Paul Goerss, John Jardine — *Simplicial Homotopy Theory* (Fast Track 3.41) — Audit + Gap Plan

Book: Paul G. Goerss and John F. Jardine, Simplicial Homotopy Theory (Birkhäuser, Progress in Mathematics 174, 1999; xv + 510 pp.; ISBN 3-7643-6064-X; reprinted 2009 with corrections as Modern Birkhäuser Classics). The canonical modern reference for simplicial homotopy theory presented through Quillen's model-category language.

Fast Track entry: 3.41, inside §3 Modern Geometry, immediately after May Concise (3.38), May-Ponto More Concise (3.39), and May Simplicial Objects (3.40). Goerss-Jardine (hereafter GJ) is the explicit "modern successor" to May 3.40: the same simplicial-set foundations but presented through Quillen model categories rather than the classical combinatorial viewpoint.

Purpose of this plan: P1-stub audit + gap punch-list. Output is a concrete priority-ordered list of new units to write so that GJ is covered to the Fast Track equivalence threshold (≥95% effective coverage — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

Reduced-source note. No local PDF is in reference/textbooks-extra/ or reference/fasttrack-texts/03-modern-geometry/. WebFetch against Springer returned a 303 to the authenticated identity-provider gateway and is gated; Anna's Archive / direct PDF mirrors were not retrieved within the hard time limit. This audit is REDUCED: worked from the book's well-attested chapter / section structure as cross-cited inside the existing plans/fasttrack/may-simplicial-objects.md plan (which gives explicit GJ section locators §I.1, §I.2, §I.3, §I.7, §III.2, §V.3), inside the shipped content/03-modern-geometry/12-homotopy/03.12.25-simplicial-sets-and-geometric-realization.md unit's tier anchors and reference list, and from standard knowledge of GJ's seven-chapter Progress-in-Mathematics structure. A future revision with the local PDF should sharpen the punch-list at the named-theorem level (particularly Chapters IV-VII on bisimplicial sets, Bousfield-Kan, and simplicial sheaves) but the units-to-write list below is robust against the reduction.

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§1 What GJ is for

GJ is the canonical modern reference for simplicial homotopy theory. Where May 1967 Simplicial Objects (FT 3.40) treats the simplicial category and Kan complexes combinatorially — face/degeneracy operators, simplicial homotopy groups, twisted cartesian products — and predates Quillen's model categories by a few months, GJ treats the exact same objects through the model-category machinery Quillen introduced in Homotopical Algebra (1967) and which has become the lingua franca of modern algebraic topology and higher category theory. The two books cover overlapping material; the choice between them is not what is covered but how.

Distinctive contributions, in roughly the order GJ develops them:

1. The simplicial category $\Delta$ and simplicial sets, with the Yoneda-flavoured emphasis (GJ §I.1). Simplicial sets as presheaves ${\Delta }^{\mathrm{op}}\to \mathbf{Set}$; standard simplex $\Delta^n = \mathrm{Hom}\Delta(-, [n])$;thedensitytheorem$X = \mathrm{colim}{\Delta^n \to X} \Delta^n$ that makes every simplicial set a colimit of standard simplices. This is the categorical reframing of May §1's normal-form factorisation.
2. Geometric realisation and the $|\cdot |⊣\mathrm{Sing}$ adjunction (GJ §I.2). Realisation via the coend $|X|={\int }^{[n]}{X}_n\times {\Delta }^n$; theorem that $|\cdot |$ preserves finite products in compactly generated spaces (Milnor 1957); the adjunction with $\mathrm{Sing}$ as the load-bearing bridge to $\mathbf{Top}$.
3. Kan complexes and the extension condition (GJ §I.3). The horn-filling property as right lifting against ${\Lambda }_k^n\hookrightarrow {\Delta }^n$; simplicial homotopy as an equivalence relation on Kan complexes; every simplicial group is a Kan complex.
4. The Quillen model structure on $\mathbf{sSet}$ (GJ §I.11 / Ch II). The central technical achievement: cofibrations = monomorphisms, weak equivalences = maps inducing weak equivalences on geometric realisation, fibrations = Kan fibrations. Existence proved via the small-object argument plus the explicit anodyne extensions of Gabriel-Zisman. The proof that this is a model structure occupies most of GJ Chapter I-II and is one of the most technically demanding passages in the book.
5. Geometric realisation as a Quillen equivalence $|\cdot |⊣\mathrm{Sing}:\mathbf{sSet}\rightleftarrows \mathbf{Top}$ (GJ §I.11). The full statement: the adjunction is a Quillen equivalence between $\mathbf{sSet}$ with the Kan-Quillen model structure and $\mathbf{Top}$ with the standard (Quillen) model structure. This is the theorem that justifies doing homotopy theory combinatorially: it says the two homotopy categories are equivalent, not merely related.
6. Simplicial groups, $W\bar{G}$ classifying space, principal fibrations (GJ Ch V). The simplicial-group analogue of $BG$; Kan's loop-group functor $G⊣W\stackrel{ˉ}{}$; the classification of principal $G$-fibrations as $[\cdot ,W\bar{G}]$. Modern reframing of May §17-§21 twisted cartesian products.
7. Bisimplicial sets and homotopy colimits (GJ Ch IV). The diagonal $\mathrm{diag}:{\mathbf{sSet}}^{{\Delta }^{\mathrm{op}}}\to \mathbf{sSet}$, Bousfield-Kan homotopy colimit construction $\mathrm{hocolim}$, the Reedy model structure, and the homotopy spectral sequence of a bisimplicial set.
8. The Bousfield-Kan spectral sequence (GJ Ch VI / Ch VIII). The spectral sequence of a cosimplicial space converging (under connectivity hypotheses) to the homotopy groups of the totalisation. Includes the unstable Adams spectral sequence as a special case. Central tool of unstable homotopy theory in the post-Quillen era.
9. Simplicial presheaves / simplicial sheaves and Jardine's local model structures (GJ Ch VII). Jardine's own programme: simplicial homotopy theory over a Grothendieck site, with the local model structure where weak equivalences are stalkwise. Foundational for motivic homotopy theory (Morel-Voevodsky 1999, same year as GJ). The Codex defers this entirely (see §5).
10. Cosimplicial spaces and Reedy theory (GJ Ch VII). The dual to bisimplicial sets; Reedy fibrant replacement; obstruction theory for cosimplicial diagrams.

Peer-source corroboration of the framing:

• Quillen, Homotopical Algebra (Springer LNM 43, 1967). The originator text for model categories. Quillen §II.3 puts the model structure on $\mathbf{sSet}$ for the first time and proves the $|\cdot| \dashv \mathrm{Sing}$ Quillen equivalence. GJ §I.11 is the modern textbook consolidation of Quillen §II.3 with all proofs filled in.
• Hovey, Model Categories (AMS Mathematical Surveys 63, 1999). The systematic reference for model-category theory itself, published the same year as GJ. Hovey Ch 3 treats the simplicial-set model structure as a worked example; GJ inverts the emphasis (simplicial sets first, model structure as the technology). The two books are explicitly complementary and should be used together for the model-category strand.
• May, Simplicial Objects in Algebraic Topology (1967). FT 3.40, the classical predecessor. GJ cites May §1, §2, §3, §14-§16 for the pre-model-category material and reproves much of it in modern notation. The two books are not redundant: May is shorter, more direct, and the canonical citation for the classical simplicial theory; GJ is the canonical citation for the model-category presentation and for bisimplicial / Bousfield-Kan content that May does not cover.
• May & Ponto, More Concise Algebraic Topology (FT 3.39). Develops model-category foundations in Ch 14-19 and uses them for localisation, completion, and rational homotopy theory. May-Ponto and GJ overlap on the abstract model-category axioms (Ch 14 of May-Ponto, Ch II of GJ); they diverge on emphasis (May-Ponto: applications to algebraic topology; GJ: internal simplicial machinery).
• Lurie, Higher Topos Theory (Princeton AMS 170, 2009). The $\infty$-categorical successor programme. Lurie §1.1 cites GJ as the canonical reference for the Kan-Quillen model structure on $\mathbf{sSet}$ and for the equivalence of categories between Kan complexes and $\infty$-groupoids. Lurie's quasi-category model structure (the Joyal model structure, also on $\mathbf{sSet}$) is not in GJ but is the natural next step after it.

GJ is not a first introduction to simplicial sets, to homotopy theory, or to model categories — it assumes all three at the level of a working graduate student. The natural prerequisite stack is: classical algebraic topology (Hatcher or May 3.38) + simplicial sets (May 3.40, or chapters I-III of GJ read concurrently) + abstract category theory (Mac Lane Categories for the Working Mathematician). Reading GJ alongside Hovey Model Categories is the standard modern path; reading GJ instead of May 3.40 is the standard modern path but loses the classical viewpoint that older literature still uses.

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§2 Coverage table (Codex vs GJ)

Cross-referenced against the current 03-modern-geometry/12-homotopy/ chapter (24 shipped units + the freshly-shipped Cycle-2 entry-point 03.12.25-simplicial-sets-and-geometric-realization.md). The entry point already carries Goerss-Jardine §I.1-§I.3 as its Intermediate / Master tier anchor and cites Lurie HTT §1.1 for the $\infty$-categorical pointer; that unit is the foundation the rest of the GJ punch-list builds on.

✓ = covered, △ = partial / different framing, ✗ = not covered. MS-overlap = the gap is already on the May 3.40 (FT) punch-list.

Chapter I — Simplicial sets

GJ topic (section)	Codex unit(s)	Status	Note
Simplicial category $\Delta$, presheaves ${\Delta }^{\mathrm{op}}\to \mathbf{Set}$ (§I.1)	03.12.25 simplicial-sets-and-geometric-realization	✓	Shipped this cycle. Anchored on GJ §I.1.
Yoneda lemma + density theorem $X={\mathrm{colim}}_{{\Delta }^n\to X}{\Delta }^n$ (§I.1)	03.12.25 (mention)	△	Mentioned in the Master tier; full categorical statement is a deepening.
Geometric realisation $	\cdot	$ as a coend (§I.2)	03.12.25
$	\cdot	\dashv \mathrm{Sing}$ adjunction (§I.2)	03.12.25
Milnor 1957 finite-products theorem (§I.2)	03.12.25 (Master)	✓	Covered at Master tier.
Kan complexes and the extension condition (§I.3)	—	✗	Gap (P1, high-priority — MS-overlap). Already on May 3.40 punch-list as 03.12.25 (now 03.12.26 in our local numbering since 03.12.25 is taken).
Simplicial homotopy / homotopy of $n$-simplices (§I.3, §I.6)	—	✗	Gap. MS-overlap.
Simplicial homotopy groups ${\pi }_n(K,\ast )$ (§I.7)	—	✗	Gap (P1). MS-overlap.
Function complex $\mathrm{Hom}(K,L)$ and the simplicial-set enrichment of $\mathbf{sSet}$ (§I.5)	—	✗	Gap. Internal hom; load-bearing for the model-category structure.
Kan fibration (§I.4, §I.7)	—	✗	Gap (P1). MS-overlap.
Anodyne extensions and Gabriel-Zisman (§I.4)	—	✗	Gap. Technical lemma class; load-bearing for the small-object argument.
Long exact sequence of a Kan fibration (§I.7)	—	✗	Gap.

Chapter II — Model categories

GJ topic	Codex unit(s)	Status	Note
Quillen model-category axioms (§II.1)	—	✗	Gap (P1, foundational). No Codex unit on model categories at all; this is the largest single piece of the GJ punch-list.
Cofibrantly generated model categories, small-object argument (§II.1, §II.5)	—	✗	Gap (P1). Foundational technique.
Quillen functor / Quillen adjunction / Quillen equivalence (§II.1)	—	✗	Gap (P1).
The homotopy category $\mathrm{Ho}(\mathcal{M})$ (§II.1)	—	✗	Gap (P1). Already a deepening candidate from May 3.38 plan (May's h𝒯).
Standard examples: $\mathbf{Top}$, ${\mathbf{Ch}}_{\ge 0}(R)$, $\mathbf{sSet}$ (§II.2-II.3)	—	✗	Gap.
The Kan-Quillen model structure on $\mathbf{sSet}$ (§II.3)	—	✗	Gap (P1, central GJ theorem). Goerss-Jardine's distinctive content; absent from May 3.40 (predates Quillen).
$	\cdot	\dashv \mathrm{Sing}$isaQuillenequivalence$\mathbf{sSet} \simeq_{\mathrm{Qu}} \mathbf{Top}$ (§II.3)	—
Simplicial model categories (§II.3)	—	✗	Gap (P2). $\mathbf{sSet}$-enrichment compatible with the model structure.

Chapter III — Classical results and constructions

GJ topic	Codex unit(s)	Status	Note
Simplicial abelian groups, simplicial $R$-modules (§III.2)	—	✗	Gap. MS-overlap.
Normalised Moore complex $N(A_{\bullet })$ (§III.2)	—	✗	Gap. MS-overlap.
Dold-Kan correspondence (§III.2)	—	✗	Gap (P1). MS-overlap.
Eilenberg-Zilber theorem and shuffle product (§III.5)	△	△	Mentioned in 03.12.11 Synthesis; not its own unit. MS-overlap.
Spectral sequences of a filtered chain complex (§III.6)	△	△	Codex has spectral-sequence content scattered (03.13-* if it exists); GJ's filtered-chain-complex set-up is a unit-level gap.
Eilenberg-MacLane simplicial groups $K(A,n)$ (§III.3-§III.4)	△	△	03.12.05 covers the space $K(\pi ,n)$; the simplicial-abelian-group construction is a deepening. MS-overlap.

Chapter IV — Bisimplicial sets

GJ topic	Codex unit(s)	Status	Note
Bisimplicial set $X:{\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}\to \mathbf{Set}$ (§IV.1)	—	✗	Gap (P2). No Codex bisimplicial content.
Diagonal $\mathrm{diag}:{\mathbf{sSet}}^{{\Delta }^{\mathrm{op}}}\to \mathbf{sSet}$ (§IV.1)	—	✗	Gap (P2).
The Bousfield-Kan homotopy colimit / homotopy limit (§IV.2-§IV.4)	—	✗	Gap (P2). Foundational construction of post-1972 homotopy theory; cited in every modern algebraic-topology paper.
Reedy model structure on ${\mathbf{sSet}}^{{\Delta }^{\mathrm{op}}}$ (§IV.3)	—	✗	Gap (P3). Technical infrastructure for hocolim/holim.
Homotopy spectral sequence of a bisimplicial set (§IV.4)	—	✗	Gap (P3).
Realisation lemma (the diagonal of a bisimplicial set is weakly equivalent to its bar-construction realisation) (§IV.1)	—	✗	Gap. Used silently in many later constructions.

Chapter V — Simplicial groups

GJ topic	Codex unit(s)	Status	Note
Simplicial group, simplicial group action (§V.1)	—	✗	Gap (P2). MS-overlap with May 3.40 §17.
Every simplicial group is a Kan complex (§V.1)	—	✗	Gap. Foundational theorem.
Kan's loop-group functor $G:{\mathbf{sSet}}_{\ast }\to \mathbf{sGrp}$ (§V.5)	—	✗	Gap (P3). MS-overlap.
$W\stackrel{ˉ}{}$ classifying-space functor (§V.4)	—	✗	Gap (P2). Simplicial-set analogue of $BG$.
$G⊣W\stackrel{ˉ}{}$ adjunction; $G\bar{W}G\to G$ weak equivalence (§V.5)	—	✗	Gap (P3). Kan's theorem.
Classification of principal $G$-bundles as $[X,W\bar{G}]$ (§V.3)	△	△	03.08.04 classifying-space covers topologically; GJ's simplicial version is a deepening.

Chapter VI — The homotopy theory of towers

GJ topic	Codex unit(s)	Status	Note
Postnikov tower of a Kan complex (§VI.3)	—	✗	Gap (P2). MS-overlap. Codex has Whitehead-tower (03.12.07, dual); Postnikov is missing.
$k$-invariants as cohomology classes (§VI.5)	—	✗	Gap. MS-overlap.
Tower of fibrations, ${\lim }^1$ exact sequence (§VI.2)	—	✗	Gap. ${\lim }^1$ already flagged in May 3.38 punch-list.
Obstruction theory for lifting through a Postnikov tower (§VI.5)	△	△	Briefly mentioned in 03.12.05 Master tier; not its own unit.

Chapter VII — Cosimplicial spaces

GJ topic	Codex unit(s)	Status	Note
Cosimplicial space $X^{\bullet }:\Delta \to \mathbf{sSet}$, totalisation $\mathrm{Tot}$ (§VII.1)	—	✗	Gap (P3).
Bousfield-Kan spectral sequence $E_2={\pi }^s{\pi }_t\Rightarrow {\pi }_{t-s}\mathrm{Tot}(X^{\bullet })$ (§VII.6)	—	✗	Gap (P2, distinctive GJ). Central tool of unstable homotopy theory; absent from May 3.40 entirely.
The unstable Adams spectral sequence as a BK spectral sequence (§VII.6)	—	✗	Gap (P3). Connection to the stable-homotopy strand (03.08.06).
Cosimplicial resolutions, $R$-completion (§VIII.3)	—	✗	Gap (P3). Bousfield-Kan 1972 monograph; pointer-only treatment recommended.

Chapter VIII (when present in the 2009 reprint) — Simplicial sheaves and presheaves

GJ topic	Codex unit(s)	Status	Note
Simplicial presheaves on a site, local weak equivalences (§VIII.1)	—	✗	Out of scope — defer to motivic-homotopy-theory pass (Morel-Voevodsky); see §5.
Jardine local model structure (§VIII.2)	—	✗	Out of scope. Same.
Stalks, Brown-Gersten descent (§VIII.4)	—	✗	Out of scope. Same.

Aggregate coverage estimate

Theorem layer. ~10% of GJ's named theorems map to Codex units. The freshly-shipped 03.12.25 covers the §I.1-§I.2 foundations and the $|\cdot |⊣\mathrm{Sing}$ adjunction. Everything model-category-theoretic (GJ Ch II), every named model-structure result on $\mathbf{sSet}$, the Dold-Kan correspondence, Postnikov towers, bisimplicial sets, the Bousfield-Kan spectral sequence — all missing. After the priority-1 punch-list (model categories + Kan complexes + Dold-Kan + Quillen equivalence) this rises to ~50%. After priority-2 to ~75%.

Framing layer. ~5% covered. The model-category framing is the GJ contribution and Codex has no model-category units at all. This is the dominant gap.

Sequencing layer. Codex follows May 3.40's classical sequencing (simplicial sets → Kan complexes → simplicial homotopy groups → realisation). GJ inverts: (simplicial sets → realisation → adjunction → Kan complexes → model structure → everything else as model-category corollary). The sequencing decision is a notation/framing decision (record per §3 below) rather than a re-ordering of shipped units.

Notation layer. ~70% aligned with GJ's modern conventions (Codex already adopted $|K|$, $\mathrm{Sing}(X)$, $d_i/s_i$ per the May 3.40 crosswalk — these are GJ's notations, which Codex inherited).

Application layer. ~5% covered. Bisimplicial / hocolim, BK spectral sequence, simplicial-group classifying-space machinery are all gaps.

The flag in the task prompt is correct: expect substantial new-unit punch-list. Model-category language is largely absent from Codex.

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§3 Gap punch-list — units to write, priority-ordered

Priority 0 — strict prerequisites (already in place or in flight):

• 03.12.25-simplicial-sets-and-geometric-realization — shipped this cycle. GJ §I.1-§I.2. All P1 units below build on it.
• May 3.40 (FT) priority-1 punch-list units 03.12.26 (Kan complex) through 03.12.30 (Dold-Kan) — see plans/fasttrack/may-simplicial-objects.md §3. These are shared prerequisites between the May 3.40 and GJ 3.41 plans (same units, both plans depend on them). Recommendation: produce them once as the joint May/GJ batch and cite both originator texts, then build the GJ-distinctive model-category superstructure on top.

Priority 1 — high-leverage, captures GJ's distinctive content (estimate: 5 new units, all new; ~5 of these are not on May 3.40 plan):

1. 03.12.31 Quillen model category. Definition of a model category (Quillen's original three-axiom presentation; the modern five-axiom factorisation presentation; equivalence of the two). Standard examples stated: $\mathbf{Top}$ with Serre fibrations + weak equivalences + Hurewicz cofibrations; ${\mathbf{Ch}}_{\ge 0}(R)$ with quasi-iso + epimorphism + monomorphism with projective cokernel; $\mathbf{sSet}$ pointer-only at this stage (full statement in unit 3 below). The homotopy category $\mathrm{Ho}(\mathcal{M})=\mathcal{M}[W^{-1}]$ and Quillen's theorem that it is locally small. Three-tier, ~2000 words. Anchor: GJ §II.1; Quillen 1967 §I.1-§I.3; Hovey Model Categories Ch 1; May-Ponto Ch 14. Foundational for everything below.
2. 03.12.32 Quillen functor and Quillen equivalence. Left/right Quillen functor (preserves cofibrations + trivial cofibrations / fibrations + trivial fibrations); Quillen adjunction; Quillen equivalence (the derived functors are an equivalence of homotopy categories). Standard examples: the singular-chains functor $\mathbf{Top}\to {\mathbf{Ch}}_{\ge 0}$ is left Quillen; the $|\cdot |⊣\mathrm{Sing}$ adjunction will be the example in unit 3. Three-tier, ~1800 words. Anchor: GJ §II.1; Hovey §1.3.
3. 03.12.33 The Kan-Quillen model structure on $\mathbf{sSet}$. The model structure: cofibrations = monomorphisms; weak equivalences = maps that induce weak equivalences on geometric realisation; fibrations = Kan fibrations (right lifting against horns). Existence proof outline via the small-object argument and anodyne extensions. The fibrant objects are exactly the Kan complexes. Three-tier; Beginner tier states the structure, Intermediate tier sketches the existence proof, Master tier walks through the anodyne-extension classification. ~2500 words. Anchor: GJ §I.11, §II.3; Quillen 1967 §II.3.
4. 03.12.34 Geometric realisation as a Quillen equivalence $\mathbf{sSet}{\simeq }_{\mathrm{Qu}}\mathbf{Top}$. Statement that $(|\cdot |,\mathrm{Sing})$ is a Quillen equivalence between the Kan-Quillen model structure on $\mathbf{sSet}$ and the Quillen (Serre) model structure on $\mathbf{Top}$. Sketch of proof via the unit / counit triangle identities and the comparison theorem $|\mathrm{Sing}(X)|\to X$ being a weak equivalence (May 3.40 §16). Master tier discusses the $k$-space subtlety and the choice of $\mathbf{CGWH}$ vs $\mathbf{Top}$. Three-tier, ~2000 words. Anchor: GJ §I.11 / §II.3; Quillen 1967 §II.3 Theorem 3. This is the load-bearing GJ theorem.
5. 03.12.35 Simplicial model category and the function complex. Definition of a simplicial model category (Quillen's SM7 axiom on the pushout-product of cofibrations and fibrations being a Kan fibration); the simplicial-set enrichment $\mathrm{Hom}(K,L)$ on $\mathbf{sSet}$ and its compatibility with the model structure. Three-tier, ~1800 words. Anchor: GJ §II.3 (definition), §I.5 (function complex); Hovey §4.2.

Priority 2 — bisimplicial sets, Bousfield-Kan, simplicial groups (estimate: 4 new units):

6. 03.12.36 Bisimplicial set, diagonal, and the realisation lemma. Bisimplicial set $X: \Delta^{\mathrm{op}} \times \Delta^{\mathrm{op}} \to \mathbf{Set}$; the two derived simplicial-set functors (diagonal, row-wise / column-wise realisation); the realisation lemma: $\mathrm{diag}(X)\simeq |X{|}_{\mathrm{row}}{|}_{\mathrm{col}}$. Three-tier; Master required for the proof. ~1800 words. Anchor: GJ §IV.1.
7. 03.12.37 Homotopy colimit (Bousfield-Kan construction). Definition of ${\mathrm{hocolim}}_{I}F$ for a small-category-indexed diagram $F: I \to \mathbf{sSet}$,viathebarconstruction$B(*, I, F)$. Universal property as the homotopy-invariant replacement of the ordinary colimit. Examples: homotopy pushout, mapping telescope. Dual: $\mathrm{holim}$. Three-tier, ~2000 words. Anchor: GJ §IV.2-§IV.4; Bousfield-Kan 1972 Homotopy Limits, Completions and Localizations (Springer LNM 304) as originator text.
8. 03.12.38 Bousfield-Kan spectral sequence. Cosimplicial space $X^{\bullet }$, totalisation $\mathrm{Tot}(X^{\bullet })$, BK spectral sequence $E_2^{s,t}={\pi }^s{\pi }_t(X^{\bullet })\Rightarrow {\pi }_{t-s}\mathrm{Tot}(X^{\bullet })$. Convergence under connectivity hypotheses. Pointer to the unstable Adams spectral sequence as a special case. Master-only, ~1800 words. Anchor: GJ §VII.6; Bousfield-Kan 1972 (originator).
9. 03.12.39 Simplicial group and the $W\stackrel{ˉ}{}$ classifying functor. Simplicial group $G$, the canonical Kan-complex structure; the $W\bar{G}$ classifying simplicial set; classification of principal-$G$ Kan fibrations as homotopy classes $[X,W\bar{G}]$. Cross-links to 03.08.04 classifying-space (topological side) and to the May 3.40 03.12.31 TCP unit (classical side). Three-tier, ~2200 words. Anchor: GJ §V.1-§V.4.

Priority 3 — Postnikov and Reedy / cosimplicial deepenings (estimate: 2 new units + 1 cross-link deepening):

10. 03.12.40 Postnikov tower of a Kan complex. Postnikov tower $P_{n}X$ with ${\pi }_i(P_{n}X)={\pi }_i(X)$ for $i\le n$, zero above; $k$-invariants as cohomology classes $[k_n]\in H^{n+2}(P_{n}X;{\pi }_{n+1}X)$. Cross-link to 03.12.07 Whitehead tower (dual construction). Three-tier, ~1800 words. Anchor: GJ §VI.3-§VI.5. MS-overlap — this is the simplicial-Kan-complex version of the Postnikov unit on the May 3.40 plan (03.12.33); produce once and cite both.
11. 03.12.41 Reedy model structure on cosimplicial / bisimplicial spaces. Reedy categories; latching and matching objects; the Reedy model structure on ${\mathcal{M}}^{{\Delta }^{\mathrm{op}}}$ for $\mathcal{M}$ a model category. Master-only, ~1800 words. Anchor: GJ §VII.2; Hirschhorn Model Categories and Their Localizations Ch 15 as the canonical modern treatment.
12. Deepening of 03.12.25 simplicial-sets-and-geometric-realization — add a Master section "Yoneda lemma and density" giving the formal categorical statement $X={\mathrm{colim}}_{{\Delta }^n\to X}{\Delta }^n$ and its role as the basis for every coend construction in the book (realisation, function complex, hocolim). ~600 words. GJ §I.1 anchor.

Priority 4 — survey-level pointers, exercise pack (optional, defer):

13. Exercise pack 03.12.E3-simplicial-homotopy-theory-exercises. GJ has exercises at the end of every chapter (~80-100 total). The Codex pack should cover: explicit horn-filling in $\mathrm{Sing}(X)$; the nerve of a category as a Kan-complex check; explicit Quillen adjunction verification for ${\pi }_0:\mathbf{sSet}\rightleftarrows \mathbf{Set}$; a worked Bousfield-Kan spectral sequence (e.g. cosimplicial ${\mathbb{F}}_p$-resolution of a finite space). ~20 exercises across three tiers.
14. Cofibrantly generated model categories (deepening). The small-object argument in full; $I$-cell complexes and the Quillen-Smith recognition theorem. Add as a Master section to 03.12.31; ~700 words. Anchor: Hovey §2.1, GJ §II.5.
15. Pointer to Joyal model structure / quasi-categories. Add as a Synthesis pointer in 03.12.33: the other model structure on $\mathbf{sSet}$ (Joyal 2008, Lurie HTT §2.2-§2.4) whose fibrant objects are quasi-categories rather than Kan complexes. Out of scope for full treatment; pointer only.

Notation crosswalk (record in the §Notation paragraph of each new unit):

• GJ uses $\mathbf{S}$ for the category of simplicial sets; Codex uses $\mathbf{sSet}$. Codex convention wins (matches modern nLab / Lurie / May-Ponto).
• GJ uses $|X|$ for geometric realisation, $\mathrm{Sing}$ for the singular complex (or $S$); $d_i/s_i$ for face/degeneracy. All match Codex conventions from the May 3.40 crosswalk.
• GJ uses $\mathrm{Hom}(K,L)$ for the function complex (the internal hom of $\mathbf{sSet}$). Codex's 03.12.25 uses $\mathrm{Map}(K,L)$ informally; reconcile to ${\mathrm{Hom}}_{\mathbf{sSet}}(K,L)$ (full notation) in the new units 3 and 5.
• GJ uses $W\bar{G}$ for the classifying simplicial set of a simplicial group $G$; Codex topology side uses $BG$. Keep both: $BG$ for the topological classifying space, $W\bar{G}$ for the simplicial-set version, with an explicit comparison statement in 03.12.39 ($|W\bar{G}|\simeq BG$ for $G$ a topological group with simplicial-set model).
• GJ writes $\mathrm{Ho}(\mathcal{M})$ for the homotopy category; matches the modern convention. May's h𝒯 notation (May 3.38 plan) is the same object; the notation/may.md crosswalk should record both.
• GJ uses $\mathrm{hocolim},\mathrm{holim}$ for homotopy colimit/limit; standard.

Three-tier scheduling. P1 units 1-5 require all three tiers (the foundational ones especially — model categories needs a Strogatz-level intuition section). P2 units 6-9 can omit Beginner for the more technical ones (bisimplicial sets, BK spectral sequence) but should keep Intermediate + Master. P3 units 10-11 are Master-only.

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§4 Implementation sketch

For a full GJ coverage pass, items 1-9 are the minimum P1+P2 set (9 new units). Realistic production estimate:

• P1 (5 units): ~4 hours each = ~20 hours. Model-category units skew higher than the corpus average because the abstraction level requires careful exposition and the proofs (small-object argument, Quillen equivalence) are nontrivial.
• P2 (4 units): ~3.5 hours each = ~14 hours.
• P3 (2 new units + 1 deepening): ~3 hours each + 1 hour = ~7 hours.
• P4 (exercise pack + 2 deepenings): ~5 hours combined.

Total: ~45-50 hours for full GJ equivalence. Fits a 6-8 day focused window. Slightly larger than the May 3.40 batch (~38-40 hours) because of the model-category abstraction layer.

Strict prerequisite: the May 3.40 priority-1 punch-list (Kan complex, simplicial homotopy groups, Kan fibration, Dold-Kan) must ship first. Recommendation: dispatch the May 3.40 P1 batch and the GJ 3.41 P1 batch as one composite "simplicial-homotopy bookshelf" batch with 10 new units total (5 from each plan), since they all live in the same chapter and share several originator citations. Run Pass-W once at the end of the composite batch.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, GJ-distinctive content has clearly identified originators:

• Quillen, D. G. Homotopical Algebra. Springer Lecture Notes in Mathematics 43 (1967). Originating reference for model categories, including the original axioms (slightly different from the modern form) and the proof that $\mathbf{sSet}$ has a model structure Quillen-equivalent to $\mathbf{Top}$. Cite in 03.12.31, 03.12.33, 03.12.34.
• Kan, D. M. "Functors involving c.s.s. complexes," Transactions of the AMS 87 (1958) 330-346. Originator of Kan complexes, the extension condition, simplicial homotopy groups, and the loop-group functor $G$. Already on the May 3.40 plan originator list; co-cite in GJ-side units 03.12.33, 03.12.39.
• Kan, D. M. "A combinatorial definition of homotopy groups," Annals of Mathematics 67 (1958) 282-312. Cite in 03.12.33.
• Bousfield, A. K., and Kan, D. M. Homotopy Limits, Completions and Localizations. Springer Lecture Notes in Mathematics 304 (1972). Originating reference for $\mathrm{hocolim}$ / $\mathrm{holim}$ and the Bousfield-Kan spectral sequence. Cite in 03.12.37, 03.12.38.
• Bousfield, A. K. "The localization of spaces with respect to homology," Topology 14 (1975) 133-150. The companion paper on Bousfield localisation; cite in 03.12.37 Synthesis.
• Goerss, P. G., and Jardine, J. F. Simplicial Homotopy Theory. Birkhäuser Progress in Mathematics 174 (1999). The book itself — definitive modern consolidation. Cite in every unit on this punch-list.
• Milnor, J. W. "The geometric realization of a semi-simplicial complex," Annals of Mathematics 65 (1957) 357-362. Already cited in 03.12.25; co-cite in 03.12.34 for the finite-products theorem.
• Hovey, M. Model Categories. AMS Mathematical Surveys and Monographs 63 (1999). Modern systematic reference; cite as the canonical "see also" in 03.12.31, 03.12.32, 03.12.35.

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§5 What this plan does NOT cover

• Simplicial presheaves / simplicial sheaves / Jardine local model structures (GJ Ch VIII in the 2009 reprint). Out of scope. This is the foundation of motivic homotopy theory (Morel-Voevodsky 1999) and the home territory of the second author's later research. Defer to a dedicated motivic-homotopy audit, which would also pull in Morel-Voevodsky A^1-Homotopy Theory of Schemes (Publ. IHÉS 90, 1999).
• Quasi-categories / $\infty$-categories and the Joyal model structure (Lurie Higher Topos Theory, Joyal 2008). Out of scope; pointer only in 03.12.33 Synthesis. The Codex Fast Track lists Lurie HTT as a separate (deferred) audit; that audit subsumes the $\infty$-categorical machinery.
• May-Ponto (FT 3.39) deep content — localisation at a set of primes, formal groups, completion, the Adams spectral sequence applications, and the Hopf-algebra structure on ${\pi }_{\ast }$ of ring spectra. Defer to the May-Ponto audit. GJ touches some of this material at the simplicial level (e.g. BK spectral sequence) but the applications live in May-Ponto.
• Model categories beyond the simplicial-set example. GJ does not develop the general theory in depth (Hovey does). Codex's model-category units 03.12.31, 03.12.32, 03.12.35 should state the general theory but illustrate it with $\mathbf{sSet}$ and $\mathbf{Top}$ only; deeper examples (chain complexes, spectra, operads) live in the May-Ponto and Hovey audits.
• Line-number-level inventory of every theorem in GJ. Done at the section-level above; this audit is reduced (no local PDF, no fetched PDF within the time budget). A revision pass with the local PDF would sharpen the named-theorem audit, especially in Ch IV-VII, but the units-to-write list is robust against the reduction.
• The full proof of the Kan-Quillen model structure on $\mathbf{sSet}$ in unit 03.12.33 — only the structure of the proof (anodyne extensions + small-object argument) is required at Master tier; the full technical proof of the anodyne-extension classification is a "see Gabriel- Zisman 1967 §IV" reference rather than a Codex unit.
• Algebraic K-theory of rings via simplicial methods (Quillen's $+$-construction and $Q$-construction). Out of scope; cite Quillen's Higher algebraic K-theory I (LNM 341, 1973) as the originator pointer in 03.12.39 Synthesis.
• Cubical sets / cubical homotopy theory. Out of scope. See the Brown-Higgins-Sivera 1.05a plan for cubical pointer units.
• Operads / $E_{\infty }$-structures on simplicial chain complexes. Out of scope; defer to a future May Geometry of Iterated Loop Spaces / Hinich audit pass.
• Figures and large commutative diagrams. Curriculum-wide deferred item; same flag as in the Hatcher / May plans.

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§6 Acceptance criteria for FT equivalence (GJ)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, GJ is at equivalence-coverage when:

• The May 3.40 P1 punch-list (Kan complex, simplicial homotopy groups, Kan fibration, Dold-Kan) has shipped (strict prereq).
• All 5 GJ priority-1 units have shipped (03.12.31-03.12.35). This alone raises GJ coverage from ~10% to ~55%.
• At least 3 of the 4 GJ priority-2 units have shipped (the bisimplicial / hocolim / BK spectral sequence triad is the highest-leverage). Brings coverage to ~80%.
• Both P3 units (Postnikov + Reedy) plus the 03.12.25 deepening have shipped. Brings coverage to ~92%.
• Notation decisions are recorded in each unit's §Notation paragraph per the crosswalk in §3 above, and a notation/model-category.md master-list is started (covering 𝒞, $W$, $\mathrm{Cof}$, $\mathrm{Fib}$, $\mathrm{Ho}(\mathcal{M})$, $\mathbb{L}F$, $\mathbb{R}G$, derived adjunction, Quillen equivalence).
• Pass-W weaving connects the new units to 03.12-homotopy/ (especially to the just-shipped 03.12.25 and to 03.12.07 Whitehead tower for the Postnikov / Whitehead duality), to 03.08-* (classifying-space units), and forward to the May 3.40, May-Ponto 3.39, and Lurie HTT audits.
• Originator-prose paragraphs present in 03.12.31 (Quillen 1967), 03.12.34 (Quillen 1967 + Milnor 1957), 03.12.37 (Bousfield-Kan 1972), 03.12.38 (Bousfield-Kan 1972), per the citation list in §4.
• Exercise pack 03.12.E3 ships (P4) — required for the ≥95% threshold.

The 5 P1 units close the model-category gap. The 4 P2 units close the bisimplicial / Bousfield-Kan / simplicial-group gaps. The 2 P3 units close the Postnikov / Reedy gaps. The P4 exercise pack closes the residual gap to ≥95%.

Composite recommendation. The GJ P1 batch (5 units) is most efficient dispatched together with the May 3.40 P1 batch (5 units) as a single "simplicial-homotopy bookshelf" composite batch of 10 units — they share prerequisites (03.12.25), share originator citations (May 1967, Kan 1958, Eilenberg-Zilber, Milnor 1957), and live in adjacent units of the same chapter. The GJ-distinctive P1 content (model-category framing) builds directly on the May-3.40 P1 content (Kan-complex foundations), so producing them in two passes risks notation drift between the batches. Run Pass-V continuity once on the composite batch and Pass-W weaving once on the full ten-unit set.

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§7 Sourcing

• Not free. Goerss-Jardine is published by Birkhäuser (Progress in Mathematics 174, 1999; reprinted 2009 Modern Birkhäuser Classics). Available at SpringerLink for institutional subscribers (https://link.springer.com/book/10.1007/978-3-0346-0189-4; auth-gated). No author-hosted free PDF; this differs from May 3.38 / 3.40 and Brown 1.05a.
• No local copy. reference/textbooks-extra/ and reference/fasttrack-texts/03-modern-geometry/ checked — neither contains a Goerss-Jardine PDF. Action item: acquire institutional-access PDF or paper copy and add to reference/fasttrack-texts/03-modern-geometry/Goerss-Jardine-SimplicialHomotopyTheory.pdf before the P1 batch ships. A revision pass on this audit at that point will sharpen Ch IV-VII coverage (currently the weakest part of the audit given the reduced sourcing).
• License. For educational use cite as P. G. Goerss and J. F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics 174, Birkhäuser 1999 (reprinted in Modern Birkhäuser Classics 2009).
• Comparison reading for P1 production. Since GJ itself is not immediately available, the P1 batch should be produced against:
  • Hovey, Model Categories (1999) — the systematic model-category reference; covers the same Ch II content as GJ in greater abstraction. Often clearer than GJ on the small-object argument.
  • May, Simplicial Objects in Algebraic Topology (1967) — local copy at reference/fasttrack-texts/03-modern-geometry/May-SimplicialObjects.pdf. Covers the same Ch I, III, V content combinatorially (no model structure).
  • May-Ponto, More Concise Algebraic Topology (2012) — Ch 14-19 cover the model-category material with worked applications.
  • Lurie, Higher Topos Theory (2009) Ch 1 — the modern $\infty$- categorical view; cite for the Kan-Quillen and Joyal model structures.
• Tertiary sources. Dwyer-Hirschhorn-Kan-Smith Homotopy Limit Functors on Model Categories and Homotopical Categories (AMS Math. Surv. & Monog. 113, 2004) and Hirschhorn Model Categories and Their Localizations (AMS Math. Surv. & Monog. 99, 2003) are the canonical secondary references for the bisimplicial / Reedy / hocolim content (GJ Ch IV, VII). Cite these in 03.12.36, 03.12.37, 03.12.41.