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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 10th 2021

am finally giving this its own entry (this used to be treated within the entry on Elmendorf’s theorem)

but just a stub for the moment

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeSep 13th 2021

Have added the observation (here) that $Maps(X,-)$ out of a $G$-CW-complex $X$ is a right Quillen endofunctor on $G$-spaces equipped with the fine model structure:

$G Act\big( TopSp_{Qu}\big)_{fine} \underoverset {\underset{Maps(X,-)}{\longrightarrow}} {\overset{X \times (-)}{\longleftarrow}} {\bot_{\mathrlap{Qu}}} G Act\big( TopSp_{Qu}\big)_{fine}$

Can this be cited directly from the literature? I haven’t yet found a reference that makes it explicit.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 14th 2021

A kind soul on the AlgTop Discord chat (here) kindly points me to proof that the fine model structure in fact does satisfy the pushout product axiom. Have made a brief note and generalized the statement about the internal Hom Quillen adjunction accordingly.

Will expand further, but first need to chase through an airport now…

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 14th 2021

I have now fleshed out remarks on the cartesian monoidal model structure a little more (here).

Also added more references establishing properness and cofibrant generation and/or topological enrichment.