Entries by Urs Schreiber

Mathematical Quantum Field Theory – Renormalization

February 8, 2018/9 Comments/in /by

This is one chapter in a series on Mathematical Quantum Field Theory The previous chapter is: 15. Interacting quantum fields. 16. Renormalization In this chapter we discuss the following topics: Epstein-Glaser normalization Stückelberg-Petermann re-normalization UV-Regularization via Counterterms Wilson-Polchinski effective QFT flow Renormalization group flow Gell-Mann & Low RG Flow In the previous chapter we have […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Interacting Quantum Fields

January 29, 2018/9 Comments/in /by

This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 14. Free quantum fields. The next chapter is 16. Renormalization. 15. Interacting quantum fields In this chapter we discuss the following topics: Free field vacua Perturbative S-matrices Conceptual remarks Interacting field observables Time-ordered products (“Re”-)Normalization Feynman perturbation series Effective […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Free Quantum Fields

January 29, 2018/1 Comment/in /by

This is one chapter in a series on Mathematical Quantum Field Theory The previous chapter is 13. Quantization. The next chapter is 15. Interacting quantum fields. 14. Free quantum fields In this chapter we discuss the following topics: Wick algebra Time-ordered product Operator product notation Hadamard vacuum state Free quantum BV-differential Schwinger-Dyson equation In the […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Quantization

January 8, 2018/15 Comments/in /by

The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 12. Gauge fixing. The next chapter is 14. Free quantum fields. 13. Quantization In this chapter we discuss the following topics: Motivation from Lie theory Geometric quantization Moyal star products Moyal star product as deformation quantization Moyal star […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Gauge Fixing

December 28, 2017/1 Comment/in /by

The following is one chapter in a series of Mathematical Quantum Field Theory. The previous chapter is 11. Reduced phase space. The next chapter is 13. Quantization. 12. Gauge fixing While in the previous chapter we had constructed the reduced phase space of a Lagrangian field theory, embodied by the local BV-BRST complex (example 11.21), […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Reduced Phase Space

December 18, 2017/4 Comments/in /by

The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 10. Gauge symmetries. The next chapter is 12. Gauge fixing. 11. Reduced phase space In this chapter we discuss these topics: Global gauge reduction for strictly invariant functions (action functionals): Derived critical loci inside Lie algebroids Schouten bracket […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Gauge Symmetries

December 6, 2017/17 Comments/in /by

This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 9. Propagators. The next chapter is 11. Reduced phase space. 10. Gauge symmetries In this chapter we discuss these topics: compactly supported infinitesimal symmetries obstruct covariant phase space Infinitesimal gauge symmetries Lie algebra action and Lie algebroids BRST complex […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Propagators

November 28, 2017/6 Comments/in /by

This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 8. Phase space. The next chapter is 10. Gauge symmetries. 9. Propagators In this chapter we discuss the following topics: Background Fourier analysis and Plane wave modes Microlocal analysis and UV-Divergences Cauchy principal values Propagators for the free scalar […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Phase Space

November 24, 2017/7 Comments/in /by

The following is one chapter of a series on Mathematical Quantum Field Theory. The previous chapter is 7. Observables. The next chapter is 9. Propagators. 8. Phase space It might seem that with the construction of the local observables (def. 7.36) on the on-shell space of field histories (prop. 5.10) the field theory defined by […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Observables

November 19, 2017/8 Comments/in /by

The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 6. Symmetries. The next chapter is 8. Phase space. 7. Observables In this chapter we discuss these topics: General observables Polynomial off-shell observables and Distributions Polynomial on-shell observables and Distributional solutions to PDEs Local observables and Transgression Infinitesimal […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Symmetries

November 16, 2017/9 Comments/in /by

The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 5. Lagrangians. The next chapter is 7. Observables.   6. Symmetries We have introduced the concept of Lagrangian field theories ##(E,\mathbf{L})## in terms of a field bundle ##E## equipped with a Lagrangian density ##\mathbf{L}## on its jet bundle […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Lagrangians

November 14, 2017/42 Comments/in /by

This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 4. Field Variations. The next chapter is 6. Symmetries.   5. Lagrangians Given any type of fields (def. 3.1), those field histories that are to be regarded as “physically realizable” (if we think of the field theory as a […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Field Variations

November 12, 2017/6 Comments/in /by

This is chapter 4 of a series on Mathematical Quantum Field Theory. The previous chapter is 3. Fields. The next chapter is 5. Lagrangians.   4. Field variations Given a field bundle as in def. 3.1 above, then we know what type of quantities the corresponding field histories assign to a given spacetime point (a […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Fields

November 9, 2017/82 Comments/in /by

The following is one chapter in a series on Mathematical Quantum Field Theory The previous chapter is 2. Spacetime. The next chapter is 4. Field variations.   3. Fields A field history on a given spacetime ##\Sigma## (a history of spatial field configurations, see remark 3.2 below) is a quantity assigned to each point of […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Spacetime

November 7, 2017/6 Comments/in /by

The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 1. Geometry. The next chapter is 3. Fields   2. Spacetime Relativistic field theory takes place on spacetime. The concept of spacetime makes sense for every dimension ##p+1## with ##p \in \mathbb{N}##. The observable universe has macroscopic dimension […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Mathematical Quantum Field Theory – Geometry

November 5, 2017/32 Comments/in /by

This is the first chapter in a series on Mathematical Quantum Field Theory. The next chapter is 2. Spacetime. 1. Geometry The geometry of physics is differential geometry. This is the flavor of geometry which is modeled on Cartesian spaces ##\mathbb{R}^n## with smooth functions between them. Here we briefly review the basics of differential geometry […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

A First Idea of Quantum Field Theory – 20 Part Series

November 2, 2017/8 Comments/in /by

These notes mean to give an expository but rigorous introduction to the basic concepts of relativistic perturbative quantum field theories, specifically those that arise as the perturbative quantization of a Lagrangian field theory — such as quantum electrodynamics, quantum chromodynamics, and perturbative quantum gravity appearing in the standard model of particle physics. Table of Contents […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Introduction to Perturbative Quantum Field Theory

September 12, 2017/108 Comments/in /by

This is the beginning of a series that gives an introduction to perturbative quantum field theory (pQFT) on Lorentzian spacetime backgrounds in its rigorous formulation as locally covariant perturbative algebraic quantum field theory. This includes the theories of quantum electrodynamics (QED) and electroweak dynamics, quantum chromodynamics (QCD), and perturbative quantum gravity (pQG) — hence the […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Super p-Brane Theory Emerging from Super Homotopy Theory

July 14, 2017/6 Comments/in /by

  A notorious open problem: What is the non-perturbative theory formerly known as Strings? We still have no fundamental formulation of “M-theory” – the hypothetical theory of which 11-dimensional supergravity and the five string theories are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned. […]. If history is […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Why Higher Category Theory in Physics?

January 4, 2017/30 Comments/in /by

  This here is my personal story. For an alternative introduction see my talk: Higher Structures in Mathematics and Physics. Initially I discovered higher category theory and higher homotopy theory for myself in my PhD work, in the course of analyzing the supersymmetric quantum mechanics of the superstring on loop space. Driven, as I am, […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Spectral Standard Model and String Compactifications

September 26, 2016/24 Comments/in /by

  The Connes-Lott-Chamseddine-Barrett model is the observation that the standard model of particle physics — as a classical action functional, but including its coupling to gravity and subsuming a fair bit of fine detail —  may succinctly be encoded in terms of operator algebraic data called a “spectral triple“. This involves some non-commutative algebra,  and […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

11d Gravity From Just the Torsion Constraint

September 6, 2016/16 Comments/in /by

  It is familiar that Einstein gravity may be formulated  in terms of a vielbein field together with a “spin connection”, subject to the constraint that the torsion vanishes. There is a little industry trying to suggestively rewrite Einstein’s field equations in this “first-order formalism” and speculating about how this might shed light on the […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Supersymmetry and Deligne’s Theorem

August 21, 2016/44 Comments/in /by

  In 2002, Pierre Deligne proved a remarkable theorem on what mathematically is called Tannakian reconstruction of tensor categories. Here I give an informal explanation what what this theorem says and why it has profound relevance for theoretical particle physics: Deligne’s theorem on tensor categories combined with Wigner’s classification of fundamental particles implies a strong […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Examples of Prequantum Field Theories IV: Wess-Zumino-Witten-type Theories

November 1, 2015/4 Comments/in /by

    At the end of this page, we will have come full circle back to the first article in the series 20 years ago — The M-Theory Conjecture. We now construct prequantum field theories — in the sense discussed at Higher Prequantum Geometry I, II, III, IV, V — of “WZW-type”, using the methods from Examples of Prequantum […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Examples of Prequantum Field Theories III: Chern-Simons-type Theories

October 28, 2015/1 Comment/in /by

    After having constructed gauge fields and higher gauge fields in the previous article by systems of ##L_\infty##-algebroid-valued differential forms on simplex bundles, we now use the same method to construct prequantum field theories — in the sense discussed at Higher Prequantum Geometry I, II, III, IV, V — of higher Chern-Simons type with such higher gauge fields. We begin with […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Examples of Prequantum Field Theories II: Higher Gauge Fields

October 24, 2015/1 Comment/in /by

      After having recalled ordinary gauge fields from a dg-algebraic perspective in the previous article, here I discuss how in these terms we easily get the concept of higher (nonabelian) gauge fields, i.e. of gauge fields whose “vector potential” is not just a 1-form but involves differential forms of higher degree. These will be […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Examples of Prequantum Field Theories I: Gauge Fields

October 22, 2015/1 Comment/in /by

    After having motivated the need for prequantum field theory and having laid out its principles (i. extremal action, ii. global action, iii. covariant phase space, iv. local observables), it is time to look at examples. The key classes of examples — which are considerably larger than one may think — are 1) field theories of Chern-Simons type […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Higher Prequantum Geometry V: The Local Observables – Lie Theoretically

October 14, 2015/1 Comment/in /by

  This article discusses how from the previous considerations naturally follow the concepts of local observables of local field theories and of the Poisson bracket on them; as well as that of conserved currents and the variational Noether theorem relating them to symmetries. At the same time all these concepts are promoted to prequantum local field theory. […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively

October 9, 2015/12 Comments/in /by

    The Euler-Lagrange ##p##-gerbes discussed in the previous article are singled out as being exactly the right coherent refinement of locally defined local Lagrangians that may be integrated over a ##(p+1)##-dimensional spacetime/worldvolume to produce a function, the action functional. In a corresponding manner there are further refinements of locally defined Lagrangians by differential cocycles that are adapted to integration […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Higher Prequantum Geometry III: The Global Action Functional – Cohomologically

October 7, 2015/9 Comments/in /by

    The previous article ended with the concept of classical locally variational field theories, of which a class of examples are field theories of higher WZW type. To recall, in the diagrammatics that we established, such theories were concisely expressed like so: Here we discuss how asking such theories to have a well-defined global action functional […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Higher Prequantum Geometry II: The Principle of Extremal Action – Comonadically

October 1, 2015/1 Comment/in /by

    The previous article motivated the importance of considering “pre-quantum field theory” in between classical and quantum field theory. This article here reviews modern classical field theory from a perspective that will be useful for this purpose. Most field theories of relevance in theory and in nature are local Lagrangian field theories (and those that are […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Higher Prequantum Geometry I: The Need for Prequantum Geometry

September 29, 2015/15 Comments/in /by

  Before proceeding with discussion of the super p-brane sigma models, whose emergence from the superpoint I discussed in the previous article, we need to speak a bit about the fundamentals of local Lagrangian field theory in general, of which these sigma-models are examples. The geometry that underlies the physics of Hamilton and Lagrange’s classical mechanics […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Emergence From the Superpoint

August 22, 2015/6 Comments/in /by

  In the previous article we saw that the generalization of super-Lie algebras to homotopy super-Lie n-algebras (super L-infinity algebras) has been found, decades back, to be at the heart of supergravity and M-theory — somewhat secretly so, in the dual guise of “FDA”s. Lie algebras are infinitesimal symmetries. Super Lie algebras are infinitesimal supersymmetries. Lie […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

Homotopy Lie-n Algebras in Supergravity

August 18, 2015/1 Comment/in /by

  The previous article in this series claimed that the mathematics of the 21st century that had fallen into the 1970s in the form of string theory is the same mathematics that Grothendieck had dreamed about in his pursuit of stacks around that same time, and which meanwhile has come to full existence: higher geometry […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber

It Was 20 Years Ago Today — the M-theory Conjecture

August 17, 2015/2 Comments/in /by

    While the world didn’t end, after all, 15 years back at the turn of the millenium, in hindsight it is curious that, almost unnoticed, something grand did come to a halt around that time. Or almost. The 90s had seen a firework of structural insight into the mathematical nature of string theory, an unprecendented global […]

Urs Schreiber

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

http://ncatlab.org/nlab/show/Urs+Schreiber