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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
has a diagonal filler $d:B\to X$,
We shall denote this relation by $u\,\pitchfork\, f$. Notice that the condition $u\,\pitchfork\, f$ means that the following commutative square $Sq(u,f)$,
is epicartesian, or equivalently if the map
is surjective.
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.
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.
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 $\{0\}\subset J$.
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 .
Every factorisation system is a weak factorisation system by the theorem here.
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.
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 $\mathcal{M}$.
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 $B\backslash \mathbf{E}$.
Left to the reader.
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
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.
The following conditions on a morphism $f:X\to Y$ in a category $\mathbf{E}$ are equivalent:
($1\Rightarrow 2$) If $f$ is invertible, then the square
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
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
and such that $p i=1_A$ and $q j=1_B$.
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,
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,
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.
We shall say that a class of maps $\mathcal{M}$ in a category $\mathbf{E}$ is closed under coproducts if the coproduct
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.
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.
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 $Sq(v u,f)$
can be obtained by composing horizontally the squares $Sq(u,f)$ and $Sq(v,f)$,
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
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,
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.
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.
This follows from Lemma 2 and Lemma 1 since $\mathcal{R}=\mathcal{L}^\pitchfork$ and $\mathcal{L}={}^\pitchfork\mathcal{R}$.
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.
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.
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:
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
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 $\mathbf{E}^{[1]}$.
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.
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.
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
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.
The class $\mathcal{M}^\pitchfork$ is closed under transfinite op-compositions for any class of maps $\mathcal{M}$ in a complete category $\mathbf{E}$.
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 $Sq(u,c(j,i))$,
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.
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,
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 $\Sigma$.
We shall see in Proposition 4 below that the left class of a weak factorisation system in a cocomplete category is saturated.
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.
The class of split monomorphisms in any cocomplete category is saturated. The class of monomorphisms in a Grothendieck topos is saturated.
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.
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.
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.
A cellular class of maps is closed under coproducts.
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
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
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
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
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
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 $i\in I$,
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 $u\in \mathcal{M}$.
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,
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 $\Sigma$.
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.
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.
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
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
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 $\alpha$.
(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
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.
We first explain the rough idea of proof in the case $\alpha=\omega$. We begin by constructing a functor
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
The object $R(X)$ is then taken to be the colimit of the infinite sequence,
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
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
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
For every map $x:A\to X$, the composite of the squares
is a square
We now give a full proof in the case $\alpha=\omega$. For every object $X\in \mathbf{E}$ let us put
where $s(\sigma)$ is the source of the map $\sigma$. This defines a functor $S:\mathbf{E}\to \mathbf{E}.$ The counits
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
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
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
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
is a square
The colimit $R(X)$ of the infinite sequence
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 $\alpha=\omega$.
Let us now consider the case where $\alpha\gt \omega$. The sequence
can be extended cocontinuously through all the ordinals $\leq \alpha$ by putting
for every limit ordinal $j \le \alpha$ and by putting $F^{j+1}(X)=F(F^{j}(X))$ and
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
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
which associates to a composable pair $A\to B\to C$ its composite $A\to B$. We shall say that a section
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
to a commutative diagram,
Moreover, we have $F(v,v')F(u,u')=F(v u,v'u')$ for any pair of composable squares,
(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
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.
We shall use Proposition 6. For any map $u:A\to B$ in $\mathbf{E}$, let us denote by $\lambda(u)$ the square
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
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
together with a natural transformation $\rho:Id\to R$. This yields a commutative square
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 $\mathbf{E}^{[1]}$
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.
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 $\Sigma$.
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 $Cell^{\alpha}(\Sigma)$.
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
has a diagonal filler $s:B\to E$, since we have $u\pitchfork r$. This shows that $u$ is a codomain retract of $v$.
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.
This follows from theorem 1, since every object of a locally presentable category is small.
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
are equivalent.
The adjunction $\theta:F\dashv G$ induces a bijection between the following commutative squares and their diagonal fillers,
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
are equivalent.
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 $\mathcal{R}=\mathcal{L}^\pitchfork$.
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.
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 $in_1:A\to A\sqcup B$.
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.
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.
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.
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.
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
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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 $[I,\mathbf{Set}]$.
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 $\mathbf{E}'$.
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).
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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)
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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)