Generalizations of tangent $\infty$-topos

Question

If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\mathbf{H}/X$. Then the category $T \mathbf{H}$ is an $\infty$-topos (see this page for a proof). The question is: can this fact be generalized to other Cartesian fibrations constructed in a similar way? For example, if we have an essentially algebraic $\infty$-theory $S$, then we can define in a similar way a Cartesian fibration $p_S : T_S \mathbf{H} \to \mathbf{H}$ such that the fiber of $p_S$ over $X$ is the $\infty$-category of models of $S$ in $\mathbf{H}/X$. Is it true that $T_S \mathbf{H}$ is an $\infty$-topos?

Answer 1

This is rarely true. For example, the axioms for ∞-topoi imply that, if $T_S\mathbf H$ is an ∞-topos, then the fibers of $p_S$ must have van Kampen pushouts (more generally van Kampen weakly contractible colimits). In particular, the fibers cannot be $n$-categories for any finite $n$, which rules out essentially algebraic ∞-theories that are $n$-categories. This also fails for most Lawvere theories: $A_\infty$-spaces, simplicial commutative rings, ... For a concrete example, $\mathbb Z[x,y]=\mathbb Z[x]\otimes_{\mathbb Z}\mathbb Z[y]$ is not a van Kampen pushout of simplicial commutative rings, since both $\mathbb Z$ and $\mathbb Z[xy]$ have the same intersections with $\mathbb Z[x]$ and $\mathbb Z[y]$.

Sufficient and almost necessary conditions for a Cartesian fibration over an ∞-topos to be an ∞-topos are described in my paper http://www-bcf.usc.edu/~hoyois/papers/loci.pdf. In particular, if you consider the Cartesian fibration $\mathbf E\to\mathbf H$ whose fiber over $X\in\mathbf H$ is $\mathbf H/X \otimes C$ for some fixed presentable ∞-category $C$ (which I think includes your example), then $\mathbf E$ is an ∞-topos iff weakly contractible colimits in $C$ are van Kampen. Joyal calls such an ∞-category $C$ an ∞-locus (see his IHES talk). Presentable stable ∞-categories are examples of such.

Answer 2

Most directly, this fact generalizes to the whole Goodwillie tower, which are certainly "constructed in a similar way".

I don't think it generalizes to all algebraic theories.

However, I think one can at least get local cartesian closure under fairly general conditions. Let $\mathbf H$ be a locally cartesian closed presentable $\infty$-category, and let $F: \mathrm{Pres}^L \to \mathrm{Pres}^L$ be an $(\infty,2)$-functor where $\mathrm{Pres}^L$ is the $(\infty,2)$-category of presentable $\infty$-categories and left adjoint functors. The construction $X \mapsto \mathbf H/X$ is contravariant in $f$, and for $f: X \to Y$, the pullback $f^\ast: \mathbf H/Y \to \mathbf H/X$ has a right adjoint $\Pi_f$ by local cartesian closure. Since $F$ is 2-functorial, the composite functor $X \mapsto F(\mathbf H/X), \mathbf H^\mathrm{op} \to \mathrm{Pres}$ is classified by a cartesian fibration $\mathbf H^F \to \mathbf H$ for which the reindexing functors have left adjoints (because they are in $\mathrm{Pres}^L$) and right adjoints (coming from the right adjoints $\Pi_f$ -- here the 2-functoriality of $F$ is crucial). This is sometimes called a trifibration. Since pullback along a morphism in the total category $\mathbf H^F$ is a composite of a pullback in $\mathbf H$ and reindexing in $\mathbf H^F$, it has both left and right adjoints. So $\mathbf H^F$ is a locally cartesian closed presentable $\infty$-category. It's also presentable by a theorem of Gepner, Haugseng, and Nikolaus.

Notes:

• Clearly, $F$ need only be defined on a suitable sub-2-category of $\mathrm{Pres}^L$ containing the categories $\mathbf H/X$ for this to work.

• Moreover, for this to work, $F$ only needs a small amount of 2-functoriality -- it needs to preserve adjunctions. Equivalently, it needs to send limit-preserving functors to limit-preserving functors. For example, when you describe the stabilization functor in terms of spectrum objects, it's clear that it has this property, so you can recover that the tangent $\infty$-category is locally cartesian closed this way.

• Perhaps considerations along these lines could give criteria for $\mathbf H^F$ to have object classifiers and hence be an $\infty$-topos.