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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 …

David has answered 1-3, and I agree with him in the abstract. However, I would like to say more and specifically address 4, since there is a huge difference between the fibrant objects in the two mai …

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 …

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 …

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 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 …

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 …

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 …

There is a general bit of category theory that was applied to ring spectra in EKMM ([83] on my website) and I'll refer to that for details. Unless I am missing something, the discussion surely speci …

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 …

Hi Chris, here is just an answer to your "Main question". I hope it responds to what you had in mind. It is Lemma 1.2 in the current draft of "Enriched model categories and presheaf categories", by …

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 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 …

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. …

I'm rushed, so this might not be right, but it feels right. Consider reduced operads in a cartesian monoidal category, so that $C(0) = \ast$. A monoid $M$ in our category gives a reduced operad $R(M …