Joyal's CatLab
Factorisation systems

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

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Definition

Definition

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

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

  • The classes \mathcal{L} and \mathcal{R} contain the isomorphisms and are closed under composition.

The class \mathcal{L} is called the left class and the class \mathcal{R} the right class of the factorisation system. We shall say that a factorisation f=pu:AEBf=p u:A\to E\to B with uu\in \mathcal{L} and pp\in \mathcal{R} is a (,)(\mathcal{L},\mathcal{R})-factorisation of the morphism ff. The uniqueness condition in the definition means that for any pair of (,)(\mathcal{L},\mathcal{R})-factorisations f=pu:AEBf=p u:A\to E\to B and f=qv:AFBf=q v:A\to F\to B of the same morphism, there exists a unique isomorphism i:EFi:E\to F such that the following diagram commutes,

Duality

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

If BB is an object of a category E\mathbf{E} and \mathcal{M} is a class of maps, we shall denote by /B\mathcal{M}/B the class of maps in the slice category E/B\mathbf{E}/B whose underlying map (in E\mathbf{E}) belongs to \mathcal{M}. Dually, we shall denote by B\B\backslash \mathcal{M} the class of maps in the coslice category B\EB\backslash \mathbf{E} whose underlying map belongs to \mathcal{M}.

Slice and coslice

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

Proof

Left to the reader.

Example

If IsoIso is the class of isomorphisms of a category E\mathbf{E} and if MapMap is the class of all maps, then the pairs (Iso,Map)(Iso,Map) and (Map,Iso)(Map,Iso) are trivial examples of factorisation systems.

Example

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

Example

The category of groups Grp\mathbf{Grp} admits a factorisation system (Surj,Mono)(Surj,Mono), where SurjSurj the class of surjective homomorphisms and InjInj is the class of injective homomorphisms. More generally, this is true for the category of models of any algebraic theory.

Example

Let CRing\mathbf{CRing} be the category of commutative rings. We shall say that a ring homomorphism u:ABu:A\to B inverts an element fAf\in A if u(f)u(f) is invertible in BB. We shall say that the morphism u:ABu:A\to B is conservative if every element fAf\in A which is inverted by uu is invertible in AA. For any subset SAS\subseteq A, there is a commutative ring S 1AS^{-1}A together with a ring homomorphism l:AS 1Al:A\to S^{-1}A which inverts universally every elements of SS. The universality means that for any ring homomorphism u:ABu:A\to B which inverts every element of SS there exists a unique homomorphism u:S 1ABu':S^{-1}A\to B such that ul=uu' l=u.

Every homorphism u:ABu:A\to B admits a canonical factorisation u=ul:AS 1ABu=u' l:A\to S^{-1}A\to B, where SAS\subseteq A is the set of elements inverted by uu. The homomorphism uu' is alaways conservative; we shall say that uu is a localisation if uu' is an isomorphism. The category CRing\mathbf{CRing} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of localisations and \mathcal{R} is the class of conservative homomorphisms.

For more examples of factorisation systems in algebra, see Example 19

Cancellation properties

Definition

We shall say that a class of maps \mathcal{M} in a category has the left cancellation property if the implication

vuandvuv u\in \mathcal{M} \quad \mathrm{and} \quad v\in \mathcal{M} \Rightarrow u\in \mathcal{M}

is true for any pair of maps u:ABu:A\to B and v:BCv:B\to C. Dually, we shall say that \mathcal{M} has the right cancellation property if the implication

vuanduvv u\in \mathcal{M} \quad \mathrm{and} \quad u\in \mathcal{M} \Rightarrow v\in \mathcal{M}

is true.

Proposition

The intersection of the classes of a factorisation system (,)(\mathcal{L},\mathcal{R}) in a category E\mathbf{E} is the class of isomorphisms. Moreover, the left class \mathcal{L} has the right cancellation property and the right class \mathcal{R} has the left cancellation property.

Proof

Let f:ABf:A\to B be a map in \mathcal{L}\cap \mathcal{R}. The trivial factorisations f=f1 Af=f 1_A and f=1 Bff=1_B f are both (,)(\mathcal{L},\mathcal{R})-factorisations, since the classes \mathcal{L} and \mathcal{R} contain the units. It follows by uniqueness that there exists an isomorphism i:ABi:A\to B such that f=i1 Af=i 1_A and f=1 Bif=1_B i,

This shows that ff is invertible. Let us prove that the left class \mathcal{L} has the right cancellation property. Let u:ABu:A\to B and v:BCv:B\to C be two maps. If uu and vuv u belong to \mathcal{L}, let us show that vv belongs to \mathcal{L}. For this, let us choose a (,)(\mathcal{L},\mathcal{R})-factorisation v=ps:BECv=p s:B\to E\to C. We then have two (,)(\mathcal{L},\mathcal{R})-factorisations w=p(su)w=p(s u) and w=1 C(vu)w=1_C(v u) of the composite w=vuw=v u,

Hence there exists an isomorphism i:ECi:E\to C such that i(su)=wi(s u)=w and 1 Ci=p1_C i=p. This shows that pp is invertible, and hence pp\in \mathcal{L}, since every isomorphism belongs to \mathcal{L}. It follows that v=psv=p s\in \mathcal{L}, since \mathcal{L} is closed under composition.

Example

Let C I\mathbf{C}^{I} be the arrow category of a category C\mathbf{C}. Then a morphism f:XYf:X\to Y in C I\mathbf{C}^{I} is a commutative square of maps in C\mathbf{C},

If C\mathbf{C} has pullbacks, then C I\mathbf{C}^{I} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of pullback squares. A square f:XYf:X\to Y belongs to \mathcal{L} iff the map f 1f_1 is invertible. See Example 21. Suppose that we have a commutative diagram in C\mathbf{C}

in which the right hand square is cartesian. Then the left hand square is cartesian iff the composite square is cartesian by Proposition 1. See the page on cartesian squares.

Orthogonality

Definition

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

has a unique diagonal filler d:BXd:B\to X (du=xd u=x and fd=yf d=y),

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

is cartesian in the category of sets. If 𝒞\mathcal{C} and \mathcal{F} are two classes of maps in E\mathbf{E}, we shall write 𝒞\mathcal{C} \bot \mathcal{F} to indicate that we have ufu\bot f for every u𝒞u\in \mathcal{C} and ff\in \mathcal{F} .

Notation

If \mathcal{M} is a class of maps in a category E\mathbf{E}, we shall denote by {}^\bot\!\mathcal{M} (resp. \mathcal{M}^\bot) the class of maps left (resp. right) orthogonal to every map in \mathcal{M}. We shall say that {}^\bot\!\!\mathcal{M} is the left orthogonal complement of \mathcal{M}, and that \mathcal{M}^\bot is its right orthogonal complement.

If 𝒞\mathcal{C} and \mathcal{F} are two classes of maps in E\mathbf{E}, then the conditions

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

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

Example

In the category Cat\mathbf{Cat}, a functor is fully faithful iff it is right orthogonal to the inclusion i:IIi:\partial I \subset I, where I\partial I denotes the discrete category with two objects 00 and 11.

Example

If J=π 1IJ=\pi_1I is the groupoid generated by one isomorphism 010\simeq 1, then a functor is conservative iff it is right orthogonal to the inclusion IJI\subset J.

Example

A functor p:XYp:X\to Y is a discrete Conduché fibration if it is right orthogonal to the inclusion d 1:[1][2]d_1:[1]\to [2]. This condition means that for every morphism f:abf:a\to b in XX and every factorisation p(f)=vu:p(a)ep(b)p(f)=vu:p(a)\to e\to p(b) of the morphism p(f)p(f), there exists a unique factorisation f=vu:aebf=v'u':a\to e'\to b of the morphism ff such that p(v)=vp(v')=v and p(u)=up(u')=u. Discrete fibrations and a discrete opfibrations are examples discrete Conduché fibrations.

Lemma

Let \mathcal{M} be a class of maps in a category E\mathbf{E}. Then the class \mathcal{M}^\bot is closed under limits, composition, base changes and it has the left cancellation property. Dually, the class {}^\bot\mathcal{M} is closed under colimits, composition, cobase changes and it has the right cancellation property.

Proof

Let us show that the class \mathcal{M}^\bot is closed under limits. We shall use the fact that the functor Sq(u,):E I[I×I,Set]Sq(u,-):\mathbf{E}^{I}\to [I\times I, \mathbf{Set}] preserves limits for any map u:ABu:A\to B in E\mathbf{E}. Let us suppose that a map f:XYf:X\to Y in E\mathbf{E} is the limit of a diagrams of maps D:KE ID:K\to \mathbf{E}^{I}. Let us put D(k)=f k:X kY kD(k)=f_k:X_k\to Y_k for every object kKk\in K. Then the square Sq(u,f)Sq(u,f) is the limit of the diagram of squares Sq(u,f k)Sq(u,f_k) for kKk\in K. If f kf_k belongs to \mathcal{M}^\bot for every kKk\in K, let us show that ff belongs to \mathcal{M}^\bot. The assumption means that the square Sq(u,f k)Sq(u,f_k) is cartesian for every map uu\in \mathcal{M}. Hence also the limit square Sq(u,f)Sq(u,f), since the category of cartesian squares is a full reflexive subcategory of the category of all squares [I×I,Set][I\times I, \mathbf{Set}] by here. This proves that f f\in \mathcal{M}^\bot. Let us now prove that the class {}^\pitchfork\mathcal{M} is closed under composition and that it has the right cancellation property. Let u:ABu:A\to B and v:BCv:B\to C be two maps in E\mathbf{E}. If uu belongs to {}^\pitchfork\mathcal{M} let us show that v vu v\in {}^\pitchfork\mathcal{M}\Leftrightarrow v u\in {}^\pitchfork\mathcal{M}. For any morphism f:XYf:X\to Y, the square Sq(vu,f)Sq(v u,f)

is the composite of the squares Sq(u,f)Sq(u,f) and Sq(v,f)Sq(v,f),

The square Sq(u,f)Sq(u,f) is cartesian for every ff\in \mathcal{M}, since u u\in {}^\pitchfork\mathcal{M}. It follows from the lemma here that the square Sq(v,f)Sq(v,f) is cartesian iff the square Sq(vu,f)Sq(v u,f) is cartesian. Thus, v vu v\in {}^\pitchfork\mathcal{M}\Leftrightarrow v u\in {}^\pitchfork\mathcal{M}. The remaining properties can be proved similarly, see the proposition here.

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

Theorem

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

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

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

  • \mathcal{L}\bot\mathcal{R}.

Moreover, in this case the pair (,)(\mathcal{L},\mathcal{R}) is a weak factorisation system and we have

= = and= = .\mathcal{L}={}^\bot\mathcal{R}={}^\pitchfork\mathcal{R} \quad \mathrm{and} \quad \mathcal{R}=\mathcal{L}^\bot=\mathcal{L}^\pitchfork.
Proof

(\Rightarrow) If (,)(\mathcal{L},\mathcal{R}) is a factorisation system, let us prove that we have \mathcal{L}\bot \mathcal{R}. If a:AAa:A\to A' belongs to \mathcal{L} and b:BBb:B\to B' belongs to \mathcal{R}, let us show that every commutative square

(1)

has a unique diagonal filler. For this, let us choose two (,)(\mathcal{L}, \mathcal{R})-factorisations u=ps:AEBu=p s:A\to E\to B and u=ps:AEBu'=p's':A'\to E'\to B'. Then from the commutative diagram

we obtain two (,)(\mathcal{L}, \mathcal{R})-factorisations of the same map ABA\to B',

(2)

It follows that there exists a unique isomorphism i:EEi:E'\to E such that isa=si s'a =s and bpi=pb p i =p'. Hence the following diagram commutes,

and the composite d=pis:ABd=p i s':A'\to B is a diagonal filler of the square (1). It remains to prove the uniqueness of dd. If d:ABd':A'\to B is another diagonal filler of the same square,

let us choose a (,)(\mathcal{L}, \mathcal{R})-factorisation d=qt:AFBd'=q t:A'\to F\to B. From the commutative diagram

we can construct two commutative diagrams,

where the first is representing two (,)(\mathcal{L}, \mathcal{R})-factorisations of a map ABA'\to B' and the second two (,)(\mathcal{L}, \mathcal{R})-factorisations of a map ABA\to B. Hence there exists a unique isomorphism k:EFk':E'\to F such that ks=tk's'=t and bqk=pb q k'=p' and unique isomorphism k:FEk:F\to E such that kta=sk t a=s and pk=qp k=q.

It follows from these relations that the following diagram commutes,

Hence also the diagram

The uniqueness of the isomorphism between two (,)(\mathcal{L}, \mathcal{R})-factorisations implies that we have kk=ik k'=i, where ii is the isomorphism in the diagram (2). Thus, d=qt=(pk)(ks)=p(kk)s=pis=dd'=q t =(p k)(k's')=p (k k') s'=p i s'=d. The relation \mathcal{L}\bot \mathcal{R} is proved. (\Leftarrow) If the three conditions are satisfied, let us show that the pair (,)(\mathcal{L},\mathcal{R}) is a factorisation system. We shall first prove that it is a weak factorisation system by showing that we have,

= = and= = .\mathcal{L}={}^\bot \mathcal{R}={}^\pitchfork\mathcal{R} \quad \mathrm{and} \quad \mathcal{R}=\mathcal{L}^\bot=\mathcal{L}^\pitchfork.

We have \mathcal{R}\subseteq \mathcal{L}^\bot since we have \mathcal{L}\bot\mathcal{R} by assumption. Obviously, \mathcal{L}^\bot\subseteq \mathcal{L}^\pitchfork. Let us then show that \mathcal{L}^\pitchfork\subseteq \mathcal{R}. If f:XYf:X\to Y belongs to \mathcal{L}^\pitchfork, let us choose a factorisation f=pu:XEYf=p u:X\to E\to Y with uu\in \mathcal{L} and pp\in \mathcal{R}. Then the square

has a diagonal filler d:EXd:E\to X, since uu\in \mathcal{L} and f f\in \mathcal{L}^\pitchfork. The relations fd=pf d=p and du=1 Xd u=1_X implies that the map ud:EEu d:E\to E is a diagonal filler of the square

But this square has a unique diagonal filler, since we have \mathcal{L}\bot \mathcal{R}. It follows that ud=1 Eu d=1_E. Thus, uu is invertible since du=1 Xd u=1_X. It follows f=puf=p u\in \mathcal{R}, since the class \mathcal{R} is replete. The eqality = = \mathcal{R}=\mathcal{L}^\bot=\mathcal{L}^\pitchfork is proved. The equality = = \mathcal{L}={}^\bot \mathcal{R}={}^\pitchfork \mathcal{R} follows by duality. It follows that the pair (,)(\mathcal{L},\mathcal{R}) is a weak factorisation system. Hence the classes \mathcal{L} and \mathcal{R} contain the isomorphisms and they are closed under composition by the proposition here. It remains to prove the uniqueness of the (,)(\mathcal{L},\mathcal{R})-factorisation of a map f:ABf:A\to B. Suppose then that we have two (,)(\mathcal{L}, \mathcal{R})-factorisations, f=pu:AEBf=p u:A\to E\to B and f=qv:AFBf=q v:A\to F\to B. Then each of the following squares

has a unique diagonal filler, respectively d:FEd:F\to E and r:EFr:E\to F. The composite dr:EEd r:E\to E is then a diagonal filler of the square

It follows that we have dr=1 Ed r=1_E by uniqueness of a diagonal filler. Similarly, we have rd=1 Fr d=1_F. This shows that dd is invertible.

Corollary

A weak factorisation system (,)(\mathcal{L},\mathcal{R}) is a factorisation system iff we have \mathcal{L}\bot\mathcal{R}.

Proof

The implication (\Rightarrow) follows from Theorem 1. Conversely, let (,)(\mathcal{L},\mathcal{R}) be a weak factorisation system for which we have \mathcal{L}\bot\mathcal{R}. The classes \mathcal{L} and \mathcal{R} are replete, since they are closed under composition and they contain the isomorphisms by the proposition here. It then follows from Theorem Theorem 1 that the pair (,)(\mathcal{L},\mathcal{R}) is a factorisation system.

Corollary

A factorisation system (,)(\mathcal{L},\mathcal{R}) is determined by any one of its two classes. The class \mathcal{R} is closed under limits, composition, base changes and it has the left cancellation property. Dually, the class \mathcal{L} is closed under colimits, composition, cobase changes and it has the right cancellation property.

Proof

This follows from Theorem 1 and the Lemma 1.

Proposition

Let (,)(\mathcal{L},\mathcal{R}) be a factorisation system in a category E\mathbf{E}. Then the full subcategory of E I=[I,E]\mathbf{E}^{I}=[I,\mathbf{E}] spanned by the arrows in \mathcal{L} is coreflective, and the full subcategory spanned by the arrows in \mathcal{R} is reflective. Hence the left class of a factorisation system is closed under colimits in the category E I\mathbf{E}^{I} and the right class is closed under limits.

Proof

Let us denote by \mathcal{R}' the full subcategory of [I,E][I,\mathbf{E}] spanned by the arrows in \mathcal{R}. Every map u:ABu:A\to B admits a (,)(\mathcal{L},\mathcal{R})-factorisation u=pi:AEBu=p i:A\to E\to B. The pair (i,1 B)(i,1_B) is a morphism upu\to p in the category [I,E][I,\mathbf{E}],

Let us show that the morphism (i,1 B)(i,1_B) is reflecting the arrow uu in the subcategory \mathcal{R}'. For this, it suffices to show that for every arrow f:XYf:X\to Y in \mathcal{R} and every commutative square

there exists a unique arrow z:EXz:E\to X such that zi=xz i=x and yp=fzy p=f z,

But this is clear, since the square

has a unique diagonal filler z:EXz:E\to X by Theorem \ref{FS4}.

Diagonals and codiagonals

The diagonal of a map f:XYf:X\to Y in a category with pullbacks is defined to be the map δ(f):XX× YX\delta(f):X\to X\times_Y X in the commutative diagram,

Dually, the codiagonal of a map u:ABu:A\to B in a category with pushouts is defined to be the map δ o(u):B ABB\delta^o(u):B\sqcup_A B\to B in the commutative diagram

Lemma

If the diagonal of a map f:XYf:X\to Y in a category E\mathbf{E} exists, then the condition ufu\bot f is equivalent to the conjunction of the conditions ufu\,\pitchfork\, f and uδ(f)u\,\pitchfork\, \delta(f) for any map u:ABu:A\to B.

Proof

If ufu\bot f, let us show that we have ufu\,\pitchfork\, f and uδ(f)u\,\pitchfork\, \delta(f). Obviously, it suffices to show that we have uδ(f)u\,\pitchfork\, \delta(f). For this, let us show that every commutative square

(3)

has a diagonal filler. We have fx 1=fx 2f x_1=f x_2, since the following diagram commutes,

Let us put y=fx 1=fx 2y=f x_1=f x_2. We have (x 1u,x 2u)=(x 1,x 2)u=δ(f)a=(a,a)(x_1u,x_2u)=(x_1,x_2)u=\delta(f)a=(a,a), since the square (3) commutes. This shows that the maps x 1,x 2:BXx_1,x_2:B\to X are both filling the diagonal of the following square,

(4)

Thus, x 1=x 2x_1=x_2 since we have ufu\bot f by assumption. The map x=x 1=x 2x=x_1=x_2 is then filling the diagonal of the square (3). The relation uδ(f)u\,\pitchfork\, \delta(f) is proved. Conversely, if ufu\,\pitchfork\, f and uδ(f)u\,\pitchfork\, \delta(f), let us show that we have ufu\bot f. For this, it suffices to prove the uniqueness of a diagonal filler of a square (4), since the existence follows from the condition ufu\pitchfork f. Suppose then that we have two maps x 1,x 2:BXx_1,x_2:B\to X filling the diagonal of the square (4). The relation fx 1=y=fx 2f x_1=y=f x_2 implies that we can define a map (x 1,x 2):BX× YX(x_1,x_2):B\to X\times_Y X. Moreover, the square (3) commutes, since (x 1,x 1)u=(x 1u,x 2u)=(a,a)=δ(f)(a)(x_1,x_1)u=(x_1 u,x_2 u)=(a,a)=\delta(f)(a). The square (3) has then a diagonal filler x:BXx:B\to X, since we have uδ(f)u\pitchfork \delta(f) by assumption. The relation δ(f)x=(x 1,x 2)\delta(f) x=(x_1,x_2) implies that x 1=x=x 2x_1=x=x_2, since δ(f)x=(x,x)\delta(f) x=(x,x).

Lemma*

(Dual to Lemma 2). If the codiagonal of a map u:ABu:A\to B in a category E\mathbf{E} exists, then the condition ufu\bot f is equivalent to the conjunction of the conditions ufu\,\pitchfork\, f and δ o(u)f\delta^o(u)\,\pitchfork\, f for any map f:XYf:X\to Y.

Definition

If the category E\mathbf{E} has pullbacks, we shall say that a class of maps \mathcal{M} in E\mathbf{E} is closed under diagonals if the implication fδ(f)f\in \mathcal{M}\Rightarrow \delta(f)\in \mathcal{M} is true. Dually, if the category E\mathbf{E} has pushouts, we shall say that \mathcal{M} is closed under codiagonals if the implication fδ o(f)f\in \mathcal{M}\Rightarrow \delta^o(f)\in \mathcal{M} is true.

Proposition

In a category with pullbacks, a weak factorisation system (,)(\mathcal{L},\mathcal{R}) is a factorisation system iff the class \mathcal{R} is closed under diagonals iff it has the left cancellation property

Proof

The implication (1)\Rightarrow(3) was proved in Proposition 1. Let us prove the implication (3)\Rightarrow(2). If a map f:XYf:X\to Y belongs to \mathcal{R} then so is the first projection pr 1:X× YXXpr_1:X\times_Y X\to X, since the right class of a weak factorisation system is closed under base change by the proposition here. But we have pr 1δ(f)=1 Xpr_1\delta(f)=1_X, and it follows that δ(f)\delta(f) belongs to \mathcal{R}, since the class \mathcal{R} has the left cancellation property by assumption. Let us prove the implication (2)\Rightarrow(1). For this, it suffices to show that we have \mathcal{L}\bot \mathcal{R} by Theorem \ref{FS4}. But if uu\in \mathcal{L} and ff\in \mathcal{R}, then we have ufu\pitchfork f and uδ(f)u \pitchfork \delta(f), since the class \mathcal{R} is closed under diagonals by assumption. It then follows by Lemma 2 that we have ufu\bot f.

Proposition*

(Dual to Proposition 3). In a category with pushouts, a weak factorisation system (,)(\mathcal{L},\mathcal{R}) is a factorisation system iff the class \mathcal{L} it is closed under codiagonals iff it has the right cancellation property.

Corollary

In a finitely bicomplete category, a weak factorisation system (,)(\mathcal{L},\mathcal{R}) is a factorisation system iff the class \mathcal{R} is closed under diagonals iff the class \mathcal{L} is closed under codiagonals.

Lemma

If u:ABu:A\to B is an epimorphism, then the conditions ufu \bot f and ufu \pitchfork f are equivalent for any map f:XYf:X\to Y.

Proof

If ufu \pitchfork f, let us show that ufu \bot f. For this we have to show that every commutative square

has a unique diagonal filler. The existence is clear since we have ufu \pitchfork f by hypothesis. Let us prove the uniquess. But if d 1,d 2:BXd_1,d_2: B\to X are two diagonal fillers of the square, then we have d 1u=a=d 2ud_1u=a=d_2u. Thus, d 1=d 2d_1=d_2, since uu is an epimorphism.

Existence

Lemma

Let Σ\Sigma be a set of maps in a category with pushout E\mathbf{E}. Then Σ =(Σδ oΣ) \Sigma^\bot =\bigl(\Sigma \,\cup\, \delta^o\Sigma\bigr)^\pitchfork .

Proof

By Lemma 3 we have Σ =Σ (δ oΣ) .\Sigma^\bot =\Sigma^\pitchfork \, \cap \, (\delta^o\Sigma)^\pitchfork. This proves the result, since

(Σδ oΣ) =Σ (δ oΣ) .\bigl(\Sigma\, \cup\, \delta^o\Sigma\bigr)^\pitchfork=\Sigma^\pitchfork\, \cap\, (\delta^o\Sigma)^\pitchfork.

Recall from here that class of maps 𝒞\mathcal{C} in a cocomplete category E\mathbf{E} is said to be saturated if it satisfies the following conditions: * 𝒞\mathcal{C} contains the isomorphisms and is closed under composition and transfinite compositions; * 𝒞\mathcal{C} is closed under cobase changes; * 𝒞\mathcal{C} is closed under retracts.

Theorem

Let Σ\Sigma be a set of maps between small objects in a cocomplete category E\mathbf{E}. Then the pair

(,)=(Sat(Σδ oΣ),Σ )(\mathcal{L},\mathcal{R})= (Sat\bigl(\Sigma\, \cup\, \delta^o\Sigma \bigr),\Sigma^\bot)

is a factorisation system.

Proof

The codiagonal of a map between small objects is a map between small objects. Thus, Σ˜=Σδ oΣ\tilde \Sigma=\Sigma\, \cup\, \delta^o\Sigma is a set of maps between small objects. It then follows from the theorem here that the pair (,)=(Sat(Σ˜),Σ˜ )(\mathcal{L},\mathcal{R})=(Sat(\tilde \Sigma), \tilde \Sigma^\pitchfork) is a weak factorisation system. But we have Σ˜ =Σ \tilde \Sigma^\pitchfork=\Sigma^\bot by Lemma 5. Hence it remains to show that the pair (,)(\mathcal{L},\mathcal{R}) is a factorisation system. For this, it suffices to show that we have \mathcal{L}\bot\mathcal{R} by Theorem \ref{FS4}. But we have

Σ (Σ )= ,\Sigma\subseteq {}^\bot(\Sigma^\bot) ={}^\bot\mathcal{R},

since Σ =\Sigma^\bot=\mathcal{R}. Thus,

Σ˜=Σδ oΣ \tilde \Sigma=\Sigma\, \cup\,\delta^o\Sigma\subseteq {}^\bot\mathcal{R}

by Lemma 3. It follows that Sat(Σ˜) Sat(\tilde \Sigma)\subseteq {}^\bot\mathcal{R}, since the class {}^\bot\mathcal{R} is saturated by Proposition \ref{saturatedleftorthogonal}. This proves that \mathcal{L}\bot\mathcal{R}, and hence that the pair (,)(\mathcal{L},\mathcal{R}) is a factorisation system.

Corollary

Let Σ\Sigma be a set of maps in a locally presentable category E\mathbf{E}. Then the pair

(,)=(Sat(Σδ oΣ),Σ )(\mathcal{L},\mathcal{R})= (Sat\bigl(\Sigma\, \cup\, \delta^o\Sigma \bigr),\Sigma^\bot)

is a factorisation system.

Proof

This follows from theorem 2, since every object of a locally presentable category is small.

Examples

In algebra

Example

Recall that if ABA\to B is a monomorphism of commutative ring, then an element bBb\in B is said to be integral over AA if it is the root of a monic polynomial pA[x]p\in A[x]. We shall say that a monomorphism of commutative rings ϕ:AB\phi:A\to B is integrally closed if every element bBb\in B integral over AA belongs to AA. The category of commutative rings CRing\mathbf{CRing} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of integrally closed monomorphisms. We shall say that an homomorphism in the class \mathcal{L} is an integral homomorphism.

Example

If RR is a commutative ring, we shall say that an element rRr\in R is a simple root of a polynomial p(x)R[x]p(x)\in R[x] if p(r)=0p(r)=0 and p(r)p'(r) is invertible. Let us denote by Z(p,R)Z(p,R) the set of simple roots in RR of a polynomial pp. We shall say that a ring homomorphism ϕ:AB\phi:A\to B is separably closed if it induces a bijection Z(p,A)Z(ϕ(p),B)Z(p,A)\to Z(\phi(p),B) for every polynomial p(x)A[x]p(x)\in A[x]. For example, a local ring AA with maximal ideal mm is henselien iff the quotient map AA/mA\to A/m is separably closed. The category of commutative rings CRing\mathbf{CRing} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of separably closed homomorphisms. We shall say that an homomorphism in the class \mathcal{L} is a separable algebraic extension. We conjecture that a ring homormorphism is a separable algebraic extension iff it is formally etale.

In Cat

Definition

Let (,)(\mathcal{L},\mathcal{R}) be a factorisation system in the category Cat\mathbf{Cat}. We shall say that a functor f:ABf:A\to B is essentially in \mathcal{L} (resp \mathcal{R}) if the functor pp (resp. uu) of an (,)(\mathcal{L}, \mathcal{R})-factorisation f=pu:AEBf=p u:A\to E\to B is an equivalence of categories.

Example

Recall that a functor f:XYf:X\to Y is said to be full (resp. faithful, fully faithful) if the map X(a,b)Y(f(a),f(b))X(a,b)\to Y(f(a),f(b)) induced by ff is surjective (resp. injective, bijective) for every pair of objects à a,bXa,b\in X. We shall say that a functor f:XYf:X\to Y is monic (resp. surjective, bijective) on objects if the induced map Ob(X)Ob(Y)Ob(X)\to Ob(Y) is injective (resp. surjective, bijective). The category Cat\mathbf{Cat} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} the class of full functors bijective on objects and \mathcal{R} is the class of faithful functors.

Example

The category Cat\mathbf{Cat} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} the class of functors bijective on objects and \mathcal{R} is the class of fully faithful functors. A (,)(\mathcal{L},\mathcal{R})-factorisation of a functor f:ABf:A\to B is the Gabriel factorisation f=pu:AEBf=p u:A\to E\to B constructed as follows: Ob(E)=Ob(A)Ob(E)=Ob(A) and E(a,b)=B(f(a),f(b))E(a,b)=B(f(a),f(b)) for every pair a,bOb(A)a,b\in Ob(A). The composition law is obvious. The functors uu and pp are induced by ff. A functor is essentially in \mathcal{L} iff it is essentially surjective.

Example

We shall say that a fully faithful functor f:XYf:X\to Y is replete if every object of YY which is isomorphic to to an object in the image of ff also belongs to this image. The category Cat\mathbf{Cat} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} the class of essentialy surjective functors and \mathcal{R} is the class of replete fully faithful functors monic on objects. A functor is essentially in \mathcal{R} iff it is fully faithful.

Example

The category of small categories Cat\mathbf{Cat} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} the class of functors surjective on objects and \mathcal{R} is the class of fully faithful functors monic on objects. A functor is essentially in \mathcal{L} iff it is essentially surjective, and a functor is essentially in \mathcal{R} iff it is fully faithful.

Example

(Street and Walters) Recall that a functor between small categories p:XYp:X\to Y is said to be a discrete fibration if for every object xXx\in X and every arrow gYg\in Y with target p(x)p(x), there exists a unique arrow fXf\in X with target xx such that p(f)=gp(f)=g. A functor is a discrete fibration iff it is right orthogonal to the map d 0:[0][1]d_0:[0]\to [1]. The category Cat\mathbf{Cat} admits a factorisation system (,)(\mathcal{L}, \mathcal{R}) in which \mathcal{R} is the class of discrete fibrations and \mathcal{L} is the class of final functors. Recall that a functor between small categories u:ABu:A \to B is final (but we shall eventually say 0-final) iff the category

b\A=(b\B)× BAb \backslash A= (b \backslash B) \times_{B} A

defined by the pullback square

is connected for every object bBb\in B.

Example*

A functor p:XYp:X\to Y is called a discrete op-fibration if the opposite functor p o:X oY op^o:X^o\to Y^o is a discrete fibration. A functor p:XYp:X\to Y is a discrete opfibration iff for every object xXx\in X and every arrow gYg\in Y with source p(x)p(x), there exists a unique arrow fXf\in X with source xx such that p(f)=gp(f)=g. A functor is a discrete opfibration iff it is right orthogonal to the map d 1:[0][1]d_1:[0]\to [1]. The category Cat\mathbf{Cat} admits a factorisation system (,)(\mathcal{L}, \mathcal{R}) in which \mathcal{R} is the class of discrete opfibrations and \mathcal{L} is the class of initial functors. Recall that a functor between small categories u:ABu:A \to B is said to be initial (but we shall eventually say 0-initial) if the opposite functor u o:A oB ou^o:A^o\to B^o is final. A functor u:ABu:A \to B is initial iff the category A/b=(B/b)× BAA/b= (B/b) \times_{B} A defined by the pullback square

is connected for every object bBb\in B.

Example

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 Grpd\mathbf{Grpd} of small groupoids admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of coverings and \mathcal{L} is the class of connected functors. A functor between groupoids is essentially a covering iff it is faithful.

Example

We shall say that a functor u:ABu:A\to B in catcat is a discrete bifibration, or a 0-covering, if it is both a discrete fibration and a discrete opfibration. If Grpd\mathbf{Grpd} is the category of groupoids, then the inclusion functor GrpdCat\mathbf{Grpd}\subset \mathbf{Cat} admits a left adjoint

π 1:CatGrpd\pi_1:\mathbf{Cat} \to \mathbf{Grpd}

which associates to a category AA its fundamental groupoid π 1(A)\pi_1(A). By construction, π 1(A)\pi_1(A) is obtained by inverting universally every arrow in AA. We shall say that a functor u:ABu:A\to B is 0-connected if the functor π 1(u):π 1(A)π 1(B)\pi_1(u):\pi_1(A)\to \pi_1(B) is connected. See Example 11. The category Cat\mathbf{Cat} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} the class of connected functors and \mathcal{R} is the class of 0-coverings.

Example

We shall say that a functor u:ABu:A\to B inverts an arrow fAf\in A if the arrow u(f)u(f) is invertible in BB. A functor u:ABu:A\to B is conservative iff every arrow fAf\in A which is inverted by uu is invertible in AA. For any subset SS of arrows in small category AA, there is a small category S 1AS^{-1}A together with a functor l:AS 1Al:A\to S^{-1}A which inverts universally every arrow in SS. The universality means that for any functor u:ABu:A\to B which inverts every arrow in SS there exists a unique functor u:S 1ABu':S^{-1}A\to B such that ul=uu'l=u.

Every functor u:ABu:A\to B admits a canonical factorisation u=ul:AS 1ABu=u'l:A\to S^{-1}A\to B, where SAS\subseteq A is the set of arrows inverted by uu. We shall say that uu is a localisation if uu' is an isomorphism. Beware that the functor uu' is not conservative in general. Hence the factorisation u=ulu=u'l can be repeated with u:S 1ABu':S^{-1}A\to B instead of uu. Let us put A 1=S 1AA_1=S^{-1}A, u 1=uu_1=u' and let S 1S_1 be the set of arrows in A 1A_1 inverted by u 1:A 1Bu_1:A_1\to B. We then obtain a factorisation u 1=u 2l 1:A 1S 1 1A 1Bu_1=u_2l_1:A_1\to S_1^{-1}A_1 \to B where l 1:A 1S 1 1A 1l_1:A_1\to S_1^{-1}A_1 is the canonical functor. Let us put A 2=S 1 1A 1A_2=S_1^{-1}A_1. By iterating this process, we obtain an infinite sequence of categories and functors,

The category EE in this diagram is defined to be the colimit of the sequence of categories A nA_n, and the functor v:EBv:E\to B to be the colimit of the functor u n:A nBu_n:A_n\to B. It is easy to see that functor vv is conservative. We shall say that uu is an iterated localisation if vv is an isomorphism. The category Cat\mathbf{Cat} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of iterated localisations and \mathcal{R} is the class of conservative functors.

Remark

We saw above that every functor u:ABu:A\to B admits a canonical factorisation u=ul:AS 1ABu=u'l:A\to S^{-1}A\to B, where SAS\subseteq A is the set of arrows inverted by uu. We shall say that the uu is essentially a localisation if uu' is an equivalence. We saw also that uu admits a factorisation u=vi:AEBu=v i:A\to E\to B, where ii is an iterated localisation and vv is conservative. We shall say that uu is essentially an iterated localisation if vv is an equivalence.

In SSet

Example

We shall say that a map of simplicial sets is discrete if it is right orthogonal to every surjection Δ[m]Δ[n]\Delta[m]\to \Delta[n]. Every monomorphism is discrete. The category of simplicial sets SSet\mathbf{SSet} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of discrete maps. We shall say that a map in \mathcal{L} is a collapse.

Example

We shall say that a map of simplicial sets is a discrete right fibration if it is right orthogonal to the maps u:Δ[m]Δ[n]u:\Delta[m]\to \Delta[n] with u(m)=nu(m)=n. The category of simplicial sets SSet\mathbf{SSet} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of discrete right fibrations. We shall say that a map in \mathcal{L} is 0-final.

Example

We shall say that a map of simplicial sets is a discrete left fibration if it is right orthogonal to the maps u:Δ[m]Δ[n]u:\Delta[m]\to \Delta[n] with u(0)=0u(0)=0. The category of simplicial sets SSet\mathbf{SSet} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of discrete left fibrations. We shall say that a map in \mathcal{L} is 0-initial.

Example

We shall say that a map of simplicial sets is a discrete Kan fibration if it is right orthogonal to every map Δ[m]Δ[n]\Delta[m]\to \Delta[n]. The category of simplicial sets SSet\mathbf{SSet} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of discrete Kan fibrations. We shall say that a map in \mathcal{L} is 0-connected.

More examples

Example

We shall say that a map f:XYf:X\to Y between two presheaves on a category AA is etale if the naturality square

is cartesian for every morphism aba\to b in AA. Then the category [A o,Set][A^o,\mathbf{Set}] admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of etale morphisms. We shall say that a morphism in the class \mathcal{L} is connected.

Example

If p:EBp:\mathbf{E}\to \mathbf{B} is a Grothendieck fibration, then the category E\mathbf{E} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of cartesian morphisms and \mathcal{L} is the class of morphisms inverted by pp.

Example

Dually, if p:EBp:\mathbf{E}\to \mathbf{B} is a Grothendieck opfibration, then the category E\mathbf{E} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{L} is the class of cocartesian morphisms and \mathcal{R} is the class of morphisms inverted by pp.

Example

If C\mathbf{C} is a category with pullbacks, then the target functor p 1:C [1]Cp_1:\mathbf{C}^{[1]}\to \mathbf{C} which associates to a map X 0X 1X_0\to X_1 its target X 1X_1 is a Grothedieck fibration. A morphism f:XYf:X\to Y in the category C [1]\mathbf{C}^{[1]} is cartesian with respect to p 1p_1 iff the corresponding square

is cartesian in C\mathbf{C}. It then follows from Example 19 that the category C [1]\mathbf{C}^{[1]} admits a factorisation system (,)(\mathcal{L},\mathcal{R}) in which \mathcal{R} is the class of pullback squares. A square f:XYf:X\to Y belongs to \mathcal{L} iff the map f 1f_1 is invertible.

Exercises

Exercise

Let p:EEp:\mathbf{E}'\to \mathbf{E} be a discrete Conduché fibration (see Example 3). Recall that this means that for every morphism f:ABf:A\to B in E\mathbf{E} and every factorisation p(f)=vu:p(A)Ep(B)p(f)=vu:p(A)\to E\to p(B) of the morphism p(f)p(f), there exists a unique factorisation f=vu:AEBf=v'u':A\to E'\to B of the morphism ff such that p(v)=vp(v')=v and p(u)=up(u')=u. Discrete fibrations and a discrete opfibrations are examples of discrete Conduché fibrations. If \mathcal{M} is a class of maps in E\mathbf{E}, let us denote by \mathcal{M}' the class of maps p 1()p^{-1}(\mathcal{M}) in E\mathbf{E}'. Show that if (,)(\mathcal{L},\mathcal{R}) is a factorisation system in the category E\mathbf{E}, then the pair (,)(\mathcal{L}',\mathcal{R}') is a factorisation system in the category E\mathbf{E}'. In particular, let us denote by E/F\mathbf{E}/F the category of elements of a presheaf FF on a category E\mathbf{E}. If \mathcal{M} is a class of maps in E\mathbf{E}, let us denote by /F\mathcal{M}/F the class of maps in E/F\mathbf{E}/F whose underlying map in E\mathbf{E} belongs to \mathcal{M}. Deduce that if (,)(\mathcal{L},\mathcal{R}) is a factorisation system in E\mathbf{E}, then the pair (/F,/F)(\mathcal{L}/F,\mathcal{R}/F) is a factorisation system in the category E/F\mathbf{E}/F. If F=Hom(,B)F=Hom(-,B) for some object BEB\in \mathbf{E}, then E/F=E/B\mathbf{E}/F=\mathbf{E}/B.

References

Papers:

Lecture Notes and Textbooks:

Revised on January 16, 2013 at 00:06:47 by Tom Hirschowitz