Do we still need model categories?

I find some of this exchange truly depressing. There is a subject of ``brave new algebra''and there are myriads of past and present constructions and calculations that depend on having concrete and specific constructions. People who actually compute anything do not use $(\infty,1)$ categories when doing so. To lay down a challenge, they would be of little or no use there. One can sometimes use $(\infty,1)$ categories to construct things not easily constructed otherwise, and then one can compute things about them (e.g. work of Behrens and Lawson). But the tools of computation are not $(\infty,1)$ categorical, and often not even model categorical. People should learn some serious computations, do some themselves, before totally immersing themselves in the formal theory. Note that $(\infty,1)$ categories are in principle intermediate between the point-set level and the homotopy category level. It is easy to translate into $(\infty,1)$ categories from the point-set level, whether from model categories or from something weaker. Then one can work in $(\infty,1)$ categories. But the translation back out to the "old-fashioned'' world that some writers seem to imagine expendable lands in homotopy categories. That is fine if that is all that one needs, but one often needs a good deal more. One must be eclectic. Just one old man's view.

Here are some rough analogies:

• Model Category :: $(\infty, 1)$-category
• Basis :: Vector space
• Local coordinates :: Manifold

I especially like the last one. When you do, say, differential geometry in a coordinate free way you often end up with more conceptual, cleaner, and more beautiful proofs. But sometimes you just have to roll up your sleeves, get your hands dirty, and compute in local coordinates. Sometimes different local coordinates are useful for different calculations.

One modern point of view is that Quillen Model categories are the local coordinates of a homotopy theory (i.e. presentable $(\infty,1)$-category, c.f. this MO answer). When you can compute without choosing local coordinate (a model category structure), that is great. But sometimes it can be very helpful (maybe even necessary) to pick a Model structure to do a specific computation. Sometimes it is helpful to jump between different Model structures which describe the same underlying $(\infty,1)$-category.

Model structures are still extremely useful, even essential to the general theory. But they are a tool, just like local coordinates in differential geometry.

One nice feature of model categories is that you can speak also of the non-bifibrant objects (which is not longer possible, once you passed to the corresponding infinity-category). A few examples where this is useful:

• Simplicial sets: non-Kan simplicial sets appear again and again. For example, the n-simplex itself.
• Spectra: in most models for spectra all fibrant objects are $\Omega$-spectra. One often wants to consider non-$\Omega$-spectra like Thom spectra or suspension spectra.
• Diagram categories: The process of replacing a map (say, between topological spaces) by a fibration can be seen as a fibrant replacement in the arrow category.
• Chain complexes: not all modules are projective...

For proving abstract theorems, the framework of $\infty$-categories seems to be in many senses very convenient. But model categories are (in my opinion) often nicer if you want to deal with concrete examples (which are often not bifibrant) and want to see how to compute derived functors of them. Also, concrete models of spectra (like symmetric spectra) where $E_\infty$-rings are modeled by strictly commutative monoids are really nice to write down concrete examples.