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 be a category (locally small).
For , write
if has the left lifting property against , or equivalently if has the right lifting property against .
For an object, write
to indicate that for the morphism the induced hom set morphism
is surjective, dually for
In the case that has a terminal object we have equivalently
and if has an initial object we have equivalently
Accordingly, for write and for the class of morphisms with left or right lifting property against all elements of , respectively.
If is a pair of adjoint functors, then
A pair of classes of morphisms in is a weak factorization system precisely if
every morphism in factors as the composition of a morphism in followed by a morphism in ;
Let , , be three categories.
is called divisible on the left if for every the functor has a right adjoint, to be denoted
is called divisible on the right if for every the functor has a right adjoint, to be denoted
If is divisble on both sides, then there are natural isomorphisms between the collections of morphisms
For every , and we have
If is a closed symmetric monoidal category, then its tensor product functor is divisible on both sides, the two divisions coincide and are given by the internal hom
Let now have finite colimits and let be a functor.
for in and in , 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 denotes the arrow category of .
If in the above situation and have finite limits and is divisble on both sides, def. 3, then also is divisible on both sides:
for in and in , the left quotient is
for in and in , 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 , , have all finite limits and colimits. For all , , we have
Let be a model category. Write for the simplex category and sSet for the category of simplicial sets. In the Reedy model structure on the presheaf category the following constructions are central.
for the functor given by
for the functor given by the coend
(Here on the right we have the canonical tensoring of over Set, where .)
The functor is divisible on both sides.
Let . Then
the object is the matching object of at stage ;
the morphism is the canonical morphism from into the -matching object.
Let be a morphism in . Then
the relative matching morphism of at stage is
the object is the latching object at stage ;
the morphism is the canonical morphism out of the latching object into ;
the morphism is the relative latching morphism of .