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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 …

(This is an answer to a question below from Akhil Mathew; he wanted examples of ``explicit computations'' since all he knew were classical 1950s calculations and abstract theory. My answer is too lon …

The monoid axiom for symmetric and orthogonal spectra of spaces is Proposition 12.5 of Mandell, May, Schwede, and Shipley's paper ``Model categories of diagram spectra''.

A very interesting example: consider semi-simplicial sets (alias $\Delta$-sets). These are simplicial sets without degeneracies, and there is an ``adjoin degeneracies'' functor from semi-simplicial se …

While Emily was too modest to say so, the history is that Garner developed a beautiful refined small object argument for the construction of algebraic weak factorization systems (his paper Understand …

There are several other papers, I think earlier ones, that cover this. [32] M. Cole. The homotopy category of chain complexes is a homotopy category. Preprint (1990's) [29] J. Daniel Christensen an …

The equivalence (P) is a deep and subtle property of the smash product of spectra in modern symmetric monoidal models for the stable homotopy category. It is very unlikely to hold in other contexts. …

As another shameless advertisement for the forthcoming book ``More concise algebraic topology: localization, completion, and model categories'', by Kate Ponto and myself, the book will contain a proof …

It misses the point to think of model category as a tool for proving that homotopy categories are equivalent. In the case of simplicial sets and topological spaces, that equivalence long preceded the …

This result has nothing special to do with compact Lie groups: it works for arbitrary topological groups $G$. And as Karol gently points out, the "expected $G$-homotopy extension property" actually …

I'll take your question as license to advertise a relatively recent paper in a slightly more specialized but concretely calculational direction: http://nyjm.albany.edu/j/2014/20-53p.pdf. Its title …

See my answer to the question "Do we still need model categories?" here.

John, your question is an advertisement for Johann Sigurdsson's thesis and our book ``Parametrized homotopy theory'', http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf, which is where the results …

There is a very illuminating paper I did not know about when I last answered a similar question: Tyler Lawson. ``Commutative $\Gamma$ rings do not model all commutative ring spectra". Homology, Homoto …

There are old-fashioned classical ways to think about excision, which can easily be translated into model theoretical language of homotopy pushouts as desired. Any excisive triad can be approximated …