Freyd was the first to formalize a striking comparison between abelian categories and topoi, showing that their exactness properties can be jointly captured by the axioms of AT categories, and the difference between the two is simply modulated by the initial objects: in every AT category, there is an initial object $0$. An object $X$ is of type A if $\pi_X : X\times 0 \to X$ is an isomorphism, and of type T if $\pi_0 : X\times 0 \to 0 $ is an isomorphism. It turns out that an AT category with all objects type A is an abelian category, and one with all objects type T is a topos. In this sense, if the initial object thinks its in a commutative ring, you have a topos, and if it thinks its in an abelian group, you have an abelian category.
On the homotopic side of things, topoi correspond to $\infty$-topoi and abelian categories to stable $\infty$-categories. I was wondering if there was any known work on whether Freyd's observations for the $1$-categorical case lift to the homotopic case? Specifically, is there a class of $\infty$-categories with some homotopy-coherent exactness properties that subsumes stable $\infty$-categories and $\infty$-topoi, and such that categories with all objects "type T" are $\infty$-topoi, and likewise for "type A" objects and stable $\infty$-categories, analogous to the above result?
To give some context, I am investigating the Goodwillie calculus, and I think the process of stabilizing an $\infty$-category corresponds to something like taking the K-theory of a commutative ring. I think $\mathbb{F}_{1}$ plays a special role in it all, as decategorifying AT categories yields some sort of "AT algebras," and the unique abelian group which is also a ring is $\mathbb{F}_{1}$, just as the unique abelian topos is $\ast$. Finding out just how it fits into the picture will help me generalize the Goodwillie calculus further, and allow for the correspondence between analysis and homotopy to be fleshed out a bit more.
