J. Peter May — *Simplicial Objects in Algebraic Topology* (Fast Track 3.40) — Audit + Gap Plan

Book: J. Peter May, Simplicial Objects in Algebraic Topology (Van Nostrand Mathematical Studies, D. Van Nostrand, 1967; reprinted in the Chicago Lectures in Mathematics series, University of Chicago Press, 1982 / 1992; ISBN 0-226-51181-2). 161 pp. Author-hosted PDF distributed freely; local copy at reference/fasttrack-texts/03-modern-geometry/May-SimplicialObjects.pdf (scanned 1982 reprint, 90 PDF pages = the 161-page reprint with two book-pages per A4 scan).

Fast Track entry: 3.40, inside §3 Modern Geometry, between May Concise Course (3.38), May-Ponto More Concise (3.39), and Goerss-Jardine Simplicial Homotopy Theory (3.41).

Purpose of this plan: P1 audit + gap punch-list. Output is a priority-ordered list of new units to write so that Simplicial Objects (SOAT hereafter) is covered to the Fast Track equivalence threshold (≥95% of named theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

SOAT is the foundational reference for the classical (pre-model-category) simplicial-set machinery. It is short, dense, and now over half a century old, but every later text — Goerss-Jardine, Lurie's HTT, the nLab — points back to it for the canonical statements of the Kan-complex homotopy theory, geometric realisation, twisted cartesian products, and the simplicial Eilenberg-MacLane construction.

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§1 What SOAT is for

SOAT is the classical foundation document for simplicial sets as a combinatorial model of homotopy types. Where Hatcher (Fast Track 3.x) and May Concise (3.38) treat simplicial complexes and singular homology as auxiliary tools, SOAT treats the simplicial category ${\Delta }^{\ast }$ and its contravariant functors as the primary objects: spaces are reconstructed from simplicial sets by geometric realisation $|K|=T(K)$, and homotopy theory is done combinatorially on Kan complexes before being transported back to topology.

Distinctive contributions, in roughly the order SOAT develops them:

1. The simplicial category $\Delta$ (May's ${\Delta }^{\ast }$) and the face / degeneracy presentation. SOAT §1 gives the canonical "simplicial identities" presentation and the unique normal form $\mu ={\delta }_{i_1}\cdots {\delta }_{i_s}{\sigma }_{j_1}\cdots {\sigma }_{j_t}$ for every monotonic map. This is the technical bedrock for the rest of the book.
2. Kan complexes and the extension condition (SOAT §2-3). May's §3 proves the basic combinatorial homotopy theory: $\sim$ is an equivalence relation on $n$-simplices of a Kan complex, simplicial homotopy groups ${\pi }_n(K,\ast )$ are well-defined, function complexes are Kan when the target is.
3. Postnikov systems combinatorially (SOAT §8). The Postnikov tower is constructed directly inside the simplicial category, without passing through topology. This is one of SOAT's signature moves and is still the cleanest exposition.
4. Minimal complexes and minimal fibrations (SOAT §9-10). Two minimal complexes that are deformation retracts of the same Kan complex are isomorphic; every Kan fibration retracts onto a minimal fibration. This is the combinatorial analogue of Postnikov decomposition and underpins the homotopy classification of fibrations.
5. Geometric realisation $T:\mathbf{sSet}\to \mathbf{Top}$ and its adjoint $S$ (SOAT §14-16). Theorem 14.1: $T(K)$ is a CW complex with one $n$-cell per non-degenerate $n$-simplex. Theorem 14.3: $T$ preserves finite products (after passing to $k$-spaces). $(T,S)$ is the canonical Quillen adjunction between simplicial sets and topological spaces.
6. Twisted cartesian products (TCP) and principal fibrations (SOAT §17-21). May's TCP machinery is the simplicial-set analogue of classifying spaces for principal bundles: a TCP with structure group a simplicial group $G$ and base $B$ is classified by a twisting function $\tau :B_n\to G_{n-1}$. This is the combinatorial source of the Serre spectral sequence in the simplicial setting.
7. Eilenberg-MacLane simplicial complexes $K(\pi ,n)$ (SOAT §22-23). Constructed as the simplicial abelian group whose normalised Moore complex is concentrated in degree $n$ with value $\pi$. Cohomology operations are read off as $[K(\pi ,n),K(\rho ,n+q)]$ via simplicial homotopy classes.
8. The Dold-Kan correspondence (SOAT §22). The category of simplicial abelian groups is equivalent to the category of non-negatively-graded chain complexes via the normalised Moore complex; SOAT proves both the equivalence and the normalisation theorem in §22.
9. Acyclic models and Eilenberg-Zilber theorem (SOAT §29). The classical Eilenberg-Zilber chain equivalence $C(K\times L)\simeq C(K)\otimes C(L)$ is proved by the acyclic-models method on simplicial sets, which is the general technique for proving chain-level natural equivalences.

SOAT is not a first course in algebraic topology. It assumes singular homology (Eilenberg-Steenrod), basic homotopy groups, CW complexes, and spectral sequences. It is the canonical combinatorial-foundations reference; the modern homotopical / model-category foundations live in Goerss-Jardine (3.41) and Hovey's Model Categories, both of which use SOAT as their starting point. Curtis's 1971 Advances in Mathematics survey "Simplicial homotopy theory" gives a contemporaneous reader's guide and pinpoints SOAT as the standard reference.

Peer-source corroboration:

• Goerss-Jardine, Simplicial Homotopy Theory (2009). Cites SOAT as the canonical source for the simplicial identities (their §I.1), for Kan complexes and simplicial homotopy groups (their §I.6-7), and for the Dold-Kan correspondence (their §III.2).
• May, A Concise Course in Algebraic Topology (1999). May's own later text (Fast Track 3.38) defers to SOAT for the simplicial-set foundations and the proof of $|S(X)|\simeq X$ (May Concise §16.5).
• E. B. Curtis, "Simplicial homotopy theory," Advances in Mathematics 6 (1971) 107-209. Survey article contemporaneous with SOAT; identifies SOAT and Lamotke's Semisimpliziale algebraische Topologie (1968) as the two standard references for simplicial-set foundations. SOAT is the more category-theoretic and concise of the two.

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§2 Coverage table (Codex vs SOAT)

Cross-referenced against the current 03-modern-geometry/12-homotopy/ chapter (24 shipped units + 03.12.22-delta--complex-semi-simplicial-set currently in production / live queue). ✓ = covered, △ = partial / different framing, ✗ = not covered.

SOAT topic (chapter, section)	Codex unit(s)	Status	Note
Simplicial category ${\Delta }^{\ast }$, face / degeneracy operators (§1)	03.12.22 (in queue)	△	The queued $\Delta$-complex unit covers semi-simplicial sets; the full simplicial category with degeneracies is a separate gap.
Simplicial identities and normal form $\mu ={\delta }_{i_1}\cdots {\sigma }_{j_t}$ (§1)	—	✗	Gap. Foundational; needed before anything else.
Simplicial object in a category, ${\mathcal{C}}^s=\mathrm{Fun}({\Delta }^{\mathrm{op}},\mathcal{C})$ (§2)	—	✗	Gap. The general notion (simplicial groups, simplicial modules, etc.) is currently absent.
Chain complex $C(K)$ of a simplicial set (§2)	03.12.11, 03.12.12	△	Singular homology covers $C(\mathrm{Sing}(X))$; simplicial homology covers the $\Delta$-complex case. The general $C(K)$ for a simplicial set is implicit but not its own unit.
Kan complex / Kan extension condition (§3)	—	✗	Gap (P1, high-priority). SOAT's central object.
Simplicial homotopy / homotopy of simplices (§3)	—	✗	Gap. $x\sim x^{\prime }$ in $K_n$, well-defined on Kan complexes.
Simplicial homotopy groups ${\pi }_n(K,\ast )$ (§3-5)	—	✗	Gap. Combinatorial definition independent of topology.
Simplicial group $G$ (§4, §17)	—	✗	Gap. Every simplicial group is a Kan complex; foundational.
Function complex $\mathrm{Map}(K,L)$ (§6)	—	✗	Gap. Internal hom on $\mathbf{sSet}$.
Kan fibration (§7)	—	✗	Gap (P1). Right lifting against horn inclusions.
Postnikov system, simplicial construction (§8)	—	✗	Gap. Codex has Whitehead tower (03.12.07) which is dual to Postnikov; Postnikov itself is missing.
Minimal complex / minimal fibration (§9-10)	—	✗	Gap. Signature SOAT construction.
Fibre products / fibre bundles, simplicial (§11)	—	✗	Gap.
Weak homotopy type (§12)	△	△	Implicit in Whitehead theorem unit 03.12.20; not its own unit.
Hurewicz theorem, simplicial form (§13)	03.12.19	△	Hurewicz is covered topologically; simplicial version is a deepening.
Geometric realisation $T(K) =	K	$ (§14)	—
Adjoint $(T,S)$: realisation vs total singular complex (§15)	—	✗	Gap (P1). $
Comparison theorem: $	S(X)	\to X$ is a weak equivalence (§16)	—
Twisted cartesian product (TCP) (§17-19)	—	✗	Gap (P2). Pre-history of classifying-space machinery.
Eilenberg subcomplexes (§20)	—	✗	Gap.
Classification theorem for principal fibrations (§21)	△	△	03.08.05-universal-bundle covers $BG$ topologically and mentions the simplicial bar construction; SOAT's TCP classification is the simplicial origin and is not its own unit.
Eilenberg-MacLane simplicial complex $K(\pi ,n)$ (§22)	△	△	03.12.05 covers $K(\pi ,n)$ as a space; SOAT's simplicial-abelian-group construction is missing.
Dold-Kan correspondence (§22 normalisation theorem)	—	✗	Gap (P1). Currently only mentioned in passing in 03.12.11 and 03.12.12.
Cohomology operations via $K(\pi ,n)$ (§23)	—	✗	Gap.
$k$-invariants of Postnikov systems (§24)	—	✗	Gap.
Loop groups (§25-26)	—	✗	Gap (P3). Simplicial Kan loop-group functor $G$, $\bar{W}$ construction.
Acyclic models theorem (§28)	—	✗	Gap (P2). Method, not just statement.
Eilenberg-Zilber theorem (§29)	△	△	Mentioned in 03.12.11 Synthesis; not its own unit. The chain-level shuffle / Alexander-Whitney maps are missing.

Aggregate coverage estimate: ~5% of SOAT has corresponding Codex units (only the topological $K(\pi ,n)$, topological Hurewicz, and partial mentions of Dold-Kan / Eilenberg-Zilber). This is a near-total gap. Closing it is the largest single piece of the simplicial-methods curriculum.

The flag in the task prompt is correct: expect a big new-unit punch-list. No deepening-only outcome.

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

Priority 0 — prerequisite, already in flight:

• 03.12.22-delta--complex-semi-simplicial-set (currently in production queue). Ships the semi-simplicial side (face operators only, no degeneracies). All SOAT P1 units below sit on top of this and on top of the full simplicial category $\Delta$ — so 03.12.22 either needs expansion to cover the full simplicial category, or a sibling unit 03.12.22b-simplicial-set needs to be written. Recommend the latter to keep 03.12.22 focused.

Priority 1 — high-leverage core simplicial machinery (5 units):

1. 03.12.24 Simplicial set and the simplicial category $\Delta$. Definition of $\Delta$ (objects $[n]$, morphisms order-preserving maps), face ${\delta }_i$ / degeneracy ${\sigma }_i$ generators, simplicial identities, unique normal-form factorisation $\mu ={\delta }_{i_1}\cdots {\delta }_{i_s}{\sigma }_{j_1}\cdots {\sigma }_{j_t}$. Definition of simplicial set $K:{\Delta }^{\mathrm{op}}\to \mathbf{Set}$ and simplicial object in a general category. Anchor: SOAT §1-2; Goerss-Jardine §I.1. Three-tier, ~2000 words. Foundational — everything else in the SOAT punch-list assumes it.
2. 03.12.25 Kan complex and the extension condition. Definition via horn-filling, examples ($\mathrm{Sing}(X)$, simplicial groups, nerve of a groupoid), non-examples (nerve of a non-groupoid category). Simplicial homotopy as an equivalence relation on Kan complexes. Anchor: SOAT §3; Goerss-Jardine §I.3. Three-tier, ~2000 words.
3. **03.12.26 Simplicial homotopy groups ${\pi }_n(K,\ast )$.** Combinatorial definition via $\sim$ on $K_n$ relative to the basepoint. Theorem: for $K=\mathrm{Sing}(X)$, ${\pi }_n^{\mathrm{simp}}(K)\cong {\pi }_n(X)$. Long exact sequence of a Kan fibration. Anchor: SOAT §3-5, §7; Goerss-Jardine §I.7. Three-tier, ~1800 words.
4. 03.12.27 Geometric realisation $|K|$ and the $|\cdot| \dashv \mathrm{Sing}$ adjunction. Definition of $|K|$ as the coend ${\int }^{[n]}{K}_n\times {\Delta }^n$ (or as the quotient construction May uses in §14). Theorem 14.1: $|K|$ is a CW complex with one $n$-cell per non-degenerate $n$-simplex. Theorem 14.3: $|K \times L| \cong |K| \times |L|$ in compactly-generated spaces. The adjunction $|\cdot |⊣\mathrm{Sing}$. Anchor: SOAT §14-16; Goerss-Jardine §I.2. Master tier covers the $k$-space subtlety. Three-tier, ~2000 words.
5. 03.12.28 Dold-Kan correspondence. Normalised Moore complex $N(A_{\bullet })$, equivalence of categories $\mathbf{sAb} \simeq \mathbf{Ch}{\geq 0}(\mathbf{Ab})$, normalisation theorem ($N(A\bullet)$ chain-homotopy equivalent to the alternating-sum complex). Application: simplicial homotopy groups of a simplicial abelian group are the homology of its Moore complex. Anchor: SOAT §22; Goerss-Jardine §III.2; Weibel §8.4. Master tier required for the full normalisation proof. Three-tier, ~2200 words.

Priority 2 — fibrations, classifying-space pre-history, acyclic models (4 units):

6. 03.12.29 Kan fibration. Right lifting property against horn inclusions ${\Lambda }_k^n\hookrightarrow {\Delta }^n$. Long exact sequence. Theorem (Quillen): geometric realisation of a Kan fibration is a Serre fibration. Anchor: SOAT §7; Goerss-Jardine §I.7. Intermediate + Master.
7. 03.12.30 Minimal complex and minimal fibration. SOAT §9-10. Existence of minimal subcomplex as deformation retract; isomorphism-up-to-isomorphism of any two minimal models. This is the combinatorial Postnikov-tower input. Master tier; ~1800 words.
8. 03.12.31 Twisted cartesian product (TCP) and the simplicial bar construction. SOAT §17-21. TCPs $E=F{\times }_{\tau }B$ with twisting function $\tau :B_n\to G_{n-1}$; classifying simplicial set $\bar{W}G$; classification of principal $G$-fibrations as homotopy classes of maps $B\to \bar{W}G$. Connect to the existing 03.08.05-universal-bundle unit with a lateral arrow. Anchor: SOAT §17-21; Goerss-Jardine §V.3. Master tier; ~2200 words.
9. 03.12.32 Acyclic models theorem and Eilenberg-Zilber. SOAT §28-29. Statement + proof outline of the acyclic-models theorem; application to Eilenberg-Zilber chain equivalence $C(K\times L)\simeq C(K)\otimes C(L)$ via shuffle and Alexander-Whitney maps. Anchor: SOAT §28-29; Mac Lane Homology §VIII.6. Intermediate + Master.

Priority 3 — Postnikov + Eilenberg-MacLane combinatorial (2 units):

10. 03.12.33 Postnikov tower, simplicial construction. SOAT §8, §24. Postnikov approximation of a Kan complex; $k$-invariants as cohomology classes $[k_n]\in H^{n+1}(P_{n-1};{\pi }_n)$. Defer the full proof of $k$-invariant naturality to the Master tier. Connect to existing 03.12.07-whitehead-tower (dual construction) with a lateral arrow + comparison paragraph. Intermediate + Master.
11. 03.12.34 Eilenberg-MacLane simplicial complex $K(\pi ,n)$ and cohomology operations. SOAT §22-23. Construction as the simplicial abelian group whose Moore complex is $\pi [n]$ (i.e. $\pi$ in degree $n$, zero elsewhere). Cohomology operations as $[K(\pi ,n),K(\rho ,n+q)]$. Connect to existing 03.12.05-eilenberg-maclane (the space) with a lateral arrow: same homotopy type, different category. Master tier.

Priority 4 — loop groups, deepenings, survey (1-2 units, optional):

12. 03.12.35 Simplicial loop group $G$ and the $G⊣\bar{W}$ adjunction. SOAT §25-26. Kan's loop-group construction; adjunction with the bar construction. Master-only deepening; ~1500 words.
13. Exercise pack 03.12.E2-simplicial-objects-exercises. SOAT is sparse on exercises (it is a Lectures volume) but a Codex exercise pack should include: the simplicial-set ${\Delta }^n$ and its non-degenerate simplices; computation of ${\pi }_n({\Delta }^n/\partial {\Delta }^n)$; verification that $\mathrm{Sing}(X)$ is Kan; explicit Eilenberg-Zilber shuffle for ${\Delta }^1\times {\Delta }^1$. Roughly 15-20 exercises across the three tiers.

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

• SOAT writes ${\Delta }^{\ast }$ for the simplicial category and $\Delta$ for its opposite. Modern convention (Goerss-Jardine, nLab, Codex): $\Delta$ is the simplicial category itself; simplicial sets are functors ${\Delta }^{\mathrm{op}}\to \mathbf{Set}$. Codex adopts the modern convention.
• SOAT writes $T(K)$ for geometric realisation; modern $|K|$. Codex uses $|K|$.
• SOAT writes $S(X)$ for the total singular complex; modern $\mathrm{Sing}(X)$. Codex uses $\mathrm{Sing}(X)$.
• SOAT writes ${\partial }_i,s_i$ for face / degeneracy; some sources use $d_i,s_i$. Codex uses $d_i,s_i$ to match Goerss-Jardine, with a one-line note in 03.12.24 mentioning SOAT's ${\partial }_i$.
• SOAT writes $C(K)$ for the unnormalised chain complex and $\bar{C}(K)$ or $N(K)$ for the normalised. Codex uses $C(K)$ (unnormalised) and $N(K)$ (normalised) following Weibel.

Three-tier scheduling. All P1 units require all three tiers (Beginner intuition + Intermediate definitions + Master proofs). P2 fibration / acyclic-models units can omit the Beginner tier (too technical for Strogatz-level analogy) but must keep Intermediate + Master. P3-P4 units are Master-only unless flagged otherwise.

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§4 Implementation sketch (P3 → P4)

For a full SOAT coverage pass, items 1-5 (P1) are the minimum set. Realistic production estimate (mirroring earlier Hatcher / Bott-Tu batches):

• P1 (5 units): ~3.5 hours each = ~17-18 hours. The Dold-Kan unit and the geometric-realisation unit will both run long because of the technical normalisation / $k$-space proofs.
• P2 (4 units): ~3 hours each = ~12 hours.
• P3 (2 units): ~2.5 hours each = ~5 hours.
• P4 (loop group + exercise pack): ~4 hours combined.

Total: ~38-40 hours for full SOAT equivalence. Fits a 5-7 day focused window. The Brown-Higgins-Sivera and Lawson-Michelsohn batches were similar in size.

Strict prerequisite: the queued 03.12.22-delta--complex-semi-simplicial-set must ship first. The P1.1 unit (03.12.24 simplicial set) builds directly on it.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, the simplicial-set machinery has clearly identified originators that should be cited:

• Eilenberg, S., and Zilber, J. A. "Semi-simplicial complexes and singular homology," Annals of Mathematics 51 (1950) 499-513. Originating paper for semi-simplicial sets (no degeneracies). Cite in 03.12.22 and 03.12.24.
• Eilenberg, S., and Zilber, J. A. "On products of complexes," American Journal of Mathematics 75 (1953) 200-204. Originating Eilenberg-Zilber theorem. Cite in 03.12.32.
• Kan, D. M. "Abstract homotopy. I-IV," Proceedings of the National Academy of Sciences 41 (1955) 1092-1096; 42 (1956) 255-258, 419-421, 542-544. Kan's foundational sequence.
• Kan, D. M. "Functors involving c.s.s. complexes," Transactions of the AMS 87 (1958) 330-346. The canonical reference for Kan complexes, the extension condition, simplicial homotopy groups, and the loop-group functor. Cite in 03.12.25, 03.12.26, 03.12.35.
• Kan, D. M. "A combinatorial definition of homotopy groups," Annals of Mathematics 67 (1958) 282-312. Originating combinatorial ${\pi }_n$ via Kan complexes. Cite in 03.12.26.
• Dold, A. "Homology of symmetric products and other functors of complexes," Annals of Mathematics 68 (1958) 54-80, and Kan, D. M. "Functors involving c.s.s. complexes," 1958 (above). Joint origin of the Dold-Kan correspondence. Cite in 03.12.28.
• May, J. P. Simplicial Objects in Algebraic Topology. Van Nostrand 1967. The book itself, definitive consolidation of the pre-Quillen simplicial theory. Cite in every unit that lands on this punch-list.
• Eilenberg, S., and MacLane, S. "On the groups $H(\Pi ,n)$, I-III," Annals of Mathematics 58 (1953) 55-106; 60 (1954) 49-139, 513-557. Originating Eilenberg-MacLane spaces and the simplicial-abelian-group $K(\pi ,n)$ construction. Cite in 03.12.34.

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§5 What this plan does NOT cover

• Model-category structure on $\mathbf{sSet}$ (Quillen's model structure, the trivial-cofibration / fibration factorisation, the proof that $|\cdot |$ is a Quillen equivalence to $\mathbf{Top}$). Defer to the Goerss-Jardine 3.41 plan. SOAT predates Quillen's model categories (Quillen Homotopical Algebra, Springer LNM 43, 1967, appeared the same year) and treats simplicial homotopy theory combinatorially, not model-categorically. The Codex should follow SOAT's treatment for the combinatorial foundations and defer model structure to the dedicated Goerss-Jardine pass.
• Quasi-categories / $\infty$-categories (Joyal, Lurie). Out of scope; pointer only in the Synthesis section of 03.12.25 (Kan complexes as the $(\infty ,0)$-categorical case of $\infty$-categories).
• Simplicial presheaves / sheaves. Out of scope; pointer in 03.12.31 toward Jardine's later work.
• Cubical sets / cubical homotopy theory. Out of scope. See the Brown-Higgins-Sivera 1.05a plan for cubical pointer units (03.12.15 in that plan).
• Higher simplicial / multisimplicial objects. Out of scope.
• Operads and $E_{\infty }$ structures on simplicial chain complexes. Out of scope; defer to May Geometry of Iterated Loop Spaces (if it lands on a future Fast Track entry).
• Line-number-level inventory of every theorem in SOAT. Done at the section-level above; deeper granularity is a P3-after-P2 follow-up.
• Sullivan minimal models / rational homotopy via simplicial sets. Already covered topologically in 03.12.06; the simplicial-set presentation is a deepening of 03.12.06 rather than a new unit.

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§6 Acceptance criteria for FT equivalence (SOAT)

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

• 03.12.22-delta--complex-semi-simplicial-set has shipped (strict prereq).
• All 5 P1 units have shipped. This alone raises SOAT coverage from ~5% to ~70%.
• At least 3 of the 4 P2 units have shipped. Brings coverage to ~85%. The Kan-fibration, geometric-realisation-adjunction, and Dold-Kan units are the highest-leverage of P2.
• Both P3 units have shipped or are flagged with explicit pointers from 03.12.05 and 03.12.07. Brings coverage to ~92%.
• Notation decisions are recorded in each unit's §Notation paragraph per the crosswalk in §3 above.
• Pass-W weaving connects the new units to the existing 03.12-homotopy/ chapter (especially to 03.12.05, 03.12.07, 03.12.10-cw-complex, 03.12.11-singular-homology, 03.12.12-simplicial-homology, 03.12.19-hurewicz-theorem).
• Originator-prose paragraph is present in 03.12.25, 03.12.27, 03.12.28, 03.12.34 per the citation list in §4.
• Exercise pack 03.12.E2 ships (P4) — required for the ≥95% threshold.

The 5 P1 units close most of the equivalence gap; the 4 P2 units close the fibration / classifying-space / acyclic-models gaps; the 2 P3 units close the Postnikov / cohomology-operations gaps; the P4 exercise pack plus the loop-group deepening close the residual gap to ≥95%.

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§7 Sourcing

• Free. Author-hosted PDF of the 1982 Chicago reprint; redistributed in various places online with May's blessing.
• Local copy. reference/fasttrack-texts/03-modern-geometry/May-SimplicialObjects.pdf — 1982 reprint scan, 90 PDF pages = 161 printed pages. iLovePDF-produced, scanned image-only (no embedded text layer); audit done by page-image rendering at 110-150 dpi. Cite as May, J. P., Simplicial Objects in Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press, 1967 / reprinted 1982 / 1992 (ISBN 0-226-51181-2).
• UChicago note. May has hosted a clean PDF of the reprint on his University of Chicago faculty page for two decades; see https://www.math.uchicago.edu/~may/ (book list). The Codex local copy is the canonical source for this plan.
• Comparison reading. When producing the P1-P2 units, consult Goerss-Jardine Simplicial Homotopy Theory (3.41) for the modern notation and Curtis (1971) "Simplicial homotopy theory," Advances in Mathematics 6, 107-209 for a contemporaneous reader's-guide framing. Lamotke's Semisimpliziale algebraische Topologie (Springer 1968) is the alternative classical reference and is occasionally clearer than SOAT on TCPs but harder to obtain.