In the context of factorization systems such as they appear notably in enriched model category one frequently needs to handle iterated lifting problems. In the appendix of (Joyal–Tierney, 06) a symbolic calculus is introduced to facilitate these computations.
A central point of it is to have the statement of prop. 6 below be easily expressible in terms of “division on both sides”-operations.
Let $\mathcal{E}$ be a category (locally small).
For $f, g \in Mor(\mathcal{E})$, write
if $f$ has the left lifting property against $g$, or equivalently if $g$ has the right lifting property against $f$.
For $S \in \mathcal{E}$ an object, write
to indicate that for the morphism $f : X \to Y$ the induced hom set morphism
is surjective, dually for
In the case that $\mathcal{E}$ has a terminal object $*$ we have equivalently
and if $\mathcal{E}$ has an initial object $\emptyset$ we have equivalently
Accordingly, for $Q \subset Mor(\mathcal{E})$ write ${}^{\pitchfork}Q$ and $Q^{\pitchfork}$ for the class of morphisms with left or right lifting property against all elements of $Q$, respectively.
If $(L \dashv R) : \mathcal{E} \to \mathcal{F}$ is a pair of adjoint functors, then
A pair of classes of morphisms $(L,R)$ in $\mathcal{E}$ is a weak factorization system precisely if
every morphism in $\mathcal{E}$ factors as the composition of a morphism in $L$ followed by a morphism in $R$;
$R = L^\pitchfork$ and $L = {}^\pitchfork R$.
Let $\mathcal{E}_1$, $\mathcal{E}_2$, $\mathcal{E}_3$ be three categories.
A functor
is called divisible on the left if for every $A \in \mathcal{E}_1$ the functor $A \otimes (-)$ has a right adjoint, to be denoted
is called divisible on the right if for every $A \in \mathcal{E}_2$ the functor $(-) \otimes A$ has a right adjoint, to be denoted
If $\otimes$ is divisble on both sides, then there are natural isomorphisms between the collections of morphisms
and
and
For every $f \in Mor(\mathcal{E}_1)$, $g \in Mor(\mathcal{E}_2)$ and $X \in \mathcal{E}_3$ we have
If $\mathcal{E}$ is a closed symmetric monoidal category, then its tensor product functor $\otimes : \mathcal{E} \times \mathcal{E} \to \mathcal{E}$ is divisible on both sides, the two divisions coincide and are given by the internal hom $[-,-] : \mathcal{E}^{op} \times \mathcal{E} \to \mathcal{E}$
Let now $\mathcal{E}_3$ have finite colimits and let $\otimes : \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3$ be a functor.
for $f : A \to B$ in $\mathcal{E}_1$ and $g : X \to Y$ in $\mathcal{E}_2$, write
for the induced pushout-product morphism, the canonical morphism out of the pushout induced from the commutativity of the diagram
The pushout-product extends to a functor
where $C^I$ denotes the arrow category of $C$.
If in the above situation $\mathcal{E}_1$ and $\mathcal{E}_2$ have finite limits and $\otimes$ is divisble on both sides, def. 3, then also $\bar{\otimes}$ is divisible on both sides:
for $f : A \to B$ in $\mathcal{E}_1$ and $g : X \to Y$ in $\mathcal{E}_3$, the left quotient is
for $f : S \to T$ in $\mathcal{E}_2$ and $g : X \to Y$ in $\mathcal{E}_3$, the right quotient is
A key statement now is the following, characterizing the right lifting property again pushout product morphisms:
In the above situation, let $\mathcal{E}_1$, $\mathcal{E}_2$, $\mathcal{E}_3$ have all finite limits and colimits. For all $u \in Mor(\mathcal{E}_1)$, $v \in Mor(\mathcal{E}_2)$, $f \in Mor(\mathcal{E}_3)$ we have
Let $\mathcal{E}$ be a model category. Write $\Delta$ for the simplex category and sSet for the category of simplicial sets. In the Reedy model structure on the presheaf category $[\Delta^{op}, \mathcal{E}]$ the following constructions are central.
Write
for the functor given by
Write
for the functor given by the coend
(Here on the right we have the canonical tensoring of $\mathcal{E}$ over Set, where $S_n \cdot X \simeq \coprod_{s \in S_n} X$.)
The functor $\Box$ is divisible on both sides.
Let $X \in [\Delta^{op}, sSet]$. Then
the object $\partial \Delta[n] \backslash X$ is the matching object of $X$ at stage $n$;
the morphism $(\partial \Delta[n] \hookrightarrow \Delta[n]) \backslash X$ is the canonical morphism from $X_n$ into the $n$-matching object.
Let $f : X \to Y$ be a morphism in $[\Delta^{op}, sSet]$. Then
the relative matching morphism of $f$ at stage $n$ is
the object $(\partial \Delta^c) \otimes X$ is the latching object at stage $n$;
the morphism $(\partial \Delta^c \to \Delta)\otimes X$ is the canonical morphism out of the latching object into $X_n$;
the morphism $(\partial \Delta^c \to \Delta) \bar \otimes f$ is the relative latching morphism of $f$.