A solid functor (also called a semi-topological functor) is a forgetful functor for which the structure of an -object can be universally lifted along sinks. One can also say that has not just a left adjoint but all possible “relative” left adjoints.
When is solid, colimits in can be constructed in a natural way out of colimits in , and inherits strong cocompleteness properties from .
Let be a faithful functor. A -structured sink is a sink in of the form . Note that the indexing family need not be a set, it can be a proper class. A semi-final lift of such a -structured sink consists of a morphism in such that
Every composite is in the image of , i.e. is of the form for some (necessarily unique, since is faithful), and
is initial with this property, i.e. for any other morphism such that each is in the image of , there exists a unique in such that .
Finally, is called solid if every -structured sink has a semi-final lift.
Any topological functor is solid. Indeed, a functor is topological just when it has final lifts for all -structured sinks, where a final lift is a semi-final lift for which is an isomorphism.
Any monadic functor into is solid.
A fully faithful functor is solid if and only if it has a left adjoint.
Suppose that is solid and let be a diagram such that has a colimit in , consisting of a cocone . Let be a semi-final lift of this -structured sink, for which we have induced morphisms in . Since is faithful, these morphisms are a cocone under , and the semi-finality makes it into a colimit in .
Therefore, if is solid over , then it admits all colimits which does. Moreover, if we understand colimits in , and we understand the semi-final lifts, then we understand colimits in .
then has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by . Using an argument of (Nikolaus) we can show:
Let be an accessible solid functor, and assume that has a cofibrantly generated model structure and the following acyclicity condition:
Then the transfer theorem applies, so that has a cofibrantly generated model structure in which the weak equivalences and fibrations are created by .
We have remarked above that has a left adjoint, and we assumed it to be accessible, so it remains to show that the given acyclicity condition implies the standard one.
We first show that pushouts in of images under of generating acyclic cofibrations become acyclic cofibrations (not just weak equivalences) upon applying . Let be a generating acyclic cofibration, and
a diagram in of which we would like to take the pushout. Consider the pushout of the corresponding diagram in :
Since is a model category, is an acyclic cofibration. Therefore, if is a semifinal lift of the singleton sink , by assumption, is also an acyclic cofibration and thus so is the composite . But it is straightforward to verify that in fact, the map of which this is the image (which exists by assumption) gives a pushout diagram in :
If is not just accessible but finitary, then it preserves all transfinite composites, so any transfinite composite of such pushouts in maps to a transfinite composite in , and we know that transfinite composites of acyclic cofibrations in are acyclic cofibrations, so the desired acyclicity condition follows. In general, we can argue as follows: given a transfinite sequence in of such pushouts, its colimit (= composition) can be constructed as above by forgetting down to , taking the colimit there, and then taking the semifinal lift. But since acyclic cofibrations in are closed under transfinite composites, the legs of the colimiting cocone in are acyclic cofibrations. Hence by the assumed acyclicity condition, so is the semifinal lift, and hence (by composition) so are the images in of the legs of the colimiting cone in . This completes the proof.
Walter Tholen, Semitopological functors. I
The example of algebraic fibrant objects and the argument entering the above lifting theorem appears in