We shall say that a pair of classes of maps in a category is a factorisation system if the following conditions are satisfied:
every morphism admits a factorisation with and and this factorisation is unique up to unique isomorphism;
The classes and contain the isomorphisms and are closed under composition.
The class is called the left class and the class the right class of the factorisation system. We shall say that a factorisation with and is a -factorisation of the morphism . The uniqueness condition in the definition means that for any pair of -factorisations and of the same morphism, there exists a unique isomorphism such that the following diagram commutes,
If is a factorisation system in a category , then the pair is a factorisation system in the opposite category .
If is an object of a category and is a class of maps, 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 factorisation system in a category , then the pair is a factorisation system in the category for any object in . Dually, the pair is a factorisation system in the category .
Left to the reader.
If is the class of isomorphisms of a category and if is the class of all maps, then the pairs and are trivial examples of factorisation systems.
The category of sets admits a factorisation system , where the class surjections and is the class of injections.
The category of groups admits a factorisation system , where the class of surjective homomorphisms and is the class of injective homomorphisms. More generally, this is true for the category of models of any algebraic theory?.
Let be the category of commutative rings. We shall say that a ring homomorphism inverts an element if is invertible in . We shall say that the morphism is conservative if every element which is inverted by is invertible in . For any subset , there is a commutative ring together with a ring homomorphism which inverts universally every elements of . The universality means that for any ring homomorphism which inverts every element of there exists a unique homomorphism such that . Every homorphism admits a canonical factorisation , where is the set of elements inverted by . The homomorphism is alaways conservative; we shall say that is a localisation if is an isomorphism. The category admits a factorisation system in which is the class of localisations and is the class of conservative homomorphisms.
For more examples of factorisation systems in algebra, see Example
We shall say that a class of maps in a category has the left cancellation property if the implication
is true for any pair of maps and . Dually, we shall say that has the right cancellation property if the implication
is true.
The intersection of the classes of a factorisation system in a category is the class of isomorphisms. Moreover, the left class has the right cancellation property and the right class has the left cancellation property.
Let be a map in . The trivial factorisations and are both -factorisations, since the classes and contain the units. It follows by uniqueness that there exists an isomorphism such that and , This shows that is invertible. Let us prove that the left class has the right cancellation property. Let and be two maps. If and belong to , let us show that belongs to . For this, let us choose a -factorisation . We then have two -factorisations and of the composite ,
Hence there exists an isomorphism such that and . This shows that is invertible, and hence , since every isomorphism belongs to . It follows that , since is closed under composition.
Let be the arrow category of a category . Then a morphism in is a commutative square of maps in , If has pullbacks, then admits a factorisation system in which is the class of pullback squares. A square belongs to iff the map is invertible. See Example . Suppose that we have a commutative diagram in in which the right hand square is cartesian. Then the left hand square is cartesian iff the composite square is cartesian by Proposition . See the page on cartesian squares.
We shall say that a map in a category is left orthogonal to a map , or that is right orthogonal to , if every commutative square has a unique diagonal filler ( and ), We shall denote this relation by . Notice that the condition means that the following square is cartesian in the category of sets. If and are two classes of maps in , we shall write to indicate that we have for every and .
If is a class of maps in a category , we shall denote by (resp. ) the class of maps left (resp. right) orthogonal to every map in . We shall say that is the left orthogonal complement of , and that is its right orthogonal complement.
If and are two classes of maps in , then the conditions
are equivalent. The operations and on classes of maps are contravariant and mutually adjoint. It follows that the operations and are closure operators.
In the category , a functor is fully faithful iff it is right orthogonal to the inclusion , where denotes the discrete category with two objects and .
If is the groupoid generated by one isomorphism , then a functor is conservative iff it is right orthogonal to the inclusion .
A functor is a discrete Conduché fibration? if it is right orthogonal to the inclusion . This condition means that for every morphism in and every factorisation of the morphism , there exists a unique factorisation of the morphism such that and . Discrete fibrations and a discrete opfibrations are examples discrete Conduché fibrations.
Let be a class of maps in a category . Then the class is closed under limits, composition, base changes and it has the left cancellation property. Dually, the class is closed under colimits, composition, cobase changes and it has the right cancellation property.
Let us show that the class is closed under limits. We shall use the fact that the functor preserves limits for any map in . Let us suppose that a map in is the limit of a diagrams of maps . Let us put for every object . Then the square is the limit of the diagram of squares for . If belongs to for every , let us show that belongs to . The assumption means that the square is cartesian for every map . Hence also the limit square , since the category of cartesian squares is a full reflexive subcategory of the category of all squares by here. This proves that . Let us now prove that the class is closed under composition and that it has the right cancellation property. Let and be two maps in . If belongs to let us show that . For any morphism , the square is the composite of the squares and , The square is cartesian for every , since . It follows from the lemma here that the square is cartesian iff the square is cartesian. Thus, . The remaining properties can be proved similarly, see the proposition here.
Recall that a class of objects in a category is said to be replete if every object isomorphic to an object of 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 .
A pair of classes of maps in a category is a factorisation system iff the following three conditions are satisfied:
every map admits a -factorisation ;
the classes and are replete;
.
Moreover, in this case the pair is a weak factorisation system and we have
() If is a factorisation system, let us prove that we have . If belongs to and belongs to , let us show that every commutative square
has a unique diagonal filler. For this, let us choose two -factorisations and . Then from the commutative diagram we obtain two -factorisations of the same map ,
It follows that there exists a unique isomorphism such that and . Hence the following diagram commutes, and the composite is a diagonal filler of the square (1). It remains to prove the uniqueness of . If is another diagonal filler of the same square, let us choose a -factorisation . From the commutative diagram we can construct two commutative diagrams, where the first is representing two -factorisations of a map and the second two -factorisations of a map . Hence there exists a unique isomorphism such that and and unique isomorphism such that and .
It follows from these relations that the following diagram commutes, Hence also the diagram The uniqueness of the isomorphism between two -factorisations implies that we have , where is the isomorphism in the diagram (2). Thus, . The relation is proved. () If the three conditions are satisfied, let us show that the pair is a factorisation system. We shall first prove that it is a weak factorisation system by showing that we have,
We have since we have by assumption. Obviously, . Let us then show that . If belongs to , let us choose a factorisation with and . Then the square has a diagonal filler , since and . The relations and implies that the map is a diagonal filler of the square But this square has a unique diagonal filler, since we have . It follows that . Thus, is invertible since . It follows , since the class is replete. The eqality is proved. The equality follows by duality. It follows that the pair is a weak factorisation system. Hence the classes and contain the isomorphisms and they are closed under composition by the proposition here. It remains to prove the uniqueness of the -factorisation of a map . Suppose then that we have two -factorisations, and . Then each of the following squares has a unique diagonal filler, respectively and . The composite is then a diagonal filler of the square It follows that we have by uniqueness of a diagonal filler. Similarly, we have . This shows that is invertible.
A weak factorisation system is a factorisation system iff we have .
The implication () follows from Theorem . Conversely, let be a weak factorisation system for which we have . The classes and are replete, since they are closed under composition and they contain the isomorphisms by the proposition here. It then follows from Theorem Theorem that the pair is a factorisation system.
A factorisation system is determined by any one of its two classes. The class is closed under limits, composition, base changes and it has the left cancellation property. Dually, the class is closed under colimits, composition, cobase changes and it has the right cancellation property.
Let be a factorisation system in a category . Then the full subcategory of spanned by the arrows in is coreflective, and the full subcategory spanned by the arrows in is reflective. Hence the left class of a factorisation system is closed under colimits in the category and the right class is closed under limits.
Let us denote by the full subcategory of spanned by the arrows in . Every map admits a -factorisation . The pair is a morphism in the category , Let us show that the morphism is reflecting the arrow in the subcategory . For this, it suffices to show that for every arrow in and every commutative square there exists a unique arrow such that and , But this is clear, since the square has a unique diagonal filler by Theorem .
The diagonal of a map in a category with pullbacks is defined to be the map in the commutative diagram,
Dually, the codiagonal of a map in a category with pushouts is defined to be the map in the commutative diagram
If the diagonal of a map in a category exists, then the condition is equivalent to the conjunction of the conditions and for any map .
If , let us show that we have and . Obviously, it suffices to show that we have . For this, let us show that every commutative square
has a diagonal filler. We have , since the following diagram commutes, Let us put . We have , since the square (3) commutes. This shows that the maps are both filling the diagonal of the following square,
Thus, since we have by assumption. The map is then filling the diagonal of the square (3). The relation is proved. Conversely, if and , let us show that we have . For this, it suffices to prove the uniqueness of a diagonal filler of a square (4), since the existence follows from the condition . Suppose then that we have two maps filling the diagonal of the square (4). The relation implies that we can define a map . Moreover, the square (3) commutes, since . The square (3) has then a diagonal filler , since we have by assumption. The relation implies that , since .
(Dual to Lemma ). If the codiagonal of a map in a category exists, then the condition is equivalent to the conjunction of the conditions and for any map .
If the category has pullbacks, we shall say that a class of maps in is closed under diagonals if the implication is true. Dually, if the category has pushouts, we shall say that is closed under codiagonals if the implication is true.
In a category with pullbacks, a weak factorisation system is a factorisation system iff the class is closed under diagonals iff it has the left cancellation property
The implication (1)(3) was proved in Proposition . Let us prove the implication (3)(2). If a map belongs to then so is the first projection , since the right class of a weak factorisation system is closed under base change by the proposition here. But we have , and it follows that belongs to , since the class has the left cancellation property by assumption. Let us prove the implication (2)(1). For this, it suffices to show that we have by Theorem . But if and , then we have and , since the class is closed under diagonals by assumption. It then follows by Lemma that we have .
(Dual to Proposition ). In a category with pushouts, a weak factorisation system is a factorisation system iff the class it is closed under codiagonals iff it has the right cancellation property.
In a finitely bicomplete category, a weak factorisation system is a factorisation system iff the class is closed under diagonals iff the class is closed under codiagonals.
If is an epimorphism, then the conditions and are equivalent for any map .
If , let us show that . For this we have to show that every commutative square has a unique diagonal filler. The existence is clear since we have by hypothesis. Let us prove the uniquess. But if are two diagonal fillers of the square, then we have . Thus, , since is an epimorphism.
Let be a set of maps in a category with pushout . Then .
Recall from here that class of maps in a cocomplete category is said to be saturated if it satisfies the following conditions: * contains the isomorphisms and is closed under composition and transfinite compositions; * is closed under cobase changes; * is closed under retracts.
Let be a set of maps between small objects in a cocomplete category . Then the pair
is a factorisation system.
The codiagonal of a map between small objects is a map between small objects. Thus, is a set of maps between small objects. It then follows from the theorem here that the pair is a weak factorisation system. But we have by Lemma . Hence it remains to show that the pair is a factorisation system. For this, it suffices to show that we have by Theorem . But we have
since . Thus,
by Lemma . It follows that , since the class is saturated by Proposition . This proves that , and hence that the pair is a factorisation system.
Let be a set of maps in a locally presentable category? . Then the pair
is a factorisation system.
Recall that if is a monomorphism of commutative ring, then an element is said to be integral over if it is the root of a monic polynomial . We shall say that a monomorphism of commutative rings is integrally closed if every element integral over belongs to . The category of commutative rings admits a factorisation system in which is the class of integrally closed monomorphisms. We shall say that an homomorphism in the class is an integral homomorphism.
If is a commutative ring, we shall say that an element is a simple root of a polynomial if and is invertible. Let us denote by the set of simple roots in of a polynomial . We shall say that a ring homomorphism is separably closed if it induces a bijection for every polynomial . For example, a local ring with maximal ideal is henselien iff the quotient map is separably closed. The category of commutative rings admits a factorisation system in which is the class of separably closed homomorphisms. We shall say that an homomorphism in the class is a separable algebraic extension. We conjecture that a ring homormorphism is a separable algebraic extension iff it is formally etale?.
Let be a factorisation system in the category . We shall say that a functor is essentially in (resp ) if the functor (resp. ) of an -factorisation is an equivalence of categories.
Recall that a functor is said to be full (resp. faithful, fully faithful) if the map induced by is surjective (resp. injective, bijective) for every pair of objects à . We shall say that a functor is monic (resp. surjective, bijective) on objects if the induced map is injective (resp. surjective, bijective). The category admits a factorisation system in which the class of full functors bijective on objects and is the class of faithful functors.
The category admits a factorisation system in which the class of functors bijective on objects and is the class of fully faithful functors. A -factorisation of a functor is the Gabriel factorisation constructed as follows: and for every pair . The composition law is obvious. The functors and are induced by . A functor is essentially in iff it is essentially surjective.
We shall say that a fully faithful functor is replete if every object of which is isomorphic to to an object in the image of also belongs to this image. The category admits a factorisation system in which the class of essentialy surjective functors and is the class of replete fully faithful functors monic on objects. A functor is essentially in iff it is fully faithful.
The category of small categories admits a factorisation system in which the class of functors surjective on objects and is the class of fully faithful functors monic on objects. A functor is essentially in iff it is essentially surjective, and a functor is essentially in iff it is fully faithful.
(Street and Walters) Recall that a functor between small categories is said to be a discrete fibration if for every object and every arrow with target , there exists a unique arrow with target such that . A functor is a discrete fibration iff it is right orthogonal to the map . The category admits a factorisation system in which is the class of discrete fibrations and is the class of final functors. Recall that a functor between small categories is final (but we shall eventually say 0-final) iff the category
defined by the pullback square is connected for every object .
A functor is called a discrete op-fibration if the opposite functor is a discrete fibration. A functor is a discrete opfibration iff for every object and every arrow with source , there exists a unique arrow with source such that . A functor is a discrete opfibration iff it is right orthogonal to the map . The category admits a factorisation system in which is the class of discrete opfibrations and is the class of initial functors. Recall that a functor between small categories is said to be initial (but we shall eventually say 0-initial) if the opposite functor is final. A functor is initial iff the category defined by the pullback square is connected for every object .
A functor between groupoids is a discrete fibration iff it is a discrete opfibration, in which case we shall say that it is a covering. We shall say that a functor between groupoids is connected if it is essentially surjective and full. A functor between groupoids is connected iff it is final iff it is initial. The category of small groupoids admits a factorisation system in which is the class of coverings and is the class of connected functors. A functor between groupoids is essentially a covering iff it is faithful.
We shall say that a functor in is a discrete bifibration, or a 0-covering, if it is both a discrete fibration and a discrete opfibration. If is the category of groupoids, then the inclusion functor admits a left adjoint
which associates to a category its fundamental groupoid? . By construction, is obtained by inverting universally every arrow in . We shall say that a functor is 0-connected if the functor is connected. See Example . The category admits a factorisation system in which the class of connected functors and is the class of 0-coverings.
We shall say that a functor inverts an arrow if the arrow is invertible in . A functor is conservative iff every arrow which is inverted by is invertible in . For any subset of arrows in small category , there is a small category together with a functor which inverts universally every arrow in . The universality means that for any functor which inverts every arrow in there exists a unique functor such that . Every functor admits a canonical factorisation , where is the set of arrows inverted by . We shall say that is a localisation if is an isomorphism. Beware that the functor is not conservative in general. Hence the factorisation can be repeated with instead of . Let us put , and let be the set of arrows in inverted by . We then obtain a factorisation where is the canonical functor. Let us put . By iterating this process, we obtain an infinite sequence of categories and functors, The category in this diagram is defined to be the colimit of the sequence of categories , and the functor to be the colimit of the functor . It is easy to see that functor is conservative. We shall say that is an iterated localisation if is an isomorphism. The category admits a factorisation system in which is the class of iterated localisations and is the class of conservative functors.
We saw above that every functor admits a canonical factorisation , where is the set of arrows inverted by . We shall say that the is essentially a localisation if is an equivalence. We saw also that admits a factorisation , where is an iterated localisation and is conservative. We shall say that is essentially an iterated localisation if is an equivalence.
We shall say that a map of simplicial sets is discrete if it is right orthogonal to every surjection . Every monomorphism is discrete. The category of simplicial sets admits a factorisation system in which is the class of discrete maps. We shall say that a map in is a collapse.
We shall say that a map of simplicial sets is a discrete right fibration? if it is right orthogonal to the maps with . The category of simplicial sets admits a factorisation system in which is the class of discrete right fibrations. We shall say that a map in is 0-final.
We shall say that a map of simplicial sets is a discrete left fibration if it is right orthogonal to the maps with . The category of simplicial sets admits a factorisation system in which is the class of discrete left fibrations. We shall say that a map in is 0-initial.
We shall say that a map of simplicial sets is a discrete Kan fibration if it is right orthogonal to every map . The category of simplicial sets admits a factorisation system in which is the class of discrete Kan fibrations. We shall say that a map in is 0-connected.
We shall say that a map between two presheaves on a category is etale if the naturality square is cartesian for every morphism in . Then the category admits a factorisation system in which is the class of etale morphisms. We shall say that a morphism in the class is connected.
If is a Grothendieck fibration, then the category admits a factorisation system in which is the class of cartesian morphisms and is the class of morphisms inverted by .
Dually, if is a Grothendieck opfibration, then the category admits a factorisation system in which is the class of cocartesian morphisms and is the class of morphisms inverted by .
If is a category with pullbacks, then the target functor which associates to a map its target is a Grothedieck fibration. A morphism in the category is cartesian with respect to iff the corresponding square is cartesian in . It then follows from Example that the category admits a factorisation system in which is the class of pullback squares. A square belongs to iff the map is invertible.
Let be a discrete Conduché fibration (see Example ). Recall that this means that for every morphism in and every factorisation of the morphism , there exists a unique factorisation of the morphism such that and . Discrete fibrations and a discrete opfibrations are examples of discrete Conduché fibrations. If is a class of maps in , let us denote by the class of maps in . Show that if is a factorisation system in the category , then the pair is a factorisation system in the category . In particular, let us denote by the category of elements of a presheaf on a category . If is a class of maps in , let us denote by the class of maps in whose underlying map in belongs to . Deduce that if is a factorisation system in , then the pair is a factorisation system in the category . If for some object , then .
Papers:
Bousfield, A.K.: Constructions of factorization systems in categories. J. Pure and Applied Algebra 9 (2-3), 207-220 (1977) 287-329.
Cassidy, C., Hebert, M., Kelly, G.M.:_Reflexive subcategories, localisations and factorisation systems_. J. Australian Math. Soc.(Series A) 38(1985)
Freyd, P.J., Kelly, G.M.: Categories of continuous functors. I. J. Pure Appl. Algebra 2, 169-191 (1972)
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)
Korostenski, M., Tholen, W.: Factorization systems as Eilenberg-Moore algebras. J. Pure and Appl. Algebra 85 (1993) 57-72.
Pultr, A, Tholen, W.: Free Quillen factorization systems. Georgian Mathematical Journal, Volume 9 (2002), Number 4, 805-818.
Rosicky, J., Tholen, W.: Factorization, fibration and torsion. J. Homotopy Theory and Related Structures. to appear (here)
Tholen, W.: Factorisation, localisation and the orthogonal subcategory problem. Math. Nachr. 114 (1983) 63-85.
Wood, R.J., Rosebrugh, R.: Coherence for factorization algebras. Theory and Applications of Categories 10 (2002) 134-147 (TAC)
Lecture Notes and Textbooks: