Comments:
We shall say that a pair $(\mathcal{L},\mathcal{R})$ of classes of maps in a category $\mathbf{E}$ is a factorisation system if the following conditions are satisfied:
every morphism $f:A\to B$ admits a factorisation $f=p u:A\to E\to B$ with $u\in \mathcal{L}$ and $p\in \mathcal{R}$ 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 $f=p u:A\to E\to B$ with $u\in \mathcal{L}$ and $p\in \mathcal{R}$ is a $(\mathcal{L},\mathcal{R})$-factorisation of the morphism $f$. The uniqueness condition in the definition means that for any pair of $(\mathcal{L},\mathcal{R})$-factorisations $f=p u:A\to E\to B$ and $f=q v:A\to F\to B$ of the same morphism, there exists a unique isomorphism $i:E\to F$ such that the following diagram commutes,
If $( \mathcal{L},\mathcal{R})$ is a factorisation system in a category $\mathbf{E}$, then the pair $(\mathcal{R}^o,\mathcal{L}^o)$ is a factorisation system in the opposite category $\mathbf{E}^o$.
If $B$ is an object of a category $\mathbf{E}$ and $\mathcal{M}$ is a class of maps, 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 factorisation system in a category $\mathbf{E}$, then the pair $(\mathcal{L}/B,\mathcal{R}/B)$ is a factorisation system in the category $\mathbf{E}/B$ for any object $B$ in $\mathbf{E}$. Dually, the pair $(B\backslash \mathcal{L},B\backslash \mathcal{R})$ is a factorisation system in the category $B\backslash \mathbf{E}$.
Left to the reader.
If $Iso$ is the class of isomorphisms of a category $\mathbf{E}$ and if $Map$ is the class of all maps, then the pairs $(Iso,Map)$ and $(Map,Iso)$ are trivial examples of factorisation systems.
The category of sets $\mathbf{Set}$ admits a factorisation system $(Surj,Mono)$, where $Surj$ the class surjections and $Inj$ is the class of injections.
The category of groups $\mathbf{Grp}$ admits a factorisation system $(Surj,Mono)$, where $Surj$ the class of surjective homomorphisms and $Inj$ is the class of injective homomorphisms. More generally, this is true for the category of models of any algebraic theory.
Let $\mathbf{CRing}$ be the category of commutative rings. We shall say that a ring homomorphism $u:A\to B$ inverts an element $f\in A$ if $u(f)$ is invertible in $B$. We shall say that the morphism $u:A\to B$ is conservative if every element $f\in A$ which is inverted by $u$ is invertible in $A$. For any subset $S\subseteq A$, there is a commutative ring $S^{-1}A$ together with a ring homomorphism $l:A\to S^{-1}A$ which inverts universally every elements of $S$. The universality means that for any ring homomorphism $u:A\to B$ which inverts every element of $S$ there exists a unique homomorphism $u':S^{-1}A\to B$ such that $u' l=u$.
Every homorphism $u:A\to B$ admits a canonical factorisation $u=u' l:A\to S^{-1}A\to B$, where $S\subseteq A$ is the set of elements inverted by $u$. The homomorphism $u'$ is alaways conservative; we shall say that $u$ is a localisation if $u'$ is an isomorphism. The category $\mathbf{CRing}$ 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
We shall say that a class of maps $\mathcal{M}$ in a category has the left cancellation property if the implication
is true for any pair of maps $u:A\to B$ and $v:B\to C$. Dually, we shall say that $\mathcal{M}$ has the right cancellation property if the implication
is true.
The intersection of the classes of a factorisation system $(\mathcal{L},\mathcal{R})$ in a category $\mathbf{E}$ 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.
Let $f:A\to B$ be a map in $\mathcal{L}\cap \mathcal{R}$. The trivial factorisations $f=f 1_A$ and $f=1_B f$ 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 $i:A\to B$ such that $f=i 1_A$ and $f=1_B i$,
This shows that $f$ is invertible. Let us prove that the left class $\mathcal{L}$ has the right cancellation property. Let $u:A\to B$ and $v:B\to C$ be two maps. If $u$ and $v u$ belong to $\mathcal{L}$, let us show that $v$ belongs to $\mathcal{L}$. For this, let us choose a $(\mathcal{L},\mathcal{R})$-factorisation $v=p s:B\to E\to C$. We then have two $(\mathcal{L},\mathcal{R})$-factorisations $w=p(s u)$ and $w=1_C(v u)$ of the composite $w=v u$,
Hence there exists an isomorphism $i:E\to C$ such that $i(s u)=w$ and $1_C i=p$. This shows that $p$ is invertible, and hence $p\in \mathcal{L}$, since every isomorphism belongs to $\mathcal{L}$. It follows that $v=p s\in \mathcal{L}$, since $\mathcal{L}$ is closed under composition.
Let $\mathbf{C}^{I}$ be the arrow category of a category $\mathbf{C}$. Then a morphism $f:X\to Y$ in $\mathbf{C}^{I}$ is a commutative square of maps in $\mathbf{C}$,
If $\mathbf{C}$ has pullbacks, then $\mathbf{C}^{I}$ admits a factorisation system $(\mathcal{L},\mathcal{R})$ in which $\mathcal{R}$ is the class of pullback squares. A square $f:X\to Y$ belongs to $\mathcal{L}$ iff the map $f_1$ is invertible. See Example 21. Suppose that we have a commutative diagram in $\mathbf{C}$
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.
We shall say that a map $u:A\to B$ in a category $\mathbf{E}$ is left orthogonal to a map $f:X\to Y$, or that $f$ is right orthogonal to $u$, if every commutative square
has a unique diagonal filler $d:B\to X$ ($d u=x$ and $f d=y$),
We shall denote this relation by $u\bot f$. Notice that the condition $u\bot f$ means that the following square $Sq(u,f)$
is cartesian in the category of sets. If $\mathcal{C}$ and $\mathcal{F}$ are two classes of maps in $\mathbf{E}$, we shall write $\mathcal{C} \bot \mathcal{F}$ to indicate that we have $u\bot f$ for every $u\in \mathcal{C}$ and $f\in \mathcal{F}$.
If $\mathcal{M}$ is a class of maps in a category $\mathbf{E}$, 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 $\mathcal{C}$ and $\mathcal{F}$ are two classes of maps in $\mathbf{E}$, then the conditions
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.
In the category $\mathbf{Cat}$, a functor is fully faithful iff it is right orthogonal to the inclusion $i:\partial I \subset I$, where $\partial I$ denotes the discrete category with two objects $0$ and $1$.
If $J=\pi_1I$ is the groupoid generated by one isomorphism $0\simeq 1$, then a functor is conservative iff it is right orthogonal to the inclusion $I\subset J$.
A functor $p:X\to Y$ is a discrete Conduché fibration if it is right orthogonal to the inclusion $d_1:[1]\to [2]$. This condition means that for every morphism $f:a\to b$ in $X$ and every factorisation $p(f)=vu:p(a)\to e\to p(b)$ 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 discrete Conduché fibrations.
Let $\mathcal{M}$ be a class of maps in a category $\mathbf{E}$. 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.
Let us show that the class $\mathcal{M}^\bot$ is closed under limits. We shall use the fact that the functor $Sq(u,-):\mathbf{E}^{I}\to [I\times I, \mathbf{Set}]$ preserves limits for any map $u:A\to B$ in $\mathbf{E}$. Let us suppose that a map $f:X\to Y$ in $\mathbf{E}$ is the limit of a diagrams of maps $D:K\to \mathbf{E}^{I}$. Let us put $D(k)=f_k:X_k\to Y_k$ for every object $k\in K$. Then the square $Sq(u,f)$ is the limit of the diagram of squares $Sq(u,f_k)$ for $k\in K$. If $f_k$ belongs to $\mathcal{M}^\bot$ for every $k\in K$, let us show that $f$ belongs to $\mathcal{M}^\bot$. The assumption means that the square $Sq(u,f_k)$ is cartesian for every map $u\in \mathcal{M}$. Hence also the limit square $Sq(u,f)$, since the category of cartesian squares is a full reflexive subcategory of the category of all squares $[I\times I, \mathbf{Set}]$ by here. This proves that $f\in \mathcal{M}^\bot$. Let us now prove that the class ${}^\pitchfork\mathcal{M}$ is closed under composition and that it has the right cancellation property. Let $u:A\to B$ and $v:B\to C$ be two maps in $\mathbf{E}$. If $u$ belongs to ${}^\pitchfork\mathcal{M}$ let us show that $v\in {}^\pitchfork\mathcal{M}\Leftrightarrow v u\in {}^\pitchfork\mathcal{M}$. For any morphism $f:X\to Y$, the square $Sq(v u,f)$
is the composite of the squares $Sq(u,f)$ and $Sq(v,f)$,
The square $Sq(u,f)$ is cartesian for every $f\in \mathcal{M}$, since $u\in {}^\pitchfork\mathcal{M}$. It follows from the lemma here that the square $Sq(v,f)$ is cartesian iff the square $Sq(v u,f)$ is cartesian. Thus, $v\in {}^\pitchfork\mathcal{M}\Leftrightarrow v u\in {}^\pitchfork\mathcal{M}$. The remaining properties can be proved similarly, see the proposition here.
Recall that a class $\mathcal{C}$ of objects in a category $\mathbf{E}$ is said to be replete if every object isomorphic to an object of $\mathcal{C}$ 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 $\mathbf{E}^{I}$.
A pair $(\mathcal{L},\mathcal{R})$ of classes of maps in a category $\mathbf{E}$ is a factorisation system iff the following three conditions are satisfied:
every map $f:X\to Y$ admits a $(\mathcal{L},\mathcal{R})$-factorisation $f=p u:X\to E\to Y$;
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
($\Rightarrow$) If $(\mathcal{L},\mathcal{R})$ is a factorisation system, let us prove that we have $\mathcal{L}\bot \mathcal{R}$. If $a:A\to A'$ belongs to $\mathcal{L}$ and $b:B\to B'$ belongs to $\mathcal{R}$, let us show that every commutative square
has a unique diagonal filler. For this, let us choose two $(\mathcal{L}, \mathcal{R})$-factorisations $u=p s:A\to E\to B$ and $u'=p's':A'\to E'\to B'$. Then from the commutative diagram
we obtain two $(\mathcal{L}, \mathcal{R})$-factorisations of the same map $A\to B'$,
It follows that there exists a unique isomorphism $i:E'\to E$ such that $i s'a =s$ and $b p i =p'$. Hence the following diagram commutes,
and the composite $d=p i s':A'\to B$ is a diagonal filler of the square (1). It remains to prove the uniqueness of $d$. If $d':A'\to B$ is another diagonal filler of the same square,
let us choose a $(\mathcal{L}, \mathcal{R})$-factorisation $d'=q t:A'\to F\to B$. From the commutative diagram
we can construct two commutative diagrams,
where the first is representing two $(\mathcal{L}, \mathcal{R})$-factorisations of a map $A'\to B'$ and the second two $(\mathcal{L}, \mathcal{R})$-factorisations of a map $A\to B$. Hence there exists a unique isomorphism $k':E'\to F$ such that $k's'=t$ and $b q k'=p'$ and unique isomorphism $k:F\to E$ such that $k t a=s$ and $p k=q$.
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 $k k'=i$, where $i$ is the isomorphism in the diagram (2). Thus, $d'=q t =(p k)(k's')=p (k k') s'=p i s'=d$. 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,
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 $f:X\to Y$ belongs to $\mathcal{L}^\pitchfork$, let us choose a factorisation $f=p u:X\to E\to Y$ with $u\in \mathcal{L}$ and $p\in \mathcal{R}$. Then the square
has a diagonal filler $d:E\to X$, since $u\in \mathcal{L}$ and $f\in \mathcal{L}^\pitchfork$. The relations $f d=p$ and $d u=1_X$ implies that the map $u d:E\to E$ 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 $u d=1_E$. Thus, $u$ is invertible since $d u=1_X$. It follows $f=p u\in \mathcal{R}$, 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 $f:A\to B$. Suppose then that we have two $(\mathcal{L}, \mathcal{R})$-factorisations, $f=p u:A\to E\to B$ and $f=q v:A\to F\to B$. Then each of the following squares
has a unique diagonal filler, respectively $d:F\to E$ and $r:E\to F$. The composite $d r:E\to E$ is then a diagonal filler of the square
It follows that we have $d r=1_E$ by uniqueness of a diagonal filler. Similarly, we have $r d=1_F$. This shows that $d$ is invertible.
A weak factorisation system $(\mathcal{L},\mathcal{R})$ is a factorisation system iff we have $\mathcal{L}\bot\mathcal{R}$.
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.
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.
Let $(\mathcal{L},\mathcal{R})$ be a factorisation system in a category $\mathbf{E}$. Then the full subcategory of $\mathbf{E}^{I}=[I,\mathbf{E}]$ 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 $\mathbf{E}^{I}$ and the right class is closed under limits.
Let us denote by $\mathcal{R}'$ the full subcategory of $[I,\mathbf{E}]$ spanned by the arrows in $\mathcal{R}$. Every map $u:A\to B$ admits a $(\mathcal{L},\mathcal{R})$-factorisation $u=p i:A\to E\to B$. The pair $(i,1_B)$ is a morphism $u\to p$ in the category $[I,\mathbf{E}]$,
Let us show that the morphism $(i,1_B)$ is reflecting the arrow $u$ in the subcategory $\mathcal{R}'$. For this, it suffices to show that for every arrow $f:X\to Y$ in $\mathcal{R}$ and every commutative square
there exists a unique arrow $z:E\to X$ such that $z i=x$ and $y p=f z$,
But this is clear, since the square
has a unique diagonal filler $z:E\to X$ by Theorem \ref{FS4}.
The diagonal of a map $f:X\to Y$ in a category with pullbacks is defined to be the map $\delta(f):X\to X\times_Y X$ in the commutative diagram,
Dually, the codiagonal of a map $u:A\to B$ in a category with pushouts is defined to be the map $\delta^o(u):B\sqcup_A B\to B$ in the commutative diagram
If the diagonal of a map $f:X\to Y$ in a category $\mathbf{E}$ exists, then the condition $u\bot f$ is equivalent to the conjunction of the conditions $u\,\pitchfork\, f$ and $u\,\pitchfork\, \delta(f)$ for any map $u:A\to B$.
If $u\bot f$, let us show that we have $u\,\pitchfork\, f$ and $u\,\pitchfork\, \delta(f)$. Obviously, it suffices to show that we have $u\,\pitchfork\, \delta(f)$. For this, let us show that every commutative square
has a diagonal filler. We have $f x_1=f x_2$, since the following diagram commutes,
Let us put $y=f x_1=f x_2$. We have $(x_1u,x_2u)=(x_1,x_2)u=\delta(f)a=(a,a)$, since the square (3) commutes. This shows that the maps $x_1,x_2:B\to X$ are both filling the diagonal of the following square,
Thus, $x_1=x_2$ since we have $u\bot f$ by assumption. The map $x=x_1=x_2$ is then filling the diagonal of the square (3). The relation $u\,\pitchfork\, \delta(f)$ is proved. Conversely, if $u\,\pitchfork\, f$ and $u\,\pitchfork\, \delta(f)$, let us show that we have $u\bot f$. For this, it suffices to prove the uniqueness of a diagonal filler of a square (4), since the existence follows from the condition $u\pitchfork f$. Suppose then that we have two maps $x_1,x_2:B\to X$ filling the diagonal of the square (4). The relation $f x_1=y=f x_2$ implies that we can define a map $(x_1,x_2):B\to X\times_Y X$. Moreover, the square (3) commutes, since $(x_1,x_1)u=(x_1 u,x_2 u)=(a,a)=\delta(f)(a)$. The square (3) has then a diagonal filler $x:B\to X$, since we have $u\pitchfork \delta(f)$ by assumption. The relation $\delta(f) x=(x_1,x_2)$ implies that $x_1=x=x_2$, since $\delta(f) x=(x,x)$.
(Dual to Lemma 2). If the codiagonal of a map $u:A\to B$ in a category $\mathbf{E}$ exists, then the condition $u\bot f$ is equivalent to the conjunction of the conditions $u\,\pitchfork\, f$ and $\delta^o(u)\,\pitchfork\, f$ for any map $f:X\to Y$.
If the category $\mathbf{E}$ has pullbacks, we shall say that a class of maps $\mathcal{M}$ in $\mathbf{E}$ is closed under diagonals if the implication $f\in \mathcal{M}\Rightarrow \delta(f)\in \mathcal{M}$ is true. Dually, if the category $\mathbf{E}$ has pushouts, we shall say that $\mathcal{M}$ is closed under codiagonals if the implication $f\in \mathcal{M}\Rightarrow \delta^o(f)\in \mathcal{M}$ is true.
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
The implication (1)$\Rightarrow$(3) was proved in Proposition 1. Let us prove the implication (3)$\Rightarrow$(2). If a map $f:X\to Y$ belongs to $\mathcal{R}$ then so is the first projection $pr_1:X\times_Y X\to X$, since the right class of a weak factorisation system is closed under base change by the proposition here. But we have $pr_1\delta(f)=1_X$, and it follows that $\delta(f)$ 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 $u\in \mathcal{L}$ and $f\in \mathcal{R}$, then we have $u\pitchfork f$ and $u \pitchfork \delta(f)$, since the class $\mathcal{R}$ is closed under diagonals by assumption. It then follows by Lemma 2 that we have $u\bot f$.
(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.
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.
If $u:A\to B$ is an epimorphism, then the conditions $u \bot f$ and $u \pitchfork f$ are equivalent for any map $f:X\to Y$.
If $u \pitchfork f$, let us show that $u \bot f$. For this we have to show that every commutative square
has a unique diagonal filler. The existence is clear since we have $u \pitchfork f$ by hypothesis. Let us prove the uniquess. But if $d_1,d_2: B\to X$ are two diagonal fillers of the square, then we have $d_1u=a=d_2u$. Thus, $d_1=d_2$, since $u$ is an epimorphism.
Let $\Sigma$ be a set of maps in a category with pushout $\mathbf{E}$. Then $\Sigma^\bot =\bigl(\Sigma \,\cup\, \delta^o\Sigma\bigr)^\pitchfork$.
By Lemma 3 we have $\Sigma^\bot =\Sigma^\pitchfork \, \cap \, (\delta^o\Sigma)^\pitchfork.$ This proves the result, since
Recall from here that class of maps $\mathcal{C}$ in a cocomplete category $\mathbf{E}$ is said to be saturated if it satisfies the following conditions: * $\mathcal{C}$ contains the isomorphisms and is closed under composition and transfinite compositions; * $\mathcal{C}$ is closed under cobase changes; * $\mathcal{C}$ is closed under retracts.
Let $\Sigma$ be a set of maps between small objects in a cocomplete category $\mathbf{E}$. Then the pair
is a factorisation system.
The codiagonal of a map between small objects is a map between small objects. Thus, $\tilde \Sigma=\Sigma\, \cup\, \delta^o\Sigma$ is a set of maps between small objects. It then follows from the theorem here that the pair $(\mathcal{L},\mathcal{R})=(Sat(\tilde \Sigma), \tilde \Sigma^\pitchfork)$ is a weak factorisation system. But we have $\tilde \Sigma^\pitchfork=\Sigma^\bot$ 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
since $\Sigma^\bot=\mathcal{R}$. Thus,
by Lemma 3. It follows that $Sat(\tilde \Sigma)\subseteq {}^\bot\mathcal{R}$, 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.
Let $\Sigma$ be a set of maps in a locally presentable category $\mathbf{E}$. Then the pair
is a factorisation system.
This follows from theorem 2, since every object of a locally presentable category is small.
Recall that if $A\to B$ is a monomorphism of commutative ring, then an element $b\in B$ is said to be integral over $A$ if it is the root of a monic polynomial $p\in A[x]$. We shall say that a monomorphism of commutative rings $\phi:A\to B$ is integrally closed if every element $b\in B$ integral over $A$ belongs to $A$. The category of commutative rings $\mathbf{CRing}$ 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.
If $R$ is a commutative ring, we shall say that an element $r\in R$ is a simple root of a polynomial $p(x)\in R[x]$ if $p(r)=0$ and $p'(r)$ is invertible. Let us denote by $Z(p,R)$ the set of simple roots in $R$ of a polynomial $p$. We shall say that a ring homomorphism $\phi:A\to B$ is separably closed if it induces a bijection $Z(p,A)\to Z(\phi(p),B)$ for every polynomial $p(x)\in A[x]$. For example, a local ring $A$ with maximal ideal $m$ is henselien iff the quotient map $A\to A/m$ is separably closed. The category of commutative rings $\mathbf{CRing}$ 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.
Let $(\mathcal{L},\mathcal{R})$ be a factorisation system in the category $\mathbf{Cat}$. We shall say that a functor $f:A\to B$ is essentially in $\mathcal{L}$ (resp $\mathcal{R}$) if the functor $p$ (resp. $u$) of an $(\mathcal{L}, \mathcal{R})$-factorisation $f=p u:A\to E\to B$ is an equivalence of categories.
Recall that a functor $f:X\to Y$ is said to be full (resp. faithful, fully faithful) if the map $X(a,b)\to Y(f(a),f(b))$ induced by $f$ is surjective (resp. injective, bijective) for every pair of objects à $a,b\in X$. We shall say that a functor $f:X\to Y$ is monic (resp. surjective, bijective) on objects if the induced map $Ob(X)\to Ob(Y)$ is injective (resp. surjective, bijective). The category $\mathbf{Cat}$ 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.
The category $\mathbf{Cat}$ 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 $f:A\to B$ is the Gabriel factorisation $f=p u:A\to E\to B$ constructed as follows: $Ob(E)=Ob(A)$ and $E(a,b)=B(f(a),f(b))$ for every pair $a,b\in Ob(A)$. The composition law is obvious. The functors $u$ and $p$ are induced by $f$. A functor is essentially in $\mathcal{L}$ iff it is essentially surjective.
We shall say that a fully faithful functor $f:X\to Y$ is replete if every object of $Y$ which is isomorphic to to an object in the image of $f$ also belongs to this image. The category $\mathbf{Cat}$ 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.
The category of small categories $\mathbf{Cat}$ 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.
(Street and Walters) Recall that a functor between small categories $p:X\to Y$ is said to be a discrete fibration if for every object $x\in X$ and every arrow $g\in Y$ with target $p(x)$, there exists a unique arrow $f\in X$ with target $x$ such that $p(f)=g$. A functor is a discrete fibration iff it is right orthogonal to the map $d_0:[0]\to [1]$. The category $\mathbf{Cat}$ 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 $u:A \to B$ is final (but we shall eventually say 0-final) iff the category
defined by the pullback square
is connected for every object $b\in B$.
A functor $p:X\to Y$ is called a discrete op-fibration if the opposite functor $p^o:X^o\to Y^o$ is a discrete fibration. A functor $p:X\to Y$ is a discrete opfibration iff for every object $x\in X$ and every arrow $g\in Y$ with source $p(x)$, there exists a unique arrow $f\in X$ with source $x$ such that $p(f)=g$. A functor is a discrete opfibration iff it is right orthogonal to the map $d_1:[0]\to [1]$. The category $\mathbf{Cat}$ 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 $u:A \to B$ is said to be initial (but we shall eventually say 0-initial) if the opposite functor $u^o:A^o\to B^o$ is final. A functor $u:A \to B$ is initial iff the category $A/b= (B/b) \times_{B} A$ defined by the pullback square
is connected for every object $b\in B$.
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 $\mathbf{Grpd}$ 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.
We shall say that a functor $u:A\to B$ in $cat$ is a discrete bifibration, or a 0-covering, if it is both a discrete fibration and a discrete opfibration. If $\mathbf{Grpd}$ is the category of groupoids, then the inclusion functor $\mathbf{Grpd}\subset \mathbf{Cat}$ admits a left adjoint
which associates to a category $A$ its fundamental groupoid $\pi_1(A)$. By construction, $\pi_1(A)$ is obtained by inverting universally every arrow in $A$. We shall say that a functor $u:A\to B$ is 0-connected if the functor $\pi_1(u):\pi_1(A)\to \pi_1(B)$ is connected. See Example 11. The category $\mathbf{Cat}$ 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.
We shall say that a functor $u:A\to B$ inverts an arrow $f\in A$ if the arrow $u(f)$ is invertible in $B$. A functor $u:A\to B$ is conservative iff every arrow $f\in A$ which is inverted by $u$ is invertible in $A$. For any subset $S$ of arrows in small category $A$, there is a small category $S^{-1}A$ together with a functor $l:A\to S^{-1}A$ which inverts universally every arrow in $S$. The universality means that for any functor $u:A\to B$ which inverts every arrow in $S$ there exists a unique functor $u':S^{-1}A\to B$ such that $u'l=u$.
Every functor $u:A\to B$ admits a canonical factorisation $u=u'l:A\to S^{-1}A\to B$, where $S\subseteq A$ is the set of arrows inverted by $u$. We shall say that $u$ is a localisation if $u'$ is an isomorphism. Beware that the functor $u'$ is not conservative in general. Hence the factorisation $u=u'l$ can be repeated with $u':S^{-1}A\to B$ instead of $u$. Let us put $A_1=S^{-1}A$, $u_1=u'$ and let $S_1$ be the set of arrows in $A_1$ inverted by $u_1:A_1\to B$. We then obtain a factorisation $u_1=u_2l_1:A_1\to S_1^{-1}A_1 \to B$ where $l_1:A_1\to S_1^{-1}A_1$ is the canonical functor. Let us put $A_2=S_1^{-1}A_1$. By iterating this process, we obtain an infinite sequence of categories and functors,
The category $E$ in this diagram is defined to be the colimit of the sequence of categories $A_n$, and the functor $v:E\to B$ to be the colimit of the functor $u_n:A_n\to B$. It is easy to see that functor $v$ is conservative. We shall say that $u$ is an iterated localisation if $v$ is an isomorphism. The category $\mathbf{Cat}$ 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.
We saw above that every functor $u:A\to B$ admits a canonical factorisation $u=u'l:A\to S^{-1}A\to B$, where $S\subseteq A$ is the set of arrows inverted by $u$. We shall say that the $u$ is essentially a localisation if $u'$ is an equivalence. We saw also that $u$ admits a factorisation $u=v i:A\to E\to B$, where $i$ is an iterated localisation and $v$ is conservative. We shall say that $u$ is essentially an iterated localisation if $v$ is an equivalence.
We shall say that a map of simplicial sets is discrete if it is right orthogonal to every surjection $\Delta[m]\to \Delta[n]$. Every monomorphism is discrete. The category of simplicial sets $\mathbf{SSet}$ 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.
We shall say that a map of simplicial sets is a discrete right fibration if it is right orthogonal to the maps $u:\Delta[m]\to \Delta[n]$ with $u(m)=n$. The category of simplicial sets $\mathbf{SSet}$ 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.
We shall say that a map of simplicial sets is a discrete left fibration if it is right orthogonal to the maps $u:\Delta[m]\to \Delta[n]$ with $u(0)=0$. The category of simplicial sets $\mathbf{SSet}$ 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.
We shall say that a map of simplicial sets is a discrete Kan fibration if it is right orthogonal to every map $\Delta[m]\to \Delta[n]$. The category of simplicial sets $\mathbf{SSet}$ 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.
We shall say that a map $f:X\to Y$ between two presheaves on a category $A$ is etale if the naturality square
is cartesian for every morphism $a\to b$ in $A$. Then the category $[A^o,\mathbf{Set}]$ 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.
If $p:\mathbf{E}\to \mathbf{B}$ is a Grothendieck fibration, then the category $\mathbf{E}$ 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 $p$.
Dually, if $p:\mathbf{E}\to \mathbf{B}$ is a Grothendieck opfibration, then the category $\mathbf{E}$ 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 $p$.
If $\mathbf{C}$ is a category with pullbacks, then the target functor $p_1:\mathbf{C}^{[1]}\to \mathbf{C}$ which associates to a map $X_0\to X_1$ its target $X_1$ is a Grothedieck fibration. A morphism $f:X\to Y$ in the category $\mathbf{C}^{[1]}$ is cartesian with respect to $p_1$ iff the corresponding square
is cartesian in $\mathbf{C}$. It then follows from Example 19 that the category $\mathbf{C}^{[1]}$ admits a factorisation system $(\mathcal{L},\mathcal{R})$ in which $\mathcal{R}$ is the class of pullback squares. A square $f:X\to Y$ belongs to $\mathcal{L}$ iff the map $f_1$ is invertible.
Let $p:\mathbf{E}'\to \mathbf{E}$ be a discrete Conduché fibration (see Example 3). Recall that this means that for every morphism $f:A\to B$ in $\mathbf{E}$ and every factorisation $p(f)=vu:p(A)\to E\to p(B)$ 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 factorisation system in the category $\mathbf{E}$, then the pair $(\mathcal{L}',\mathcal{R}')$ is a factorisation system in the category $\mathbf{E}'$. In particular, let us denote by $\mathbf{E}/F$ the category of elements of a presheaf $F$ on a category $\mathbf{E}$. If $\mathcal{M}$ is a class of maps in $\mathbf{E}$, let us denote by $\mathcal{M}/F$ the class of maps in $\mathbf{E}/F$ whose underlying map in $\mathbf{E}$ belongs to $\mathcal{M}$. Deduce that if $(\mathcal{L},\mathcal{R})$ is a factorisation system in $\mathbf{E}$, then the pair $(\mathcal{L}/F,\mathcal{R}/F)$ is a factorisation system in the category $\mathbf{E}/F$. If $F=Hom(-,B)$ for some object $B\in \mathbf{E}$, then $\mathbf{E}/F=\mathbf{E}/B$.
Papers:
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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: