# Joyal's CatLab Weak factorisation systems

**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

# Contents

## Main definitions

###### Definition

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

$\xymatrix{ A \ar[d]_u \ar[r]^{x} & X \ar[d]^{f} \\ B \ar[r]_{y} & Y }$

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

$\xymatrix{ A \ar[d]_u \ar[r]^{x} & X \ar[d]^{f} \\ B \ar[r]_{y} \ar[ur]^d & Y }$

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

(1) 
$\xymatrix{ hom(B,X) \ar[d]_{hom(B,f)} \ar[rr]^{hom(u,X)} & & hom(A,X) \ar[d]^{hom(A,f)} \\ hom(B,Y) \ar[rr]_{hom(u,Y)} & & hom(A,Y), }$

is epicartesian, or equivalently if the map

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

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 $h^k_n: \Lambda^k[n] \subset \Delta[n]$ for every $n\gt 0$ and $0\le k\le n$. A simplicial set $X$ is a Kan complex iff the map $X\to 1$ is a Kan fibration.

###### Example

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

###### Definition

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

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

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

We shall say that a factorisation $f=p u:A\to E\to B$ with $u\in \mathcal{L}$ and $p\in \mathcal{R}$ is a $(\mathcal{L},\mathcal{R})$-factorisation of the map $f$. 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 $\mathbf{Set}$ admits a weak factorisation system $(Inj,Surj)$, where $Inj$ the class of injections and $Surj$ 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 $\mathbf{E}$, then the pair $(\mathcal{R}^o,\mathcal{L}^o)$ is a weak factorisation system in the opposite category $\mathbf{E}^o$.

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

###### Slice and coslice

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

## Closure properties

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

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

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

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 $f:X\to Y$ in a category $\mathbf{E}$ are equivalent:

$f \mathrm{is}\:\mathrm{invertible}, \quad \mathbf{E}\,\pitchfork\, f, \quad f\,\pitchfork\, f, \quad f\,\pitchfork\, \mathbf{E}.$
###### Proof

($1\Rightarrow 2$) If $f$ is invertible, then the square

$\xymatrix{ A \ar[d]_u \ar[r]^{x} & X \ar[d]^{f} \\ B \ar[r]_{y} & Y }$

has a diagonal filler $f^{-1}y:B\to X$. Thus, $u\,\pitchfork\, f$ for any arrow $u$, and hence $\mathbf{E}\,\pitchfork\, f$. ($2\Rightarrow 3$) If $\mathbf{E}\,\pitchfork\, f$, then $f\,\pitchfork\,f$. ($3\Rightarrow 1$) If $f\,\pitchfork\,f$, then the square

$\xymatrix{ X \ar[d]_f \ar@{=}[r] & X \ar[d]^f \\ Y \ar@{=}[r] & Y }$

has a diagonal filler $g:Y\to X$ and this shows that $f$ is invertible. The equivalences ($1\Leftrightarrow 2\Leftrightarrow 3$) are proved. The equivalences ($1\Leftrightarrow 4\Leftrightarrow 3$) are proved similarly.

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

$\xymatrix{A \ar[d]^u\ar[r]^i& C \ar[d]^v\ar[r]^p & A \ar[d]^u \\ B \ar[r]^j & D \ar[r]^q & B}$

and such that $p i=1_A$ and $q j=1_B$.

###### Definition

We shall say that a class of maps $\mathcal{M}$ in a category $\mathbf{E}$ 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 $f:X\to Y$ along a map $v:V\to X$ is the map $g:U \to V$ in a pullback square,

$\xymatrix{ U \ar[d]_g \ar[r] & X \ar[d]^{f} \\ V \ar[r]_{v} & Y }$

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

$\xymatrix{ A \ar[d]_{u} \ar[r]^{c} & C\ar[d]^{v}\\ B \ar[r] & D.}$
###### Definition

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

$\sqcup_{i\in I}u_i: \sqcup_{i\in I} A_i\to \sqcup_{i\in I} B_i$

of any family of maps $(u_i:i\in I)$ 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 $\mathbf{E}$. 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 $u:A\to B$ and $v:B\to C$ belongs to ${}^\pitchfork\mathcal{M}$, then so does their composite $v u:A\to C$. For any morphism $f:X\to Y$, the square $Sq(v u,f)$

$\xymatrix{ hom(C,X) \ar[rr]^{hom(v u,X)} \ar[d]_{hom(C,f)} & & hom(A,X) \ar[d]^{hom(A,f)} \\ hom(C,Y) \ar[rr]_{hom(v u,Y)} & & hom(A,Y), }$

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

$\xymatrix{ hom(C,X) \ar[rr]^{hom(v ,X)} \ar[d]_{hom(C,f)} & & hom(B,X) \ar[d]_{hom(B,f)} \ar[rr]^{hom(u,X)} & & hom(A,X) \ar[d]^{hom(A,f)} \\ hom(C,Y) \ar[rr]_{hom(v,Y)} & & hom(B,Y) \ar[rr]_{hom(u,Y)} & & hom(A,Y). }$

The squares $Sq(u,f)$ and $Sq(v,f)$ are epicartesian if $f\in \mathcal{M}$; hence their composite is epicartesian by the lemma here. This shows that $v u\in {}^\pitchfork\mathcal{M}$. 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 $f:X\to Y$ is a retract of a map $g:U\to V$, then the square $Sq(u,f)$ is a retract of the square $Sq(u,g)$ for any map $u:A\to B$. 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 $f:X\to Y$ is the product of a family of maps $f_i:X_i\to Y_i$ ($i\in I$), then the square $Sq(u,f)$ is the product of the family of squares $Sq(u,f_i)$. 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) 
$\xymatrix{ U \ar[d]_g \ar[r]^u & X \ar[d]^f \\ V \ar[r]^v & Y }$

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

$\xymatrix{ hom(B,U) \ar[rr] \ar[dd] \ar[dr] & & hom(A,U) \ar[dr] \ar '[d] [dd] & \\ & hom(B,X)\ar[dd]\ar[rr] & & hom(A,X) \ar[dd] \\ hom(B,V) \ar[dr] \ar '[r] [rr]& & hom(A,V) \ar[dr] & \\ & hom(B,Y) \ar[rr] & & hom(A,Y) . }$

The left and the right faces of the cube are cartesian, since the square (2) is cartesian and the functors $hom(A,-)$ and $hom(B,-)$ preserve limits. The front face is epicartesian since we have $u\pitchfork p$. 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 $u:A\to B$ in a category $\mathbf{E}$ is said to be a domain retract of a map $v:C\to B$, if the object $(A,u)$ of the category $\mathbf{E}/B$ is a retract of the object $(C,v)$. There is a dual notion of codomain retract.

###### Definition

We shall say that a class of maps $\mathcal{M}$ in a category $\mathbf{E}$ 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 $\mathbf{E}$ is a weak factorisation system iff the following conditions are satisfied:

• every map $f:X\to Y$ admits a $(\mathcal{L},\mathcal{R})$-factorisation $f=p u:X\to E\to Y$;
• $\mathcal{L}\,\pitchfork\,\mathcal{R}$;
• 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 $f:X\to Y$ belongs to $\mathcal{L}^\pitchfork$, let us choose a $(\mathcal{L},\mathcal{R})$-factorisation $f=p u:X\to E\to Y$. The square

$\xymatrix{ X \ar[d]_u \ar@{=}[r] & X \ar[d]^f\\ E \ar[r]^p & Y }$

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

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

###### Corollary

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

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

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

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

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

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

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

###### Proof

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

$\xymatrix{ hom(B,C(j)) \ar[rr]^{hom(u,C(j))} \ar[d]_{hom(B,c(j,i))} & & hom(A,C(j)) \ar[d]^{hom(A,c(j,i))} \\ hom(B,C(i)) \ar[rr]_{hom(u,C(i))} & & hom(A,C(i)). }$

Beware that here the square $Sq(u,c(j,i))$ 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 $Sq(u,C):[\alpha]\to \mathbf{Set}^I$ is continuous, since $C$ is continuous. If $u\in \mathcal{M}$, then the square $Sq(u,c(j,i))$ is epicartesian for every $i\leq j\lt \alpha$ by the assumption on $C$. It follows that the square $Sq(u,c(\alpha,0))$ is epicartesian by the lemma here. This show that $c(\alpha,0)$ 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 $\mathcal{C}$ in a cocomplete category $\mathbf{E}$ is cellular if it satisfies the following conditions: * $\mathcal{C}$ contains the isomorphisms and is closed under composition, * $\mathcal{C}$ is closed under transfinite compositions; * $\mathcal{C}$ is closed under cobase changes.

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

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

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

###### 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 $\partial \Delta[n] \subset \Delta[n]$ ($n\geq 0$).

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

###### Proposition

The class ${}^\pitchfork\mathcal{M}$ is saturated for any class of maps $\mathcal{M}$ in a cocomplete category $\mathbf{E}$. 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 $\mathbf{E}$. We shall say that an object $A\in \mathbf{E}$ cofibrant, if the map $\bot \to A$ belongs to $\mathcal{M}$, where $\bot$ is the initial object of $\mathbf{E}$. 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 $A$ and $B$ are cofibrant, consider the pushout square

$\xymatrix{ \bot \ar[r]\ar[d]& B \ar[d]^{i_B} \\ A \ar[r]^(0.4){i_A}& A\sqcup B . }$

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

$A= \bigsqcup_{i\in I} A_i$

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

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

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

$C(j,k)=\bigsqcup_{j\le i\lt k} A_i$

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

$\xymatrix{ \bot \ar[r]\ar[d]& C(j) \ar[d] \\ C(j,k) \ar[r]& C(k). }$

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

$\xymatrix{ A_i\ar[r]\ar[d]_{u_i}& A \ar[d]^{v_i} \\ B_i \ar[r] & E_i. }$

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

###### Definition

If $\alpha$ is a regular cardinal, we shall say that a class of maps $\mathcal{M}$ in a complete category $\mathbf{E}$ is $\alpha$-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 $\le \alpha$; * $\mathcal{M}$ is closed under cobase changes; * $\mathcal{M}$ is closed under coproducts.

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

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

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

###### Example

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

###### Example

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

### Small object argument

###### Definition

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

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

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

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

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

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

###### Proposition

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

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

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

Moreover, the functor $R$ preserves $\alpha$-directed colimits.

###### Proof(Part 0)

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

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

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

$\xymatrix{ A \ar[d]_{\sigma} \ar[r]^{x} & X \ar[d]^{\theta_X}\\ B \ar[r]^(0.4){x^\sigma} & F(X). }$

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

$\xymatrix{ X \ar[r]^{\theta^0} & F(X) \ar[r]^{\theta^1}& F^2(X) \ar[r]^{\theta^2} & F^3(X)\ar[r] &\cdots }$

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

$\xymatrix{ F^n(X) \ar[d]_{\theta^n} \ar[r]^{v_n} & R(X) \\ F^{n+1}(X) \ar[ur]_{v_{n+1}} & }$

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

$\xymatrix{ A \ar[d]_{\sigma} \ar[r]^{y} & F^nX \ar[d]^{\theta^n} \\ B \ar[r]^(0.4){y^\sigma} & F^{n+1}(X). }$

If $z=v_{n+1}y^\sigma$, then $z\sigma=v_{n+1}y^\sigma \sigma =v_{n+1} \theta^n y= v_{n} y=x.$ This shows that $R(X)$ is $\Sigma$-fibrant. Let us describe the construction of the functor $F$ in the case where $\Sigma$ consists of a single map $\sigma:A\to B$. If $E$ is a set we shall denote by $E\times A$ the coproduct of $E$ copies of $A$. The functor $E\mapsto E\times A$ is left adjoint to the functor $X\mapsto hom(A,X)$. Let $\epsilon(A,X):hom(A,X)\times A\to X$ be the counit of the adjunction. By definition, we have $\epsilon(A,X)i_x=x$ for every $x:A\to X$, where $i_x:A\to hom(A,X)\times A$ is the inclusion indexed by $x$. The object $F(X)$ and the map $\theta_X:X\to F(X)$ are then defined by a pushout square

$\xymatrix{ hom(A,X)\times A \ar[d]_{hom(A,X)\times \sigma} \ar[rr]^(0.7){\epsilon(A,X)} & & X \ar[d]^{\theta_X} \\ hom(A,X)\times B \ar[rr] & & F(X), }$

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

$\xymatrix{ A \ar[rr]^(0.3){i_x}\ar[d]_\sigma & & hom(A,X)\times A \ar[d]_{hom(A,X)\times \sigma} \ar[rr]^(0.7){\epsilon(A,X)} & & X \ar[d]^{\theta_X} \\ B\ar[rr]^(0.3){i_x}& & hom(A,X)\times B \ar[rr] & & F(X), }$

is a square

$\xymatrix{ A \ar[d]_{\sigma} \ar[r]^{x} & X \ar[d]^{\theta_X}\\ B \ar[r]^(0.4){x^\sigma} & F(X). }$
###### Proof(Part 1)

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

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

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

$\epsilon(s(\sigma),X):hom(s(\sigma), X)\,\times\, s(\sigma) \to X$

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

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

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

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

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

$\xymatrix{ S(X) \ar[d]_{\phi_X} \ar[rr]^{\epsilon_X} & & X \ar[d]^{\theta_X}\\ T(X) \ar[rr] & & F(X). }$

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

$\xymatrix{ A \ar[rr]^{i_{(\sigma,x)}}\ar[d]_\sigma & & S(X) \ar[d]_{\phi_X} \ar[rr]^{\epsilon_X} & & X \ar[d]^{\theta_X} \\ B\ar[rr]^{i_{(\sigma,x)}}& & T(X) \ar[rr] & & F(X), }$

is a square

$\xymatrix{ A \ar[d]_{\sigma} \ar[r]^{x} & X \ar[d]^{\theta_X}\\ B \ar[r]^{x^\sigma} & F(X). }$

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

$\xymatrix{ X \ar[r]^{\theta^0} & F(X) \ar[r]^{\theta^1}& F^2(X) \ar[r]^{\theta^2} & F^3(X)\ar[r] &\cdots }$

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

###### Proof (Part 2)

Let us now consider the case where $\alpha\gt \omega$. The sequence

$\xymatrix{ X \ar[r]^{\theta^0} & F(X) \ar[r]^{\theta^1}& F^2(X) \ar[r]^{\theta^2} & F^3(X)\ar[r] &\cdots }$

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

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

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

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

for every ordinal $j \lt \alpha$. Let us then put $R(X)=F^\alpha(X)$ and let $v_i:F^i(X)\to R(X)$ be the canonical map for $i \lt \alpha$. This defines a functor $R:\mathbf{E}\to \mathbf{E}$ equipped with a natural transformation $\rho_X=v_0:X\to R(X)$. Let us show that the object $R(X)$ is $\Sigma$-fibrant. For every map $\sigma:A\to B$ in $\Sigma$ and every map $x:A\to X$, there exist an ordinal $i \lt \alpha$ together with a map $y:A\to F^i(X)$ such that $x=v_i y$, since the object $A$ is $\alpha$-compact. But there is then a map $y^\sigma:B\to F^{i+1}(X)$ fitting in a commutative square

$\xymatrix{ A \ar[d]_{\sigma} \ar[r]^{y} & F^iX \ar[d]^{\theta^i} \\ B \ar[r]^(0.4){y^\sigma} & F^{i+1}(X). }$

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

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

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

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

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

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

$\xymatrix{ A \ar[d]_{f} \ar[r]^{u} & B \ar[d]^{g}\\ A' \ar[r]^{u'} & B' }$

to a commutative diagram,

$\xymatrix{ A \ar[d]_{f_0} \ar[rr]^{u} & & B \ar[d]^{g_0}\\ F(f) \ar[d]_{f_1} \ar[rr]^{F(u,u')} & & F(g) \ar[d]^{g_1}\\ A' \ar[rr]^{u'} & & B'. }$

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

$\xymatrix{ A \ar[d]_{f} \ar[r]^{u} & B \ar[d]^{g}\ar[r]^{v} & C\ar[d]_{h} \\ A' \ar[r]^{u'} & B'\ar[r]^{v'} & C' .}$
###### Proposition

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

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

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

###### Proof

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

$\xymatrix{ A \ar[d]_{u} \ar[r]^{u} & B \ar[d]^{1_B}\\ B \ar@{=}[r] & B }$

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

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

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

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

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

$\xymatrix{ X \ar[d]_{f} \ar[rr]^{\rho_0(f)} & & R_0(f) \ar[d]^{R(f)}\\ Y \ar[rr]^{\rho_1(f)} & & R_1(f). }$

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

$\xymatrix{ A_0 \ar[d]_{u} \ar[r]^{u_0} & B_0 \ar[d]\\ A_1 \ar[r]^{u_1} & B_1 }$

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

###### Remark

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

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

###### Theorem

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

###### Proof

We shall apply Proposition 2 to the classes $\mathcal{L}=Sat(\Sigma)$ and $\mathcal{R}=\Sigma^\pitchfork$. 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 $\Sigma^\pitchfork$. Let us show that we have $\mathcal{L}\,\pitchfork\, \mathcal{R}$. We have $\Sigma \subseteq {}^\pitchfork(\Sigma^\pitchfork)=\mathcal{R}^\pitchfork$. Thus, $Sat(\Sigma)\subseteq \mathcal{R}^\pitchfork$, 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 $\alpha$ for which $\Sigma$ is a set of maps between $\alpha$-compact objects. It then follows from Proposition 7 that every map $f:X\to Y$ admits a factorisation $f=p u:X\to E\to Y$ with $u\in Cell^{\alpha}(\Sigma)$ and $p\in \mathcal{R}$. But we have $Cell^{\alpha}(\Sigma)\subseteq Sat(\Sigma)$ since a saturated class is $\alpha$-cellular for any regular cardinal $\alpha$ by Lemma 5 This shows that $u\in \mathcal{L}$ 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 $Cell^{\alpha}(\Sigma)$. If $u:A\to B$ belongs to $\mathcal{L}$, let us choose a factorisation $u=p v:X\to E\to Y$ with $v\in Cell^{\alpha}(\Sigma)$ and $r\in \mathcal{R}$. The square

$\xymatrix{ A \ar[d]_{u} \ar[r]^{v} & E \ar[d]^r\\ B \ar@{=}[r] & B }$

has a diagonal filler $s:B\to E$, since we have $u\pitchfork r$. This shows that $u$ is a codomain retract of $v$.

###### Corollary

Let $\Sigma$ be a set of maps in a locally presentable category $\mathbf{E}$. Then the pair $(Sat(\Sigma),\Sigma^\pitchfork)$ 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 $F:\mathbf{C}\leftrightarrow \mathbf{D}:G$ be a pair of adjoint functors between two categories $\mathbf{C}$ and $\mathbf{D}$. If $u:A\to B$ is an arrow in $\mathbf{C}$ and $f:X\to Y$ is an arrow in $\mathbf{D}$, then the two conditions

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

are equivalent.

###### Proof

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

$\xymatrix{ FA \ar[d]_{Fu} \ar[r]^x & X \ar[d]^{f} \\ FB \ar[r]_{y} \ar@{-- />}[ur]^{d} & Y ,} \quad \quad \quad \xymatrix{ A \ar[d]_u \ar[r]^{\theta(x)}& GX \ar[d]^{Gf} \\ B \ar[r]_{\theta(y)} \ar@{-->}[ur]^{\theta(d)} & GY. }$
###### Proposition

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

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

are equivalent.

###### Proof

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

## Examples

### In algebra

###### Example

If $R$ is a ring, we shall say that a morphism of (left) $R$-modules is projective if it has the left lifting property with respect to the the epimorphisms. An $R$-module $M$ is projective iff the morphism $0\to M$ is projective. More generally, a map of $R$-modules $u:M\to N$ is projective iff it is monic and its cokernel is a projective $R$-module. The category of $R$-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 $\mathbf{E}$ admits a factorisation system $(\mathcal{L},\mathcal{R})$ in which $\mathcal{R}$ is the class of split epimorphisms. A morphism $u:A\to B$ belongs to $\mathcal{L}$ iff it is a codomain retract of an inclusion $in_1:A\to A\sqcup B$.

###### 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 $\mathbf{Grp}$ 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 $\mathbf{V}$ is a variety of algebras, we shall say that a morphism in $\mathbf{V}$ 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 $in_1:A\to A\sqcup C$, where $C$ is a free algebra. Then the category $\mathbf{V}$ 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 $R$ is a ring, we shall say that a homomorphism of (left) $R$-modules is a trivial fibration if it has the right lifting property with respect to every monomorphism. The category of left $R$-modules $mathbf{RMod}$ 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 $X$ is said to be injective if the map $X\to 1$ 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 $\mathbf{Cat}$ admits a natural model structure $(\mathcal{C},\mathcal{W},\mathcal{F})$ in which $\mathcal{C}$ is the class of functors monic on objects, $\mathcal{W}$ is the class of equivalences of categories and $\mathcal{F}$ is the class of isofibrations. Hence the category $\mathbf{Cat}$ admits two weak factorisation systems $( \mathcal{C}\cap \mathcal{W},\mathcal{F})$ and $(\mathcal{C},\mathcal{W}\cap \mathcal{F})$. In the first, $\mathcal{C}\cap \mathcal{W}$ is the class of equivalences monic on objects and $\mathcal{F}$ is the class of isofibrations. In the second, $\mathcal{C}$ is the class of functors monic on objects and $\mathcal{W}\cap \mathcal{F}$ is the class of equivalences surjective on objects.

###### Example

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

$\xymatrix{ b \backslash A \ar[d] \ar[r] & A\ar[d]^u\\ b \backslash B\ar[r] & B. }$

is 1-connected for every object $b\in B$. We shall say that a Grothendieck fibration is a 1-fibration if its fibers are groupoids. The category $\mathbf{Cat}$ 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 $u:A \to B$ is 1-initial if the category $b \backslash A= (B/b) \times_{B} A$ defined by the pullback square

$\xymatrix{ A/b \ar[d] \ar[r] & A\ar[d]^u\\ B/b\ar[r] & B. }$

is connected for every object $b\in B$. We shall say that a Grothendieck opfibration is a 1-opfibration if its fibers are groupoids. The category $\mathbf{Cat}$ 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 $u:A\to B$ is 1-connected if the map of simplicial sets $N(u):N(A)\to N(B)$ is 1-connected, where $N:\mathbf{Cat} \to \mathbf{SSet}$ is the nerve functor. We shall say that a Grothendieck bifibration is a 1-bifibration if its fibers are groupoids. The category $\mathbf{Cat}$ 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 $\Sigma$ be the set of inclusions $h^k_n: \Lambda^k[n] \subset \Delta[n]$ ($n\gt 0, 0\le k\le n$) in the category of simplicial sets. A map of simplicial sets is a Kan fibration if it belongs to $\Sigma^\pitchfork$, and a map is anodyne if it belongs to $Sat(\Sigma)$ ($=\Sat^\omega(\Sigma)$). The category of simplicial sets $\mathbf{SSet}$ 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 $\Sigma$ be the set of inclusions $h^k_n: \Lambda^k[n] \subset \Delta[n]$ ($0\lt k\le n$) in the category of simplicial sets. A map of simplicial sets is a right fibration if it belongs to $\Sigma^\pitchfork$, and a map is right anodyne if it belongs to $Sat(\Sigma)$ ($=\Sat^\omega(\Sigma)$). The category of simplicial sets $\mathbf{SSet}$ 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 $\Sigma$ be the set of inclusions $h^k_n: \Lambda^k[n] \subset \Delta[n]$ ($0\le k\lt n$) in the category of simplicial sets. A map of simplicial sets is a left fibration if it belongs to $\Sigma^\pitchfork$, and a map is left anodyne if it belongs to $Sat(\Sigma)$ ($=\Sat^\omega(\Sigma)$). The category of simplicial sets $\mathbf{SSet}$ 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 $\Sigma$ be the set of inclusions $h^k_n: \Lambda^k[n] \subset \Delta[n]$ ($0\lt k\lt n$) in the category of simplicial sets. A map of simplicial sets is a mid fibration if it belongs to $\Sigma^\pitchfork$, and a map is mid anodyne if it belongs to $Sat(\Sigma)$ ($=\Sat^\omega(\Sigma)$). The category of simplicial sets $\mathbf{SSet}$ 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 $\mathbf{E}$ is a triple $(\mathcal{C},\mathcal{W},\mathcal{F})$ of classes of maps in $\mathbf{E}$ satisfying the following two axioms: * the class $\mathcal{W}$ has the three-for-two property; * The pairs $( \mathcal{C}\cap \mathcal{W},\mathcal{F})$ and $(\mathcal{C},\mathcal{W}\cap \mathcal{F})$ 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 $X$ in the topos is said to be injective if the map $X\to 1$ is a trivial fibration.

###### Example

If $\mathbf{C}$ is a category with pullbacks, then to every weak factorisation system $(\mathcal{L},\mathcal{R})$ in $\mathbf{C}$ is associated a weak factorisation system $(\mathcal{L}',\mathcal{R}')$ in the category $[I,\mathbf{C}]$ (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 $[I,\mathbf{Set}]$.

## Exercises

###### Exercise

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

###### Exercise

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

## References

Papers:

• Adamek, J., Herrlich, H., Rosicky, J., Tholen, W.: Weak factorisation systems and topological functors. Appl. Categorical Structures 10 (2002) 237-249.

• Adamek, J., Herrlich, H., Rosicky, J., Tholen, W.: On a generalised small-objects argument for the injective subcategory problem. Cah. Topol. Géom. Différ. Catég. 43(2), 83-106 (2002)

• Bousfield, A.K.: Constructions of factorization systems in categories. J. Pure and Applied Algebra 9 (2-3), 207-220 (1977)

• Freyd, P.J., Kelley, G.M.: Categories of continuous functors. I. J. Pure Appl. Algebra 2, 169-191 (1972)

• Garner, R.: Understanding the small objects argument. Applied Categorical Structure.(pdf)

• Grandis, M., Tholen, W.: Natural weak factorisation systems. Arch. Math. 42, 397-408 (2006)(website)

• Kelly, G.M.: A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on. Bull. Austral. Math. Soc. 22(1), 1-83 (1980)

Lecture Notes and Textbooks:

• Gabriel, P., Ulmer, F.: Lokal präsentierbare Kategorien. Lecture Notes in Mathematics, vol.221. Springer-Verlag, Berlin (1971)

• Gabriel, P., Zisman, M.: Calculus of fractions and homotopy theories. Ergeb. der Math. undihrer Grenzgebiete, vol 35, Springer-Verlag, New-York (1967)

• Hirschhorn, Philip S.: Model categories and their localization. AMS Math. Survey and Monographs Vol 99 (2002)

• Hovey, Mark: Model categories. AMS Math. Survey and Monographs Vol 63 (1999)

• Quillen, Daniel: Homotopical algebra Lecture Notes in Mathematics, vol. 43. Springer Verlag, Berlin (1967)

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