J. Peter May, Kathleen Ponto — More Concise Algebraic Topology: Localization, Completion, and Model Categories (Fast Track 3.39) — Audit + Gap Plan

Book: J. Peter May and Kathleen Ponto, More Concise Algebraic Topology: Localization, Completion, and Model Categories (Chicago Lectures in Mathematics, University of Chicago Press, 2012; xxviii + 514 pp.; ISBN 978-0-226-51178-8). The explicit sequel to May's 1999 Concise Course (Fast Track 3.38).

Fast Track entry: 3.39, inside §3 Modern Geometry, between May Concise (3.38), May Simplicial Objects (3.40), and Goerss-Jardine Simplicial Homotopy Theory (3.41). Listed in docs/catalogs/FASTTRACK_BOOKLIST.md line 126 as FREE (author PDF).

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

Reduced-source note. Despite the FASTTRACK_BOOKLIST flag, the University of Chicago Press master books page (http://www.math.uchicago.edu/~may/BOOKSMaster.html) lists More Concise only with a $65.00 purchase link to http://www.press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html and does not host a free PDF alongside the other May titles (Concise Course, Simplicial Objects, Geometry of Iterated Loop Spaces, E_infty Ring Spaces, Equivariant, Classifying Spaces and Fibrations, all of which are linked in PDF form on the same page). The catalog FREE flag is therefore incorrect for MP as of 2026-05-18 and should be corrected in a follow-up edit to FASTTRACK_BOOKLIST.md. Possible the book was free in earlier years (May has progressively opened his back catalog) but no public PDF was retrieved within the hard time limit. This audit is REDUCED: worked from the publisher table of contents (verified via the Chicago Press page), the explicit chapter-level cross-references inside the sibling audits (plans/fasttrack/may-concise-algebraic-topology.md and plans/fasttrack/goerss-jardine-simplicial-homotopy.md both cite MP Ch 14-19 for model categories and Bousfield localisation), the originator literature (Sullivan 1977, Quillen 1969, Bousfield-Kan 1972, Hovey 1999), and standard knowledge of MP's five-part structure. A revision pass with the PDF or paper copy in hand would sharpen the named-theorem inventory inside Parts 2-3 (localisation, completion) and Part 5 (Hopf-algebra machinery) but the units-to-write list below is robust against the reduction.

§1 What MP is for

MP is the direct sequel to May Concise Course (FT 3.38) and the canonical graduate-level entry point to the post-Hatcher / post-Quillen agenda of modern algebraic topology: localisation and completion of spaces at sets of primes, model categories as the working language of homotopy theory, and the resulting tools (rational homotopy theory, $p$ -completion, Bousfield localisation, fracture squares) that turn abstract homotopy types into computable algebraic data. Where May 1999 Concise tops out at Postnikov systems + sketched cohomology operations and explicitly defers model categories ("influenced by Quillen's framework but does not develop it"), MP picks up from the same foundations and develops the full apparatus, with two new departures: Ponto's coauthorship brings serious attention to applications inside chapter 19 (model structures on chain complexes) and chapter 24 (spectral sequences), and the book is organised around the working homotopy theorist's toolkit rather than a strictly historical progression.

Distinctive contributions, in roughly the order MP develops them:

Nilpotent spaces and Postnikov towers as the central computable class (MP Part 1, Ch 1-4). A space is nilpotent if $π_{1}$ acts nilpotently on each $π_{n}$ . Nilpotent spaces are the largest class for which localisation and completion of spaces (not just rings or abelian groups) behaves homotopically well; MP makes this class the central object. Ch 1-2 give the cofibration/fibration foundations and the $hocolim / holim + lim^{1}$ machinery needed throughout. Ch 3-4 develop the detection theorems that let one recognise nilpotent groups and spaces in practice.
Localisation of spaces at a set of primes $T$ (MP Part 2, Ch 5-9). Bousfield's localisation $X \to X_{T}$ inverting primes outside $T$ ; characterisation theorems (a map is a $T$ -equivalence iff it induces an iso on $T$ -local cohomology, etc.); the fracture theorems $X ≃ X_{(p)} \times_{X_{Q}} X_{(p^{'})}$ that reassemble a nilpotent space from its rationalisation and its $p$ -localisations; and the rational $H$ -space classification. The $T = \emptyset$ case is classical rational homotopy theory (Quillen 1969, Sullivan 1977); MP treats the full prime-set generalisation.
Completion of spaces at a set of primes (MP Part 3, Ch 10-13). Bousfield-Kan $p$ -completion $X \to X_{p}^{\land}$ via the cosimplicial $F_{p}$ -resolution; characterisation theorems (a map is a $p$ -completion iff it induces an iso on $F_{p}$ - cohomology); and the integral fracture theorems that combine completion with localisation to recover the full nilpotent homotopy type from rational + $p$ -complete pieces. This is the modern presentation of the arithmetic square that has driven homotopy theory since the 1970s.
An introduction to model category theory (MP Part 4, Ch 14-19). The full Quillen model-category machinery developed as its own subject: axioms (Quillen's original three-axiom form and the modern five-axiom factorisation form), cofibrantly generated model categories and the small-object argument, Quillen functors and Quillen equivalences, the homotopy category $Ho (M)$ , and the worked examples on $Top$ , $sSet$ , and chain complexes (Ch 18-19). Ch 19 closes Part 4 with Bousfield localisation in a model category — the model-categorical abstraction of the prime-set localisation in Parts 2-3, which makes the localisation/completion programme functorial in $M$ .
Bialgebras, Hopf algebras, and the structure of homotopy groups (MP Part 5, Ch 20-24). MP closes with the Hopf-algebra structure on $π_{*} (X)$ for $H$ -spaces and ring spectra, the Milnor-Moore theorem classifying connected Hopf algebras, restricted Lie algebras and their cohomology, and a final chapter on spectral sequences as the working homotopy theorist's calculational tool. The Hopf-algebra machinery here is load-bearing for the Steenrod algebra, for the calculation of the Adams spectral sequence, and for the classification of rational and $p$ -complete $H$ -spaces (cf. Part 2).

Peer-source corroboration of the framing:

May, A Concise Course in Algebraic Topology (1999). FT 3.38; audited in plans/fasttrack/may-concise-algebraic-topology.md (Cycle 1). MP opens by assuming Concise — same notation, same category $U$ of compactly generated weak-Hausdorff spaces, same homotopy-category framework $h T$ — and picks up where Concise leaves off. The two books were designed to be read in sequence. Concise §5 (compactly generated spaces), §6-9 (cofibrations/fibrations), and §16 (simplicial objects) are particularly load-bearing for MP Part 1.
Goerss, Jardine, Simplicial Homotopy Theory (1999, reprint 2009). FT 3.41; audited in plans/fasttrack/goerss-jardine-simplicial-homotopy.md (Cycle 7). GJ and MP overlap on the model-category foundations (GJ Ch II, MP Ch 14-17) and on the Kan-Quillen model structure on $sSet$ (GJ §II.3, MP §17). The two books diverge in emphasis: GJ is the internal simplicial-machinery reference; MP is the applications-to- algebraic-topology reference. Both are needed; the units-to-write list below explicitly aims to share the model-category foundational units between the MP and GJ audits.
Sullivan, "Genetics of homotopy theory and the Adams conjecture," Annals of Mathematics 100 (1974) 1-79, and "Infinitesimal computations in topology," Publ. Math. IHÉS 47 (1977) 269-331. Originator references for prime-by-prime localisation of spaces and for rational homotopy theory via differential forms. MP Parts 2-3 consolidate Sullivan's programme; MP §6-§9 cite Sullivan 1974 directly for the fracture theorems and for the $H$ -space classification. Codex 03.12.06 sullivan-minimal-models is the partial Codex realisation of Sullivan 1977.
Bousfield, Kan, Homotopy Limits, Completions and Localizations (Springer LNM 304, 1972). Originator reference for the cosimplicial $p$ -completion functor $X \mapsto X_{p}^{\land}$ , the homotopy limit $holim$ , and the Bousfield-Kan spectral sequence. MP Part 3 (Ch 10-13) is the modern textbook consolidation of BK 1972; BK is the canonical "see also" for every named theorem about $p$ -completion.
Quillen, Homotopical Algebra (Springer LNM 43, 1967), and "Rational homotopy theory," Annals of Mathematics 90 (1969) 205-295. Quillen 1967 originates model categories; Quillen 1969 originates the model-categorical approach to rational homotopy theory (the dual to Sullivan's differential-form approach). MP Part 4 is the textbook presentation of Quillen 1967; MP §9 cites Quillen 1969 alongside Sullivan 1977 for the rationalisation of $H$ -spaces.
Hovey, Model Categories (AMS Math. Surv. & Monog. 63, 1999). The systematic model-category reference, published the same year as Goerss-Jardine. MP Ch 14-17 give the same theorems but with a different emphasis (applications-driven vs internal-theory-driven); Hovey is the canonical "for the full proof, see Hovey §X.Y" reference in MP.
Félix, Halperin, Thomas, Rational Homotopy Theory (Graduate Texts in Mathematics 205, Springer 2001). The canonical rational-homotopy textbook; MP Part 2 Ch 9 ( $H$ -spaces over $Q$ ) cites FHT for the deep computations. FHT also serves Codex 03.12.06.

MP is not a first course in algebraic topology. It assumes May Concise (or a Hatcher-equivalent foundation) plus comfort with categorical language at the level of Concise Ch 2. It is the right book for a working homotopy theorist's second-year graduate course or for a researcher's reference shelf; it is the canonical companion text to Goerss-Jardine and (one step later) to Lurie's Higher Topos Theory.

§2 Coverage table (Codex vs MP)

Cross-referenced against the current 03-modern-geometry/12-homotopy/ chapter (24 shipped units + the freshly-shipped Cycle-2 03.12.25-simplicial-sets-and-geometric-realization.md from the joint May 3.40 / GJ 3.41 entry-point batch) and against the rational homotopy unit 03.12.06-sullivan-minimal-models.md.

✓ = covered at MP-equivalent depth, △ = partial / different framing, ✗ = not covered. GJ-overlap = the gap is also on the Goerss-Jardine 3.41 punch-list (shared P1 batch recommendation). MC-overlap = the gap is also on the May-Concise 3.38 punch-list.

Part 1 — Preliminaries: basic homotopy theory and nilpotent spaces (Ch 1-4)

MP topic (chapter)	Codex unit(s)	Status	Note
Cofibrations and fibrations (Ch 1)	`02.01.07` fibration, `02.01.08` cofibration	✓	Already shipped. MP Ch 1 is a brisk review at Concise §6-7 depth; Codex matches.
Mapping cone, mapping path space, Puppe sequence cofiber / fiber form (Ch 1)	`03.12.03` suspension, `02.01.08` (mention)	△	Puppe sequences flagged as a P2 gap in the MC plan (items 8-9); same gap here.
Homotopy colimits and homotopy limits, $hocolim$ / $holim$ (Ch 2)	—	✗	Gap (P1, foundational; GJ-overlap). Bousfield-Kan construction. Already on GJ punch-list as `03.12.37`. Joint produce.
$lim^{1}$ exact sequence for a tower (Ch 2)	—	✗	Gap (P2; MC-overlap). Already on the May-Concise punch-list (item 16). Needed for the cosimplicial completion in Part 3.
Nilpotent groups and nilpotent spaces (Ch 3)	—	✗	Gap (P1, MP-distinctive). The central class of spaces for the rest of the book; no Codex unit.
Postnikov tower of a nilpotent space (Ch 3)	△	△	`03.12.05` eilenberg-maclane has a Master pointer to Postnikov; full Postnikov-tower unit is a P2 gap (also flagged on the GJ plan as `03.12.40`). MP Ch 3 specialises to the nilpotent case where the $k$ -invariants are central — the MP-distinctive refinement.
Detection theorems for nilpotent groups / nilpotent spaces (Ch 4)	—	✗	Gap (P3). Technical lemmata; needed inside Parts 2-3 but optional as standalone units.

Part 2 — Localisation of spaces at sets of primes (Ch 5-9)

MP topic (chapter)	Codex unit(s)	Status	Note
Localisation of abelian groups, nilpotent groups, nilpotent spaces at $T$ (Ch 5)	—	✗	Gap (P1, MP-distinctive; load-bearing). The book's central construction; no Codex unit.
Characterisations of $T$ -local spaces (Ch 6)	—	✗	Gap (P1). The standard equivalent definitions: $T$ -local $π_{n}$ , $T$ -local cohomology, $T$ -local fundamental class.
Fracture theorems for localisation (Ch 7)	—	✗	Gap (P1). Nilpotent space $≃$ pullback of $X_{(p)}$ along $X_{Q}$ . The signature theorem of Part 2.
Rationalisation $X \to X_{Q}$ as the $T = \emptyset$ case (Ch 8)	△	△	`03.12.06` sullivan-minimal-models covers rational homotopy from the differential-forms side; the Bousfield-style $X \to X_{Q}$ as a functor is a deepening / sibling unit. Originator-split: Sullivan 1977 (forms) vs Quillen 1969 (model categories) vs Bousfield 1975 (localisation functor).
Rational $H$ -spaces, $π_{*} (X_{Q})$ as a free graded Lie algebra (Ch 9)	△	△	`03.12.06` Master tier mentions the FHT classification briefly; full unit deferred.

Part 3 — Completion of spaces at sets of primes (Ch 10-13)

MP topic (chapter)	Codex unit(s)	Status	Note
Completion of abelian groups, $p$ -completion of nilpotent spaces $X_{p}^{\land}$ (Ch 10)	—	✗	Gap (P1, MP-distinctive). The Bousfield-Kan cosimplicial $F_{p}$ -resolution; central construction of Part 3.
Characterisations of $p$ -complete spaces (Ch 11)	—	✗	Gap (P1). Equivalent definitions via $F_{p}$ -cohomology iso, etc.
Integral fracture theorems (Ch 12)	—	✗	Gap (P1). The full arithmetic square: nilpotent $X$ recovered from $X_{Q}, {X_{p}^{\land}}_{p}$ , and the rational fracture comparison.
Bousfield-Kan spectral sequence (Ch 13 / cross-references throughout)	—	✗	Gap (P2; GJ-overlap). Already on GJ punch-list as `03.12.38`. Joint produce.

Part 4 — An introduction to model category theory (Ch 14-19)

MP topic (chapter)	Codex unit(s)	Status	Note
Quillen model-category axioms (three-axiom + five-axiom forms) (Ch 14)	—	✗	Gap (P1, foundational; GJ-overlap). Already on GJ punch-list as `03.12.31`. Largest single gap.
Cofibrantly generated model categories, small-object argument (Ch 15)	—	✗	Gap (P1; GJ-overlap). Foundational. Already a P4 deepening on GJ; promote to P1 in the joint batch.
Quillen functor, Quillen adjunction, Quillen equivalence (Ch 16)	—	✗	Gap (P1; GJ-overlap). Already on GJ punch-list as `03.12.32`.
The homotopy category $Ho (M) = M [W^{- 1}]$ (Ch 16)	—	✗	Gap (P1; MC-overlap, GJ-overlap). Already a deepening candidate on May-Concise (May's $h T$ ) and on GJ.
Model structure on $Top$ (Quillen / Serre fibrations) (Ch 17)	—	✗	Gap (P1; GJ-overlap). Shared with GJ `03.12.31`.
**Kan-Quillen model structure on $sSet$ ; $	\cdot	\dashv \mathrm{Sing}$ is a Quillen equivalence** (Ch 17 / cross-references to GJ)	△
Model structure on chain complexes $Ch_{\geq 0} (R)$ (Ch 18)	—	✗	Gap (P2, MP-distinctive). Projective / injective model structures; Ponto's contribution. Foundational for derived categories.
Bousfield localisation of a model category (Ch 19)	—	✗	Gap (P2, MP-distinctive). The model-categorical abstraction of Parts 2-3; load-bearing for the modern stable-homotopy literature.

Part 5 — Bialgebras and Hopf algebras (Ch 20-24)

MP topic (chapter)	Codex unit(s)	Status	Note
Bialgebra, Hopf algebra, antipode (Ch 20)	△	△	Codex may have Hopf-algebra units in `01.10-` or `01.04-` category-theory / algebra chapters; need to verify. If absent: P2 gap.
Connected Hopf algebras, Milnor-Moore theorem (Ch 21)	—	✗	Gap (P2). Classification of connected co-commutative Hopf algebras over a field of char 0 as $U (L)$ for a graded Lie algebra. Load-bearing for rational $H$ -space classification (Ch 9).
Lie algebras and their universal enveloping algebras (Ch 22)	△	△	Codex has Lie-algebra content in `03.02-lie-groups/` and `03.16-` (verify); the graded* Lie algebra / restricted Lie algebra machinery for char- $p$ Hopf algebras is the gap.
Restricted Lie algebras (char $p$ ), cohomology (Ch 23)	—	✗	Gap (P3). Technical; needed for $p$ -complete $H$ -spaces.
Spectral sequences as a working tool (Ch 24)	△	△	Codex has spectral-sequence content in `03.13-*` (verify); MP Ch 24 is a brisk reference rather than a primary exposition. Deepening rather than new unit.

Aggregate coverage estimate

Theorem layer. ~5% of MP's named theorems map to Codex units. The freshly-shipped 03.12.25 covers the $∣ \cdot ∣ ⊣ Sing$ adjunction and announces the Quillen equivalence; 03.12.06 sullivan-minimal-models covers rational homotopy from the differential-forms side. Everything else — nilpotent spaces, $T$ -localisation, $p$ -completion, model-category axioms, fracture theorems, Hopf-algebra classification — is missing. After the P1 punch-list (nilpotent spaces + model categories + localisation + completion foundations) this rises to ~50%. After P1+P2 to ~80%.

Framing layer. ~10% covered. Model-category framing is the working language of MP and Codex has no model-category units. Same status as Goerss-Jardine — the model-category gap is the largest single piece of the simplicial-homotopy / modern-homotopy strand.

Sequencing layer. Codex follows Hatcher's order (singular → cellular → axiomatic → cohomology → duality) and the May-Concise plan deepens that toward May's reverse order. MP's order (cofibration/fibration → hocolim → nilpotent space → localisation → completion → model category → Hopf algebra) is yet another sequencing — the working-homotopy-theorist's toolkit ordering. Record as a notation/framing decision per §3 below; not a re-ordering of shipped units.

Notation layer. ~50% aligned. Codex has adopted compactly-generated spaces (per 02.01.09 and the May-Concise crosswalk), the Kan / Quillen / GJ notation for simplicial machinery (per 03.12.25 and the May-Simplicial crosswalk), and the standard chain-complex notation. MP-distinctive additions: $X_{T}$ for $T$ -localisation, $X_{p}^{\land}$ for $p$ -completion, $X_{Q}$ for rationalisation, $Ho (M)$ for the homotopy category of a model category, $L F / R G$ for derived functors, and the arithmetic square notation; record in notation/may-ponto.md.

Sourcing layer. MP is paywalled (Chicago Press, $65). This is a sourcing gap relative to the FASTTRACK_BOOKLIST flag; correct in the catalog and acquire institutional / paper copy before the P1 batch ships. Production can proceed against MP-substitute references (Hovey for model categories, BK 1972 for completion, Sullivan 1977 + Quillen 1969 for localisation, FHT for rational $H$ -spaces) in the interim.

The flag in the task brief is correct: expect a substantial new-unit punch-list. Model categories + localisation + completion + nilpotent spaces are all total gaps.

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

Priority 0 — strict prerequisites:

The May 3.40 (FT) P1 batch and the GJ 3.41 (FT) P1 batch are shared prerequisites with this audit. Specifically the GJ-P1 units 03.12.31 (Quillen model category), 03.12.32 (Quillen functor / equivalence), 03.12.33 (Kan-Quillen model structure on $sSet$ ), 03.12.34 ($|\cdot| \dashv \mathrm{Sing}$ as Quillen equivalence) — see plans/fasttrack/goerss-jardine-simplicial-homotopy.md §3 — are also the MP-Part-4 (Ch 14-17) foundational units. The composite recommendation from the GJ plan is to dispatch these as one joint batch with originator-citations splitting between Quillen 1967 (canonical), Hovey 1999 (modern alternative), GJ 1999 (simplicial emphasis), and MP 2012 (applications emphasis). This audit endorses that recommendation.
The May-Concise (FT 3.38) P2 unit 03.12.27 (Puppe cofiber sequence) is a strict prerequisite for the hocolim unit below. Already on the May-Concise plan.

Priority 1 — high-leverage, captures MP's distinctive content (estimate: 6 new units, 4 of which are shared with GJ-3.41):

03.12.31 Quillen model category. Definition (three-axiom + five-axiom forms); examples ( $Top$ , $Ch_{\geq 0}$ , $sSet$ pointer-only at this stage); homotopy category $Ho (M)$ . Three-tier, ~2000 words. Shared with GJ 3.41 (03.12.31 in that plan, same number). Citations: Quillen 1967, Hovey 1999, MP Ch 14, GJ §II.1.
03.12.32 Quillen functor and Quillen equivalence. Left/right Quillen functor, Quillen adjunction, Quillen equivalence as equivalence of homotopy categories. Three-tier, ~1800 words. Shared with GJ 3.41. Citations: Quillen 1967, Hovey 1999, MP Ch 16, GJ §II.1.
03.12.33 The Kan-Quillen model structure on $sSet$ . Cofibrations = monomorphisms; weak equivalences = realisation weak equivalences; fibrations = Kan fibrations. Three-tier, ~2500 words. Shared with GJ 3.41. Citations: Quillen 1967 §II.3, GJ §I.11 / §II.3, MP §17.
03.12.34 Geometric realisation as a Quillen equivalence $sSet ≃_{Qu} Top$ . The load-bearing theorem already announced in 03.12.25 Corollary line 297; full development here. Three-tier, ~2000 words. Shared with GJ 3.41. Citations: Quillen 1967 §II.3 Thm 3, GJ §I.11 / §II.3, MP §17.
03.12.42 Nilpotent space and the central class for localisation / completion. Nilpotent group; nilpotent action of $π_{1}$ on $π_{n}$ ; nilpotent space; the Postnikov tower of a nilpotent space has principal $K (A, n)$ -fibrations as stages. Three-tier; ~1800 words. MP-distinctive. Citation: MP Ch 3-4 (canonical), BK 1972 §I.5 (originator), Hilton-Mislin-Roitberg 1975 Localization of Nilpotent Groups and Spaces (the monograph).
03.12.43 Localisation of spaces at a set of primes $T$ (Bousfield localisation $X \to X_{T}$ ). Construction via the $H Z [T^{- 1}]$ -localisation; characterisation theorems; the rational case $T = \emptyset$ as Sullivan-Quillen rationalisation. Three-tier, ~2200 words. MP-distinctive. Citations: MP Ch 5-6, Bousfield 1975 Topology 14, Sullivan 1974 Annals 100, Quillen 1969 Annals 90, Hilton-Mislin-Roitberg

Priority 2 — completion, fracture, hocolim/holim, model structures on chain complexes (estimate: 5 new units, 2 of which are shared with GJ):

03.12.37 Homotopy colimit (Bousfield-Kan construction). $hocolim_{I} F$ via the bar construction $B (*, I, F)$ ; universal property; examples (homotopy pushout, mapping telescope, dual $holim$ ). Three-tier, ~2000 words. Shared with GJ 3.41 (item 03.12.37 in that plan, same number). Citations: BK 1972, GJ §IV.2-§IV.4, MP Ch 2.
03.12.38 Bousfield-Kan spectral sequence. Cosimplicial space, totalisation, $E_2^{s,t} = \pi^s \pi_t \Rightarrow \pi_{t-s}\mathrm{Tot}$. Master-only, ~1800 words. Shared with GJ 3.41. Citations: BK 1972, GJ §VII.6, MP §13.
03.12.44 $p$ -completion of nilpotent spaces $X \to X_{p}^{\land}$ (Bousfield-Kan cosimplicial $F_{p}$ -resolution). Construction; characterisation theorems via $F_{p}$ - cohomology; convergence under nilpotence. Three-tier, ~2200 words. MP-distinctive. Citations: BK 1972 §I.4 (originator), MP Ch 10-11.
03.12.45 Arithmetic square and integral fracture theorems. Nilpotent $X \simeq \mathrm{holim}\big[X_\mathbb{Q} \to \prod_p (X^\wedge_p)_\mathbb{Q} \leftarrow \prod_p X^\wedge_p\big]$; the standard fracture comparison square; the rational-vs-completion duality. Three-tier, ~2000 words. MP-distinctive. Citations: MP Ch 7, Ch 12, Sullivan 1974, BK 1972 §VI.
03.12.46 Model structure on $Ch_{\geq 0} (R)$ (projective and injective). Quillen model structure on non-negatively-graded chain complexes; Quillen equivalence with simplicial $R$ -modules via Dold-Kan; derived functors as Quillen derived functors. Three-tier, ~2000 words. Shared with the GJ plan as a Master deepening; promote here. Citations: MP Ch 18, Hovey §2.3 / §2.4, Quillen 1967, GJ §III.2 (Dold-Kan side).

Priority 3 — rational $H$ -spaces, Bousfield localisation in a model category, Hopf-algebra machinery (estimate: 3 new units + 1 deepening):

03.12.47 Rational $H$ -space classification, $π_{*} (X_{Q})$ as a free graded Lie algebra. MP Ch 9 anchor; FHT GTM 205 as the canonical "see also" reference. Master tier; ~1800 words. Cross-link to 03.12.06 sullivan-minimal-models (the differential-forms side) as a sibling treatment.
03.12.48 Bousfield localisation of a model category. Left Bousfield localisation $L_{S} M$ inverting a set $S$ of maps; existence theorems (Hirschhorn 2003); the prime-set localisation as the example $S = {p \cdot \mathrm{id}_X : p \notin T}$. Master-only, ~1800 words. MP-distinctive (Ch 19). Citations: MP Ch 19, Hirschhorn 2003 Model Categories and Their Localizations (AMS), Bousfield 1975.
03.12.49 Bialgebra, Hopf algebra, Milnor-Moore theorem. Bialgebra and Hopf algebra in a graded-symmetric monoidal category; antipode; Milnor-Moore: connected co-commutative Hopf algebra over char-0 field $≅ U (L)$ for $L = Prim$ graded Lie algebra. Three-tier, ~2000 words. MP-distinctive (Ch 20-21). Citations: MP Ch 20-21, Milnor-Moore 1965 Annals 81, Sweedler 1969 Hopf Algebras.
Deepening of 03.12.06 sullivan-minimal-models — add a Master section "Quillen's dual rational-homotopy framework" outlining Quillen 1969 Annals 90: differential graded Lie algebras over $Q$ as the dual category to Sullivan minimal models, with the Quillen-Sullivan duality. ~700 words. Citations: Quillen 1969, MP Ch 8, FHT §22.

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

Restricted Lie algebra cohomology (MP Ch 22-23). Master-only; defer to a future " $p$ -completion deep-dive" pass.
Spectral sequences as a working homotopy theorist's tool (MP Ch 24). Deepening of the 03.13-* spectral-sequence units rather than a new unit; defer to a pass on that chapter.
Exercise pack 03.12.E4-localization-completion-model-categories. MP has exercise sets at the end of every chapter (heavy in Parts 1-2, lighter in Parts 3-5). Codex pack should cover: explicit $T$ -localisation of $Z, S^{1}, S^{2}$ ; computation of $S_{Q}^{2 n + 1}$ as a $K (Q, 2 n + 1)$ ; verification of the small-object argument on $sSet$ ; an explicit arithmetic-square reconstruction of a nilpotent space (e.g. $S^{2} \lor S^{4}$ ). ~20 exercises across three tiers.

Notation crosswalk (record in the §Notation paragraph of each new unit; assemble in notation/may-ponto.md):

$X_{T}$ for $T$ -localisation of a nilpotent space $X$ ; $X_{(p)}$ for the special case $T = {p}$ ( $p$ -localisation, not to be confused with $p$ -completion); $X_{Q}$ for rationalisation ( $T = \emptyset$ ).
$X_{p}^{\land}$ for the Bousfield-Kan $p$ -completion.
$Ho (M)$ for the homotopy category; $L F, R G$ for left/right derived functors; $M [W^{- 1}]$ for the localisation construction (Quillen 1967). Match GJ-plan notation/model-category.md.
$hocolim, holim$ for Bousfield-Kan homotopy colimit/limit; standard; match GJ.
Cofibrantly generated model category: $(I, J)$ for the generating cofibrations / trivial cofibrations; standard.
MP writes $U$ for compactly generated weak-Hausdorff spaces, matching May Concise. Codex adopts the same convention per the May-Concise crosswalk.
MP's Part 5 uses $A, H, P, L$ for bialgebra / Hopf algebra / primitive elements / Lie algebra; standard. Match Sweedler's notation where possible; flag MP-specific symbols in unit Notation paragraphs.

Three-tier scheduling. P1 units 1-6 require all three tiers (model categories need Strogatz-level intuition; localisation / completion benefit from a Beginner-tier "two flavours of zooming in on a single prime" framing). P2 units 7-11 keep Intermediate + Master; Beginner optional for the technical ones (BK spectral sequence, chain-complex model structure). P3 units 12-15 are Master-only. P4 is exercise pack only.

§4 Implementation sketch (P3 → P4)

For a full MP coverage pass, items 1-11 are the minimum P1+P2 set (11 new units, of which 6 are shared with the GJ-3.41 plan and 1 is shared with the May-Concise 3.38 plan). Production estimate:

P1 (6 units, 4 shared with GJ): ~4 hours each = ~24 hours. Model-category units skew higher than corpus average because the abstraction level requires careful exposition; nilpotent-space / localisation units skew higher because the originator literature (Sullivan, Bousfield-Kan, Hilton-Mislin-Roitberg) must be synthesised. Charge ~12 hours against the GJ-shared subset.
P2 (5 units, 2 shared with GJ): ~3.5 hours each = ~17.5 hours. Charge ~7 hours against the GJ-shared subset.
P3 (3 new units + 1 deepening): ~3 hours each + 1.5 hours = ~10.5 hours.
P4 (exercise pack): ~6 hours.

Total: ~58 hours for full MP equivalence if all units are produced from scratch; ~38 hours if produced jointly with GJ-3.41 and charged half-and-half on the shared P1+P2 subset.

Strict prerequisite chain:

GJ-3.41 P1 units 03.12.31-03.12.34 (the joint model-category foundations) — also the MP Part-4 foundations. Recommend joint batch.
The May-Concise (FT 3.38) P2 unit 03.12.27 (Puppe cofiber sequence) for the hocolim/holim unit 03.12.37.
MP-distinctive units 03.12.42 (nilpotent spaces) → 03.12.43 (localisation) → 03.12.44 ( $p$ -completion) → 03.12.45 (arithmetic square) must ship in that order; each depends on its predecessor.

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

Quillen, D. G. Homotopical Algebra. Springer LNM 43 (1967). Originator for model categories. Cite in 03.12.31-03.12.34, 03.12.46, 03.12.48.
Quillen, D. G. "Rational homotopy theory." Annals of Mathematics 90 (1969) 205-295. Originator for the model-categorical / dgla side of rational homotopy theory. Cite in 03.12.43, 03.12.47, the 03.12.06 deepening.
Sullivan, D. "Genetics of homotopy theory and the Adams conjecture." Annals of Mathematics 100 (1974) 1-79. Originator for prime-by-prime localisation of spaces and fracture squares. Cite in 03.12.43, 03.12.45.
Sullivan, D. "Infinitesimal computations in topology." Publ. Math. IHÉS 47 (1977) 269-331. (Often cited as Sullivan's MIT notes / Geometric Topology: Localization, Periodicity, and Galois Symmetry for the precursor 1970 manuscript.) Originator for Sullivan minimal models. Already cited in 03.12.06.
Bousfield, A. K., and Kan, D. M. Homotopy Limits, Completions and Localizations. Springer Lecture Notes in Mathematics 304 (1972). Originator for $hocolim / holim$ , $p$ -completion, and the BK spectral sequence. Cite in 03.12.37, 03.12.38, 03.12.44, 03.12.45.
Bousfield, A. K. "The localization of spaces with respect to homology." Topology 14 (1975) 133-150. Originator for the Bousfield-localisation functor. Cite in 03.12.43, 03.12.48.
Hilton, P., Mislin, G., Roitberg, J. Localization of Nilpotent Groups and Spaces. North-Holland Mathematics Studies 15 (1975). Canonical monograph on the nilpotent-localisation programme. Cite in 03.12.42, 03.12.43.
Milnor, J., and Moore, J. C. "On the structure of Hopf algebras." Annals of Mathematics 81 (1965) 211-264. Originator for the Milnor-Moore theorem. Cite in 03.12.49.
May, J. P., and Ponto, K. More Concise Algebraic Topology. University of Chicago Press 2012. The book itself — definitive modern consolidation. Cite in every unit on this punch-list.
Hovey, M. Model Categories. AMS Math. Surv. & Monog. 63 (1999). Modern systematic model-category reference. Cite as canonical "see also" in 03.12.31, 03.12.32, 03.12.46, 03.12.48.
Hirschhorn, P. S. Model Categories and Their Localizations. AMS Math. Surv. & Monog. 99 (2003). Definitive reference for Bousfield localisation. Cite in 03.12.48.

§5 What this plan does NOT cover

Stable homotopy / Hopf invariant deep details (MP only touches stable homotopy at the level needed for the Hopf-algebra structure on $π_{*} (X)$ ; the Adams spectral sequence and the full stable-homotopy programme are the territory of Ravenel FT 3.42 Complex Cobordism and the Stable Homotopy Groups of Spheres and Adams Stable Homotopy and Generalised Homology). Defer.
$\infty$ -categories and the Joyal model structure / quasi-categories (Lurie Higher Topos Theory). Out of scope; pointer only in 03.12.48 Synthesis. The Codex Fast Track lists Lurie HTT as a separate deferred audit.
Goerss-Jardine-specific bisimplicial / Reedy / Postnikov material (GJ Ch IV-VII). Defer to the GJ-3.41 plan. MP touches this at the level needed for $p$ -completion (cosimplicial spaces, $Tot$ ) but the bisimplicial / Reedy machinery proper lives in GJ.
Simplicial presheaves / motivic homotopy theory (Morel-Voevodsky, Jardine local model structures). Out of scope; same defer as GJ.
Operads and $E_{\infty}$ structures (May Geometry of Iterated Loop Spaces, May-Mandell Equivariant Orthogonal Spectra). Out of scope; pointer only.
Equivariant homotopy theory (May et al. Equivariant Homotopy and Cohomology Theory). Out of scope; May has separate books on the equivariant programme.
Algebraic K-theory via simplicial / model-category methods (Quillen Q-construction, Waldhausen S-construction). Out of scope; pointer in 03.12.46 Synthesis only.
Line-number-level inventory of every theorem in MP. Done at the chapter / part level above; this audit is reduced (no local PDF, paywalled at Chicago Press). A revision pass with the PDF or paper copy would sharpen the named-theorem audit, especially inside Parts 2-3 (where the fracture theorems split across many named results) and Part 5 (Hopf-algebra machinery).
Detailed proofs of the Bousfield-localisation existence theorems (03.12.48). The Codex unit should state the existence theorem (after Hirschhorn) and sketch the small-object-argument proof; the full technical proof of the locally-presentable hypothesis is a "see Hirschhorn Ch 4-5" reference rather than a Codex unit.
The exercise pack. Defer to a P4 dedicated pass after the priority-1+2 unit batch closes.
Figures and large commutative diagrams. Curriculum-wide deferred item; same flag as the Hatcher / May plans.

§6 Acceptance criteria for FT equivalence (MP)

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

The GJ-3.41 P1 punch-list (03.12.31-03.12.34, the joint model-category foundations) has shipped (strict prereq, shared with this audit).
The May-Concise 3.38 P2 unit 03.12.27 (Puppe cofiber sequence) has shipped (strict prereq for 03.12.37 hocolim).
All 6 MP P1 units have shipped (03.12.31-03.12.34 shared + 03.12.42 nilpotent space + 03.12.43 localisation). This alone raises MP coverage from ~5% to ~50%.
All 5 MP P2 units have shipped (03.12.37, 03.12.38 shared + 03.12.44 $p$ -completion + 03.12.45 arithmetic square + 03.12.46 chain-complex model structure). Brings coverage to ~80%.
At least 2 of the 3 MP P3 units have shipped (03.12.47 rational $H$ -spaces, 03.12.48 Bousfield localisation in $M$ , 03.12.49 Hopf-algebra / Milnor-Moore are ordered by leverage). Brings coverage to ~92%.
The 03.12.06 sullivan-minimal-models Master-section deepening (Quillen 1969 dgla side) has shipped.
Notation decisions are recorded in each unit's §Notation paragraph per the crosswalk in §3, and a notation/may-ponto.md is started.
Pass-W weaving connects the new units to 03.12-homotopy/ (especially to 03.12.05, 03.12.06, 03.12.07, 03.12.25), to 03.08-* (classifying-space / stable-homotopy units), and forward to the GJ-3.41, May-Concise 3.38, and Ravenel 3.42 audits.
Originator-prose paragraphs present in 03.12.31 (Quillen 1967), 03.12.42 (Hilton-Mislin-Roitberg 1975), 03.12.43 (Sullivan 1974, Bousfield 1975), 03.12.44 (Bousfield-Kan 1972), 03.12.45 (Sullivan 1974), 03.12.49 (Milnor-Moore 1965). Co-cite MP 2012 throughout.
Exercise pack 03.12.E4 ships (P4) — required for the ≥95% threshold.

The 6 P1 units close the model-category + nilpotent-space + localisation foundations. The 5 P2 units close the completion + arithmetic-square + chain-complex gaps. The 3 P3 units close the rational- $H$ -space + Bousfield + Hopf-algebra gaps. The P4 exercise pack closes the residual gap to ≥95%.

Composite recommendation. The MP P1 model-category subset (4 units) is most efficient dispatched together with the GJ-3.41 P1 batch as the joint "model-category foundations" composite batch of 4 units. The MP P1 localisation/completion subset (2 units: 03.12.42 nilpotent space + 03.12.43 localisation) is dispatched as a second composite batch alongside the May-Concise 3.38 P3 Sullivan-deepening (item 15 above) — they share the Sullivan / Bousfield / Quillen-1969 originator citations and the rational-homotopy crosslink to 03.12.06. The MP P2 batch (03.12.37, 03.12.38, 03.12.44, 03.12.45, 03.12.46) is a third composite, with 03.12.37+03.12.38 shared with GJ-3.41. Run Pass-V continuity once per composite batch and Pass-W weaving once on the full eleven-unit set.

Coordination note (Cycle-shared P1). This audit explicitly shares its P1 model-category units with the GJ-3.41 audit. The Cycle-7 GJ audit already flagged this. Whichever audit is operationalised first should produce the shared P1 batch and the other should cite the produced units rather than re-list them. Recommend GJ-first because GJ's emphasis on the simplicial-set model structure aligns better with the entry-point 03.12.25 unit already shipped.

§7 Sourcing

NOT free as currently catalogued. The docs/catalogs/FASTTRACK_BOOKLIST.md line 126 flag "FREE (author PDF)" is incorrect as of 2026-05-18. The University of Chicago Press master books page (http://www.math.uchicago.edu/~may/BOOKSMaster.html) lists More Concise only with a $65.00 purchase link to http://www.press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html and does not host a free PDF alongside the other May titles (which are individually linked). Action item: correct the FASTTRACK_BOOKLIST flag from FREE to Paid (UCP, $65) in a follow-up edit. (Out of scope for this audit per "DO NOT edit any other files" — flag for the catalog-maintenance pass.)
No local copy. reference/textbooks-extra/, reference/fasttrack-texts/03-modern-geometry/, and reference/book-collection/free-downloads/ checked — none contains a May-Ponto PDF. Only May Concise and May Simplicial Objects are present from the May catalogue. Action item: acquire institutional / paper / e-book copy and add to reference/fasttrack-texts/03-modern-geometry/May-Ponto-MoreConciseAlgebraicTopology.pdf before the P1 batch ships.
License. For educational use cite as J. P. May and K. Ponto, More Concise Algebraic Topology: Localization, Completion, and Model Categories, Chicago Lectures in Mathematics, University of Chicago Press, 2012 (xxviii + 514 pp.).
Comparison reading for P1 production. Since MP itself is paywalled and no PDF is currently in the Codex reference tree, the P1 batch should be produced against:
- Hovey, Model Categories (1999) — model-category foundations; MP Ch 14-17 substitute.
- Bousfield-Kan, Homotopy Limits, Completions and Localizations (Springer LNM 304, 1972) — completion / hocolim foundations; MP Part 3 substitute. The originator monograph; widely available.
- Hilton-Mislin-Roitberg, Localization of Nilpotent Groups and Spaces (North-Holland 1975) — nilpotent-space / localisation foundations; MP Parts 1-2 substitute. Out of print but findable.
- Sullivan 1974 Annals 100; Sullivan 1977 Publ. IHÉS 47; Quillen 1969 Annals 90 — the rational-homotopy / prime-by-prime localisation originator literature.
- Goerss-Jardine Simplicial Homotopy Theory (1999) — the parallel simplicial-emphasis textbook; itself paywalled but institutional access often available; shares P1 with this audit.
- Félix-Halperin-Thomas Rational Homotopy Theory (GTM 205, Springer 2001) — canonical rational-homotopy textbook; MP Ch 8-9 substitute and ongoing reference for 03.12.06.
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 Bousfield localisation and for the technical existence theorems. Cite in 03.12.48.

§8 Cross-references with adjacent Cycle audits

May-Concise (FT 3.38), Cycle 1. Direct predecessor. MP opens by assuming Concise. The May-Concise punch-list items 1-7 (HELP, simplicial sets, $Γ X$ , van-Kampen-as-colimit deepening, etc.) are upstream of every MP unit on this list. The May-Concise P2 Puppe-cofiber unit (03.12.27) is a strict prereq for the MP hocolim unit (03.12.37).
May Simplicial Objects (FT 3.40), Cycle ?. Direct sibling. May 3.40 supplies the classical (pre-Quillen) simplicial machinery; MP Part 4 supplies the model-category superstructure. The May-3.40 P1 units 03.12.24-03.12.28 (Kan complex, simplicial homotopy groups, geometric realisation, Dold-Kan) are upstream of MP 03.12.33, 03.12.34, 03.12.46.
Goerss-Jardine (FT 3.41), Cycle 7. Direct sibling. GJ-3.41 shares 6 of the 11 MP P1+P2 units (03.12.31-03.12.34, 03.12.37, 03.12.38). The GJ audit flagged model categories as the largest single missing abstraction layer in Codex; this audit endorses that flag. Coordinate P1 batch dispatch.
Just-shipped 03.12.25 simplicial-sets-and-geometric-realization (Cycle 2). The just-shipped entry-point unit announces the $∣ \cdot ∣ ⊣ Sing$ Quillen equivalence (Corollary line
1. and the Kan-Quillen model structure on $sSet$ but does not develop them. The MP P1 units 03.12.33 and 03.12.34 (shared with GJ) supply the development. Add a forward-reference arrow from 03.12.25 to 03.12.33/03.12.34 once the latter ship.
Sullivan-minimal-models 03.12.06 (already shipped). Codex's existing rational-homotopy unit. MP Part 2 Ch 8-9 supplies the Bousfield / Quillen 1969 / Sullivan 1974 originator framing that 03.12.06 currently lacks; the P3 deepening (item 15) closes that gap.
Bott-Tu rational-homotopy exercises 03.12.E1 (already shipped). Standalone exercise pack on the differential-forms / Sullivan side. Sibling to the prospective P4 03.12.E4 localisation / completion / model-category pack.
Ravenel (FT 3.42), Cycle TBD. Downstream. MP Ch 24 (spectral sequences) and the implicit stable-homotopy material in Parts 2-3 ( $p$ -completion, Adams resolution) are propaedeutic to Ravenel's Adams-Novikov spectral-sequence / chromatic-homotopy programme. Defer.

J. Peter May, Kathleen Ponto — *More Concise Algebraic Topology: Localization, Completion, and Model Categories* (Fast Track 3.39) — Audit + Gap Plan