This is a real issue, in the sense that I really don't know what to do.
Homotopy (co)limits can be discussed at several levels. A part from the two different motivations I will give (classical and (\infty,1)-categorical), I could discuss the Bousfield-Kan formulas in general or use the model structures on the categories of diagrams.
Advantages of the Bousfield-Kan formulas:
they work absolutely in general;
they are rather explicit formulas, giving to a certain extent a feeling of what's going on and a possibility to do computations (see Chapter 6 in E. Riehl notes for example);
Drawbacks of the Bousfield-Kan formulas:
to prove that they solve the orginal problem (i.e. to obtain the homotopy invariance) it is needed a lot of work in enriched category theory, which is beyond the scope of the notes.
Advantages of the approach via model structures on diagrams:
it is quick to completely prove the existence of the global projective model structure;
the homotopy invariance is an easy consequence of the existence of the model structure;
it should be perfectly clear what's going on.
the same reasonings apply also in the case of a Reedy diagram.
Drawbacks of the approach via model structures on diagrams:
it works only for cofibrantly generated model categories (not a serious drawback);
it works only for homotopy colimits (for homotopy limits the injective model structure is needed, but to get the injective model structure the Lurie cardinality argument is needed);
no computations are available.
I am thinking to use the global projective model structure, to say that the Bousfield-Kan formulas exist and to give a proper reference (e.g. E. Riehl's notes), to say that the homotopy limit case can be dealt with using the global injective model structure (but without proving its existence).
At the end, I'll do a few computations (*), mainly concerning suspensions and loop spaces for spaces and for chain complexes. These computations will be needed later on, for example in the discussion of spectra and (hence) in the chapter about formal deformation theory.
What does the rest of the world say?
P.S. If you agree with me to avoid Bousfield-Kan formulas (I'm not really happy because they are a cornerstone of the theory, but whatever...), then the appendix on enriched categories is essentially useless. To keep an appendix to give a definition is meaningless, my only purpose was to arrive to the bar-cobar constructions in the enriched setting, which allow to show the homotopy invariance of the Bousfield-Kan formulas.
(*) I said that no computations are available. In fact, I'll introduce the well known criteria for computing homotopy pullbacks and homotopy pushouts in the "computations section".
This is a real issue, in the sense that I really don't know what to do.
Homotopy (co)limits can be discussed at several levels. A part from the two different motivations I will give (classical and (\infty,1)-categorical), I could discuss the Bousfield-Kan formulas in general or use the model structures on the categories of diagrams.
Advantages of the Bousfield-Kan formulas:
Drawbacks of the Bousfield-Kan formulas:
Advantages of the approach via model structures on diagrams:
Drawbacks of the approach via model structures on diagrams:
I am thinking to use the global projective model structure, to say that the Bousfield-Kan formulas exist and to give a proper reference (e.g. E. Riehl's notes), to say that the homotopy limit case can be dealt with using the global injective model structure (but without proving its existence).
At the end, I'll do a few computations (*), mainly concerning suspensions and loop spaces for spaces and for chain complexes. These computations will be needed later on, for example in the discussion of spectra and (hence) in the chapter about formal deformation theory.
What does the rest of the world say?
P.S. If you agree with me to avoid Bousfield-Kan formulas (I'm not really happy because they are a cornerstone of the theory, but whatever...), then the appendix on enriched categories is essentially useless. To keep an appendix to give a definition is meaningless, my only purpose was to arrive to the bar-cobar constructions in the enriched setting, which allow to show the homotopy invariance of the Bousfield-Kan formulas.
(*) I said that no computations are available. In fact, I'll introduce the well known criteria for computing homotopy pullbacks and homotopy pushouts in the "computations section".