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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 …

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 …

They are exactly such. One point is that one can use classical cell complexes, stop at $\omega$ with no transfinite nonsense, in setting up the Quillen model structure: it is a compactly generated (a …

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 …

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 …

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 …

This is dealt with in the book ``More concise algebraic topology'' by Kate Ponto and myself, and of course elsewhere, I am sure. We used compactly generated spaces (for us, that means weak Hausdorff …

I haven't thought about this hard (no time) but here are quick observations. Your homotopy colimit is the bar construction $B(\ast,K,X)$, the geometric realization of the simplicial space with $n$-sim …

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 …

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 …

Megan, I imagine that what you have in mind is comparisons of $G$-spectra as $G$ varies, not restricting attention just to subgroups of a given $G$ as in the references cited so far. There are some th …

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

(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''.

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 …