@Tyler, Thanks! I meant a reference for the [-j:0] truncations preserving symmetric monoidal structure

@DylanWilson, sorry, I should have put up a co-trigger warning

@TylerLawson, thanks! Do you have a reference?

The notation $K(A)$ for the homotopy category is unusual, and there is no reasonable functor from the homotopy category to $D(A)$. By $K(A)$ do you mean the category of cofibrant objects?

Did I get my arrows backwards? The map $X^i\wedge \Sigma^\infty BS_i\to X$ should canonically preserve the connective part

Thanks! This is a great example that I should have thought of before posting. I suppose it shows that the dga $\text{End}^*_{\mathbb{P}^1}(\mathcal{O}_P)$ is a deformation of $\text{End}^*_{\mathbb{P}^1}(\mathcal{O}\oplus \mathcal{O}(-1)[1]),$ but they are not deformations in the opposite order (which is why I was curious about this)

I am open to interpretation. But a simple working definition on an affine variety $\text{Spec}(R)$ would be a compact object in the category of objects of $\mathcal{C}$ with $R$-action.

You mean non-existence of nontrivial deformations? Thanks :)