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John Klein
John Klein
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Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric) realization of a simplicial group (re-topologize this using the compactly generated topology).

Let $R^G(\ast)$ be the category of based left $G$-spaces (where spaces are taken in $T$). Then $R^G(\ast)$ is a model category in which a fibration and weak equivalence are defined via the forgetful functor to $T$. Cofibrations are defined by the lifting property. It's well-known that this gives a model structure, so I'll take that for granted.

Let $R(BG)$ be the category of spaces containing $BG$ as a retract. Objects are spaces $Y$ equipped with maps $r:Y \to BG$, $s: BG \to Y$ such that $r\circ s : BG \to BG$ is the identity (call $r$ and $s$ structure maps). A morphism $Y \to Y'$ is a map of underlying spaces preservingthat preserves the structure maps.

Then $R(BG)$ is a model category in which a fibration, cofibration and weak equivalence are defined using the forgetful functor to $T$. This is due to Quillen.

I think the following is a folklore result:

Assertion: $R^G(\ast)$ and $R(BG)$ are Quillen equivalent.$R^G(\ast)$ and $R(BG)$ are Quillen equivalent.

My question: Does anyone know a concrete reference for this?

Remark: A statement suggesting that the assertion is true in the context of Waldhausen categories appears in Waldhausen's foundational paper in LNM 1126.

Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric) realization of a simplicial group (re-topologize this using the compactly generated topology).

Let $R^G(\ast)$ be the category of based left $G$-spaces (where spaces are taken in $T$). Then $R^G(\ast)$ is a model category in which a fibration and weak equivalence are defined via the forgetful functor to $T$. Cofibrations are defined by the lifting property. It's well-known that this gives a model structure, so I'll take that for granted.

Let $R(BG)$ be the category of spaces containing $BG$ as a retract. Objects are spaces $Y$ equipped with maps $r:Y \to BG$, $s: BG \to Y$ such that $r\circ s : BG \to BG$ is the identity (call $r$ and $s$ structure maps). A morphism is a map of spaces preserving the structure maps.

Then $R(BG)$ is a model category in which a fibration, cofibration and weak equivalence are defined using the forgetful functor to $T$. This is due to Quillen.

I think the following is a folklore result:

Assertion: $R^G(\ast)$ and $R(BG)$ are Quillen equivalent.

My question: Does anyone know a concrete reference for this?

Remark: A statement suggesting that the assertion is true in the context of Waldhausen categories appears in Waldhausen's foundational paper in LNM 1126.

Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric) realization of a simplicial group (re-topologize this using the compactly generated topology).

Let $R^G(\ast)$ be the category of based left $G$-spaces (where spaces are taken in $T$). Then $R^G(\ast)$ is a model category in which a fibration and weak equivalence are defined via the forgetful functor to $T$. Cofibrations are defined by the lifting property. It's well-known that this gives a model structure, so I'll take that for granted.

Let $R(BG)$ be the category of spaces containing $BG$ as a retract. Objects are spaces $Y$ equipped with maps $r:Y \to BG$, $s: BG \to Y$ such that $r\circ s : BG \to BG$ is the identity (call $r$ and $s$ structure maps). A morphism $Y \to Y'$ is a map of underlying spaces that preserves the structure maps.

Then $R(BG)$ is a model category in which a fibration, cofibration and weak equivalence are defined using the forgetful functor to $T$. This is due to Quillen.

I think the following is a folklore result:

Assertion: $R^G(\ast)$ and $R(BG)$ are Quillen equivalent.

My question: Does anyone know a concrete reference for this?

Remark: A statement suggesting that the assertion is true in the context of Waldhausen categories appears in Waldhausen's foundational paper in LNM 1126.

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John Klein
John Klein
  • 18.8k
  • 53
  • 109
Source Link
John Klein
John Klein
  • 18.8k
  • 53
  • 109

Equivariant versus retractive spaces: a reference request

Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric) realization of a simplicial group (re-topologize this using the compactly generated topology).

Let $R^G(\ast)$ be the category of based left $G$-spaces (where spaces are taken in $T$). Then $R^G(\ast)$ is a model category in which a fibration and weak equivalence are defined via the forgetful functor to $T$. Cofibrations are defined by the lifting property. It's well-known that this gives a model structure, so I'll take that for granted.

Let $R(BG)$ be the category of spaces containing $BG$ as a retract. Objects are spaces $Y$ equipped with maps $r:Y \to BG$, $s: BG \to Y$ such that $r\circ s : BG \to BG$ is the identity (call $r$ and $s$ structure maps). A morphism is a map of spaces preserving the structure maps.

Then $R(BG)$ is a model category in which a fibration, cofibration and weak equivalence are defined using the forgetful functor to $T$. This is due to Quillen.

I think the following is a folklore result:

Assertion: $R^G(\ast)$ and $R(BG)$ are Quillen equivalent.

My question: Does anyone know a concrete reference for this?

Remark: A statement suggesting that the assertion is true in the context of Waldhausen categories appears in Waldhausen's foundational paper in LNM 1126.