I'm a novice to homotopical algebra, but I've found myself confronted with it by necessity and have some basic questions...

I'm going to consider chain complexes over a field $F := \mathbb{F}_2$. Given a chain complex $C$, I'm interested in two operations:

- the ``homotopy Sym'', where I form $(C \otimes C \otimes E (\mathbb{Z}/2))_{\mathbb{Z}/2}$ where the $\mathbb{Z}/2$ acts diagonally on the tensor product. I'll call this $hSym^2 C$.
- if $C$ has a $\mathbb{Z}$-action, then I can form ``homotopy quotient'' $C/\mathbb{Z}$, which is $(C \otimes E \mathbb{Z})_{\mathbb{Z}}$, with $\mathbb{Z}$ acting diagonally. I don't know if there is "official" notation for this; I'll just call it $hC/\mathbb{Z}$.

(Edit: I thought I wrote this down but must have deleted it accidentally; $E G$ is a projective $F[G]$-resolution of the complex $F$ in degree $0$, which is supposed to represent a point; thus $EG$ is morally to be the chain complex of some contractible space on which $G$ acts freely.)

So my question is about how the composition of these two operations in either order are related. If $C$ has a $\mathbb{Z}$-action, then I think $hSym^2 C$ still has a $\mathbb{Z}$-action, so I could form

$$ h( hSym^2 C )/\mathbb{Z} $$

or I could do things in the opposite order:

$$ hSym^2 (hC/\mathbb{Z}). $$

Based on naive intuition about how ordinary quotients work, I guess that there should be an induced map

$$ h( hSym^2 C )/\mathbb{Z} \rightarrow hSym^2 (hC/\mathbb{Z}) $$

- Is this right?
- And if it is, then is the above map an (edit:quasi-)isomorphism? (I guess probably not in general)
- How can I understand this map explicitly? For instance, if I choose an explicit model for $E \mathbb{Z}/2$ and $E \mathbb{Z}$, like the standard ones that spit out $\mathbb{RP}^{\infty}$ and $S^1$, then I should in principle be able to write it down explicitly, but I'm confused about how that goes.

homotopy fixed pointsand usually denoted $C^{h\mathbb{Z}}$ (and in fact it behaves more like fixed points than quotients). Similarly, yourhomotopy Symis usually called arestricted powerand denoted $D_2(C)$ (although this notation might be much more common in homotopy theory than in homological algebra) $\endgroup$