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slightly edited AT category to make the definition/lemma/proposition-numbering and cross-referencing to them come out.
Probably Todd should have a look over it to see if he agrees.
Any new thoughts on your question of whether there’s some relation to the Goodwillie calculus?
Indeed, it was some thinking in this direction that made me come back to the entry of AT-categories. But I am not sure yet.
I need an axiomatic way to characterize an ambient $\infty$-topos such that to each object $X$ is associated a closed symmetric monoidal $\infty$-category $E Mod(X)$ and to suitable morphisms $f : X \to Y$ a Wirthmüller context $(f_! \dashv f^\ast \dashv f_\ast) : E Mod(X) \to E Mod(Y)$.
In tangent cohesion we have something close: here it is immediate to say $GL_1(E) Mod(X)$ and so forth. I have to see how much that helps me.
Specifically I am interested in $E_\infty$-rings $E$ that are obtained from free $E_\infty$-rings on an abelian $\infty$-group (connective spectrum) after inverting some elements. For these I might actually get away with $GL_1(E) Mod$. Not sure yet.
Is there any way to get at a $(\infinity, 1)$-version of Freyd’s AT result? Are stable (infinity,1)-categories the right analogue of abelian categories? Would there need to be a notion of $(\infinity, 1)$-pretoposes?
Urs #1: not only do I agree, but I thank you. I’ve taken advantage of your improved structuring by rewriting some of the proofs so that they link back in logical fashion.
There’s something I find a little ungainly though in Freyd’s development; intriguing though the result is, the axioms look pretty ad hoc, and that probably explains why there hasn’t been much work on them. Perhaps this should be revisited.
I think David has a good point, in that in the context of $\infty$-category theory the role of abelian categories is played by stable $\infty$-categories, and these already share some key abstract properties with $\infty$-toposes.
The most striking one is probably the “stable Giraud theorem” which says that presentable stable $\infty$-categories are precisely the lex reflective acessible localizations of categories of presheaves of spectra, just like sheaf $\infty$-toposes are precisely the lex reflective accessible localizations of the categories of presheaves of homotopy types.
As a slogan this means: stable $\infty$-categories are to $\infty$-toposes as stable homotopy types are to homotopy types.
Now for a given $\infty$-site of definition, the tangent (infinity,1)-category of the $\infty$-topos over that site, unifies both of these: it has both the $\infty$-topos as well as the stable $\infty$-category over that site as full sub-$\infty$-categories.
At this point one seems to be pretty close to what Freyd was after, maybe one should play with this a bit more.
Since at AT category there’s a topos version of the idea TA category after the main pretopos version, does Mike’s new definition of elementary (infinity,1)-topos suggest anything for my question in #4?
Another thought, should there be a concept of $(\infty, 1)$-pretopos?
Suppose $\mathcal{A}$ is an abelian category, $\mathcal{T}$ is a pretopos, and $G:\mathcal{A} \to \mathcal{T}$ is a left exact functor (such as the forgetful functor $Ab\to Set$). Then unless I am mistaken, then Artin gluing $(\mathcal{T} \downarrow G)$ is a “weak AT category”, i.e. it satisfies Freyd’s universal Horn axioms but not the existential AE axiom. So that axiom is actually doing something to cut down the class of models, and there are interesting models of the rest of the axioms that aren’t products. I wouldn’t be surprised if there are other axioms we can impose to characterize such gluings among the AT categories, similarly to how every quasitopos is the Artin gluing of a Heyting algebra with an extensive quasitopos (A2.6.7 in the Elephant).
This also suggests to me that AT categories are not doing quite the same thing as tangent $\infty$-categories: both of them are a way to put together an abelian thing with a topos thing, but the ways of putting them together are different. I wonder what elementary axioms are satisfied by categories like $Fam(Ab)$, the obvious 1-categorical analogue of parametrized spectra?
Actually, here is one additional axiom on a weak AT category that suffices to enable reconstruction of the functor $G$ from $(\mathcal{T}\downarrow G)$: the object $0$ is exponentiable (in the cartesian sense). This is true in a topos, of course, but also in any pretopos since $A^0=1$, and also in any abelian category where $A^0=A$. And in $(\mathcal{T}\downarrow G)$, for $A = (A_0, A_1, A_1 \to G(A_0))$ we have $A^0 = (A_0, G(A_0), id_{G(A_0)})$. So for a type A object $A$, which is of the form $(A_0, 0, 0\to G(A_0))$, we have $T(A^0)$ being the type T object corresponding to $G(A_0)$, namely $(1, G(A_0), !)$.
In $(\mathcal{T}\downarrow G)$, the operation $A\mapsto A^0$ is a right adjoint modality to $A\mapsto A\times 0$, giving two different ways to embed $\mathcal{A}$ into the gluing category. In general, once this is true, if we define $G(A) = T(A^0)$, then the map $A^0 \to T(A^0)$ induces a functor from $\mathcal{C}$ to the gluing category of this $G$ (as a functor from the abelian category of type A objects to the pretopos of type T objects), and so there should be axioms ensuring that this functor is an equivalence.
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