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Urs Schreiber
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Urs Schreiber

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Manifolds and Orbifolds... and  Étale ∞-Groupoids, that's the topic of the next session of the Higher Cartan Geometry seminar [1]. Lecture notes are here:

http://ncatlab.org/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds

We'll first see how to reformulate the definition of manifolds in terms of a modal operator, then see that passing this to smooth groupoids gives  étale groupoids and hence orbifolds, and then further passing all the way to smooth infinity-stacks gives what elsewhere are called higher or derived schemes, in much generality. We'll see how from the neat axiomatization via that model operator follow immediately, by some elementary categorical yoga, that these higher manifolds/étale ∞-groupoids have frame ∞-bundles and that the frame ∞-bundle of a cohesive ∞-group is trivialized by ∞-left translation.

That's the structure that one needs in order to set up higher Cartan geometry proper, which will be the topic of the session after the next one.

[1] http://ncatlab.org/schreiber/show/Higher+Cartan+Geometry 
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German Mathematical Society meeting 2015 takes place in Hamburg this September. I'll be giving two talks:

One in the "Algebra"-section and one in the "Homotopy Type Theory"- mini-symposium:

1) Obstruction Theory for Parameterized Higher WZW-Terms. I present a general characterization of the obstructions for higher WZW-terms (higher gerbes with connection) defined on some higher (or derived) group stack G/H to have a parameterization over higher Cartan geometries locally modeled on G/H. Applied to the canonical Kostant-Souriau line bundle the construction reproduces metaplectic pre-quantization. For the traditional degree-3 WZW term it reproduces the Green-Schwarz anomaly; for the degree-7 WZW term we get a Fivebrane-analog, for the degree-11 term a Ninebrane-analog. Applied to the exceptional cocycles on extended super-Minkowski spacetimes the construction yields a forgetful infinity-functor on globally dened (classical anomaly free) Green-Schwarz super p-brane sigma models propagating on higher super etale stacks, which sends these to G-structures on these super stacks, for G the higher Heisenberg group stack of the higher WZW term. Specically for the super-5-brane sigma-model this yields a forgetful infinity-functor from its classical anomaly free backgrounds to super etale 3-stacks satisfying the equations of motion of 11-dimensional supergravity and satisfying a further topological constraint. Notes are here: http://ncatlab.org/schreiber/show/Obstruction+theory+for+parameterized+higher+WZW+terms


2) Some thoughts on the future of modal homotopy type theory. In 1991 Lawvere suggested a) that the future of category theory revolves around toposes equippped with adjoint system of idempotent (co-)monads [1] and that b) this is formalization of what the ancients had called the "objective logic" [2]. While for 1-toposes this seems inconclusive, one finds [3] that internal to infinity-toposes equipped with such adjoint systems much of higher differential geometry and of modern physics has a succinct and useful synthetic formalization. But here the syntax of this internal language is modal homotopy type theory [4]. In this talk I survey the immensely rich semantics and the potential prospects of  its full syntactic formalization, in the hope to motivate the type theory community to further look into this fascinating but under-explored aspect of their theory.

[1] http://ncatlab.org/nlab/show/Some+Thoughts+on+the+Future+of+Category+Theory
[2] http://ncatlab.org/nlab/show/Tools+for+the+advancement+of+objective+logic
[3] http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos
[4] http://ncatlab.org/schreiber/show/Quantum+gauge+field+theory+in+Cohesive+homotopy+type+theory
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I see what you are getting at. I may very well be naive about how much of the conference cost is siphoned off by para-conference profiteers. And you also have a valid point that not all potential participants would have a local travel/expense budget to cover these costs.
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Groups, representations and associated bundles, that's the topic of the next session of the Higher Cartan Geometry seminar [1]. I will indicate how to set this up such that it seamlessly, really verbatim, generalizes to infinity-groups, infinity-respresentations and associated infinity-bundles. Lecture notes for this sesssion are at [2] and [3]. Oh, and last time we already talked about principal bundles along these lines [4].

[1] http://ncatlab.org/schreiber/show/Higher+Cartan+Geometry
[2] http://ncatlab.org/nlab/show/geometry+of+physics+--+groups
[3] http://ncatlab.org/nlab/show/geometry+of+physics+--+representations+and+associated+bundles
[4] http://ncatlab.org/nlab/show/geometry+of+physics+--+principal+bundles
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More on the shape modality [1] for smooth infinity-stacks [2] has been worked out in a neat new article by David Carchedi 

David Carchedi,
"On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces"
http://arxiv.org/abs/1504.02394

David shows that the previous result that the left adjoint (shape) to the constant smooth infinity-stack functor sends simplicial manifolds to their geometric realization homotopy type generalizes to non-Hausdorff manifolds. He then uses this to discuss the geometric realization of Haefliger-type groupoids [3] and obtains this way some new statements about the homotopy type of classifying spaces of the correcponding geometric structures.

[1] http://ncatlab.org/nlab/show/shape+modality
[2] http://ncatlab.org/nlab/show/smooth+infinity-groupoid
[3] http://ncatlab.org/nlab/show/Haefliger+groupoid
Abstract: We describe various equivalent ways of associating to an orbifold, or more generally a higher \'etale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack X arising ...
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One vision I have is that recent advances in foundations will eventually serve to bridge the communication gap, present in our age, between physics, mathematics and philosophy. There is the real chance that with something like homotopy type theory being a practical concept logic, and with a decent and workable formalization of physics inside it, there will appear common ground on which to meet and have genuinely fruitful, technically precise discourse. In the note linked to below, David Corfield is exploring how formalized reasoning in homotopy type theory may help sort out issues that practicing philosophers these days are having inconclusive debates on. Conversely, once this gets off the ground, it has the chance to equip the abstract formal calculus with more meaning, more connection to the body of human spirit.
 
Here's a brief note I put together on what 'The structure of' might mean in homotopy type theory.
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There is a related question on PhilosophyStackExchange: http://philosophy.stackexchange.com/a/23026/5473
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#spnetwork

Gauge AQFT done right. It is a well-kept secret that even though AQFT [1] was originally motivated from Yang-Mills gauge theory, there has in all the decades not been a formulation that would account both for the gauge principle as well as for the locality principle of QFT: a simple general argument [2] shows that such a formulation necessarily must have groupoid valued nets of observables, coming from stacks of fields. But this was never considered...

...until recently when +Alexander Schenkel and coauthors started to attack this open problem. I had reported on that here earlier [3]. Now they are out with a fully-fledged article that does it right, for the case of electromagnetic gauge fields:

Marco Benini, Alexander Schenkel, Richard J. Szabo,
"Homotopy colimits and global observables in Abelian gauge theory"
http://arxiv.org/abs/1503.08839

This is the future of AQFT, check it out.


[1] http://ncatlab.org/nlab/show/AQFT
[2] http://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields
[3] https://plus.google.com/+UrsSchreiber/posts/6ASLBSjzsZA
Abstract: We study chain complexes of field configurations and observables for Abelian gauge theory on contractible manifolds, and show that they can be extended to non-contractible manifolds by using techniques from homotopy theory. The extension prescription yields functors from a category of ...
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See around example 5.5.34 here: https://dl.dropboxusercontent.com/u/12630719/dcct.pdf
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Deligne's proof of "Why supersymmetry?" [1] is well-known, but not well-known enough. 

By Wigner's classification [2] fundamental particles in a quantum field theory are equivalently the irreducible representations of the symmetry group of the local model for spacetime. Given many such particles, then the relevant representation is the tensor product of these. Hence the particle content of a field theory forms a complex-linear tensor category [3] and much of what one ever wants to do with these particles, notably writing down their Feynman diagrams, is encoded in just this tensor category structure.

Hence one may turn this around: build a scattering theory based on just a nice enough tensor category, and then ask what kind of spacetime with what kind of symmetry that is the category of irreducible representations of.

This question was solved by Pierre Deligne [4] in (or before) 2002 in 

Catégorie Tensorielle,
Moscow Math. Journal 2 (2002) no. 2, 227-248. 
https://www.math.ias.edu/files/deligne/Tensorielles.pdf
http://ncatlab.org/nlab/show/Deligne+theorem+on+tensor+categories#Statement

There he shows that, under mild assumptions (excluding the case that the tensor category is obscenely large, set theoretically), all complex-linear tensor categories are the categories of representations of supersymmetries, hence the categories of what physicists call "supermultiplets" [5].

[1] http://ncatlab.org/nlab/show/Deligne+theorem+on+tensor+categories
[2] http://ncatlab.org/nlab/show/Wigner+classification
[3] http://ncatlab.org/nlab/show/tensor+category
[4] http://ncatlab.org/nlab/show/Pierre+Deligne
[5] http://ncatlab.org/nlab/show/super+multiplet
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The Reader Monad and Random Variables. The function monad [W,-] (aka "reader monad" [1]) has the right to be thought of as producing spaces of random variables on the "probability space of possible worlds" W.

Who agrees with this, in citable print?

I see that +Olivier Verdier does [2]:

"The intuition behind the Reader monad, for a mathematician, is perhaps stochastic variables. A stochastic variable is a function from a probability space to some other space. So we see a stochastic variable as a monadic value."

Also Toronto-MacCarthy10 have this statement [3, slides 23, 24, 35]:

"you could interpret this by regarding random variables as reader monad computations".

Who else?

[1] http://ncatlab.org/nlab/show/function+monad
[2] http://www.olivierverdier.com/posts/2014/12/31/reader-writer-monad-comonad/
[3] http://jeapostrophe.github.io/home/static/toronto-2010ifl-slides.pdf
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Ah -- so necessarily is "in every world p holds the 'same way'" and possibly is "in some world, p holds" and "contextually" or "configurably" or "randomly" is "in every world, p holds, but not necessarily in the same way" or "p obeys a distribution across the worlds". (the latter being much more suggestive!). I see the idea at least, I think.
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Regarding Russell's famous attitude [1] towards Aristotle's logic [2], I have come to wonder if in the course of a justified frustration about an enormous time span of intellectual stagnation, Russell maybe missed an opportunity to recognize a few excellent aspects of Aristotle's logic, which, after dismissing them, took much effort to be rediscovered by Russell himself and then eventually by other people.

Here I am thinking of the fact that Aristotle's logic, while certainly naive and inaccurate from a modern perspective, has the exceptional feature of being a primitive kind of type theory [3].

Where Aristotle says "All A are B" we should recognize what in modern systems would be a function of types A -> B (maybe a monomorphism, if you insist, making A a subtype of B).

Where Aristotle says "Some B1 are B2" this is clearly to be read as an intersection of types, what in modern categorical logic is called a fiber product.

From this perspective, two complaints that are frequently raised against Aristotle's logic seem to be easily transmuted into virtues:

Some of Aristotle's deductions really depend on some silently assumed context. While that means that these were inaccuracies back then, today we easily know how to fix this right away: all types should be regarded as dependent types [4] that exist in some context, to be specified.

Another common complaint is that as Aristotle's types move from the subject of a judgement to the predicate, they seem to turn from types to propositions. For instance on the one hand Aristotle speaks of the collection of all mortal beings, on the other hand he speaks of the proposition "X is mortal". But there is no need to complain about this, in fact this very conflation is a famous accomplishment of modern logic, famous as the Curry-Howard isomorphism or the propositions-as-types paradigm. [5]

Summing up then, we see a logic with a concept of function types and product types formed in context. That's precisely the ingredients of locally Cartesian closed categories of types, which is what modern dependent type theory is about [6]

This is a bit ironic, because it is Russell him very self who, right after rejecting Aristotle, runs into the paradoxes of the young modern logic and is then the one to introduce the modern fix to these: types. See the references [7].

I came to think of this when following Lawvere's suggestion to keep an eye open for hidden insights in Hegel that are invisible to first-order logic but that begin to make a great deal of sense in modern categorical logic and type theory. Since Hegel likes syllogisms, that made me wonder. I have collected a few further details on what I have in mind at [8].


[0] http://philosophy.stackexchange.com/a/23060/5473

[1] http://ncatlab.org/nlab/show/Logic+as+the+Essence+of+Philosophy
[2] http://ncatlab.org/nlab/show/Aristotle's+logic
[3] http://ncatlab.org/nlab/show/type theory
[4] http://ncatlab.org/nlab/show/dependent+type+theory
[5] http://ncatlab.org/nlab/show/propositions+as+types
[6] http://ncatlab.org/nlab/show/relation between category theory and type theory
[7] http://ncatlab.org/nlab/show/type%20theory#References
[8] http://ncatlab.org/nlab/show/Science+of+Logic#FormalizationText
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+Dan Piponi , thanks for saying this, this is much the feeling I have, too.

 The criticisms raised by Russell in [1] seem to be attacking straw men more than the actual substance. But,  conversely the mainstream opinion then that "Aristotle had discovered everything there was to know about logic" [2] was just as fallacious, and I suppose Russell was rebelling really against this intellecutal laziness. But in any case, we as of today should be able to look back at it all and draw some useful conclusions. I am aware of only two articles that go in the direction of suggesting that Aristotle's logic has a useful formalization in categorical logic or type theory

Marie La Palme Reyes, John Macnamara, Gonzalo Reyes ,
"Functoriality and Grammatical Role in Syllogisms"
Notre Dame J. Formal Logic 35 no.1 (1994) pp.41-66.
http://projecteuclid.org/euclid.ndjfl/1040609293

and

Ruggero Pagnan,
"A diagrammatic calculus of syllogisms",
Journal of Logic, Language and Information July 2012, Volume 21, Issue 3, pp 347-364 
http://arxiv.org/abs/1001.1707

Both contain the proposal that Aristotle's "All A are B." is to be interpreted as a function between types. I think that's the right starting point, but then from there on both these articles continue to make it much more complicated than it really is, it seems to me. 

[1] http://ncatlab.org/nlab/show/Logic+as+the+Essence+of+Philosophy
[2] http://plato.stanford.edu/entries/aristotle-logic/
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Last week I spoke at a meeting of philosophers of science on the issue of formulating physics in homotopy type theory. In my expanded talk notes [1] I survey some basic ideas of physics in cohesive homotopy theory that you may have seen before, but also present some new material. There is now a progression of three triples of adjoint modal operators, the observation that this triple has a faithful model in super formal smooth infinity-groupoids ("the reflective subcategories of supergravity" [2]), the observation that this serves to axiomatically capture the real line continuum R as well as the superpoint R^0|1 and finally some observations to the effect that given only these two objects and universal constructions, there naturally emerges super-Minkowski spacetime (and then further the brane bouquet of string theory).

Being the first time that I was invited by philosophers, I thought I'd be entitled to highlight a bit of metaphysics beneath the mathematics. This is esoteric and I expect only a handful of humans to appreciate it.  Dies zu wissen ist nicht not.

[1] http://ncatlab.org/schreiber/show/Modern+Physics+formalized+in+Modal+Homotopy+Type+Theory

[2] http://ncatlab.org/schreiber/show/The+%28co-%29reflective+categories+of+supergravity
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Thanks, +Gershom B , that sounds all most intriuguing. I hope the people working on this will eventually find the leisure to write up a comprehensive account.
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Smooth groupoids (smooth stacks) and Examples is the topic of the upcoming session of the Higher Cartan Geometry seminar [1].

I'll first discuss the basics of smooth groupoids (smooth stacks) [2] and then look into the examples obtained from Lie integration [3] (we did not actually get to last time).

I am slowly merging all this material into the chapter "Smooth homotopy types" [4] of the "Geometry of physics"-series, but that's not done yet



[1] http://ncatlab.org/schreiber/show/Higher+Cartan+Geometry

[2] http://ncatlab.org/nlab/show/smooth+groupoid#PreSmoothGroupoids

[3] http://ncatlab.org/nlab/show/Lie+integration

[4] http://ncatlab.org/nlab/show/geometry+of+physics+--+smooth+homotopy+types
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+Dmitri Pavlov thanks for alerting me, fixed now. Thanks.
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NCG with all noncommutativity in a nilpotent ideal.  While in general non-commutative geometry behaves rather differently from commutative geometry when it comes to local-to-global properties (descent), there are versions of "mild" noncommutative geometry that behave very much like commutative geometry in this respect. The archetypical example here is supergeometry.

One may argue that the reason that the theory of supergeometry proceeds in close analogy with ordinary differential geometry is simply because a supercommutative algebra is just a commutative algebra, but internal to a nontrivially braided symmetric monoidal category. On the other hand when it comes to local properties and the fact that Grothendieck topologies work well in supergeometry, this is to do more specifically with the fact that the non-commutativity is all in the nilpotent ideals of supercommutative superalgebras, and hence irrelevant to coverings and descent.

This leads one to wonder if there is something to be gained in developing a geometry based on formal duals to those noncommutative algebras for which "all the noncommutativity is in the nilpotent ideals", e.g. such that when quotienting out the maximal two-sided nilpotent ideal they become commutative. Supergeometry would be a special case of this, but it would be more general.

Has anything like this been investigated somewhat systematically anywhere? Is there any names attached to this that one could search for to find more?

http://math.stackexchange.com/q/1214990/58526
http://mathoverflow.net/q/201643/381
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Thanks. Yes, that certainly goes in the right direction. For what I was after it would also be necessary to check that not only the monomials of generators vanish for high enough powers, but that also all polynomials (at least) in the generators  become zero when taken to a high enough power. (That may be immediate from what you are looking at, but I haven't given it a glance yet, even.)
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