# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• Copied over the formation/introduction/elimination rules, added beta and eta

• A skeleton

• Created skeleton page for reference, but I am not an expert on partial orders so independent verification that this is the right notion of order type for a partial order would be appreciated.

I was chatting with Robin Cockett yesterday at SYCO1. In a talk Robin claims to be after

The algebraic/categorical foundations for differential calculus and differential geometry.

It would be good to see how this approach compares with differential cohesive HoTT.

• Finally added to fracture theorem the basic statement of the “arithmetic fracture square”, hence the following discussion.

The number theoretic statement is the following:

+– {: .num_prop #ArithmeticFractureSquare}

###### Proposition

The integers $\mathbb{Z}$ are the fiber product of all the p-adic integers $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the rational numbers $\mathbb{Q}$ over the rationalization of the former, hence there is a pullback diagram in CRing of the form

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\underset{p\;prime}{\prod} \mathbb{Z}_p && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,.$

Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$

$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,.$

=–

In the context of a modern account of categorical homotopy theory this appears for instance as (Riehl 14, lemma 14.4.2).

+– {: .num_remark}

###### Remark

Under the function field analogy we may think of

• $Spec(\mathbb{Z})$ as an arithmetic curve over F1;

• $\mathbb{A}_{\mathbb{Z}}$ as the ring of functions on the formal disks around all the points in this curve;

• $\mathbb{Q}$ as the ring of functions on the complement of a finite number of points in the curve;

• $\mathbb{A}_{\mathbb{Q}}$ is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk.

Under this analogy the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare} says that the curve $Spec(\mathbb{Z})$ has a cover whose patches are the complement of the curve by some points, and the formal disks around these points.

This kind of cover plays a central role in number theory, see for instance thr following discussions:

=–

• for completeness (to go along with entries like SU(2), Spin(6), etc.) and to record a neat fact

• prompted by a question by email, I have expanded at homotopy pullback the section on Concrete constructions by listing and discussing the precise conditions under which ordinary pullbacks are homotopy pullbacks.

Most of this information is scattered around elswehere on the $n$Lab (such as at homotopy limit and right proper model category) and I had wrongly believed that it was already collected here. But it wasn’t.

• Adds ’freely generated’ as a redirection since ’free construction/freely generated’ is noted in some pages like ’cartesian monoidal category’ and ’free category’

• needed to be able to point to duality in physics, so I created an entry. For the moment just a glorified redirect.

• I have added to M5-brane a fairly detailed discussion of the issue with the fractional quadratic form on differential cohomology for the dual 7d-Chern-Simons theory action (from Witten (1996) with help of Hopkins-Singer (2005)).

In the new section Conformal blocks and 7d Chern-Simons dual.

• subdivided the Properties-section into subsections; added subsection for branched coverings of $n$-spheres

• Added some references on the categorical point of view on AECs. I think ideally this page would be rewritten to take that perspective from the start, in line with the nPOV, but I certainly don’t have the time (or background) to do that myself.

• a minimum, just for completeness and to make broken links work

• I pasted in something Mike wrote on sketches and accessible models to sketch. But now it needs tidying up, and I’m wondering if it might have been better placed at accessible category. Alternatively we start a new page on sketch-theoretic model theory. Ideas?

• Page created, but author did not leave any comments.

• I want this entry to have an actual section on construction of (compact) Examples, so I started one (here). But so far there is nothing in there apart from pointers to the original articles by Joyce and Kovalev, and a graphics illustrating Kovalev’s twisted connected sums.