck's Pursuing Stacks, homotopy type theory).

NAT is not a first introduction to algebraic topology. It assumes Topology and Groupoids (1.05), basic homotopy theory (mapping cones, fibrations, CW), and standard homological algebra. It is the canonical entry point to higher-dim algebraic topology if one wants the computable, strict-ω-groupoid programme rather than the ∞-categorical / Joyal-Lurie / quasi-category programme. The two programmes are equivalent in their content but very different in style; the Fast Track explicitly chooses NAT.

§2 Coverage table (Codex vs NAT)

Cross-referenced against the current 313-unit corpus. ✓ = covered, △ = partial / different framing, ✗ = not covered.

NAT topic	Codex unit(s)	Status	Note
Fundamental groupoid $π_{1} (X, A)$	—	✗	Gap — already on the Brown 1.05 punch-list (`03.12.0a`); NAT depends on it.
Seifert-van Kampen for groupoids (group + groupoid)	—	✗	Gap — already on Brown 1.05 punch-list (`03.12.0b`). NAT prerequisite.
Filtered space	—	✗	Gap. Definition + standard examples (CW filtration, skeletal filtration).
Mapping-cone construction	△	△	Touched in `03.12-homotopy/` but not its own unit; called out in Brown 1.05 punch-list as `02.01.06` quotient-topology dependency (already shipped).
Relative homotopy group $π_{n} (X, A, x_{0})$	—	✗	Gap. Foundational and load-bearing for NAT.
Whitehead's crossed module of a pair $Π_{2} (X, A, x_{0}) \to π_{1} (A, x_{0})$	—	✗	Gap. This is the 2-dim object NAT generalises to all dimensions.
Crossed complex of a filtered space	—	✗	Gap (high priority — NAT's central object).
Cubical singular complex / cubical ω-groupoid	—	✗	Gap. Cubical methods are absent from the Codex; need a dedicated stub even if Codex doesn't follow the cubical track further.
Equivalence: crossed complexes ≃ cubical ω-groupoids	—	✗	Gap. Brown-Higgins theorem; pointer unit only at FT-equivalence.
Higher Homotopy Seifert-van Kampen (HvKT)	—	✗	Gap (high priority — NAT's central theorem).
Worked computation: $π_{2}$ of the mapping cone $S^{1} \cup_{2} D^{2}$	—	✗	Gap. The signature demonstration that HvKT computes where classical methods don't.
Free crossed resolution of a group	—	✗	Gap. Connects to identities-among-relations and group cohomology.
Tensor product of crossed complexes	—	✗	Gap (low priority — internal machinery, not load-bearing for FT-equivalence).
Higher Whitehead product via crossed complexes	—	✗	Gap (Master-tier deepening).
Classifying space $B (C)$ of a crossed complex	—	✗	Gap (low priority — survey-level pointer).
Pointer to ∞-groupoid models / Grothendieck Pursuing Stacks	—	✗	Gap (low priority — Master-tier connection only).

Aggregate coverage estimate: ~0% of NAT has corresponding Codex units. The gap is total. This is unsurprising — NAT is a research-monograph extension of Brown 1.05, and Brown 1.05 itself is only ~30% covered. Closing the Brown 1.05 punch-list is a hard prerequisite for any meaningful NAT coverage.

§3 Gap punch-list (P3-lite — units to write, priority-ordered)

Priority 0 — blocked by Brown 1.05 punch-list: Items 5, 6, 7 of plans/fasttrack/brown-topology-and-groupoids.md (fundamental group, fundamental groupoid, Seifert-van Kampen) must ship before NAT units can be written. NAT inherits all 1.05 prereqs.

Priority 1 — high-leverage, captures NAT's central content:

03.12.10 Relative homotopy group $π_{n} (X, A, x_{0})$ . Standard definition, action of $π_{1} (A)$ , long exact sequence of a pair. Hatcher §4.1 anchor; NAT §B.1.6 anchor. Three-tier, ~1500 words. Foundational for HvKT.
03.12.11 Whitehead's crossed module $Π_{2} (X, A, x_{0}) \to π_{1} (A, x_{0})$ of a pair. Brown-Higgins notation. Includes the worked example of $π_{2}$ of a mapping cone in the simply-connected vs non-simply-connected cases (the latter being where classical methods stall). NAT §2.1 anchor.
03.12.12 Filtered space. Definition (sequence $X_0 \subseteq X_1 \subseteq \cdots $), s t an d a r d e x am pl es (C W s k e l e t a l f i l t r a t i o n,$ n$-skeleta). NAT §B.7 anchor. Short unit (~1000 words); mostly definitional.
03.12.13 Crossed complex of a filtered space. The central object. Construction $C (X_{*})_{n} = π_{n} (X_{n}, X_{n - 1}, *)$ for $n \geq 2$ , $C(X_*)_1 = \pi_1(X_1)$ (a groupoid), boundary operators. NAT §7.1 anchor. Master tier required; Intermediate tier covers the definition; Beginner tier gives the 2-truncation picture.
03.12.14 Higher Homotopy Seifert-van Kampen Theorem (HvKT). Statement, sketch of the proof via cubical methods, and the canonical worked computation: $π_{2}$ of $S^{1} \cup_{2} D^{2}$ (Möbius band / real-projective-plane mapping cone). Brown-Higgins 1981 Proc. London Math. Soc. as originator-citation; NAT §8.1 anchor. Three-tier; Beginner section gives only the statement and the worked computation without proof.

Priority 2 — cubical side and resolutions:

03.12.15 Cubical singular complex / cubical ω-groupoid $ρ (X_{*})$ . Pointer unit at FT-equivalence: definition and statement of the Brown-Higgins equivalence with crossed complexes. NAT §6 anchor. Master-only, ~1500 words.
03.12.16 Free crossed resolution of a group. Identities-among- relations problem; connection to group cohomology via the bar resolution comparison. NAT §10 anchor. Intermediate + Master.

Priority 3 — Brylinski-style deepenings (Master-tier, not strictly required for FT equivalence):

Tensor product and monoidal closed structure on crossed complexes. NAT §9. Add as a Master section to 03.12.13 rather than a new unit; referenced in passing.
Higher Whitehead products via crossed complexes. Add as a Master section to 03.12.10 or 03.12.13.

Priority 4 — survey pointers (optional, Master-only):

03.12.17 Classifying space of a crossed complex. Pointer unit. Connects to the further programme (∞-groupoid models, Grothendieck, homotopy type theory). NAT §11 anchor.

§4 Implementation sketch (P3 → P4)

For a full NAT coverage pass, items 1–5 are the minimum set (with the Brown 1.05 punch-list as a strict prerequisite). Realistic production estimate (mirroring earlier Brown / Lawson-Michelsohn / Bott-Tu batches):

~3–4 hours per unit. NAT units skew higher than the corpus average because the master tier requires careful crossed-complex notation and worked computations are nontrivial.
5 priority-1 units × ~3.5 hours = ~17–18 hours of focused production. Plus the Brown 1.05 priority-1 prereqs (~21 hours). Total ~40 hours. Fits a focused 5–7 day window.

Originator-prose target. Ronald Brown (with Higgins, with Loday) is the originator of the higher-dim Seifert-van Kampen programme. Units 4 and 5 should carry originator-prose treatment per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, citing:

R. Brown, P. J. Higgins, "Colimit theorems for relative homotopy groups," J. Pure Appl. Algebra 22 (1981) 11–41 — originating HvKT for crossed complexes.
R. Brown, P. J. Higgins, "On the algebra of cubes," J. Pure Appl. Algebra 21 (1981) 233–260 — originating the cubical ω-groupoid / crossed complex equivalence.
R. Brown, P. J. Higgins, R. Sivera (2011) — the book itself, definitive consolidation.

Notation crosswalk. NAT uses both $Π_{n} (X, A, x_{0})$ (Brown-Higgins crossed-module form) and $π_{n} (X, A, x_{0})$ (classical relative homotopy group) — they are the same set, but $Π$ carries the crossed-module action in the notation. NAT also writes $ρ (X_{*})$ for the cubical ω-groupoid and $C (X_{*})$ for the crossed complex. The Codex notation decision (per docs/specs/UNIT_SPEC.md §11) should: use $π_{n} (X, A, x_{0})$ for the relative homotopy group as a set, use $Π_{2} (X, A, x_{0}) \to π_{1} (A, x_{0})$ explicitly when the crossed-module structure is invoked, and adopt NAT's $ρ (X_{*})$ and $C (X_{*})$ for the cubical and crossed-complex functors. Record in a §Notation paragraph of 03.12.11 and 03.12.13.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in NAT (full P1 audit; deferred — NAT is 668 pp.).
Exercise-pack production. NAT exercises are extensive and often open-ended; exercise pack is a P3-priority-3 follow-up after the priority-1 units ship.
The cubical track beyond the single pointer unit 03.12.15. The Codex is not committing to a parallel cubical curriculum.
Connections to ∞-groupoid models / quasi-categories / Lurie. Pointer only in 03.12.17.

§6 Acceptance criteria for FT equivalence (NAT)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

The Brown 1.05 punch-list priority-1 units have shipped (strict prereq).
≥95% of NAT's named theorems in chapters 1–10 map to Codex units (currently 0%; after priority-1 units this rises to ~70%; after priority-1+2 to ~90%; full ≥95% requires priority-3 + selective priority-4).
≥90% of NAT's worked computations in chapters 1–10 have a direct unit or are referenced from a unit that covers them.
Notation decisions are recorded (see §4).
Pass-W weaving connects the new units to 03.12-homotopy/ and to the Brown 1.05 priority-1 units via lateral connections.

The 5 priority-1 units close most of the equivalence gap given the Brown 1.05 prereqs are in place. Priority-2 closes the cubical and free-resolution gaps. Priority-3+4 are deepenings.

§7 Sourcing

Free. Author-hosted PDF at https://groupoids.org.uk/pdffiles/NAT-book.pdf.
License. Author has placed the PDF freely available; for educational use cite as Brown, Higgins, Sivera, Nonabelian Algebraic Topology, EMS Tracts in Mathematics 15, European Mathematical Society 2011.
Local copy. Add to reference/fasttrack-texts/01-fundamentals/ as Brown-Higgins-Sivera-NonabelianAlgebraicTopology.pdf to mirror the pattern of other free FT texts.

Brown, Higgins, Sivera — *Nonabelian Algebraic Topology* (Fast Track 1.05a) — Audit + Gap Plan