We shall say that a map in a category has the left lifting property with respect to a map , or that has the right lifting property with respect to , if every commutative square
has a diagonal filler ,
We shall denote this relation by . Notice that the condition means that the following commutative square ,
is epicartesian, or equivalently if the map
is surjective.
If is a class of maps, we shall denote by (resp. ) the class of maps having the left (resp. the right) lifting property with respect to every map in . We shall say that is the left complement of , and that 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 for every and . A simplicial set is a Kan complex? iff the map is a Kan fibration.
Let be the groupoid generated by one isomorphism . Then a functor in the category is an isofibration? iff it has the right lifting property with respect to the inclusion .
We shall say that a pair of classes of maps in a category is a weak factorisation system if the following conditions are satisfied:
every map admits a factorisation with and ;
and ;
We shall say that a factorisation with and is a -factorisation of the map . The class is called the left class of the system, and the class is called the right class .
Every factorisation system is a weak factorisation system by the theorem here.
The category of sets admits a weak factorisation system , where the class of injections and is the class of surjections.
For more examples of weak factorisation systems, go to Example .
If is a weak factorisation system in a category , then the pair is a weak factorisation system in the opposite category .
If is a class of maps in a category , then for any object we shall denote by the class of maps in the slice category whose underlying map in belongs to . Dually, we shall denote by the class of maps in the coslice category whose underlying map belongs to .
If is a weak factorisation system in a category , then the pair is a weak factorisation system in the slice category for any object in . Dually, the pair is a weak factorisation system in the coslice category .
Left to the reader.
If and are two classes of maps in , we shall write to indicate that we have for every and . The three conditions
are equivalent. If , we shall write instead of . Similarly, we shall write instead of .
The operations and on classes of maps are contravariant and mutually adjoint. It follows that the operations and are closure operators.
The following conditions on a morphism in a category are equivalent:
() If is invertible, then the square
has a diagonal filler . Thus, for any arrow , and hence . () If , then . () If , then the square
has a diagonal filler and this shows that is invertible. The equivalences () are proved. The equivalences () are proved similarly.
Recall that a map in a category is said to be a retract of another map , if is a retract of in the category of arrows . The condition means that there exists four maps fitting in a commutative diagram
and such that and .
We shall say that a class of maps in a category is closed under retracts if every retract of a map in belongs to .
We recall that the base change of a map along a map is the map in a pullback square,
Dually, the cobase change of a map along a map is the map in a pushout square,
We shall say that a class of maps in a category is closed under base changes if the base change of a map in belongs to , 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 in a category is closed under coproducts if the coproduct
of any family of maps in belongs to , when this coproduct exists. The notion of a class of maps closed under products is defined similarly.
Let be a class of maps in a category . Then the class contains the isomorphisms and is closed under composition, retracts, products, and base changes. Dually, the class is contains the isomorphisms and is closed under composition, retracts, coproducts, and cobase changes.
That class contains the isomorphisms by Scholie . Let us show that it is closed under composition. We shall use the properties of epicartesian squares. Let us show that if two morphisms and belongs to , then so does their composite . For any morphism , the square
can be obtained by composing horizontally the squares and ,
The squares and are epicartesian if ; hence their composite is epicartesian by the lemma here. This shows that . We have proved that the class is closed under composition. Let us now show that the class is closed under retracts. If a map is a retract of a map , then the square is a retract of the square for any map . But a retract of an epicartesian square is epicartesian by the lemma here. It follows that the class is closed under retracts. Let us show that the class is closed under products. If a map is the product of a family of maps (), then the square is the product of the family of squares . But the product of a family of epicartesian squares is epicartesians by the lemma here. This shows that the class is closed under products. Let us show that the class is closed under base changes. Suppose that we have a pullback square
with and let us prove that . It suffices to show that the square is epicartesian for every morphism in . 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 and preserve limits. The front face is epicartesian since we have . Hence the back face is epicartesian by the cube lemma here.
The two classes of a weak factorisation system contain the isomorphisms and they are closed under composition and retracts. The right class is closed under base changes and products, and the left class under cobase changes and coproducts. The intersection is the class of isomorphisms.
Recall that a map in a category is said to be a domain retract of a map , if the object of the category is a retract of the object . There is a dual notion of codomain retract.
We shall say that a class of maps in a category is closed under domain retracts if every domain retract of a map in belongs to . The notion of a class closed under codomain retract is defined similarly.
A pair of classes of maps in a category is a weak factorisation system iff the following conditions are satisfied:
The implication () is clear since the classes of a weak factorisation system are closed under retracts by Proposition . Let us prove the implication (). We have since we have by the hypothesis. Let us show that we have . If a map belongs to , let us choose a -factorisation . The square
has a diagonal filler since we have . Hence we have and and this shows that is a domain retract of . Thus, , since is closed under domain retracts by hypothesis.
Recall that a class of objects in a category is said to be replete if every object isomorphic to an object in belongs to . We shall say that a class of maps in is replete, if it is replete as a class of objects of the category .
Suppose that a pair of classes of maps in a category satisfies the following three conditions:
every map admits a -factorisation ;
;
the classes and are replete.
If denotes the class of maps which are codomain retracts of maps in and denotes the class of maps which are domain retracts of maps in , then the pair is a weak factorisation system.
The condition implies the condition by Lemma . It is easy to see that is closed under codomain retracts, and that is closed under domain retracts. The result then follows from Theorem .
For any ordinal , let us put and . Let be a cocomplete category. We shall say that a functor is a chain of lentgth , or an -chain. The composite of is defined to be the canonical map . The base of is the restriction of to . The chain is cocontinuous if the canonical map
is an isomorphism for every non-zero limit ordinal . We shall say that a subcategory is closed under transfinite compositions if for any limit ordinal any cocontinuous chain with a base in has a composite in .
Dually, if is a complete category, and is an ordinal, we shall say that a contravariant functor is an opchain. The opchain is continuous if the corresponding chain is cocontinuous. We shall say that a subcategory is closed under transfinite op-compositions if the opposite subcategory is closed under transfinite compositions.
The class is closed under transfinite op-compositions for any class of maps in a complete category .
Let us show that if is a limit ordinal then every continuous op-chain with a base in has its composite in . Let us denote by the transition map defined for . For any morphism , let us denote by the contravariant functor obtained by putting for . By definition, the functor takes a pair to the square ,
Beware that here the square 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 is continuous, since is continuous. If , then the square is epicartesian for every by the assumption on . It follows that the square is epicartesian by the lemma here. This show that belongs to , and hence that is closed under transfinite op-compositions.
We shall say that a class of maps in a cocomplete category is cellular if it satisfies the following conditions: * contains the isomorphisms and is closed under composition, * is closed under transfinite compositions; * is closed under cobase changes.
We shall say that is saturated if in addition,
Every class of maps is contained in a smallest cellular class called the cellular class generated by . Similarly, is contained in a smallest saturated class called the saturated class generated by .
We shall see in Proposition 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 ().
For more examples of saturated classes of the form , go to Example .
The class is saturated for any class of maps in a cocomplete category . In particular, the left class of a weak factorisation system in a cocomplete category is saturated.
The class contains the isomorphisms and it is closed under composition, retracts and cobase changes by . And it is closed under transfinite compositions by Lemma dualised.
A cellular class of maps is closed under coproducts.
Let be a cellular class of maps in a cocomplete category . We shall say that an object cofibrant, if the map belongs to , where is the initial object of . 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 belongs to , since contains the isomorphisms. Hence the object 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 and are cofibrant, consider the pushout square
The map is a cobase change of the map . Thus, , since is cofibrant and is closed under cobase change. The map also belongs to , since is cofibrant. Hence the composite belongs to , since is closed under composition. This shows that is cofibrant. Let us now show that the coproduct
of an infinite family of cofibrant objects is cofibrant. We shall argue by induction on the ordinal . If , let us put
There is an obvious canonical map for and this defines a cocontinuous chain . Notice that and . Hence we can prove that is cofibrant by showing that the composite of belongs to . For this it suffices to show that the base of belongs to , since is closed under transfinite compositions. But the object
is cofibrant for every by the induction hypothesis, since . And the transition map is a base change of the map since we have a pushout square
This shows that the transition map belongs to for every . We have proved that the base of belongs to and hence that the object is cofibrant. Let us now show that the class is closed under coproducts. For this, let us show that the coproduct of a family of maps () in belongs to . For this, let us denote by the class of maps in the category whose underlying map in belongs to . It is easy to verify that the class satisfies the hypothesis of the proposition. Let us put for each ,
The object of is the coproduct of the family of objects for . The map belongs to , since by assumption, and since the class is closed under cobase change. Hence the object of the category is cofibrant with respect to the class . It follows that the object is cofibrant by the first part of the proof. This proves that .
If is a regular cardinal, we shall say that a class of maps in a complete category is -cellular if it satisfies the following conditions: * contains the isomorphisms and is closed under composition; * is closed under transfinite compositions of cocontinuous chains of length ; * is closed under cobase changes; * is closed under coproducts.
We shall say that an -cellular class is -saturated if in addition,
Every class of maps is contained in a smallest -cellular class called the -cellular class generated by . Similarly, is contained in a smallest -saturated class called the -saturated class generated by .
If is the set of inclusions () in the category (of simplicial sets), then is the class of monomorphisms.
If is the set of inclusions for and , then is the class of anodyne maps.
If is a map in a category , we shall say that an object in is -fibrant if the map
is surjective. More generally, if is a class of maps in , we shall say that an object is -fibrant if it is -fibrant for every . When has a terminal object , then an object is -fibrant iff the map belongs to .
Recall that an object in a cocomplete category is said to be compact? if the functor
preserves directed colimits. More generally, if is a regular cardinal, then an object is said to be -compact? if the functor preserves -directed colimits. An object is said to be small? if it is -compact for some regular cardinal .
(Small object argument) Let be a set of maps in a cocomplete category . If the domains of the maps in are -compact, then there exists a functor
together with a natural transformation such that: * the object is -fibrant for every object ; * the map belongs to for every .
Moreover, the functor preserves -directed colimits.
We first explain the rough idea of proof in the case . We begin by constructing a functor
together with a natural transformation having the following properties: for every arrow in and every map , there exists a map fitting in a commutative square
The object is then taken to be the colimit of the infinite sequence,
where , and natural transformation is defined by the canonical map . The nice properties of are deduced from the nice properties of . Let us show that the object is -fibrant. If denotes the canonical map, then we have a commutative triangle
for every . The domain of every map in is compact by hypothesis. It follows that for every map , there exist an integer together with a map such that . But there is then a map fitting in a commutative square
If , then This shows that is -fibrant. Let us describe the construction of the functor in the case where consists of a single map . If is a set we shall denote by the coproduct of copies of . The functor is left adjoint to the functor . Let be the counit of the adjunction. By definition, we have for every , where is the inclusion indexed by . The object and the map are then defined by a pushout square
For every map , the composite of the squares
is a square
We now give a full proof in the case . For every object let us put
where is the source of the map . This defines a functor The counits
induces a map . This defines a natural transformation , where denotes the identity functor. By definition, if is a map in , then for every map we have , where is the inclusion indexed by . For every object let us put
where is the target of the map . This defines a functor The coproduct over of the maps
is a map . This defines a natural transformation Let us denote by the object defined by the pushout square
This defines a functor together with a natural transformation . Observe that for every map in and every map , the composite of the squares
is a square
The colimit of the infinite sequence
is -fibrant by the part 0 of the proof, where . Let us show that the canonical map belongs to . For this it suffices to show that the maps belong to , since an -cellular class is closed under -compositions. But is a cobase change of , and is a coproduct of maps in . This shows that belongs to by the closure preperties of this class of maps. It remains to show that the functor preserves directed colimits. The functor preserves directed colimits for any compact object . Hence, also the functor for any object , since the functor is cocontinuous. The functor is by construction a colimit of functors of the form , for compact objects . It follows that preserves directed colimits. This completes the proof of the proposition in the case where .
Let us now consider the case where . The sequence
can be extended cocontinuously through all the ordinals by putting
for every limit ordinal and by putting and
for every ordinal . Let us then put and let be the canonical map for . This defines a functor equipped with a natural transformation . Let us show that the object is -fibrant. For every map in and every map , there exist an ordinal together with a map such that , since the object is -compact. But there is then a map fitting in a commutative square
If , then This shows that is -fibrant. We leave to the reader the verification that belongs to , and the verification that the functor preserves -directed colimits.
If is a category, then an object of the category is a composable pair of maps in the category There is then a composition functor
which associates to a composable pair its composite . We shall say that a section
of the functor is a factorisation functor. It associates to a map a factorisation , and it takes a commutative square
to a commutative diagram,
Moreover, we have for any pair of composable squares,
(Functorial factorisation) Let be a set of maps in a cocomplete category . If the domain and codomain of every map in is -compact then there exists a factorisation functor
which associates to every morphism a factorisation with and . Moreover, the functor preserves -directed colimits.
We shall use Proposition . For any map in , let us denote by the square
viewed as a morphism in the category . If is a map in , then the condition exactly means that the map
is surjective and hence that is -fibrant. Hence a map belongs to iff it is -fibrant as an object of the category . It is easy to verify that domain and codomain of a map in are -compact, since this is true of the maps in . It then follows from Proposition that we can construct a functor
together with a natural transformation . This yields a commutative square