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Weak factorisation systems

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**Category theory** ##Contents * Contributors * References * Introduction * Basic category theory * Weak factorisation systems * Factorisation systems * Distributors and barrels * Model structures on Cat * Homotopy factorisation systems in Cat * Accessible categories * Locally presentable categories * Algebraic theories and varieties of algebras

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Contents

Main definitions

Definition

We shall say that a map u:ABu:A\to B in a category E\mathbf{E} has the left lifting property with respect to a map f:XYf:X\to Y, or that ff has the right lifting property with respect to uu, if every commutative square

has a diagonal filler d:BXd:B\to X,

We shall denote this relation by ufu\,\pitchfork\, f. Notice that the condition ufu\,\pitchfork\, f means that the following commutative square Sq(u,f)Sq(u,f),

(1)

is epicartesian, or equivalently if the map

Hom(u,f):hom(B,X)hom(B,Y)× hom(A,Y)hom(A,X)Hom(u,f)': hom(B,X)\to hom(B,Y) \times_{hom(A,Y)}hom(A,X)

is surjective.

Notation

If \mathcal{M} is a class of maps, we shall denote by {}^\pitchfork\!\mathcal{M} (resp. \mathcal{M}^\pitchfork) the class of maps having the left (resp. the right) lifting property with respect to every map in \mathcal{M}. We shall say that {}^\pitchfork\!\!\mathcal{M} is the left complement of \mathcal{M}, and that \mathcal{M}^\pitchfork is its right complement.

Example

Recall that a map of simplicial sets is said to be a Kan fibration if it has the right lifting property with respect to the inclusion h n k:Λ k[n]Δ[n]h^k_n: \Lambda^k[n] \subset \Delta[n] for every n>0n\gt 0 and 0kn0\le k\le n. A simplicial set XX is a Kan complex iff the map X1X\to 1 is a Kan fibration.

Example

Let JJ be the groupoid generated by one isomorphism 010\simeq 1. Then a functor in the category Cat\mathbf{Cat} is an isofibration iff it has the right lifting property with respect to the inclusion {0}J\{0\}\subset J.

Definition

We shall say that a pair (,)(\mathcal{L},\mathcal{R}) of classes of maps in a category E\mathbf{E} is a weak factorisation system if the following conditions are satisfied:

  • every map f:ABf:A\to B admits a factorisation f=pu:AEBf=p u:A\to E\to B with uu\in \mathcal{L} and pp\in \mathcal{R};

  • = \mathcal{L}= {}^\pitchfork\mathcal{R} and = \mathcal{R}=\mathcal{L}^\pitchfork ;

We shall say that a factorisation f=pu:AEBf=p u:A\to E\to B with uu\in \mathcal{L} and pp\in \mathcal{R} is a (,)(\mathcal{L},\mathcal{R})-factorisation of the map ff. The class \mathcal{L} is called the left class of the system, and the class \mathcal{R} is called the right class .

Example

Every factorisation system is a weak factorisation system by the theorem here.

Example

The category of sets Set\mathbf{Set} admits a weak factorisation system (Inj,Surj)(Inj,Surj), where InjInj the class of injections and SurjSurj is the class of surjections.

For more examples of weak factorisation systems, go to Example 1.

Duality

If (,)( \mathcal{L},\mathcal{R}) is a weak factorisation system in a category E\mathbf{E}, then the pair ( o, o)(\mathcal{R}^o,\mathcal{L}^o) is a weak factorisation system in the opposite category E o\mathbf{E}^o.

If \mathcal{M} is a class of maps in a category E\mathbf{E}, then for any object BEB\in \mathbf{E} we shall denote by /B\mathcal{M}/B the class of maps in the slice category E/B\mathbf{E}/B whose underlying map in E\mathbf{E} belongs to \mathcal{M}. Dually, we shall denote by B\B\backslash \mathcal{M} the class of maps in the coslice category B\EB\backslash \mathbf{E} whose underlying map belongs to \mathcal{M}.

Slice and coslice

If (,)(\mathcal{L},\mathcal{R}) is a weak factorisation system in a category E\mathbf{E}, then the pair (/B,/B)(\mathcal{L}/B,\mathcal{R}/B) is a weak factorisation system in the slice category E/B\mathbf{E}/B for any object BB in E\mathbf{E}. Dually, the pair (B\,B\)(B\backslash \mathcal{L},B\backslash \mathcal{R}) is a weak factorisation system in the coslice category B\EB\backslash \mathbf{E}.

Proof

Left to the reader.

Closure properties

If 𝒞\mathcal{C} and \mathcal{F} are two classes of maps in E\mathbf{E}, we shall write 𝒞\mathcal{C}\,\pitchfork\, \mathcal{F} to indicate that we have ufu\pitchfork f for every u𝒞u\in \mathcal{C} and ff\in \mathcal{F} . The three conditions

𝒞 ,𝒞,𝒞 \mathcal{C}\subseteq {}^\pitchfork \mathcal{F}, \quad \quad \mathcal{C} \,\pitchfork\, \mathcal{F},\quad \quad \mathcal{F} \subseteq \mathcal{C}^\pitchfork

are equivalent. If 𝒞={u}\mathcal{C}=\{u\}, we shall write uu \,\pitchfork\, \mathcal{F} instead of {u}\{u\}\,\pitchfork\,\mathcal{F} . Similarly, we shall write 𝒞f\mathcal{C} \,\pitchfork\, f instead of 𝒞{f}\mathcal{C} \pitchfork \{f\}.

The operations \mathcal{M}\mapsto \mathcal{M}^\pitchfork and \mathcal{M}\mapsto {}^\pitchfork\mathcal{M} on classes of maps are contravariant and mutually adjoint. It follows that the operations ( ) \mathcal{M}\mapsto ({}^\pitchfork\mathcal{M})^\pitchfork and ( )\mathcal{M}\mapsto {}^\pitchfork(\mathcal{M}^\pitchfork) are closure operators.

Lemma

The following conditions on a morphism f:XYf:X\to Y in a category E\mathbf{E} are equivalent:

fisinvertible,Ef,ff,fE.f \mathrm{is}\:\mathrm{invertible}, \quad \mathbf{E}\,\pitchfork\, f, \quad f\,\pitchfork\, f, \quad f\,\pitchfork\, \mathbf{E}.
Proof

(121\Rightarrow 2) If ff is invertible, then the square

has a diagonal filler f 1y:BXf^{-1}y:B\to X. Thus, ufu\,\pitchfork\, f for any arrow uu, and hence Ef\mathbf{E}\,\pitchfork\, f. (232\Rightarrow 3) If Ef\mathbf{E}\,\pitchfork\, f, then fff\,\pitchfork\,f. (313\Rightarrow 1) If fff\,\pitchfork\,f, then the square

has a diagonal filler g:YXg:Y\to X and this shows that ff is invertible. The equivalences (1231\Leftrightarrow 2\Leftrightarrow 3) are proved. The equivalences (1431\Leftrightarrow 4\Leftrightarrow 3) are proved similarly.

Recall that a map u:ABu:A\to B in a category E\mathbf{E} is said to be a retract of another map v:CDv:C\to D, if uu is a retract of vv in the category of arrows E [1]\mathbf{E}^{[1]}. The condition means that there exists four maps p,i,q,jp,i,q,j fitting in a commutative diagram

and such that pi=1 Ap i=1_A and qj=1 Bq j=1_B.

Definition

We shall say that a class of maps \mathcal{M} in a category E\mathbf{E} is closed under retracts if every retract of a map in \mathcal{M} belongs to \mathcal{M}.

We recall that the base change of a map f:XYf:X\to Y along a map v:VXv:V\to X is the map g:UVg:U \to V in a pullback square,

Dually, the cobase change of a map u:ABu:A\to B along a map c:ACc:A\to C is the map v:CDv:C\to D in a pushout square,

Definition

We shall say that a class of maps \mathcal{M} in a category E\mathbf{E} is closed under base changes if the base change of a map in \mathcal{M} belongs to \mathcal{M}, when the base change exists. The notion of a class of maps closed under cobase changes is defined dually.

Definition

We shall say that a class of maps \mathcal{M} in a category E\mathbf{E} is closed under coproducts if the coproduct

iIu i: iIA i iIB i\sqcup_{i\in I}u_i: \sqcup_{i\in I} A_i\to \sqcup_{i\in I} B_i

of any family of maps (u i:iI)(u_i:i\in I) in \mathcal{M} belongs to \mathcal{M}, when this coproduct exists. The notion of a class of maps closed under products is defined similarly.

Lemma

Let \mathcal{M} be a class of maps in a category E\mathbf{E}. Then the class \mathcal{M}^\bot contains the isomorphisms and is closed under composition, retracts, products, and base changes. Dually, the class {}^\bot\mathcal{M} is contains the isomorphisms and is closed under composition, retracts, coproducts, and cobase changes.

Proof

That class {}^\pitchfork\mathcal{M} contains the isomorphisms by Scholie 1. Let us show that it is closed under composition. We shall use the properties of epicartesian squares. Let us show that if two morphisms u:ABu:A\to B and v:BCv:B\to C belongs to {}^\pitchfork\mathcal{M}, then so does their composite vu:ACv u:A\to C. For any morphism f:XYf:X\to Y, the square Sq(vu,f)Sq(v u,f)

can be obtained by composing horizontally the squares Sq(u,f)Sq(u,f) and Sq(v,f)Sq(v,f),

The squares Sq(u,f)Sq(u,f) and Sq(v,f)Sq(v,f) are epicartesian if ff\in \mathcal{M}; hence their composite is epicartesian by the lemma here. This shows that vu v u\in {}^\pitchfork\mathcal{M}. We have proved that the class {}^\pitchfork\mathcal{M} is closed under composition. Let us now show that the class \mathcal{M}^\pitchfork is closed under retracts. If a map f:XYf:X\to Y is a retract of a map g:UVg:U\to V, then the square Sq(u,f)Sq(u,f) is a retract of the square Sq(u,g)Sq(u,g) for any map u:ABu:A\to B. But a retract of an epicartesian square is epicartesian by the lemma here. It follows that the class \mathcal{M}^\pitchfork is closed under retracts. Let us show that the class \mathcal{M}^\pitchfork is closed under products. If a map f:XYf:X\to Y is the product of a family of maps f i:X iY if_i:X_i\to Y_i (iIi\in I), then the square Sq(u,f)Sq(u,f) is the product of the family of squares Sq(u,f i)Sq(u,f_i). But the product of a family of epicartesian squares is epicartesians by the lemma here. This shows that the class \mathcal{M}^\pitchfork is closed under products. Let us show that the class \mathcal{M}^\pitchfork is closed under base changes. Suppose that we have a pullback square

(2)

with f f\in \mathcal{M}^\pitchfork and let us prove that g g\in \mathcal{M}^\pitchfork. It suffices to show that the square Sq(u,g)Sq(u,g) is epicartesian for every morphism u:ABu:A\to B in \mathcal{M}. But the square is the back face of the following commutative cube,

The left and the right faces of the cube are cartesian, since the square (2) is cartesian and the functors hom(A,)hom(A,-) and hom(B,)hom(B,-) preserve limits. The front face is epicartesian since we have upu\pitchfork p. Hence the back face is epicartesian by the cube lemma here.

Proposition

The two classes of a weak factorisation system (,)(\mathcal{L},\mathcal{R}) contain the isomorphisms and they are closed under composition and retracts. The right class \mathcal{R} is closed under base changes and products, and the left class \mathcal{L} under cobase changes and coproducts. The intersection \mathcal{L}\,\cap \,\mathcal{R} is the class of isomorphisms.

Proof

This follows from Lemma 2 and Lemma 1 since = \mathcal{R}=\mathcal{L}^\pitchfork and = \mathcal{L}={}^\pitchfork\mathcal{R}.

Definition

Recall that a map u:ABu:A\to B in a category E\mathbf{E} is said to be a domain retract of a map v:CBv:C\to B, if the object (A,u)(A,u) of the category E/B\mathbf{E}/B is a retract of the object (C,v)(C,v). There is a dual notion of codomain retract.

Definition

We shall say that a class of maps \mathcal{M} in a category E\mathbf{E} is closed under domain retracts if every domain retract of a map in \mathcal{M} belongs to \mathcal{M}. The notion of a class closed under codomain retract is defined similarly.

Theorem

A pair (,)(\mathcal{L},\mathcal{R}) of classes of maps in a category E\mathbf{E} is a weak factorisation system iff the following conditions are satisfied:

  • every map f:XYf:X\to Y admits a (,)(\mathcal{L},\mathcal{R})-factorisation f=pu:XEYf=p u:X\to E\to Y;
  • \mathcal{L}\,\pitchfork\,\mathcal{R};
  • the class \mathcal{L} is closed under codomain retracts and the class \mathcal{R} under domain retracts.
Proof

The implication (\Rightarrow) is clear since the classes of a weak factorisation system are closed under retracts by Proposition 1. Let us prove the implication (\Leftarrow). We have \mathcal{R} \subseteq \mathcal{L}^\pitchfork since we have \mathcal{L}\,\pitchfork\, \mathcal{R} by the hypothesis. Let us show that we have \mathcal{L}^\pitchfork \subseteq \mathcal{R}. If a map f:XYf:X\to Y belongs to \mathcal{L}^\pitchfork, let us choose a (,)(\mathcal{L},\mathcal{R})-factorisation f=pu:XEYf=p u:X\to E\to Y. The square

has a diagonal filler r:EXr:E\to X since we have ufu\,\pitchfork\, f. Hence we have fr=pf r=p and ru=1 Xr u =1_X and this shows that ff is a domain retract of pp. Thus, ff\in \mathcal{R}, since \mathcal{R} is closed under domain retracts by hypothesis.

Recall that a class 𝒞\mathcal{C} of objects in a category E\mathbf{E} is said to be replete if every object isomorphic to an object in 𝒞\mathcal{C} belongs to 𝒞\mathcal{C}. We shall say that a class of maps \mathcal{M} in E\mathbf{E} is replete, if it is replete as a class of objects of the category E [1]\mathbf{E}^{[1]}.

Corollary

Suppose that a pair (,)(\mathcal{L},\mathcal{R}) of classes of maps in a category E\mathbf{E} satisfies the following three conditions:

  • every map f:XYf:X\to Y admits a (,)(\mathcal{L},\mathcal{R})-factorisation f=pu:XEYf=p u:X\to E\to Y;

  • \mathcal{L}\,\pitchfork\,\mathcal{R};

  • the classes \mathcal{L} and \mathcal{R} are replete.

If \mathcal{L}' denotes the class of maps which are codomain retracts of maps in \mathcal{L} and \mathcal{R}' denotes the class of maps which are domain retracts of maps in \mathcal{R}, then the pair (,)(\mathcal{L}',\mathcal{R}') is a weak factorisation system.

Proof

The condition \mathcal{L}\,\pitchfork\,\mathcal{R} implies the condition \mathcal{L}'\,\pitchfork\,\mathcal{R}' by Lemma 2. It is easy to see that \mathcal{L}' is closed under codomain retracts, and that \mathcal{R}' is closed under domain retracts. The result then follows from Theorem 2.

Existence

Saturated classes

Definition

For any ordinal α\alpha, let us put [α]={i:0iα}[\alpha]=\{i :0\le i \le \alpha \} and [α)={i:0i<α}[\alpha)=\{i :0\le i \lt \alpha \}. Let E\mathbf{E} be a cocomplete category. We shall say that a functor C:[α]EC:[\alpha] \to \mathbf{E} is a chain of lentgth α\alpha, or an α\alpha-chain. The composite of CC is defined to be the canonical map C(0)C(α)C(0)\to C( \alpha). The base of CC is the restriction of CC to [α)[\alpha). The chain CC is cocontinuous if the canonical map

colim i<jC(i)C(j) \mathrm{colim}_{i\lt j} C(i)\to C(j)

is an isomorphism for every non-zero limit ordinal j[α]j\in [\alpha]. We shall say that a subcategory 𝒞E\mathcal{C}\subseteq \mathbf{E} is closed under transfinite compositions if for any limit ordinal α>0\alpha\gt 0 any cocontinuous chain C:αEC:\alpha \to \mathbf{E} with a base in 𝒞\mathcal{C} has a composite in 𝒞\mathcal{C}.

Dually, if E\mathbf{E} is a complete category, and α\alpha is an ordinal, we shall say that a contravariant functor C:[α]EC:[\alpha]\to \mathbf{E} is an opchain. The opchain is continuous if the corresponding chain C o:[α]E oC^o:[\alpha]\to \mathbf{E}^o is cocontinuous. We shall say that a subcategory 𝒞E\mathcal{C}\subseteq \mathbf{E} is closed under transfinite op-compositions if the opposite subcategory 𝒞 oE o\mathcal{C}^o\subseteq \mathbf{E}^o is closed under transfinite compositions.

Lemma

The class \mathcal{M}^\pitchfork is closed under transfinite op-compositions for any class of maps \mathcal{M} in a complete category E\mathbf{E}.

Proof

Let us show that if α>0\alpha \gt 0 is a limit ordinal then every continuous op-chain C:[α]EC:[\alpha]\to \mathbf{E} with a base in \mathcal{M}^\pitchfork has its composite in \mathcal{M}^\pitchfork . Let us denote by c(j,i)c(j,i) the transition map C(j)C(i)C(j)\to C(i) defined for 0ijα0\le i\le j\le \alpha. For any morphism u:ABu:A\to B, let us denote by Sq(u,C)Sq(u,C) the contravariant functor [α]Set I[\alpha]\to \mathbf{Set}^I obtained by putting Sq(u,C)(i)=hom(u,C(i)):hom(B,C(i))hom(A,C(i))Sq(u,C)(i)=hom(u,C(i)):hom(B,C(i))\to hom(A,C(i)) for i[α]i\in [\alpha]. By definition, the functor Sq(u,C)Sq(u,C) takes a pair iji\leq j to the square Sq(u,c(j,i))Sq(u,c(j,i)),

Beware that here the square Sq(u,c(j,i))Sq(u,c(j,i)) is defining a morphism from the its top horizontal line to the bottom horizontal line; this means that we are presently using the vertical composition in the category of squares). The (vertical) op-chain Sq(u,C):[α]Set ISq(u,C):[\alpha]\to \mathbf{Set}^I is continuous, since CC is continuous. If uu\in \mathcal{M}, then the square Sq(u,c(j,i))Sq(u,c(j,i)) is epicartesian for every ij<αi\leq j\lt \alpha by the assumption on CC. It follows that the square Sq(u,c(α,0))Sq(u,c(\alpha,0)) is epicartesian by the lemma here. This show that c(α,0)c(\alpha,0) belongs to \mathcal{M}^\pitchfork , and hence that $ \mathcal{M}^\pitchfork is closed under transfinite op-compositions.

Definition

We shall say that a class of maps 𝒞\mathcal{C} in a cocomplete category E\mathbf{E} is cellular if it satisfies the following conditions: * 𝒞\mathcal{C} contains the isomorphisms and is closed under composition, * 𝒞\mathcal{C} is closed under transfinite compositions; * 𝒞\mathcal{C} is closed under cobase changes.

We shall say that 𝒞\mathcal{C} is saturated if in addition,

  • 𝒞\mathcal{C} is closed under retracts.

Every class of maps ΣE\Sigma\subseteq \mathbf{E} is contained in a smallest cellular class Cell(Σ)Cell(\Sigma) called the cellular class generated by Σ\Sigma. Similarly, Σ\Sigma is contained in a smallest saturated class Sat(Σ)Sat(\Sigma) called the saturated class generated by Σ\Sigma.

Example

We shall see in Proposition 4 below that the left class of a weak factorisation system in a cocomplete category is saturated.

Example

The class of epimorphisms in any cocomplete category is saturated. Let us say that a map in a cocomplete category is surjective if it is left orthogonal to every monomorphisms; then the class of surjective maps in a cocomplete category is saturated.

Example

The class of split monomorphisms in any cocomplete category is saturated. The class of monomorphisms in a Grothendieck topos is saturated.

Example

The class of monomorphisms in the category of simplicial sets is generated as a cellular class by the set of inclusions Δ[n]Δ[n]\partial \Delta[n] \subset \Delta[n] (n0n\geq 0).

For more examples of saturated classes of the form Sat(Σ)Sat(\Sigma), go to Example 10.

Proposition

The class {}^\pitchfork\mathcal{M} is saturated for any class of maps \mathcal{M} in a cocomplete category E\mathbf{E}. In particular, the left class of a weak factorisation system in a cocomplete category is saturated.

Proof

The class {}^\pitchfork\mathcal{M} contains the isomorphisms and it is closed under composition, retracts and cobase changes by 2. And it is closed under transfinite compositions by Lemma 3 dualised.

Lemma

A cellular class of maps is closed under coproducts.

Proof

Let \mathcal{M} be a cellular class of maps in a cocomplete category E\mathbf{E}. We shall say that an object AEA\in \mathbf{E} cofibrant, if the map A\bot \to A belongs to \mathcal{M}, where \bot is the initial object of E\mathbf{E}. We shall first prove that the coproduct of a family of cofibrant objects is cofibrant. Let us first show that the coproduct of a finite family of cofibrant objects is cofibrant. The identity map \bot \to \bot belongs to \mathcal{M}, since \mathcal{M} contains the isomorphisms. Hence the object \bot is cofibrant. This show that the coproduct of the empty family of objects is cofibrant. It remains to show that the coproduct of a finite non-empty family of cofibrant objects is cofibrant. For this it suffices to show that the coproduct of two cofibrant objects is cofibrant. If AA and BB are cofibrant, consider the pushout square

The map i Bi_B is a cobase change of the map A\bot \to A. Thus, i Bi_B\in \mathcal{M}, since AA is cofibrant and \mathcal{M} is closed under cobase change. The map B\bot \to B also belongs to \mathcal{M}, since BB is cofibrant. Hence the composite BAB\bot \to B\to A\sqcup B belongs to \mathcal{M}, since \mathcal{M} is closed under composition. This shows that AB A\sqcup B is cofibrant. Let us now show that the coproduct

A= iIA iA= \bigsqcup_{i\in I} A_i

of an infinite family of cofibrant objects (A i:iI)(A_i:i\in I) is cofibrant. We shall argue by induction on the ordinal α=Card(I)\alpha=\mathrm{Card}(I). If j<αj\lt \alpha, let us put

C(j)= i<jA i.C(j)=\bigsqcup_{i\lt j} A_i.

There is an obvious canonical map C(j)C(k)C(j)\to C(k) for jkαj\le k \le \alpha and this defines a cocontinuous chain C:[α]EC:[\alpha] \to \mathbf{E}. Notice that C(0)=C(0)=\bot and C(α)=AC(\alpha)=A. Hence we can prove that AA is cofibrant by showing that the composite of CC belongs to \mathcal{M}. For this it suffices to show that the base of CC belongs to \mathcal{M}, since \mathcal{M} is closed under transfinite compositions. But the object

C(j,k)= ji<kA iC(j,k)=\bigsqcup_{j\le i\lt k} A_i

is cofibrant for every jk<αj\le k\lt \alpha by the induction hypothesis, since k<αk\lt \alpha. And the transition map C(j)C(k)C(j)\to C(k) is a base change of the map C(j,k)\bot \to C(j,k) since we have a pushout square

This shows that the transition map C(j)C(k)C(j)\to C(k) belongs to \mathcal{M} for every jk<αj\le k\lt \alpha. We have proved that the base of CC belongs to \mathcal{M} and hence that the object AA is cofibrant. Let us now show that the class \mathcal{M} is closed under coproducts. For this, let us show that the coproduct u:ABu:A\to B of a family of maps u i:A iB iu_i:A_i\to B_i (iIi\in I) in \mathcal{M} belongs to \mathcal{M}. For this, let us denote by \mathcal{M}' the class of maps in the category A\EA\backslash \mathbf{E} whose underlying map in E\mathbf{E} belongs to \mathcal{M}. It is easy to verify that the class \mathcal{M}' satisfies the hypothesis of the proposition. Let us put E i=B i A iAE_i=B_i\sqcup_{A_i}A for each iIi\in I,

The object (B,u)(B,u) of A\EA\backslash \mathbf{E} is the coproduct of the family of objects (E i,v i)(E_i,v_i) for iIi\in I. The map v i:AE iv_i:A\to E_i belongs to \mathcal{M}, since u iu_i\in \mathcal{M} by assumption, and since the class \mathcal{M} is closed under cobase change. Hence the object (E i,v i)(E_i,v_i) of the category A\EA\backslash \mathbf{E} is cofibrant with respect to the class \mathcal{M}'. It follows that the object (B,u)(B,u) is cofibrant by the first part of the proof. This proves that uu\in \mathcal{M}.

Definition

If α\alpha is a regular cardinal, we shall say that a class of maps \mathcal{M} in a complete category E\mathbf{E} is α\alpha-cellular if it satisfies the following conditions: * \mathcal{M} contains the isomorphisms and is closed under composition; * \mathcal{M} is closed under transfinite compositions of cocontinuous chains of length α\le \alpha; * \mathcal{M} is closed under cobase changes; * \mathcal{M} is closed under coproducts.

We shall say that an α\alpha-cellular class 𝒞\mathcal{C} is α\alpha-saturated if in addition,

  • 𝒞\mathcal{C} is closed under retracts.

Every class of maps ΣE\Sigma\subseteq \mathbf{E} is contained in a smallest α\alpha-cellular class Cell α(Σ)Cell^\alpha(\Sigma) called the α\alpha-cellular class generated by Σ\Sigma. Similarly, Σ\Sigma is contained in a smallest α\alpha-saturated class Sat(Σ)Sat(\Sigma) called the α\alpha-saturated class generated by Σ\Sigma.

Example

If Σ\Sigma is the set of inclusions Δ[n]Δ[n]\partial \Delta[n] \subset \Delta[n] (n0n\geq 0) in the category SSet\mathbf{SSet} (of simplicial sets), then Cell ω(Σ)=Sat(Σ)Cell^\omega(\Sigma)=Sat(\Sigma) is the class of monomorphisms.

Example

If Σ\Sigma is the set of inclusions h n k:Λ k[n]Δ[n]h^k_n: \Lambda^k[n] \subset \Delta[n] for n>0n\gt 0 and 0kn0\le k\le n, then Sat ω(Σ)=Sat(Σ)Sat^\omega(\Sigma)=Sat(\Sigma) is the class of anodyne maps.

Small object argument

Definition

If uu is a map in a category E\mathbf{E}, we shall say that an object XX in E\mathbf{E} is uu-fibrant if the map

hom(u,X):hom(B,X)hom(A,X)hom(u,X):hom(B,X)\to hom(A,X)

is surjective. More generally, if Σ\Sigma is a class of maps in E\mathbf{E}, we shall say that an object XX is Σ\Sigma-fibrant if it is uu-fibrant for every uΣu\in \Sigma. When E\mathbf{E} has a terminal object 11, then an object XX is Σ\Sigma-fibrant iff the map X1X\to 1 belongs to Σ \Sigma^\pitchfork.

Recall that an object AA in a cocomplete category E\mathbf{E} is said to be compact if the functor

hom(A,):ESethom(A,-): \mathbf{E}\to \mathbf{Set}

preserves directed colimits. More generally, if α\alpha is a regular cardinal, then an object AA is said to be α\alpha-compact if the functor hom(A,)hom(A,-) preserves α\alpha-directed colimits. An object AA is said to be small if it is α\alpha-compact for some regular cardinal α\alpha.

Proposition

(Small object argument) Let Σ\Sigma be a set of maps in a cocomplete category E\mathbf{E}. If the domains of the maps in Σ\Sigma are α\alpha-compact, then there exists a functor

R:EER:\mathbf{E}\to \mathbf{E}

together with a natural transformation ρ:IdR\rho:Id\to R such that: * the object R(X)R(X) is Σ\Sigma-fibrant for every object XX; * the map ρ X:XR(X)\rho_X:X\to R(X) belongs to Cell α(Σ)Cell^{\alpha}(\Sigma) for every XEX\in \mathbf{E}.

Moreover, the functor RR preserves α\alpha-directed colimits.

Proof(Part 0)

We first explain the rough idea of proof in the case α=ω\alpha=\omega. We begin by constructing a functor

F:EEF:\mathbf{E}\to \mathbf{E}

together with a natural transformation θ:IdF\theta:Id\to F having the following properties: for every arrow σ:AB\sigma:A\to B in Σ\Sigma and every map x:AXx:A\to X, there exists a map x σ:BF(X)x^\sigma:B\to F(X) fitting in a commutative square

The object R(X)R(X) is then taken to be the colimit of the infinite sequence,

where θ n=θ F n(X)\theta^n=\theta_{F^n(X)}, and natural transformation ρ:IdR\rho:Id\to R is defined by the canonical map XR(X)X\to R(X). The nice properties of ρ\rho are deduced from the nice properties of θ\theta. Let us show that the object R(X)R(X) is Σ\Sigma-fibrant. If v n:F n(X)R(X)v_n:F^n(X)\to R(X) denotes the canonical map, then we have a commutative triangle

for every n0n\geq 0. The domain of every map σ:AB\sigma:A\to B in Σ\Sigma is compact by hypothesis. It follows that for every map x:AR(X)x:A\to R(X), there exist an integer n0n\ge 0 together with a map y:AF n(X)y:A\to F^n(X) such that x=v nyx=v_n y. But there is then a map y σ:BF n+1(X)y^\sigma:B\to F^{n+1}(X) fitting in a commutative square

If z=v n+1y σz=v_{n+1}y^\sigma, then zσ=v n+1y σσ=v n+1θ ny=v ny=x.z\sigma=v_{n+1}y^\sigma \sigma =v_{n+1} \theta^n y= v_{n} y=x. This shows that R(X)R(X) is Σ\Sigma-fibrant. Let us describe the construction of the functor FF in the case where Σ\Sigma consists of a single map σ:AB\sigma:A\to B. If EE is a set we shall denote by E×AE\times A the coproduct of EE copies of AA. The functor EE×AE\mapsto E\times A is left adjoint to the functor Xhom(A,X)X\mapsto hom(A,X). Let ϵ(A,X):hom(A,X)×AX\epsilon(A,X):hom(A,X)\times A\to X be the counit of the adjunction. By definition, we have ϵ(A,X)i x=x\epsilon(A,X)i_x=x for every x:AXx:A\to X, where i x:Ahom(A,X)×Ai_x:A\to hom(A,X)\times A is the inclusion indexed by xx. The object F(X)F(X) and the map θ X:XF(X)\theta_X:X\to F(X) are then defined by a pushout square

For every map x:AXx:A\to X, the composite of the squares

is a square

Proof(Part 1)

We now give a full proof in the case α=ω\alpha=\omega. For every object XEX\in \mathbf{E} let us put

S(X)= σΣhom(s(σ),X)×s(σ)S(X)=\bigsqcup_{\sigma\in \Sigma} hom(s(\sigma),X)\,\times\, s(\sigma)

where s(σ)s(\sigma) is the source of the map σ\sigma. This defines a functor S:EE.S:\mathbf{E}\to \mathbf{E}. The counits

ϵ(s(σ),X):hom(s(σ),X)×s(σ)X\epsilon(s(\sigma),X):hom(s(\sigma), X)\,\times\, s(\sigma) \to X

induces a map ϵ X:S(X)X\epsilon_X:S(X)\to X. This defines a natural transformation ϵ:SId\epsilon:S\to Id, where IdId denotes the identity functor. By definition, if σ:AB\sigma:A\to B is a map in Σ\Sigma, then for every map x:AXx:A\to X we have ϵ Xi (σ,x)=x\epsilon_X i_{(\sigma,x)}=x, where i (σ,x):AS(X)i_{(\sigma,x)}:A\to S(X) is the inclusion indexed by (σ,x)(\sigma,x). For every object XEX\in \mathbf{E} let us put

T(X)= σΣhom(s(σ),X)×t(σ),T(X)= \bigsqcup_{\sigma\in \Sigma} hom(s({\sigma}),X)\,\times\, t({\sigma}),

where t(σ)t(\sigma) is the target of the map σ\sigma. This defines a functor T:EE.T:\mathbf{E}\to \mathbf{E}. The coproduct over σΣ\sigma\in \Sigma of the maps

hom(s(σ),X)×σ:hom(s(σ),X)×s(σ)hom(s(σ),X)×t(σ)hom(s(\sigma),X)\,\times\, \sigma: hom(s(\sigma),X)\,\times\, s(\sigma) \to hom(s({\sigma}),X)\,\times\, t({\sigma})

is a map ϕ X:S(X)T(X)\phi_X:S(X)\to T(X). This defines a natural transformation ϕ:ST.\phi:S\to T. Let us denote by F(X)F(X) the object defined by the pushout square

This defines a functor F:EEF: \mathbf{E}\to \mathbf{E} together with a natural transformation θ:IdF\theta:Id\to F. Observe that for every map σ:AB\sigma:A\to B in Σ\Sigma and every map x:AXx:A\to X, the composite of the squares

is a square

The colimit R(X)R(X) of the infinite sequence

is Σ\Sigma-fibrant by the part 0 of the proof, where θ n=θ F n(X)\theta^n=\theta_{F^n(X)}. Let us show that the canonical map ρ X:XR(X)\rho_X:X\to R(X) belongs to Cell ω(Σ)Cell^{\omega}(\Sigma). For this it suffices to show that the maps θ X\theta_X belong to Cell ω(Σ)Cell^{\omega}(\Sigma), since an ω\omega-cellular class is closed under ω\omega-compositions. But θ X\theta_X is a cobase change of ϕ X\phi_X, and ϕ X\phi_X is a coproduct of maps in Σ\Sigma. This shows that θ X\theta_X belongs to Cell ω(Σ)Cell^{\omega}(\Sigma) by the closure preperties of this class of maps. It remains to show that the functor RR preserves directed colimits. The functor hom(A,)hom(A,-) preserves directed colimits for any compact object AA. Hence, also the functor hom(A,)×Bhom(A,-)\,\times\, B for any object BB, since the functor ()×B(-)\times B is cocontinuous. The functor RR is by construction a colimit of functors of the form hom(A,)×Bhom(A,-)\,\times\, B, for compact objects AA. It follows that RR preserves directed colimits. This completes the proof of the proposition in the case where α=ω\alpha=\omega.

Proof (Part 2)

Let us now consider the case where α>ω\alpha\gt \omega. The sequence

can be extended cocontinuously through all the ordinals α\leq \alpha by putting

F j(X)=colim i<jF i(X)F^j(X)=\mathrm{colim}_{i\lt j} F^i(X)

for every limit ordinal jαj \le \alpha and by putting F j+1(X)=F(F j(X))F^{j+1}(X)=F(F^{j}(X)) and

θ j=θ F j(X):F j(X)F j+1(X)\theta^j=\theta_{F^j(X)}:F^j(X)\to F^{j+1}(X)

for every ordinal j<αj \lt \alpha. Let us then put R(X)=F α(X)R(X)=F^\alpha(X) and let v i:F i(X)R(X)v_i:F^i(X)\to R(X) be the canonical map for i<αi \lt \alpha. This defines a functor R:EER:\mathbf{E}\to \mathbf{E} equipped with a natural transformation ρ X=v 0:XR(X)\rho_X=v_0:X\to R(X). Let us show that the object R(X)R(X) is Σ\Sigma-fibrant. For every map σ:AB\sigma:A\to B in Σ\Sigma and every map x:AXx:A\to X, there exist an ordinal i<αi \lt \alpha together with a map y:AF i(X)y:A\to F^i(X) such that x=v iyx=v_i y, since the object AA is α\alpha-compact. But there is then a map y σ:BF i+1(X)y^\sigma:B\to F^{i+1}(X) fitting in a commutative square

If z=v i+1y σz=v_{i+1}y^\sigma, then zσ=v i+1y σσ=v i+1θ iy=v iy=x.z\sigma=v_{i+1}y^\sigma \sigma =v_{i+1} \theta^i y= v_{i} y=x. This shows that R(X)R(X) is Σ\Sigma-fibrant. We leave to the reader the verification that ρ X\rho_X belongs to Cell α(Σ)Cell^{\alpha}(\Sigma), and the verification that the functor RR preserves α\alpha-directed colimits.

If E\mathbf{E} is a category, then an object of the category E [2]\mathbf{E}^{[2]} is a composable pair of maps ABCA\to B\to C in the category E.\mathbf{E}. There is then a composition functor

σ 1:E [2]E [1]\sigma_1:\mathbf{E}^{[2]}\to \mathbf{E}^{[1]}

which associates to a composable pair ABCA\to B\to C its composite ABA\to B. We shall say that a section

F:E [1]E [2]F:\mathbf{E}^{[1]}\to \mathbf{E}^{[2]}

of the functor σ 1\sigma_1 is a factorisation functor. It associates to a map f:AAf:A\to A' a factorisation f=f 1f 0:AF(f)Af=f_1f_0:A\to F(f)\to A', and it takes a commutative square

to a commutative diagram,

Moreover, we have F(v,v)F(u,u)=F(vu,vu)F(v,v')F(u,u')=F(v u,v'u') for any pair of composable squares,

Proposition

(Functorial factorisation) Let Σ\Sigma be a set of maps in a cocomplete category E\mathbf{E}. If the domain and codomain of every map in Σ\Sigma is α\alpha-compact then there exists a factorisation functor

F:E [1]E [2]F:\mathbf{E}^{[1]}\to \mathbf{E}^{[2]}

which associates to every morphism f:ABf:A\to B a factorisation f=f 1f 0:AF(f)Bf=f_1f_0: A\to F(f)\to B with f 0Cell α(Σ)f_0\in Cell^{\alpha}(\Sigma) and f 1Σ f_1\in \Sigma^\pitchfork. Moreover, the functor FF preserves α\alpha-directed colimits.

Proof

We shall use Proposition 6. For any map u:ABu:A\to B in E\mathbf{E}, let us denote by λ(u)\lambda(u) the square

viewed as a morphism u1 Bu\to 1_B in the category E [1]\mathbf{E}^{[1]}. If f:XYf:X\to Y is a map in E\mathbf{E}, then the condition ufu\,\pitchfork\, f exactly means that the map

Hom(λ(u),f):Hom(1 B,f)Hom(u,f)Hom(\lambda(u),f): Hom(1_B,f)\to Hom(u,f)

is surjective and hence that ff is λ(u)\lambda(u)-fibrant. Hence a map f:XYf:X\to Y belongs to Σ \Sigma^\pitchfork iff it is λ(Σ)\lambda(\Sigma)-fibrant as an object of the category E [1]\mathbf{E}^{[1]}. It is easy to verify that domain and codomain of a map in λ(Σ)\lambda(\Sigma) are α\alpha-compact, since this is true of the maps in Σ\Sigma. It then follows from Proposition 6 that we can construct a functor

R:E [1]E [1]R:\mathbf{E}^{[1]}\to \mathbf{E}^{[1]}

together with a natural transformation ρ:IdR\rho:Id\to R. This yields a commutative square

in the category E\mathbf{E} for every map f:XYf:X\to Y in E\mathbf{E}. The map R(f)R(f) belongs to Σ \Sigma^\pitchfork, since it is a λ(Σ)\lambda(\Sigma)-fibrant object of the category E [1]\mathbf{E}^{[1]}. The morphism ρ(f):fR(f)\rho(f):f\to R(f) belongs to Cell α(λ(Σ))Cell^{\alpha}(\lambda(\Sigma)) for every map fEf\in \mathbf{E} by Proposition 6. Let us show that the map ρ 0(f)\rho_0(f) belongs to Cell α(Σ)Cell^{\alpha}(\Sigma) and that the map ρ 1(f)\rho_1(f) is invertible. For this, let us denote by 𝒞\mathcal{C} the class of maps u:ABu:A\to B in E [1]\mathbf{E}^{[1]}

for which u 0Cell α(Σ)u_0\in Cell^{\alpha}(\Sigma) and for which u 1u_1 invertible. The It is easy to verify that the class 𝒞\mathcal{C} is α\alpha-cellular. Moreover, we have λ(Σ)𝒞\lambda(\Sigma)\subseteq \mathcal{C}. It follows that we have Cell α(λ(Σ))𝒞Cell^{\alpha}(\lambda(\Sigma))\subseteq \mathcal{C}. Thus, the morphism ρ(f):fR(f)\rho(f):f\to R(f) belongs to 𝒞 \mathcal{C} for every ff. Hence the map ρ 0(f)\rho_0(f) belongs to Cell α(Σ)Cell^{\alpha}(\Sigma) and the map ρ 1(f)\rho_1(f) is invertible. We can then construct a functorial factorisation f=f 1f 0:XF(f)Yf=f_1f_0:X\to F(f)\to Y by putting F(f)=R 0(f)F(f)=R_0(f), f 0=ρ 0(f)f_0=\rho_0(f) and f 1=ρ 1(f) 1R(f)f_1=\rho_1(f)^{-1}R(f). By construction, we have f 0Cell α(Σ)f_0\in \Cell^{\alpha}(\Sigma) and f 1Σ f_1\in \Sigma^\pitchfork. The functor FF preserves α\alpha-directed colimits, since the functor RR preserves α\alpha-directed colimits.

Remark

The first part of the proposition can be proved under the weaker assumption that the domains of the maps in Σ\Sigma are α\alpha-compact (but the resulting factorisation functor FF may not preserves α\alpha-directed colimits). See Exercise 2.

Recall from Definition 9 that Sat(Σ)Sat(\Sigma) denotes the saturated class generated by a class Σ\Sigma.

Theorem

Let Σ\Sigma be a set of maps between small objects in a cocomplete category E\mathbf{E}. Then the pair (Sat(Σ),Σ )(Sat(\Sigma),\Sigma^\pitchfork) is a weak factorisation system. Moreover, if Σ\Sigma is a set of maps between α\alpha-compact objects, then every morphism in Sat(Σ)Sat(\Sigma) is a codomain retract of a morphism in Cell α(Σ)Cell^{\alpha}(\Sigma).

Proof

We shall apply Proposition 2 to the classes =Sat(Σ)\mathcal{L}=Sat(\Sigma) and =Σ \mathcal{R}=\Sigma^\pitchfork. The class \mathcal{L} is closed under codomain retracts, since it is saturated. The class \mathcal{R} is closed under domain retracts by Proposition \ref{closureofcomplements} since it is a right complement Σ \Sigma^\pitchfork. Let us show that we have \mathcal{L}\,\pitchfork\, \mathcal{R}. We have Σ (Σ )= \Sigma \subseteq {}^\pitchfork(\Sigma^\pitchfork)=\mathcal{R}^\pitchfork. Thus, Sat(Σ) Sat(\Sigma)\subseteq \mathcal{R}^\pitchfork, since the class \mathcal{R}^\pitchfork is saturated by Proposition 4. This proves that \mathcal{L}\,\pitchfork\, \mathcal{R}. Let us choose a regular cardinal α\alpha for which Σ\Sigma is a set of maps between α\alpha-compact objects. It then follows from Proposition 7 that every map f:XYf:X\to Y admits a factorisation f=pu:XEYf=p u:X\to E\to Y with uCell α(Σ)u\in Cell^{\alpha}(\Sigma) and pp\in \mathcal{R}. But we have Cell α(Σ)Sat(Σ)Cell^{\alpha}(\Sigma)\subseteq Sat(\Sigma) since a saturated class is α\alpha-cellular for any regular cardinal α\alpha by Lemma 5 This shows that uu\in \mathcal{L} and hence that the pair (,)(\mathcal{L},\mathcal{R}) is a weak factorisation system by Proposition 2. It remains to prove that every morphism in \mathcal{L} is a codomain retract of a morphism in Cell α(Σ)Cell^{\alpha}(\Sigma). If u:ABu:A\to B belongs to \mathcal{L}, let us choose a factorisation u=pv:XEYu=p v:X\to E\to Y with vCell α(Σ)v\in Cell^{\alpha}(\Sigma) and rr\in \mathcal{R}. The square

has a diagonal filler s:BEs:B\to E, since we have uru\pitchfork r. This shows that uu is a codomain retract of vv.

Corollary

Let Σ\Sigma be a set of maps in a locally presentable category E\mathbf{E}. Then the pair (Sat(Σ),Σ )(Sat(\Sigma),\Sigma^\pitchfork) is a weak factorisation system.

Proof

This follows from theorem 1, since every object of a locally presentable category is small.

Functorial aspects

Lemma

Let F:CD:GF:\mathbf{C}\leftrightarrow \mathbf{D}:G be a pair of adjoint functors between two categories C\mathbf{C} and D\mathbf{D}. If u:ABu:A\to B is an arrow in C\mathbf{C} and f:XYf:X\to Y is an arrow in D\mathbf{D}, then the two conditions

F(u)fanduG(f)F(u)\,\pitchfork\, f \quad \mathrm{and}\quad u \,\pitchfork\, G(f)

are equivalent.

Proof

The adjunction θ:FG\theta:F\dashv G induces a bijection between the following commutative squares and their diagonal fillers,

Proposition

Let F:CD:GF:\mathbf{C}\leftrightarrow \mathbf{D}:G be a pair of adjoint functors between two categories C\mathbf{C} and D\mathbf{D}. If (,)(\mathcal{L},\mathcal{R}) is a weak factorisation system in the category C\mathbf{C} and (,)(\mathcal{L}',\mathcal{R}') a weak factorisation system in D\mathbf{D}, then the two conditions

F()andG()F(\mathcal{L})\subseteq \mathcal{L}'\quad \mathrm{and} \quad G(\mathcal{R}')\subseteq \mathcal{R}

are equivalent.

Proof

The condition F()F(\mathcal{L})\subseteq \mathcal{L}' is equivalent to the condition F()F(\mathcal{L})\,\pitchfork\, \mathcal{R}', since = \mathcal{L}'={}^\pitchfork\mathcal{R}'. But the condition F()F(\mathcal{L})\,\pitchfork\, \mathcal{R}' is equivalent to the condition G()\mathcal{L}\,\pitchfork\, G(\mathcal{R}') by Lemma 8. But the condition G()\mathcal{L}\,\pitchfork\, G(\mathcal{R}') is equivalent to the condition G()G(\mathcal{R}')\subseteq \mathcal{R}, since = \mathcal{R}=\mathcal{L}^\pitchfork.

Examples

In algebra

Example

If RR is a ring, we shall say that a morphism of (left) RR-modules is projective if it has the left lifting property with respect to the the epimorphisms. An RR-module MM is projective iff the morphism 0M0\to M is projective. More generally, a map of RR-modules u:MNu:M\to N is projective iff it is monic and its cokernel is a projective RR-module. The category of RR-modules admits a weak factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of projective morphisms and \mathcal{R} is the class of epimorphisms.

Example

A category with finite coproducts E\mathbf{E} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of split epimorphisms. A morphism u:ABu:A\to B belongs to \mathcal{L} iff it is a codomain retract of an inclusion in 1:AABin_1:A\to A\sqcup B.

Example

We shall say that a homomorphism of groups is projective if it has the left lifting property with respect to the surjective homomorphisms. The category of groups Grp\mathbf{Grp} admits a weak factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of projective homorphisms and \mathcal{R} is the class of surjective homomorphisms. More generally, if V\mathbf{V} is a variety of algebras, we shall say that a morphism in V\mathbf{V} is projective if it has the left lifting property with respect to surjective morphisms. A morphism is projective iff it is the codomain retract of an inclusion in 1:AACin_1:A\to A\sqcup C, where CC is a free algebra. Then the category V\mathbf{V} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of projective morphisms and \mathcal{R} is the class of surjective morphisms.

Example

If RR is a ring, we shall say that a homomorphism of (left) RR-modules is a trivial fibration if it has the right lifting property with respect to every monomorphism. The category of left RR-modules mathbfRModmathbf{RMod} admits a weak factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of monomorphisms and \mathcal{R} is the class of trivial fibrations. More generally, any Grothendieck abelian category admits such a factorisation system. An object XX is said to be injective if the map X1X\to 1 is a trivial fibration.

Example

We shall say that a homomorphism of boolean algebras is a trivial fibration if it has the right lifting property with respect to every monomorphism. The category of boolean algebras admits a weak factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of monomorphisms and \mathcal{R} is the class of trivial fibrations. A boolean algebra is injective iff it is complete.

In Cat

Example

We recall that the category of small categories Cat\mathbf{Cat} admits a natural model structure (𝒞,𝒲,)(\mathcal{C},\mathcal{W},\mathcal{F}) in which 𝒞\mathcal{C} is the class of functors monic on objects, 𝒲\mathcal{W} is the class of equivalences of categories and \mathcal{F} is the class of isofibrations. Hence the category Cat\mathbf{Cat} admits two weak factorisation systems (𝒞𝒲,)( \mathcal{C}\cap \mathcal{W},\mathcal{F}) and (𝒞,𝒲)(\mathcal{C},\mathcal{W}\cap \mathcal{F}). In the first, 𝒞𝒲 \mathcal{C}\cap \mathcal{W} is the class of equivalences monic on objects and \mathcal{F} is the class of isofibrations. In the second, 𝒞 \mathcal{C} is the class of functors monic on objects and 𝒲\mathcal{W}\cap \mathcal{F} is the class of equivalences surjective on objects.

Example

We shall say that a small category AA is 1-connected if its fundamental groupoid π 1(A)\pi_1(A) is equivalent to the terminal category 1. We shall that a functor between small categories u:ABu:A \to B is 1-final if the category b\A=b\B× BAb \backslash A=b \backslash B \times_{B} A defined by the pullback square

is 1-connected for every object bBb\in B. We shall say that a Grothendieck fibration is a 1-fibration if its fibers are groupoids. The category Cat\mathbf{Cat} admits a weak factorisation system (,)(\mathcal{L}, \mathcal{R}) in which \mathcal{L} the class of 1-final functors monic on objects and \mathcal{R} is the class of 1-fibrations.

Example

Let us say that a functor between small categories u:ABu:A \to B is 1-initial if the category b\A=(B/b)× BAb \backslash A= (B/b) \times_{B} A defined by the pullback square

is connected for every object bBb\in B. We shall say that a Grothendieck opfibration is a 1-opfibration if its fibers are groupoids. The category Cat\mathbf{Cat} admits a weak factorisation system (,)(\mathcal{L}, \mathcal{R}) in which \mathcal{L} the class of 1-initial functors monic on objects and \mathcal{R} is the class of 1-opfibrations.

Example

Let us say that a functor between small categories u:ABu:A\to B is 1-connected if the map of simplicial sets N(u):N(A)N(B)N(u):N(A)\to N(B) is 1-connected, where N:CatSSetN:\mathbf{Cat} \to \mathbf{SSet} is the nerve functor. We shall say that a Grothendieck bifibration is a 1-bifibration if its fibers are groupoids. The category Cat\mathbf{Cat} admits a weak factorisation system (,)(\mathcal{L}, \mathcal{R}) in which \mathcal{L} the class of 1-connected functors monic on objects and \mathcal{R} is the class of 1-bifibrations.

In SSet

Example

Let Σ\Sigma be the set of inclusions h n k:Λ k[n]Δ[n]h^k_n: \Lambda^k[n] \subset \Delta[n] (n>0,0knn\gt 0, 0\le k\le n) in the category of simplicial sets. A map of simplicial sets is a Kan fibration if it belongs to Σ \Sigma^\pitchfork, and a map is anodyne if it belongs to Sat(Σ)Sat(\Sigma) (=Sat ω(Σ)=\Sat^\omega(\Sigma)). The category of simplicial sets SSet\mathbf{SSet} admits a weak factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of anodyne maps and \mathcal{R} is the class of Kan fibrations.

Example

Let Σ\Sigma be the set of inclusions h n k:Λ k[n]Δ[n]h^k_n: \Lambda^k[n] \subset \Delta[n] (0<kn0\lt k\le n) in the category of simplicial sets. A map of simplicial sets is a right fibration if it belongs to Σ \Sigma^\pitchfork, and a map is right anodyne if it belongs to Sat(Σ)Sat(\Sigma) (=Sat ω(Σ)=\Sat^\omega(\Sigma)). The category of simplicial sets SSet\mathbf{SSet} admits a weak factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of right anodyne maps and \mathcal{R} is the class of right fibrations.

Example

Let Σ\Sigma be the set of inclusions h n k:Λ k[n]Δ[n]h^k_n: \Lambda^k[n] \subset \Delta[n] (0k<n0\le k\lt n) in the category of simplicial sets. A map of simplicial sets is a left fibration if it belongs to Σ \Sigma^\pitchfork, and a map is left anodyne if it belongs to Sat(Σ)Sat(\Sigma) (=Sat ω(Σ)=\Sat^\omega(\Sigma)). The category of simplicial sets SSet\mathbf{SSet} admits a weak factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of left anodyne maps and \mathcal{R} is the class of left fibrations.

Example

Let Σ\Sigma be the set of inclusions h n k:Λ k[n]Δ[n]h^k_n: \Lambda^k[n] \subset \Delta[n] (0<k<n0\lt k\lt n) in the category of simplicial sets. A map of simplicial sets is a mid fibration if it belongs to Σ \Sigma^\pitchfork, and a map is mid anodyne if it belongs to Sat(Σ)Sat(\Sigma) (=Sat ω(Σ)=\Sat^\omega(\Sigma)). The category of simplicial sets SSet\mathbf{SSet} admits a weak factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of mid anodyne maps and \mathcal{R} is the class of mid fibrations.

More examples

Example

We recall that a Quillen model structure on a category E\mathbf{E} is a triple (𝒞,𝒲,)(\mathcal{C},\mathcal{W},\mathcal{F}) of classes of maps in E\mathbf{E} satisfying the following two axioms: * the class 𝒲\mathcal{W} has the three-for-two property; * The pairs (𝒞𝒲,)( \mathcal{C}\cap \mathcal{W},\mathcal{F}) and (𝒞,𝒲)(\mathcal{C},\mathcal{W}\cap \mathcal{F}) are weak factorisation systems.

Example

Recall that a map in a topos is called a trivial fibration if it has the right lifting property with respect to every monomorphism. This terminology is non-standard but useful. If \mathcal{L} is the class of monomorphisms in the topos and \mathcal{R} is the class of trivial fibrations then the pair (,)(\mathcal{L},\mathcal{R}) is a weak factorisation system by a proposition here. An object XX in the topos is said to be injective if the map X1X\to 1 is a trivial fibration.

Example

If C\mathbf{C} is a category with pullbacks, then to every weak factorisation system (,)(\mathcal{L},\mathcal{R}) in C\mathbf{C} is associated a weak factorisation system (,)(\mathcal{L}',\mathcal{R}') in the category [I,C][I,\mathbf{C}] (by the proposition here), where \mathcal{R}' is the class of \mathcal{R}-cartesian squares. In particular, the class of epi-cartesian squares in the category of sets is the right class of a weak factorisation system in the category [I,Set][I,\mathbf{Set}].

Exercises

Exercise

Let p:EEp:\mathbf{E}'\to \mathbf{E} be a discrete Conduché fibration. Recall that this means that for every morphism f:ABf:A\to B in E\mathbf{E} and every factorisation p(f)=vu:p(A)Ep(B)p(f)=v u:p(A)\to E\to p(B) of the morphism p(f)p(f), there exists a unique factorisation f=vu:AEBf=v'u':A\to E'\to B of the morphism ff such that p(v)=vp(v')=v and p(u)=up(u')=u. Discrete fibrations and a discrete opfibrations are examples of discrete Conduché fibrations. If \mathcal{M} is a class of maps in E\mathbf{E}, let us denote by \mathcal{M}' the class of maps p 1()p^{-1}(\mathcal{M}) in E\mathbf{E}'. Show that if (,)(\mathcal{L},\mathcal{R}) is a weak factorisation system in the category E\mathbf{E}, then the pair (,)(\mathcal{L}',\mathcal{R}') is a weak factorisation system in the category E\mathbf{E}'.

Exercise

Show that the functorial factorisation of 7 can be obtained under the weaker assumption that the domains of the maps in Σ\Sigma are α\alpha-compact. (but the resulting factorisation functor FF may not preserves α\alpha-directed colimits).

References

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Revised on January 21, 2013 at 19:26:25 by Zhen Lin