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I apologize in advance if this question is too basic for this site, I tried to search online and through the literature for a few days with no success already.

I am trying to understand the concept of homotopy fixed points in its various reincarnations, and would like to understand, in particular, the relation between these definitions.

I know of 3 definitions in different "categorical" settings:

  1. 0-categorical: let $S$ be a set with a $G$ action, the fixed points, denoted $S^G$ is a subset on which $G$ acts trivially.

  2. 1-categorical: let $C$ be a category with a $G$-action ($G$ acts via endofunctors), we define $C^G$ as the functor category $Fun_G[EG, C]$ of $G$-equivariant functors from $EG$ to $C$.

This already doesn't look like the classical fixed points in a way similar to how $EG\times C/G$ doesn't seem like an orbit set, but rather an orbit groupoid. I imagine we should think of this as a generalization of the original notion of fixed points since $EG$ is "contractible". And the construction consists of datum of "embeddings" of $EG$ into $C$.

So far so good?, this is where I lost it completely: in the Scholze-Nikolaus paper, the authors introduce a definition of a homotopy fixed point $\infty$ functor, acting on stable $\infty$ categories with a $G$-action.

Let $C$ be a stable $\infty$ category, the functor category $\text{Fun}(BG, C)$, denoted $C^{BG}$ is such.

  1. The homotopy fixed point functor $(-)^{hG}$ is then a functor between $C^{BG}$ to $C$, sending an object $F: BG\longrightarrow C$ to $\lim_{BG} F$.

What is a cone over this diagram? It seems that $F$ already singles out an object in $C$ together with a bunch of $1$-arrows in $C$ attached to $F(*)$, right? So isn't $F(*)$ trivially the universal cone over this diagram? This construction seems even more restrictive than the $1$-categorical one, since it fixes a $0$-symplex with an embedding of $BG$ into $\text{Map}(F(*), F(*))$, rather than spreading the objects in $C$ via a $G$ equivariant map from $EG$.

Another question I have is about the lack of $G$-equivariance in the above definition: is it implicit in the definition of a functor from one stable $\infty$-category to another that it respects the monoidal structure on the connected components of the mapping spaces?

Finally, how can we compare this definition to the $0$, or $1$-categorical definitions? Is it possible to realize this definition of homotopy fixed points as the classical fixed points in a given module with a $G$-action?

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  • $\begingroup$ The categories $C$ in Definitions 2 and 3 play entirely different roles: $C$ in Definition 2 is the looping of $C$ in Definition 3, with respect to the basepoint given by the object $F(*)$. To see Definition 1 and 2 as a special case of Definition 3 (which makes sense for arbitrary ∞-categories), take $C$ in Definition 3 to be the ∞-category of sets respectively categories and set $F(*)=S$ respectively $F(*)=C$. Finally, $F(*)$ (with identities as injection maps) does not give a cone since a cone must satisfy a commutative diagram, so $F(*)$ is a cone only if $G$ acts via identities. $\endgroup$
    – Dmitri Pavlov
    Commented Aug 16 at 13:22
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    $\begingroup$ I think it's helpful to note that $\lim_{BG} F = S^G$ already holds for a set $S$ with a $G$-action viewed as a functor $F \colon BG \to \mathbf{Set}$ (here I'm thinking of $BG$ quite explicitly as the one-object category with endomorphism monoid $G$). As homotopy limits are a well-established concept, this gives a good definition for the higher setting as well. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Aug 16 at 16:04
  • $\begingroup$ @R.vanDobbendeBruyn that's precisely the example I was trying to understand! Your replies on MO are generally extremely insightful. Thanks! $\endgroup$
    – kindasorta
    Commented Aug 16 at 17:02

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