The n-Category Café

The last months I have been busy with writing up a kind of thesis on the topic that I have been thinking about since a long time. This here is to present what should roughly be a version 1.0, up to proof-reading.

A pdf with the current version is behind the first link on this page, which is also the title of the opus:

Differential cohomology in a cohesive $\infty$-topos.

Below the fold is the abstract. I’d be grateful for whatever comments you might have.

I haven’t written an acknowledgement yet, because it is always hard to decide where to stop thanking people. But two names must be mentioned: I greatly profited from discussion with Mike Shulman on all general abstract aspects and from discussion with Domenico Fiorenza on the decisive aspects of concrete implementation.

Can anyone help me out on the question of how Tarski’s work with McKinsey on topological semantics of modal logic connects to Tarski’s work with Jónsson on a duality between modal algebras and (descriptive) general frames.

The Tarski-McKinsey work from 1944 established that

For any consistent theory $T$ extending the modal logic S4, there exists a topological model $(X, \mathcal{O}X, [-])$ (where $[-]$ satisfies certain conditions to make it an interpretation) that validates all and only theorems of $T$.

A proposition is interpreted as a subset of $X$, and interior plays the role of the necessity operator. A proposition, $P$, is necessarily true at a world if there’s an open neighbourhood of worlds containing that world, at each of which $P$ is true.

You can read all about this result and more in Topology and modality: The topological interpretation of first-order modal logic, where Awodey and Kishida prove an analogous result for first-order logic using neighbourhood sheaf models. The worlds are the points of a space, fibres of which are models for the first-order logic.

Khovanov homology is a knot invariant that is a categorification of the Jones polynomial.

Khovanov homology has long been expected to appear as the observables in a 4-dimensional TQFT in higher analogy of how the Jones polynomial arises as an observable in 3-dimensional Chern-Simons theory. For instance for $\Sigma : K \to K'$ a cobordism between two knots there is a natural morphism

$$\Phi_\Sigma : \mathcal{K}(K) \to \mathcal{K}(K')$$

between the Khovanov homologies associated to the two knots.

In the recent

• Edward Witten, Fivebranes and knots (arXiv:1101.3216)

it is argued, following indications in

• S. Gukov, A. S. Schwarz, and C. Vafa, Khovanov-Rozansky Homology And Topological Strings , Lett. Math. Phys. 74 (2005) 53-74, (arXiv:hep-th/0412243),

that this 4d TQFT is related to the worldvolume theory of D3-branes ending on NS5-branes as they appear in the type IIA string theory spacetime. Earlier indication for this had come from the observation that Chern-Simons theory is the effective background theory for the A-model 2d TCFT (see TCFT – Worldsheet and effective background theories for details).

This summer, Hamburg hosts the meeting

• Structured Ring Spectra - TNG

  1st to 5th August 2011, Hamburg, Germany.

  website

about structured ring spectra.

This is the next in the informal series of conferences on Structured Ring Spectra in Glasgow 2002, Bonn 2004 and Banff 2008. The conference will begin Monday morning and end Friday evening.

Here’s a somewhat frivolous question which just occurred to me. A concrete category is usually defined to mean a category equipped with a functor $C\to Set$ which is faithful, which is to say, surjective on equations (= 2-morphisms) between 1-morphisms. (One sometimes adds additional conditions as well.) In the language of stuff, structure, property, it is a category whose objects can be regarded as sets equipped with structure.

It makes sense to define a concrete 2-category to be one equipped with a functor $C \to Cat$ which is locally faithful, i.e. surjective on equations (= 3-morphisms) between 2-morphisms. For instance, the 2-category of monoidal categories is concrete in this sense: two monoidal transformations are equal if their underlying natural transformations are, but we cannot say the same for monoidal functors. Thus a concrete 2-category is a 2-category whose objects are regarded as categories equipped with extra stuff.

But what about ∞-categories? When we get all the way up to the top, it seems like there’s no condition left to impose. Is a concrete ∞-category just an ∞-category equipped with a completely arbitrary functor to ∞Cat (or ∞Gpd, if we have an (∞,1)-category)?

guest post by J. Scott Carter

An undergraduate student, David Mullens, and I recently posted a paper that presents a 4-dimensional proof of Heron’s formula. (Caution: it’s about 2.5 megabytes in size.)

You may recall from your high school geometry class that Heron’s formula tells how to compute the area of a triangle given its side lengths. The form that I want to present the formula is as follows:

$$16 A \times A = (a+b+c)(a+b-c)(a-b+c)(-a+b+c)$$

where $A$ indicates the area of the triangle, and the side lengths are $a, b$, and $c$. By convention we choose variables so that $a$ is no greater than $b$ which is no greater than $c$. Traditionally, Heron’s formula is stated in terms of the semi-perimeter, but our purpose is to demonstrate that two hyper-volumes (4-dimensional analogues of volume) are actually scissors congruent.

Peter Johnstone has a new paper out in TAC yesterday called Remarks on punctual local connectedness. Here’s the abstract:

We study the condition, on a connected and locally connected geometric morphism $p : \mathcal{E} \to \mathcal {S}$, that the canonical natural transformation $p_*\to p_!$ should be (pointwise) epimorphic — a condition which F.W. Lawvere called the ‘Nullstellensatz’, but which we prefer to call ‘punctual local connectedness’. We show that this condition implies that $p_!$ preserves finite products, and that, for bounded morphisms between toposes with natural number objects, it is equivalent to being both local and hyperconnected.

David Corfield used to study statistical inference and machine learning. This led him to become interested in some variants of the ordinary notion of entropy — for example, Tsallis entropy and Rényi entropy. They’re very odd-looking concepts when you first see them, especially if you’ve spent a lot of time studying the ordinary Boltzmann–Gibbs–Shannon–von Neumann entropy. So, I’ve never been happy with them. Whenever I saw them, I thought: what do they mean?

I just got a lot happier with Renyi entropy. Thanks to conversations at the Entropy Club here at the Centre for Quantum Technologies, especially with Oscar Dahlsten, I noticed that Renyi entropy is just a slightly disguised version of a more familiar concept in thermodynamics: free energy.

Unsurprisingly, somebody already knew this.

However, further discussions revealed that Rényi entropy is just the natural q-deformation of the ordinary notion of entropy!

You can now see the paper these blog entries are based on:

• John Baez, Division algebras and quantum theory.

But the blog entries have more jokes! So far, I’ve explained how certain complex representations of groups can be seen as arising from real or quaternionic representations. This gives a sense in which ordinary complex quantum theory subsumes the real and quaternionic theories. But there’s also a sense in which all three theories have equal priority. This idea can be seen already at the level of Hilbert spaces, even before group representations enter the game.

For this we need to think about categories of Hilbert spaces. As usual, let $\mathbb{K}$ be either $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Now, let $Hilb_\mathbb{K}$ be the category where:

• an object is a $\mathbb{K}$-Hilbert space;
• a morphism is a bounded $\mathbb{K}$-linear operator.

I want to show you how any one of the categories $Hilb_\mathbb{R}$, $Hilb_\mathbb{C}$ and $Hilb_\mathbb{H}$ can be embedded in any other. This means that a Hilbert space over any one of the three normed division algebras can be seen as Hilbert space over any other, equipped with some extra structure!

So if you ask which is fundamental: real, complex or quaternionic quantum theory, there’s a certain sense in which the answer is: take your pick!

Jacob Lurie has made his new book available on his website:

after

• Higher Topos Theory

now

• Higher Algebra.

Eric Sharpe asks me to announce the following, which I gladly do:

There is planned a new conference series meant to be anlogous to the standard annual string theory meetings but concentrating on mathematical/structural aspects of string theory. The inaugural meeting of this series is

• String-Math 2011

  UPenn, Philadelphia,

  June 6-11, 2011

  conference website

Here is the description from the conference website, equipped with $n$Lab links, for your convenience:

For mathematics, string theory has been a source of many significant inspirations, ranging from Seiberg-Witten theory in four-manifolds, to enumerative geometry and Gromov-Witten theory in algebraic geometry, to work on the Jones polynomial in knot theory, to recent progress in the geometric Langlands program and the development of derived algebraic geometry and $n$-category theory. In the other direction, mathematics has provided physicists with powerful tools, ranging from powerful differential geometric techniques for solving or analyzing key partial differential equations, to toric geometry, to K-theory and derived categories in D-branes, to the analysis of Calabi-Yau manifolds and string compactifications, to the use of modular forms and other arithmetic techniques.

This is the first conference in a series of large meetings bringing together mathematicians and physicists who work on ideas related to string theory. String theory, as well as quantum field theory, have contributed a series of profound ideas which gave rise to entirely new mathematical fields and revitalized older ones. The conference aims to engage the large and rapidly growing number of mathematicians and physicists working at the string-theoretic interface between the two academic fields and to facilitate the flow of ideas with mathematical techniques and ideas contributing crucially to major advances in string theory.

Topics to be covered include but are not limited to

• Homological mirror symmetry
• Categorical constructions of topological field theories
• Mathematical string phenomenology
• F-theory
• Wall crossing formulas
• Geometric Langlands
• Arithmetic of strings
• Gromov-Witten theory and related enumerative geometry
• A-twisted Landau-Ginzburg models
• String topology
• Elliptic cohomology
• Heterotic mirror symmetry
• Topological T duality
• String measures
• Chiral de Rham complexes
• Noncommutative geometry