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nLab > Latest Changes: fine model structure on topological G-spaces

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeSep 10th 2021
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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
- 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Maps(X,-)</annotation></semantics></math>$ out of a $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ -CW-complex $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>$ is a right Quillen endofunctor on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>$ -spaces equipped with the fine model structure:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>G</mi><mi>Act</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>TopSp</mi> <mi>Qu</mi></msub><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>fine</mi></msub><munderover><mrow><msub><mo>⊥</mo> <mpadded width="0px"><mi>Qu</mi></mpadded></msub></mrow><munder><mo>⟶</mo><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></munder><mover><mo>⟵</mo><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></munderover><mi>G</mi><mi>Act</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>TopSp</mi> <mi>Qu</mi></msub><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>fine</mi></msub></mrow><annotation encoding="application/x-tex"> G Act\big( TopSp_{Qu}\big)_{fine} \underoverset {\underset{Maps(X,-)}{\longrightarrow}} {\overset{X \times (-)}{\longleftarrow}} {\bot_{\mathrlap{Qu}}} G Act\big( TopSp_{Qu}\big)_{fine} </annotation></semantics></math>$
Can this be cited directly from the literature? I haven’t yet found a reference that makes it explicit.

diff, v6, current
- 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
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeSep 14th 2021
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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.

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nLab > Latest Changes: fine model structure on topological G-spaces