Model Categories¶
Job: equip a category with weak equivalences, fibrations, and cofibrations so that homotopy theory (localization, mapping spaces, derived functors) is computable by lifting and factorizations — Quillen’s axiomatic setting for modern homotopy algebra.
Core¶
A model category (Quillen) is a complete and cocomplete category \(\mathcal{M}\) with three classes of morphisms — weak equivalences \(\mathcal{W}\), fibrations \(\mathrm{Fib}\), cofibrations \(\mathrm{Cof}\) — satisfying factorization, lifting, and 2-out-of-3 axioms. The homotopy category \(\mathrm{Ho}(\mathcal{M})=\mathcal{M}[\mathcal{W}^{-1}]\) is equivalent to the category of cofibrant–fibrant objects and homotopy classes of maps.
| Layer | Content |
|---|---|
| Axioms | (MC1)–(MC5): limits/colimits; 2-out-of-3; retracts; lifting; factorizations |
| Homotopy | cylinder/path objects; left/right homotopy; \(\pi(X,Y)\) when bifibrant |
| Replacement | \(Q\) cofibrant, \(R\) fibrant; \(RQX\) models \(\mathrm{Ho}\) |
| Functors | Quillen adjunctions; derived \(LF\), \(RG\); Quillen equivalences |
| Examples | \(\mathbf{Top}\), \(\mathrm{Ch}_{\ge 0}(R)\), \(\mathbf{sSet}\), spectra, simplicial modules |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | Quillen axioms; lifting; \(\mathrm{Ho}\); replacements; Quillen adjunctions; \(\mathbf{Top}\), \(\mathrm{Ch}\), \(\mathbf{sSet}\) |
| Bousfield & Localization | \(L_{\mathcal{S}}\), cellular Hirschhorn, simplicial, framed |
| Stable & Spectral | stable/triangulated, spectra, monoidal pushout-product, \(E_\infty\) |
| Quasicategories & ∞ | Joyal vs Quillen, model→∞ Lurie, ∞-operads, presentable ∞ |
| Combinatorial & Left-Determined | cofibrantly generated + locally presentable, left-determined, Vopěnka, transfer |
Dependency order¶
- Category theory (adjunctions, limits, localization).
- Algebraic topology (CW, homotopy groups, simplicial sets).
- Homological algebra (chain complexes, derived functors).
- Model categories \(\to\) higher categories (presentable \(\infty\)-cats as localizations) \(\to\) DAG / spectral AG.
Interfaces¶
| Area | Contact |
|---|---|
| Higher categories | quasicategories present many model cats after localization |
| Homological algebra | \(D(\mathcal{A})\simeq\mathrm{Ho}(\mathrm{Ch})\) |
| Algebraic topology | classical homotopy via Quillen model on \(\mathbf{Top}\)/\(\mathbf{sSet}\) |
| Derived AG | model cats of simplicial rings / prestacks (HAG) |
| ∞-topoi | model for sheaves of spaces; hypercompleteness |
| Category theory | calculus of adjunctions and localization |
Notation (paper-standard)¶
- \(\mathcal{W}\), \(\mathrm{Cof}\), \(\mathrm{Fib}\); \(\mathrm{Cof}\cap\mathcal{W}=\) acyclic cofibrations; \(\mathrm{Fib}\cap\mathcal{W}=\) acyclic fibrations
- \(X\twoheadrightarrow Y\) fibration; \(X\rightarrowtail Y\) cofibration; \(\xrightarrow{\sim}\) weak equivalence
- \(QX\to X\), \(X\to RX\); \(\mathrm{Map}^h\), \(\mathrm{RMap}\); \(F\dashv G\) Quillen pair; \(\mathbb{L}F\), \(\mathbb{R}G\)
See it¶
Why not just localize at weak equivalences? Naive Gabriel–Zisman localization \(\mathcal{M}[\mathcal{W}^{-1}]\) is hard to compute (morphisms are zig-zags). Model structure supplies functorial factorizations and homotopy relation on bifibrant objects so that $$ \mathrm{Ho}(\mathcal{M})(X,Y)\;\cong\;\pi\bigl(RQX,\,RQY\bigr) $$ — homotopy classes of actual maps. Derived functors become \(F(Q{-})\) or \(G(R{-})\); this is how \(\mathrm{Tor}\), \(\mathrm{Ext}\), and homotopy (co)limits are made rigorous before \(\infty\)-categories.
Check¶
What fails if one only has a category with weak equivalences (no fibrations/cofibrations)?
Localization may exist as a category but mapping sets are inaccessible; there is no systematic way to derive functors or compute homotopy limits. Model (or \(\infty\)-/derivator) structure supplies replacements and lifting to control \(\mathrm{Ho}\).
Why is the identity \(\mathbf{sSet}\to\mathbf{sSet}\) a Quillen equivalence from Quillen to Joyal model structure? (spoiler: it is not)
Different weak equivalences: Quillen weq \(=\) weak homotopy equivalences (geometric realization); Joyal weq \(=\) categorical equivalences. Same underlying category, non-equivalent homotopy theories — standard warning that “model structure on \(\mathbf{sSet}\)” is ambiguous.
Pitfalls¶
- Cofibrant/fibrant hypotheses: many constructions (\(X\otimes^L Y\), function complexes) need \(Q\)/\(R\) first.
- Left vs right Quillen: left Quillen preserves cofibrations and acyclic cofibrations (and thus weq between cofibrants after Ken Brown); do not swap.
- Model structure is extra data: many choices on the same category (e.g. injective vs projective on diagrams).
- ∞-upgrade: model cats present \((\infty,1)\)-categories; uniqueness is up to Quillen equivalence, not strict equality of \(\mathcal{W},\mathrm{Fib},\mathrm{Cof}\).
Refs¶
- Quillen, Homotopical Algebra (LNM 43)
- Hovey, Model Categories (AMS Math.\ Surveys)
- Hirschhorn, Model Categories and Their Localizations
- Dwyer–Spaliński, Homotopy theories and model categories (handbook survey)
Next¶
Foundations · Higher categories · Algebraic topology · Homological algebra · Derived AG · Category theory · ∞-topoi · Bousfield & Localization · Stable & Spectral · Quasicategories & ∞ · Combinatorial & Left-Determined