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Stream: theory: mathematics

Topic: Concrete use of the sharp modality

Alonso Perez-Lona (May 18 2026 at 16:42):

In the context of cohesive topoi, we speak of the three functors: flat, shape, and sharp. In particular, only the first two play an important role in the definition of the differential cohomology hexagon. How does one understand concretely the sharp modality and its uses? Are there some notable examples of images under it? I am particularly curious whether the hexagon further extends by considering how flat and sharp interact as adjoints/

David Corfield (May 18 2026 at 18:29):

For some context: [[differential cohomology diagram]]

Adrian Clough (May 18 2026 at 18:29):

Here are a conceptual and practical POV on the sharp modality.

Adrian Clough (May 18 2026 at 18:35):

Conceptually, the sharp modality tells you that your topos (let's call it $\mathcal{E}$) has a final object when viewed as on object in the 2-category of toposes, i.e., the unique geometric morphism $\mathcal{E} \to \mathcal{S}$ to the topos of sets / anima admits a right adjoint. (Note that in the 2-category of categories, a category $C$ has a final object iff the unique morphism $C \to 1$ has a right adjoint). This essentially tells you that the final object (in usual sense) in $\mathcal{E}$ geometrically looks like a point.

Adrian Clough (May 18 2026 at 18:37):

Practically, the sharp modality allows you to define concrete objects as objects $X$ in $\mathcal{E}$ for which the canonical morphism $X \to \sharp X$ is a monomorphism. This makes it fairly easy for example to prove that that the category of concrete objects in $\mathcal{E}$ is presentable.

Adrian Clough (May 18 2026 at 18:39):

I don't know how $\sharp$ interacts with differential cohomology.

Alonso Perez-Lona (May 18 2026 at 18:52):

Adrian Clough said:

Practically, the sharp modality allows you to define concrete objects as objects $X$ in $\mathcal{E}$ for which the canonical morphism $X \to \sharp X$ is a monomorphism.

Thanks, Adrian. So for example, in the case $X \to \sharp X$ is a monomorphism, how to understand the "extra" information in $\sharp X$? For instance, how to understand the morphisms $X \to Y$ for which there is a $\sharp X \to Y$ closing the triangle?

David Corfield (May 18 2026 at 19:15):

There's a way to understand the difference between intensive and extensive quantities via $\sharp$.

Adrian Clough (May 18 2026 at 19:25):

Alonso Perez-Lona said:

Thanks, Adrian. So for example, in the case $X \to \sharp X$ is a monomorphism, how to understand the "extra" information in $\sharp X$? For instance, how to understand the morphisms $X \to Y$ for which there is a $\sharp X \to Y$ closing the triangle?

I don't have a particularly sophisticated view on what $\sharp$ does to $X$. If $X$ is a manifold in the cohesive topos of sheaves on manifolds, then $\sharp X$ is essentially the underlying set of $X$ with the trivial topology. I guess a morphism $X \to Y$ factoring as $X \to \sharp X \to Y$ just means that the image of $X \to Y$ has the trivial topology in some sense.

Alonso Perez-Lona (May 18 2026 at 19:36):

David Corfield said:

There's a way to understand the difference between intensive and extensive quantities via $\sharp$.

I see, this and the perspective in terms of generalized homology in the same nLab page is very helpful, thanks!