Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
Let be a (cofibrantly generated) model category and let be any category, regarded as a diagram-shape in the following.
Write for the projective model structure on the functor category of functors from to , hence of -diagrams in .
A functor/diagram is a projectively cofibrant diagram in if it is a cofibrant object in the projective model structure .
This means that a diagram is projectively cofibrant precisely if the inclusion of the initial diagram has the left lifting property with respect to natural transformations of diagrams
which are projective acyclic fibrations, hence which are such that for each the component is an acyclic fibration in .
This means that is projectively cofibrant precisely if for every diagram of natural transformations
with as above, there exists a lift in
making the triangle commute.
The main point of projectively cofibrant diagrams is that the ordinary colimit over them is a presentation of the homotopy colimit:
because the (colimit constant diagram)-adjunction
is a Quillen adjunction (because is by the very definition of the projective model structure a right Quillen functor), the homotopy colimit, being the left derived functor of the colimit, is computed as the ordinary colimit evaluated on a cofibrant resolution of a diagram :
For specific diagram shapes
A span diagram is projectively cofibrant precisely if the two morphisms are cofibrations in and , hence all three objects, are cofibrant.
The colimit over such a diagram is the homotopy pushout of the span.
A cotower diagram
is projectively cofibrant precisely if every morphism is a cofibration and if the first object , and hence all objects, are cofibrant in .
The colimit over such a diagram is a homotopy sequential colimit.
A parallel morphisms diagram
is projectively cofibrant precisely if is cofibrant, and if the morphism is a cofibration.
This implies that also and are cofibrations and hence that is cofibrant.
The colimit over such a diagram is a homotopy coequalizer.
For specific ambient model categories
Let sSet be the standard model structure on simplicial sets. Then is the projective model structure on simplicial presheaves.
For the following see at model structure on simplicial presheaves the section Cofibrant objects for more details (due to Dan Dugger).
A sufficient condition for a diagram to be projectively cofibrant is:
is degreewise a coproducts of representables
the degenerate cells in each degree form a separate coproduct summand;
For any simplicial presheaf, a cofibrant resolution is given by
where the coproduct runs over all sequences of morphisms between representables , as indicated.
See the references at homotopy colimit and generally at model category.
Related discussion is at
Related discussion is for instance also in
where cofibrant cotowers are mentioned as example 2.3.15.