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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 10th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexam 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 <a href="https://ncatlab.org/nlab/revision/fine+model+structure+on+topological+G-spaces/1">v1</a>, <a href="https://ncatlab.org/nlab/show/fine+model+structure+on+topological+G-spaces">current</a>

   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

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 13th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexHave added the observation ([here](https://ncatlab.org/nlab/show/fine+model+structure+on+topological+G-spaces#InternalHomQuillenAdjunction)) 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. <a href="https://ncatlab.org/nlab/revision/diff/fine+model+structure+on+topological+G-spaces/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/fine+model+structure+on+topological+G-spaces/6">v6</a>, <a href="https://ncatlab.org/nlab/show/fine+model+structure+on+topological+G-spaces">current</a>

   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.

   diff, v6, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 14th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexA kind soul on the AlgTop Discord chat ([here](https://discord.com/channels/801815065165955143/844609732505239552)) 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... <a href="https://ncatlab.org/nlab/revision/diff/fine+model+structure+on+topological+G-spaces/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/fine+model+structure+on+topological+G-spaces/9">v9</a>, <a href="https://ncatlab.org/nlab/show/fine+model+structure+on+topological+G-spaces">current</a>

   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…

   diff, v9, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 14th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have now fleshed out remarks on the cartesian monoidal model structure a little more ([here](https://ncatlab.org/nlab/show/fine+model+structure+on+topological+G-spaces#MonoidalModelCategoryStructure)). Also added more references establishing properness and cofibrant generation and/or topological enrichment.

   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.