Return to Question

$\DeclareMathOperator\Map{Map}$Precis: I am looking for a reference/explanation of a (general) computation of homotopy fixed points $(BG)^{h\Gamma}$ of a finite group $\Gamma$ acting on a discrete (but not necessarily finite) group $G$.

I am primary interested in the case of $\Gamma = \mathbb{Z}/2\mathbb{Z}$, but don't see why the story shouldn't be completely general. Ideally, someone will explain precisely why (or why not) this is a purely algebraic computation (morally this is true because there is no topological input involved involved apart from the general gluing of simplices in the nerve construction).

Note: I can rephrase this question as one about computing a particular type of homotopy limit (homotopy fixed points, finite group) of a homotopy colimit (the classifying space $BG$) and maybe that is the "correct" way to look at this.

In more detail: Suppose given an involutive automorphism of a discrete group $G$. I will write the automorphism as $ g\mapsto \bar g$

For any (discrete) group $G$, I will write $\mathbb{B}G$ for the groupoid with one object and morphisms given by $G$. Let $\Gamma = \mathbb{Z}/2\mathbb{Z}$ and write $\mathbb{E}\Gamma$ for the groupoid with objects as well as morphisms given by $\Gamma$ (I hope it is clear in this context what the morphisms are doing and where I am going with this).

Then $\Gamma$ acts on these groupoids (on $\mathbb{B}G$ via the involution) in the evident way (in the sense that it acts on the set of objects and morphisms in a way that all structure maps are equivariant).

Let $ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) $ denote the groupoid whose objects are $\Gamma$-equivariant functors between the mentioned groupoids. Then just playing around with the definitions it is straightforward to show (hopefully, I didn't mess this up):

$$ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) \cong \bigsqcup_{[\sigma] \in H^1(\Gamma; G)} \mathbb{B}K_{\sigma} $$

where

$$ K_{\sigma} = \{ g\in G \mid \bar g = \bar{\sigma}g\sigma \} $$

and $H^1(\Gamma; G)$ is the set

$$Z^1(\Gamma; G) = \{ \sigma\in G \mid \sigma\bar \sigma = 1\}.$$

modulo the relation $\sigma \sim g\sigma\bar g^{-1}$ for all $g\in G$.

Of course, now my desire is to say that the above formula holds topologically. There is certainly a canonical map from the geometric realization of the groupoid above to $(BG)^{h\Gamma}$ but it isn't obvious to me that it is (or isn't) a weak equivalence.

Added later: I think I may have just reverse engineered the Bousfield-Kan model of the homotopy limit here (that's what I get for trying to first understand it via its formal properties before looking at an explicit model). I am staring at Ch XI, Section 3.2 of the monograph "Homotopy limits, completions and localizations", and unless I am misreading/misunderstanding, the explicit construction of a model for a homotopy limit given there is precisely the description above in terms of the groupoid of functors (in this particular situation).

Hopefully, someone with expertise can confirm or explain.

$\DeclareMathOperator\Map{Map}$Precis: I am looking for a reference/explanation of a (general) computation of homotopy fixed points $(BG)^{h\Gamma}$ of a finite group $\Gamma$ acting on a discrete (but not necessarily finite) group $G$.

I am primary interested in the case of $\Gamma = \mathbb{Z}/2\mathbb{Z}$, but don't see why the story shouldn't be completely general. Ideally, someone will explain precisely why (or why not) this is a purely algebraic computation (morally this is true because there is no topological input involved involved apart from the general gluing of simplices in the nerve construction).

Note: I can rephrase this question as one about computing a particular type of homotopy limit (homotopy fixed points, finite group) of a homotopy colimit (the classifying space $BG$) and maybe that is the "correct" way to look at this.

In more detail: Suppose given an involutive automorphism of a discrete group $G$. I will write the automorphism as $ g\mapsto \bar g$

For any (discrete) group $G$, I will write $\mathbb{B}G$ for the groupoid with one object and morphisms given by $G$. Let $\Gamma = \mathbb{Z}/2\mathbb{Z}$ and write $\mathbb{E}\Gamma$ for the groupoid with objects as well as morphisms given by $\Gamma$ (I hope it is clear in this context what the morphisms are doing and where I am going with this).

Then $\Gamma$ acts on these groupoids (on $\mathbb{B}G$ via the involution) in the evident way (in the sense that it acts on the set of objects and morphisms in a way that all structure maps are equivariant).

Let $ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) $ denote the groupoid whose objects are $\Gamma$-equivariant functors between the mentioned groupoids. Then just playing around with the definitions it is straightforward to show (hopefully, I didn't mess this up):

$$ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) \cong \bigsqcup_{[\sigma] \in H^1(\Gamma; G)} \mathbb{B}K_{\sigma} $$

where

$$ K_{\sigma} = \{ g\in G \mid \bar g = \bar{\sigma}g\sigma \} $$

and $H^1(\Gamma; G)$ is the set

$$Z^1(\Gamma; G) = \{ \sigma\in G \mid \sigma\bar \sigma = 1\}.$$

modulo the relation $\sigma \sim g\sigma\bar g^{-1}$ for all $g\in G$.

Of course, now my desire is to say that the above formula holds topologically. There is certainly a canonical map from the geometric realization of the groupoid above to $(BG)^{h\Gamma}$ but it isn't obvious to me that it is (or isn't) a weak equivalence.

Added later: I think I may have just reverse engineered the Bousfield-Kan model of the homotopy limit here (that's what I get for trying to first understand it via its formal properties before looking at an explicit model). I am staring at Ch XI, Section 3.2 of the monograph "Homotopy limits, completions and localizations", and unless I am misreading/misunderstanding, the explicit construction of a model for a homotopy limit given there is precisely the description above in terms of the groupoid of functors (in this particular situation).

Hopefully, someone with expertise can confirm or explain.

Added later: For anyone interested in this, Bertram Arnold's comment below is essentially the correct answer, as far as I can tell (it took me a while to decipher it though).

$\DeclareMathOperator\Map{Map}$Precis: I am looking for a reference/explanation of a (general) computation of homotopy fixed points $(BG)^{h\Gamma}$ of a finite group $\Gamma$ acting on a discrete (but not necessarily finite) group $G$.

I am primary interested in the case of $\Gamma = \mathbb{Z}/2\mathbb{Z}$, but don't see why the story shouldn't be completely general. Ideally, someone will explain precisely why (or why not) this is a purely algebraic computation (morally this is true because there is no topological input involved involved apart from the general gluing of simplices in the nerve construction).

Note: I can rephrase this question as one about computing a particular type of homotopy limit (homotopy fixed points, finite group) of a homotopy colimit (the classifying space $BG$), but one needs stringent conditions on both for this to have an easy answer (otherwise I am venturing into Sullivan conjecture and Adams operations type territory I think) and the current "easy" situation should have a clean answer that is hopefully obvious to the experts (I am not a homotopy theorist so please be gentle).

In more detail: Suppose given an involutive automorphism of a discrete group $G$. I will write the automorphism as $ g\mapsto \bar g$

For any (discrete) group $G$, I will write $\mathbb{B}G$ for the groupoid with one object and morphisms given by $G$. Let $\Gamma = \mathbb{Z}/2\mathbb{Z}$ and write $\mathbb{E}\Gamma$ for the groupoid with objects as well as morphisms given by $\Gamma$ (I hope it is clear in this context what the morphisms are doing and where I am going with this).

Then $\Gamma$ acts on these groupoids (on $\mathbb{B}G$ via the involution) in the evident way (in the sense that it acts on the set of objects and morphisms in a way that all structure maps are equivariant).

Let $ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) $ denote the groupoid whose objects are $\Gamma$-equivariant functors between the mentioned groupoids. Then just playing around with the definitions it is straightforward to show (hopefully, I didn't mess this up):

$$ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) \cong \bigsqcup_{[\sigma] \in H^1(\Gamma; G)} \mathbb{B}K_{\sigma} $$

where

$$ K_{\sigma} = \{ g\in G \mid \bar g = \bar{\sigma}g\sigma \} $$

and $H^1(\Gamma; G)$ is the set

$$Z^1(\Gamma; G) = \{ \sigma\in G \mid \sigma\bar \sigma = 1\}.$$

modulo the relation $\sigma \sim g\sigma\bar g^{-1}$ for all $g\in G$.

Of course, now my desire is to say that the above formula holds topologically. There is certainly a canonical map from the geometric realization of the groupoid above to $(BG)^{h\Gamma}$ but it isn't obvious to me that it is (or isn't) a weak equivalence.

Added later: I think I may have just reverse engineered the Bousfield-Kan model of the homotopy limit here (that's what I get for trying to first understand it via its formal properties before looking at an explicit model). I am staring at Ch XI, Section 3.2 of the monograph "Homotopy limits, completions and localizations", and unless I am misreading/misunderstanding, the explicit construction of a model for a homotopy limit given there is precisely the description above in terms of the groupoid of functors (in this particular situation).

Hopefully, someone with expertise can confirm or explain.

$\DeclareMathOperator\Map{Map}$Precis: I am looking for a reference/explanation of a (general) computation of homotopy fixed points $(BG)^{h\Gamma}$ of a finite group $\Gamma$ acting on a discrete (but not necessarily finite) group $G$.

I am primary interested in the case of $\Gamma = \mathbb{Z}/2\mathbb{Z}$, but don't see why the story shouldn't be completely general. Ideally, someone will explain precisely why (or why not) this is a purely algebraic computation (morally this is true because there is no topological input involved involved apart from the general gluing of simplices in the nerve construction).

Note: I can rephrase this question as one about computing a particular type of homotopy limit (homotopy fixed points, finite group) of a homotopy colimit (the classifying space $BG$) and maybe that is the "correct" way to look at this.

In more detail: Suppose given an involutive automorphism of a discrete group $G$. I will write the automorphism as $ g\mapsto \bar g$

For any (discrete) group $G$, I will write $\mathbb{B}G$ for the groupoid with one object and morphisms given by $G$. Let $\Gamma = \mathbb{Z}/2\mathbb{Z}$ and write $\mathbb{E}\Gamma$ for the groupoid with objects as well as morphisms given by $\Gamma$ (I hope it is clear in this context what the morphisms are doing and where I am going with this).

Then $\Gamma$ acts on these groupoids (on $\mathbb{B}G$ via the involution) in the evident way (in the sense that it acts on the set of objects and morphisms in a way that all structure maps are equivariant).

Let $ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) $ denote the groupoid whose objects are $\Gamma$-equivariant functors between the mentioned groupoids. Then just playing around with the definitions it is straightforward to show (hopefully, I didn't mess this up):

$$ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) \cong \bigsqcup_{[\sigma] \in H^1(\Gamma; G)} \mathbb{B}K_{\sigma} $$

where

$$ K_{\sigma} = \{ g\in G \mid \bar g = \bar{\sigma}g\sigma \} $$

and $H^1(\Gamma; G)$ is the set

$$Z^1(\Gamma; G) = \{ \sigma\in G \mid \sigma\bar \sigma = 1\}.$$

modulo the relation $\sigma \sim g\sigma\bar g^{-1}$ for all $g\in G$.

Of course, now my desire is to say that the above formula holds topologically. There is certainly a canonical map from the geometric realization of the groupoid above to $(BG)^{h\Gamma}$ but it isn't obvious to me that it is (or isn't) a weak equivalence.

Added later: I think I may have just reverse engineered the Bousfield-Kan model of the homotopy limit here (that's what I get for trying to first understand it via its formal properties before looking at an explicit model). I am staring at Ch XI, Section 3.2 of the monograph "Homotopy limits, completions and localizations", and unless I am misreading/misunderstanding, the explicit construction of a model for a homotopy limit given there is precisely the description above in terms of the groupoid of functors (in this particular situation).

Hopefully, someone with expertise can confirm or explain.

$\DeclareMathOperator\Map{Map}$Precis: I am looking for a reference/explanation of a (general) computation of homotopy fixed points $(BG)^{h\Gamma}$ of a finite group $\Gamma$ acting on a discrete (but not necessarily finite) group $G$.

I am primary interested in the case of $\Gamma = \mathbb{Z}/2\mathbb{Z}$, but don't see why the story shouldn't be completely general. Ideally, someone will explain precisely why (or why not) this is a purely algebraic computation (morally this is true because there is no topological input involved involved apart from the general gluing of simplices in the nerve construction).

Note: I can rephrase this question as one about computing a particular type of homotopy limit (homotopy fixed points, finite group) of a homotopy colimit (the classifying space $BG$), but one needs stringent conditions on both for this to have an easy answer (otherwise I am venturing into Sullivan conjecture and Adams operations type territory I think) and the current "easy" situation should have a clean answer that is hopefully obvious to the experts (I am not a homotopy theorist so please be gentle).

In more detail: Suppose given an involutive automorphism of a discrete group $G$. I will write the automorphism as $ g\mapsto \bar g$

For any (discrete) group $G$, I will write $\mathbb{B}G$ for the groupoid with one object and morphisms given by $G$. Let $\Gamma = \mathbb{Z}/2\mathbb{Z}$ and write $\mathbb{E}\Gamma$ for the groupoid with objects as well as morphisms given by $\Gamma$ (I hope it is clear in this context what the morphisms are doing and where I am going with this).

Then $\Gamma$ acts on these groupoids (on $\mathbb{B}G$ via the involution) in the evident way (in the sense that it acts on the set of objects and morphisms in a way that all structure maps are equivariant).

Let $ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) $ denote the groupoid whose objects are $\Gamma$-equivariant functors between the mentioned groupoids. Then just playing around with the definitions it is straightforward to show (hopefully, I didn't mess this up):

$$ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) \cong \bigsqcup_{[\sigma] \in H^1(\Gamma; G)} \mathbb{B}K_{\sigma} $$

where

$$ K_{\sigma} = \{ g\in G \mid \bar g = \bar{\sigma}g\sigma \} $$

and $H^1(\Gamma; G)$ is the set

$$Z^1(\Gamma; G) = \{ \sigma\in G \mid \sigma\bar \sigma = 1\}.$$

modulo the relation $\sigma \sim g\sigma\bar g^{-1}$ for all $g\in G$.

Of course, now my desire is to say that the above formula holds topologically. There is certainly a canonical map from the geometric realization of the groupoid above to $(BG)^{h\Gamma}$ but it isn't obvious to me that it is (or isn't) a weak equivalence.

$\DeclareMathOperator\Map{Map}$Precis: I am looking for a reference/explanation of a (general) computation of homotopy fixed points $(BG)^{h\Gamma}$ of a finite group $\Gamma$ acting on a discrete (but not necessarily finite) group $G$.

I am primary interested in the case of $\Gamma = \mathbb{Z}/2\mathbb{Z}$, but don't see why the story shouldn't be completely general. Ideally, someone will explain precisely why (or why not) this is a purely algebraic computation (morally this is true because there is no topological input involved involved apart from the general gluing of simplices in the nerve construction).

Note: I can rephrase this question as one about computing a particular type of homotopy limit (homotopy fixed points, finite group) of a homotopy colimit (the classifying space $BG$), but one needs stringent conditions on both for this to have an easy answer (otherwise I am venturing into Sullivan conjecture and Adams operations type territory I think) and the current "easy" situation should have a clean answer that is hopefully obvious to the experts (I am not a homotopy theorist so please be gentle).

In more detail: Suppose given an involutive automorphism of a discrete group $G$. I will write the automorphism as $ g\mapsto \bar g$

For any (discrete) group $G$, I will write $\mathbb{B}G$ for the groupoid with one object and morphisms given by $G$. Let $\Gamma = \mathbb{Z}/2\mathbb{Z}$ and write $\mathbb{E}\Gamma$ for the groupoid with objects as well as morphisms given by $\Gamma$ (I hope it is clear in this context what the morphisms are doing and where I am going with this).

Then $\Gamma$ acts on these groupoids (on $\mathbb{B}G$ via the involution) in the evident way (in the sense that it acts on the set of objects and morphisms in a way that all structure maps are equivariant).

Let $ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) $ denote the groupoid whose objects are $\Gamma$-equivariant functors between the mentioned groupoids. Then just playing around with the definitions it is straightforward to show (hopefully, I didn't mess this up):

$$ \Map_{\Gamma}(\mathbb{E}\Gamma, \mathbb{B}G) \cong \bigsqcup_{[\sigma] \in H^1(\Gamma; G)} \mathbb{B}K_{\sigma} $$

where

$$ K_{\sigma} = \{ g\in G \mid \bar g = \bar{\sigma}g\sigma \} $$

and $H^1(\Gamma; G)$ is the set

$$Z^1(\Gamma; G) = \{ \sigma\in G \mid \sigma\bar \sigma = 1\}.$$

modulo the relation $\sigma \sim g\sigma\bar g^{-1}$ for all $g\in G$.

Of course, now my desire is to say that the above formula holds topologically. There is certainly a canonical map from the geometric realization of the groupoid above to $(BG)^{h\Gamma}$ but it isn't obvious to me that it is (or isn't) a weak equivalence.

Added later: I think I may have just reverse engineered the Bousfield-Kan model of the homotopy limit here (that's what I get for trying to first understand it via its formal properties before looking at an explicit model). I am staring at Ch XI, Section 3.2 of the monograph "Homotopy limits, completions and localizations", and unless I am misreading/misunderstanding, the explicit construction of a model for a homotopy limit given there is precisely the description above in terms of the groupoid of functors (in this particular situation).

Hopefully, someone with expertise can confirm or explain.