nLab cofibrantly generated model category

Idea

A model category $C$ is cofibrantly generated if there is a set (meaning: small set, not a proper class) of cofibrations as well as one of acyclic cofibrations, such that all other cofibrations and acyclic cofibrations, respectively, are generated from these, in some sense.

Definition

We need the following general terminology:

Definition

(cells and injectives)
Let $C$ be a category with all colimits and let $S \subset Mor(C)$ a class of morphisms. We write:

• $rlp(S)$ for the collection of morphisms with the right lifting property;

• $llp(S)$ for the collection of morphisms with the left lifting property with respect to $S$.

Moreover, we also write, now for $I \subset Mor(C)$:

• $cell(I)$ for the relative cell complexes, the class of morphisms obtained by transfinite composition of pushouts of coproducts of elements in $I$;

• $cof(I)$ for the class of retracts (in the arrow category $Arr(C)$) of elements in $cell(I)$;

• $inj(I) \coloneqq rlp(I)$ for the class of morphisms with the right lifting property with respect to $I$, the $I$-injective morphisms.

Definition

A model category $\mathcal{C}$ is cofibrantly generated if there are small sets of morphisms $I, J \subset Mor(\mathcal{C})$ such that

• $cof(I)$ is precisely the collection of cofibrations of $C$;

• $cof(J)$ is precisely the collection of acyclic cofibrations in $C$; and

• $I$ and $J$ permit the small object argument.

Since $I$ and $J$ are assumed to admit the small object argument the collection of cofibrations and acyclic cofibrations has the following simpler characterization:

Proposition

In a cofibrantly generated model category (Def. ) we have

• $cof(I) = llp\big(rlp(I)\big)$,

• $cof(J) = llp\big(rlp(J)\big)$.

And therefore the fibrations form precisely $rlp(J)$ and the acyclic fibrations precisely $rlp(I)$.

Proof

The argument is the same for $I$ and $J$. So take $I$.

By definition we have $I \subset llp\big(rlp(I)\big)$ and it is readily checked that collections of morphisms given by a left lifting property are stable under pushouts, transfinite composition and retracts (see here for details). So $cof(I) \subset llp\big(rlp(I)\big)$.

For the converse inclusion we use the small object argument: let $f \colon X \to Z$ be in $llp\big(rlp(I)\big)$. The small object argument produces a factorization $f : X \stackrel{f' \in cof(I)}{\to} Y \stackrel{f''\in rlp(I)}{\to} Z$.

Finally we apply the “retract argument”: It follows that $f$ has the left lifting property with respect to $f''$ which yields a morphism $\sigma$ in

$$\array{ X &\overset{f'}{\longrightarrow}& Y \\ \mathllap{{}^{f}} \big\downarrow &{}^\sigma\nearrow& \big\downarrow \mathrlap{{}^{f''}} \\ Z &\underset{=}{\longrightarrow}& Z }$$

which exhibits $f$ as a retract of $f'$

$$\array{ X &\overset{=}{\longrightarrow}& X &\overset{=}{\longrightarrow}& X \\ \big\downarrow {\mathrlap{{}^f}} && \big\downarrow {\mathrlap{{}^{f'}}} && \big\downarrow {\mathrlap{{}^f}} \\ Z &\underset{\sigma}{\longrightarrow}& Y &\underset{f''}{\longrightarrow}& Z \,. }$$

Therefore $f \in cof(I)$.

Properties

Recognition theorem

The following theorem allows one to recognize cofibrantly generated model categories by checking fewer conditions.

Theorem

Let $C$ be a category with all small limits and colimits and $W$ a class of maps satisfying 2-out-of-3.

If $I$ and $J$ are sets of maps in $C$ such that

1. both $I$ and $J$ permit the small object argument;

2. $cof(J) \subset cof(I) \cap W$;

3. $inj(I) \subset inj(J) \cap W$;

4. one of the following holds

   1. $cof(I) \cap W \subset cof(J)$

   2. $inj(J) \cap W \subset inj(I)$

then there is the stucture of a cofibrantly generated model category on $C$ with

• weak equivalences $W_C := W$

• generating cofibrations $I$ (i.e. $cof_C := llp(rlp(I))$)

• generating acyclic cofibrations $J$.

This is originally due to Daniel Kan, reproduced for instance as (Hirschhorn 03, theorem 11.3.1).

Proof

We have to show that with weak equivalences $W$ setting $cof_C := cof(I)$ and $fib_C := inj(J)$ defines a model category structure.

The existence of limits, colimits and the 2-out-of-3 property holds by assumption. Closure under retracts of the weak equivalences will hold automatically if we check the rest of the axioms without using it, by an argument of A. Joyal. Closure under retracts of $fib$ and $cof$ follows by the general statement that classes of morphisms defined by a left or right lifting property are closed under retracts (e.g. Hirschhorn 03, 7.2.8,).

The factorization property follows by applying the small object argument to the set $I$, showing that every morphism may be factored as

$$\stackrel{\in cof(I)}{\to} \stackrel{\in inj(I) \subset fib_C}{\to}$$

and assumption 3 says that $inj(I) \subset W$. Similarly applying the small object argument to $J$ gives factorizations

$$\stackrel{\in cof(J)}{\to} \stackrel{\in inj(J) = fib_C}{\to}$$

and assumption 2 guarantees that $cof(J) \subset W$.

It remains to verify the lifting axiom. This verification depends on which of the two parts of item 4 is satisfied. Assume the first one is, the argument for the second one is analogous.

Then using the assumption $cof(I) \cap W \subset cof(J)$ and remembering that we have set $inj(J) = fib_C$ we immediately have the lifting of trivial cofibrations on the left against fibrations on the right.

To get the lifting of cofibrations on the left with acyclic fibrations on the right, we show finally that $inj(J) \cap W \subset inj(I)$. To see this, apply the factorization established before to an acyclic fibration $f : X \to Y$ to get

$$\array{ X &\stackrel{=}{\to}& X \\ \downarrow^{\mathrlap{\in cof(I) \cap W}} && \downarrow^{\mathrlap{f \in inj(J) \cap W}} \\ Z &\stackrel{inj(I)}{\to}& Y } \,.$$

With assumption 4 a this is

$$\array{ X &\stackrel{=}{\to}& X \\ \downarrow^{\mathrlap{\in cof(J) }} && \downarrow^{\mathrlap{f \in inj(J) \cap W}} \\ Z &\stackrel{inj(I)}{\to}& Y }$$

so that we have a lift

$$\array{ X &\stackrel{=}{\to}& X \\ {}^{\mathllap{\in cof(J) }} \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{f \in inj(J) \cap W}} \\ Z &\stackrel{inj(I)}{\to}& Y }$$

which establishes a retract

$$\array{ X &\to& Z &\stackrel{\sigma}{\to}& X \\ \downarrow^f && \downarrow^{\mathrlap{\in inj(I)}} && \downarrow^f \\ Y &\stackrel{=}{\to}& Y & \stackrel{=}{\to} & Y }$$

that exhibits $f$ as an element of $inj(I)$ as this is closed under retracts.

Transfer along adjunctions

Theorem

Let $\mathcal{C}$ be a cofibrantly generated model category, def. , with generating (acyclic) cofibrations $I$ (and $J$). Let $\mathcal{D}$ be any category with all small limits and colimits and consider a pair of adjoint functors

$$(F \dashv U) \;\colon\; \array{ \mathcal{D} \stackrel{\overset{F}{\longleftarrow}}{\underset{U}{\longrightarrow}} \mathcal{C} } \,.$$

Write $F I \coloneqq \{F(i) | i \in I\}$ and $F J \coloneqq \{F(j) | j \in J\}$. If

1. both $F I$ and $F J$ admit the small object argument;

2. $U$ takes $F J$-relative cell complexes to weak equivalences

then $F I$, $F J$ induce a cofibrantly generated model structure, def. , on $\mathcal{D}$. Its weak equivalences are the morphisms that are taken to weak equivalences by $U$. Moreover, the above adjunction is a Quillen adjunction for these model structures.

This is due to Daniel Kan, reproduced in (Hirschhorn 03, theorem 11.3.2). See also at transferred model structure.

Presentation and generation

Proposition

Let $C$ be a cofibrantly generated model category which is also left proper. Then there exists a small set $S \subset Obj(C)$ of cofibrant objects which detect weak equivalences:

a morphism $f : A \to B$ in $C$ is a weak equivalence, precisely if for all $s \in S$ the induced morphism of derived hom-spaces

$$\mathbb{R}Hom(s,f) : \mathbb{R}Hom(s,A) \to \mathbb{R}Hom(s,B)$$

is a weak equivalence.

This appears as (Dugger, prop. A.5).

Examples

On simplicial sets

The classical model category structure on simplicial sets is cofibrantly generated:

Proposition

$sSet_{Quillen}$ is a cofibrantly generated model category with

• generating cofibrations the boundary inclusions $\partial \Delta[n] \to \Delta[n]$;

• generating acyclic cofibrations the horn inclusions $\Lambda^i[n] \to \Delta[n]$.

On topological spaces

The category of (based, compactly generated) topological spaces has a cofibrantly generated model structure (the classical model structure on pointed topological spaces) in which the set of cells is

$$I = \{S^{n-1}_+ \rightarrow D^n_+\}_{n\geq 0}$$

and the set of acyclic cells is

$$J = \{D^n_+ \rightarrow (D^n \times I)_+\}_{n\geq 0} \,.$$

(Here $+$ means disjoint basepoint, not northern hemisphere.)

The category of unbased spaces has a similar cofibrantly generated model structure. (The classical model structure on topological spaces.)

On sequential spectra

The category of sequential prespectra has two cofibrantly generated model structures (the Bousfield-Friedlander model structures).

Let $F_d A$ denote a prespectrum whose $n$th space is $\Sigma^{n-d} A$ when $n \geq d$, and $*$ otherwise. Then the level model structure is generated by cells

$$I = \{F_d S^{n-1}_+ \rightarrow F_d D^n_+\}_{d,n\geq 0}$$

(free spectra formed on the standard cells of the classical model structure on pointed topological spaces)

and the set of acyclic cells is

$$J = \{F_d D^n_+ \rightarrow F_d (D^n \times I)_+\}_{d,n\geq 0}$$

The stable model structure has the same cells, but more acyclic cells, which in turn guarantees that the fibrant spectra are the $\Omega$-spectra.

On structured spectra

The categories of symmetric spectra and orthogonal spectra have similar cofibrantly generated level and stable model structures (model structure on symmetric spectra, model structure on orthogonal spectra).

(see Mandell, May, Schwede, Shipley: Model categories of diagram spectra.)

On diagrams

The category of diagrams indexed by a fixed small category $D$, taking values in another cofibrantly generated model category $C$.

(The model structure on functors.)

Related concepts

• A cofibrantly generated model category that is also a locally presentable category is called a combinatorial model category.

• A cofibrantly generated model category for which the domains of the morphisms in $I$ and $J$ are compact objects or small objects is a cellular model category.

• cellular category

• transferred model structure

References

The original reference is:

• William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Chapter II of: Model Categories and More General Abstract Homotopy Theory, [pdf, pdf]

In particular, the Kan recognition theorem is in §II.8 and the Kan transfer theorem is in §II.9. This manuscript draft later transformed in Homotopy Limit Functors on Model Categories and Homotopical Categories, losing the content on cofibrantly generated model categories in the process.

Textbook accounts:

• Philip Hirschhorn, Sec. 11 in: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

Further review:

• Paul Goerss, Kirsten Schemmerhorn, Section 3.1 of: Model categories and simplicial methods, Notes from lectures given at the University of Chicago in August 2004, published as: Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436 AMS (2007) [arXiv:math.AT/0609537, doi:10.1090/conm/436]

For the general case a useful reference is for instance the first section of

• Tibor Beke, Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 3, 447–475.(math.CT/0102087), Sheafifiable homotopy model categories. II, J. Pure Appl. Algebra 164 (2001), no. 3, 307–324.

For the case of a presentable category:

• Jacob Lurie, section A.1.2 in: Higher Topos Theory

See also:

• Dan Dugger, Appendix of: Replacing model categories with simplicial ones (pdf)