The n-Category Café

Here is the simple idea of infinitesimal cohesion: any coreflective embedding of categories (and then of cohesive $\infty$-toposes) $i : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$ can be understood as providing in $\mathbf{H}_{th}$ infinitesimal thickenings of the objects of $\mathbf{H}$. We think of the coreflector $Red : \mathbf{H}_{th} \to \mathbf{H}$ as contracting away any infinitesimal extension. Then if for instance $D \in \mathbf{H}_{th}$ is an infinitesimally thickened point in that $Red(D) = pt$ is the terminal object, then the reflector adjunction tells us that while there may be many morphisms $D \to i(X)$ (many $D$-shaped tangents in $X$) there is only a single map $i(X) \to D$, that which factors through the unique points $* \to D$. This reflects the fact that the infinitesimal extension of $D$ is “so small” that it is not visible to points in $C$.

Therefore it makes sense to say that a cohesive $\infty$-topos $\mathbf{H}$ is equipped with infinitesimal cohesion if it is equipped with a suitable coreflective embedding into another cohesive $\infty$-topos $\mathbf{H}_{th}$. Simple as this axiom is, it does imply a wealth of structures, and of the right structures, just as the simple but powerful axioms on a cohesive $\infty$-topos themselves do.

For instance passing to the adjoint triple induced by the adjoint quadruple of morphism that comes with the cohesive inclusion $\mathbf{H} \hookrightarrow \mathbf{H}_{th}$ is readily seen to yield the general abstract context of which Carlos Simpson and Constantin Teleman in their de Rham theorem for ∞-stacks consider some implementations.

This I knew for a while, thanks to David Ben-Zvi. What I only learned recently, thanks to Zoran Škoda persistenly pushing me, is that this abstract notion of infinitesimal cohesion is essentially (of course in different terminology and up to some slight differences in assumptions and in focus) the very content of the work

• Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces , MPI preprint 2004 (pdf)

This article aims to lay foundations for a kind of topos theory suitable for noncommutative geometry. Based on the observation (which I am not qualified to comment on at the moment) that the “right” notion of covers in noncommutative geometry does not yield a Grothendieck (pre)topology, hence not a sheaf topos, the authors suggest a modified notion of “sheaf”. The notion of sheaves on a Q-category that they propose and study is – with just a slight extra assumption that is satisfied in most of their examples – this: on a coreflective (co)site $C \hookrightarrow C_{th}$ consider the induced coreflective inclusion $u : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$ of (cohesive, in the examples) toposes. Then they consider the subcategory of objects $X \in \mathbf{H}$ for which the canonical morphism

$$u^\ast X \to u^! X$$

is an epi, a mono or an iso, depending on taste and application (see below).

But notice: this canonical morphism and its localizations is an old friend of ours, we have discussed it at some length here before: Bill Lawvere in Axiomatic cohesion had proposed that this, and its dual, is a crucial structure in any cohesive topos. In fact, he looked at the dual localization: the subcategory of objects $X$ for which not $u^\ast X \to u^! X$ but $u_\ast X \to u_! X$ is an isomorphism.

Moreover, Kontsevich and Rosenberg notice that apart from the original motivation of characterizing noncommutative spaces as such, these “Q-categorical epi/mono(pre)sheaves” or whatever we should call them, naturally serve to characterize infinitesimal extensions. They show that it is sensible to make the following general abstract definition:

for $u : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$ an infinitesimal extension of cohesive $\infty$-toposes (my paraphrase, of course, but it does apply to their classes of examples) we say that an object $X \in \mathbf{H}$ is

• formally smooth if $u^\ast X \to u^! X$ is an epimorphism – for every morphism $Y \to X$ and every infinitesimal extension of $Y$ there is at least one extension of the morphism;

• formally unramified if $u^\ast X \to u^! X$ is a monomorphism – for every morphism $Y \to X$ and every infinitesimal extension of $Y$ there is at most one extension of the morphism;

• formally étale if $u^\ast X \to u^! X$ is an isomorphism – for every morphism $Y \to X$ and every infinitesimal extension of $Y$ there is exactly one extension of the morphism;

(Or rather, there is an evident relative version of these statements, which you can see behind the links provided above.)

When you unwind all the definitions while working over the site of ordinary algebraic geometry, hence over formal duals of commutative algebras, it is eventually straightforward to see that this abstract definition does reproduce the traditional one. But the article argues that the abstract notion here also produces the right concept of formal smoothness, etc., in noncommutative geometry.

Moreover, while not mentioned in Kontsevich-Rosenberg, it is straightforward to see that $u^\ast X \to u^! X$ being epi is formally equivalent to the morphism $X \to dR(X) = \mathbf{\Pi}_{inf}(X)$ into the intrinsically defined de Rham space being an epimorphism, as it should be for formally smooth $X$.

So it seems there is a nice, simple, useful and powerful formal axiomatization of infinitesimal cohesion using cohesive $\infty$-toposes that does subsume various suggestions that are floating around in the preprint literature.