Model Categories: Foundations¶

Job: state Quillen’s axioms, construct the homotopy category via cofibrant–fibrant replacement, and record Quillen adjunctions with the standard examples $\mathbf{Top}$, $\mathrm{Ch}$, $\mathbf{sSet}$.

Core¶

Quillen model category¶

A model category is a bicomplete category $\mathcal{M}$ together with $\mathcal{W},\mathrm{Cof},\mathrm{Fib}$ such that:

Axiom	Content
(2-out-of-3)	if two of $f,g,gf$ are in $\mathcal{W}$, so is the third
(Retracts)	$\mathcal{W},\mathrm{Cof},\mathrm{Fib}$ closed under retracts
(Lifting)	acyclic cofibrations have LLP w.r.t.\ all fibrations; cofibrations have LLP w.r.t.\ all acyclic fibrations
(Factorization)	every $f$ factors as $\mathrm{Cof}\cap\mathcal{W}$ then $\mathrm{Fib}$, and as $\mathrm{Cof}$ then $\mathrm{Fib}\cap\mathcal{W}$

Modern packaging often assumes functorial factorizations. Anodyne / trivial synonyms: acyclic $=$ weak equivalence $\cap$ (co)fibration.

Lifting square. Given $$ \begin{CD} A @>>> X\ @ViVV @VVpV\ B @>>> Y \end{CD} $$ a diagonal $B\to X$ exists whenever $(i,p)$ is a (acyclic cofibration, fibration) or (cofibration, acyclic fibration) pair. This is the computational engine of the theory.

Homotopy and $\mathrm{Ho}(\mathcal{M})$¶

Cylinder object for $X$: factorization $X\sqcup X\xrightarrow{i_0\sqcup i_1} \mathrm{Cyl}(X)\xrightarrow{\sim} X$ with $i_0\sqcup i_1$ a cofibration. Path object: $X\xrightarrow{\sim} \mathrm{Path}(X)\twoheadrightarrow X\times X$.

Left homotopy $f\sim^\ell g:X\to Y$ via $\mathrm{Cyl}(X)\to Y$; right homotopy via $X\to\mathrm{Path}(Y)$. When $X$ is cofibrant and $Y$ fibrant, left and right homotopy coincide and are equivalence relations; write $\pi(X,Y)$ for the quotient set.

Theorem (Quillen). The localization $\mathrm{Ho}(\mathcal{M})=\mathcal{M}[\mathcal{W}^{-1}]$ exists (as a locally small category under standard set-theoretic care), and for cofibrant–fibrant $X,Y$, $$ \mathrm{Ho}(\mathcal{M})(X,Y)\;\cong\;\pi(X,Y). $$ Composition is induced by composition of representatives.

Cofibrant and fibrant replacement¶

Replacement	Factorization	Property
Cofibrant $QX\to X$	$\emptyset\rightarrowtail QX\xrightarrow{\sim} X$	$QX$ cofibrant, map weq
Fibrant $X\to RX$	$X\xrightarrow{\sim} RX\twoheadrightarrow *$	$RX$ fibrant, map weq
Bifibrant	$RQX$ or $QRX$	models object of $\mathrm{Ho}$

Any morphism $X\to Y$ in $\mathrm{Ho}$ is represented by an actual map $RQX\to RQY$ up to homotopy. Ken Brown’s lemma: a functor sending acyclic cofibrations between cofibrants to weq sends all weq between cofibrants to weq (dually for fibrants).

Quillen adjunctions¶

An adjunction $F:\mathcal{M}\rightleftarrows\mathcal{N}:G$ is a Quillen adjunction if $F$ preserves cofibrations and acyclic cofibrations (equivalently $G$ preserves fibrations and acyclic fibrations). Then:

Left derived $\mathbb{L}F:\mathrm{Ho}(\mathcal{M})\to\mathrm{Ho}(\mathcal{N})$, $\mathbb{L}F(X)=F(QX)$;
Right derived $\mathbb{R}G$, $\mathbb{R}G(Y)=G(RY)$;
$\mathbb{L}F\dashv\mathbb{R}G$ on homotopy categories.

Quillen equivalence: Quillen adjunction such that for cofibrant $X$ and fibrant $Y$, $FX\to Y$ is weq iff $X\to GY$ is weq. Then $\mathbb{L}F,\mathbb{R}G$ are inverse equivalences of $\mathrm{Ho}$.

Example adjunction	Role
$	{-}
Normalization / Dold–Kan	$\mathbf{sAb}\simeq\mathrm{Ch}_{\ge 0}(\mathbb{Z})$
Geometric realization of simplicial rings	input to HAG / DAG model presentations

Standard examples¶

1. Topological spaces ($\mathbf{Top}$ / $\mathbf{Top}_*$). Quillen–Serre: weak equivalences $=$ weak homotopy equivalences; fibrations $=$ Serre fibrations; cofibrations $=$ retracts of relative cell complexes. Cofibrant objects related to CW; every space is fibrant. Homotopy category $=$ classical $\mathrm{Ho}(\mathbf{Top})$.

2. Chain complexes. For $\mathrm{Ch}_{\ge 0}(R)$ (non-negative, $R$ ring): projective model structure — weq $=$ quasi-isomorphisms; fibrations $=$ degreewise epimorphisms in positive degrees (conventions vary); cofibrations $=$ monomorphisms with projective cokernel (degreewise). $\mathrm{Ho}\simeq D_{\ge 0}(R)$. Unbounded $\mathrm{Ch}(R)$: injective and projective model structures (Hovey, Spaltenstein resolutions).

3. Simplicial sets $\mathbf{sSet}$. Quillen model: weq $=$ maps inducing weak homotopy equivalences on $|{-}|$; fibrations $=$ Kan fibrations; cofibrations $=$ monomorphisms. Fibrant $=$ Kan complexes. Joyal model: weq $=$ categorical equivalences; fibrant $=$ quasicategories — presents $(\infty,1)$-categories; see higher categories.

Category	Weq	Fibrant objects	$\mathrm{Ho}$
$\mathbf{Top}$	weak htpy eq	all	classical homotopy types
$\mathbf{sSet}$ (Quillen)	$	f	$ weq
$\mathrm{Ch}(R)$	q.i.	injectives (inj.\ str.)	derived category
$\mathbf{sSet}$ (Joyal)	cat.\ eq	quasicategories	$\mathrm{Cat}_\infty$ (homotopy)

Homotopy limits / colimits (pointer)¶

Projective/injective model structures on $\mathcal{M}^I$ present homotopy (co)limits as derived $\mathrm{colim}$ / $\mathrm{lim}$. After $\infty$-categorical upgrade, these become ordinary (co)limits in the underlying quasicategory — higher categories.

See it¶

Derived tensor product via cofibrant replacement. For $A$-modules (or chain complexes), $M\otimes^L_A N := QM\otimes_A N$ with $QM\to M$ a cofibrant replacement in the projective model structure. Then $\mathrm{Tor}_i^A(M,N)=H_i(M\otimes^L_A N)$. Flat resolutions are special cofibrant replacements; the model axioms guarantee independence in $\mathrm{Ho}$. This is the same formalism underlying derived intersections in derived AG.

Function complexes. For cofibrant $X$ and fibrant $Y$, a simplicial mapping space $\mathrm{Map}(X,Y)$ (via cosimplicial frames / simplicial cotensors when $\mathcal{M}$ is simplicial) satisfies $\pi_0\mathrm{Map}\cong\mathrm{Ho}(X,Y)$ and higher $\pi_*$ encode automorphisms in the $\infty$-sense.

Check¶

Sketch why a left Quillen functor preserves weak equivalences between cofibrant objects.

Ken Brown: factor a weq $f:X\to Y$ between cofibrants as $X\rightarrowtail Z\twoheadrightarrow Y$ with $X\to Z$ acyclic cofibration and $Z\to Y$ acyclic fibration. Left Quillen $\Rightarrow$ $F(X)\to F(Z)$ acyclic cofibration hence weq; $Z\to Y$ is weq between cofibrants and is a retract of an acyclic cofibration after a standard cylinder trick / or apply 2-out-of-3 after comparing to a factorization — textbook Ken Brown argument yields $F(f)$ weq.

Give a counterexample: functor preserving weq but not a left Quillen functor.

Any functor that preserves weq but not cofibrations (e.g.\ a right Quillen functor that is not left). Concretely: $\mathrm{Sing}:\mathbf{Top}\to\mathbf{sSet}$ is right Quillen; it preserves fibrations/weq of fibrants after derivation but does not preserve cofibrations in general — so not left Quillen.

Pitfalls¶

“All objects fibrant” (as in $\mathbf{Top}$ Serre) is special; in $\mathbf{sSet}$ you must Kan-replace for homotopy classes.
Properness: left/right proper model cats needed for homotopy pushouts/pullbacks along (co)fibrations to stay well-behaved; not automatic.
Generating sets: cofibrantly generated $\Rightarrow$ small object argument builds factorizations; check domains of generating (acyclic) cofibrations.
Stable model categories present stable $\infty$-categories / triangulated $\mathrm{Ho}$; smash products need monoidal model compatibility (pushout-product).

Refs¶

Quillen, Homotopical Algebra
Hovey, Model Categories; Hirschhorn, Model Categories and Their Localizations
Goerss–Jardine, Simplicial Homotopy Theory
Lurie, HTT App.\ A (model $\leftrightarrow$ quasicategory comparison)

Next¶

Model categories index · Higher categories · Algebraic topology · Homological algebra · Derived AG · ∞-topoi · Category theory · Bousfield & Localization · Stable & Spectral · Quasicategories & ∞ · Combinatorial & Left-Determined

Replacement	Factorization	Property
Cofibrant \(QX\to X\)	\(\emptyset\rightarrowtail QX\xrightarrow{\sim} X\)	\(QX\) cofibrant, map weq
Fibrant \(X\to RX\)	\(X\xrightarrow{\sim} RX\twoheadrightarrow *\)	\(RX\) fibrant, map weq
Bifibrant	\(RQX\) or \(QRX\)	models object of \(\mathrm{Ho}\)