Higher Structures in M-Theory 2018

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This page collects material related to

* London Mathematical Society EPSRC Durham Symposium

  **Higher Structures in M-Theory**

  12 to 18 August 2018

  [www.maths.dur.ac.uk/lms/109](http://www.maths.dur.ac.uk/lms/109/index.html)

on [[schreiber:Higher Structures]] in [[M-theory]];.

Proccedings in:

* [[Branislav Jurčo]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.)

  **Proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M‐Theory** (Durham University (UK) 12–18 August 2018

  Fortschritte der Physik Special Issue

  Volume 67, Issue 8-9

  [doi:10.1002/prop.201870081](https://www.onlinelibrary.wiley.com/toc/15213978/2019/67/8-9)

\linebreak

**Related events:

* *[[M-Theory and Mathematics]]*

\linebreak

#Contents#
* table of contents
{:toc}

## Editorial

* **Editorial**

  [epdf](https://www.onlinelibrary.wiley.com/doi/epdf/10.1002/prop.201870076), [doi:10.1002/prop.201870076](https://www.onlinelibrary.wiley.com/doi/10.1002/prop.201870076)

## Higher Structures in M-Theory
 {#HigherStructuresInMTheory}

* [[Branislav Jurčo]], [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]]

  **Higher Structures in M‐Theory**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02807](https://arxiv.org/abs/1903.02807),
  [doi:10.1002/prop.201910001](https://onlinelibrary.wiley.com/doi/abs/10.1002/prop.201910001)

partly based on

* [[Urs Schreiber]],

  _[[schreiber:Introduction to Higher Supergeometry]]_

## Parametrised Homotopy Theory and Gauge Enhancement

* [[Vincent Braunack-Mayer]]

  **[[schreiber:Parametrised homotopy theory and gauge enhancement|Parametrised Homotopy Theory and Gauge Enhancement]]**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02862](https://arxiv.org/abs/1903.02862),
  [doi:10.1002/prop.201910003](https://doi.org/10.1002/prop.201910003)

based on

* [[nLab:Vincent Braunack-Mayer]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]]

  **[[schreiber:Gauge enhancement of Super M-Branes]]**

  ([arXiv:1806.01115](https://arxiv.org/abs/1806.01115))

* [[nLab:Vincent Braunack-Mayer]],

  _**[[schreiber:thesis Braunack-Mayer|Rational parameterized stable homotopy theory]]**_

  PhD thesis, Zurich 2018

## Pre‐NQ Manifolds and Correspondence Spaces

* [[Andreas Deser]]

  **Pre‐NQ Manifolds and Correspondence Spaces: the Nilmanifold Example**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02864](https://arxiv.org/abs/1903.02864), [doi:10.1002/prop.201910006](https://doi.org/10.1002/prop.201910006)

## The Rational Higher Structure of M‐theory

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:The rational higher structure of M-theory|The Rational Higher Structure of M‐theory]]**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02834](https://arxiv.org/abs/1903.02834), [doi:10.1002/prop.201910017](https://doi.org/10.1002/prop.201910017)

## Graded Geometry, Q ‐Manifolds, and Microformal Geometry

* [[Theodore Voronov]]

  **Graded Geometry, Q ‐Manifolds, and Microformal Geometry**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02884](https://arxiv.org/abs/1903.02884), [doi:10.1002/prop.201910023](https://doi.org/10.1002/prop.201910023)

## Wilson Surfaces for Surface Knots

* [[Roberto Zucchini]]:

  **Wilson Surfaces for Surface Knots: A Field Theoretic Route to Higher Knots**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02853](https://arxiv.org/abs/1903.02853), [doi:10.1002/prop.201910026](https://doi.org/10.1002/prop.201910026)

on [[higher parallel transport]]

## Higher Forms and Membranes in 4D Supergravities

* [[Igor Bandos]], [[Fotis Farakos]] Stefano Lanza, L. Martucci,  [[Dmitri Sorokin]]:

  **Higher Forms and Membranes in 4D Supergravities**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02841](https://arxiv.org/abs/1903.02841), [doi:10.1002/prop.201910020](https://doi.org/10.1002/prop.201910020)

on [[membranes]] in [[D=4 supergravity]] ([[super 2-brane in 4d]])

## Weyl Anomaly and Vacuum Magnetization Current of M5‐brane in Background Flux

* Chong‐Sun Chu,

  **Weyl Anomaly and Vacuum Magnetization Current of M5‐brane in Background Flux**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02817](https://arxiv.org/abs/1903.02817), [doi:10.1002/prop.201910005](https://doi.org/10.1002/prop.201910005)

## How Space‐Times Emerge from the Superpoint

* [[John Huerta]]:

  **How Space‐Times Emerge from the Superpoint**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02822](https://arxiv.org/abs/1903.02822), [doi:10.1002/prop.201910009](https://doi.org/10.1002/prop.201910009)

based on

* [[John Huerta]], [[Urs Schreiber]]:

  **[[schreiber:M-Theory from the Superpoint]]**

  Letters in Mathematical Physics

  **108**, pages 2695–2727 (2018)

  [arXiv:1702.01774](https://arxiv.org/abs/1702.01774), [doi:10.1007/s11005-018-1110-z](https://doi.org/10.1007/s11005-018-1110-z)

## M‐Branes: Lessons from M2's and Hopes for M5's

* [[Neil Lambert]]:

  **M‐Branes: Lessons from M2's and Hopes for M5's**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02825](https://arxiv.org/abs/1903.02825), [doi:10.1002/prop.201910011](https://doi.org/10.1002/prop.201910011)

on the [[BLG model]]/[[ABJM model]] for the [[M2-brane]] and outlook for the [[M5-brane]]

## Higher Structures, Self‐Dual Strings and 6d Superconformal Field Theories

* [[Christian Saemann]]:

  **Higher Structures, Self‐Dual Strings and 6d Superconformal Field Theories**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02888](https://arxiv.org/abs/1903.02888), [doi:10.1002/prop.201910014](https://doi.org/10.1002/prop.201910014)

based on

* [[Christian Saemann]], [[Lennart Schmidt]], _Towards an M5-Brane Model I: A 6d Superconformal Field Theory_, J. Math. Phys. 59 (2018) 043502 ([arxiv:1712.06623](https://arxiv.org/abs/1712.06623))

## Twisted Weil Algebras for the String Lie 2‐Algebra

* [[Lennart Schmidt]]:

  **Twisted Weil Algebras for the String Lie 2‐Algebra**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02873](https://arxiv.org/abs/1903.02873), [doi:10.1002/prop.201910016](https://doi.org/10.1002/prop.201910016)

on [[Weil algebras]] for the [[string Lie 2-algebra]] (as in [[schreiber:L-infinity algebra connections|SSS09]]), motivated by

* [[Christian Saemann]], [[Lennart Schmidt]], _Towards an M5-Brane Model I: A 6d Superconformal Field Theory_, J. Math. Phys. 59 (2018) 043502 ([arxiv:1712.06623](https://arxiv.org/abs/1712.06623))

## A Kaluza–Klein Approach to Double and Exceptional Field Theory

* [[David Berman]]:

  **A Kaluza–Klein Approach to Double and Exceptional Field Theory**

  [arXiv:1903.02860](https://arxiv.org/abs/1903.02860), [doi:10.1002/prop.201910002](https://doi.org/10.1002/prop.201910002)

on a [[Kaluza-Klein theory]]-perspective on [[double field theory]] and [[exceptional field theory]]

## Higher Gauge Structures in Double and Exceptional Field Theory

* [[Olaf Hohm]], [[Henning Samtleben]]:

  **Higher Gauge Structures in Double and Exceptional Field Theory**

  [arXiv:1903.02821](https://arxiv.org/abs/1903.02821), [doi:10.1002/prop.201910008](https://doi.org/10.1002/prop.201910008)

on [[higher gauge theory]] in [[double field theory]] and [[exceptional field theory]]

## Effective Actions for σ‐Models of Poisson–Lie Type

* [[Branislav Jurčo]]:

  **Effective Actions for σ‐Models of Poisson–Lie Type**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02848](https://arxiv.org/abs/1903.02848), [doi:10.1002/prop.201910024](https://doi.org/10.1002/prop.201910024)

## Supergravity Fluxes and Generalised Geometry

* [[Charles Strickland-Constable]]

  **Supergravity Fluxes and Generalised Geometry**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02842](https://arxiv.org/abs/1903.02842), [doi:10.1002/prop.201910021](https://doi.org/10.1002/prop.201910021)

## $L_\infty$-Algebras, the BV Formalism, and Classical Fields

* [[Branislav Jurčo]], Tommaso Macrelli, Lorenzo Raspollini, [[Christian Sämann]], [[Martin Wolf]]:

  **$L_\infty$-Algebras, the BV Formalism, and Classical Fields**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02887](https://arxiv.org/abs/1903.02887), [doi:10.1002/prop.201910025](https://doi.org/10.1002/prop.201910025)

## Homotopy Algebras in String Field Theory

* [[Ivo Sachs]]:

  **Homotopy Algebras in String Field Theory**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02870](https://arxiv.org/abs/1903.02870), [doi:10.1002/prop.201910013](https://doi.org/10.1002/prop.201910013)

on [[homotopy algebra]] in [[string field theory]]

## $L_\infty$-Bootstrap Approach to Non‐Commutative Gauge Theories

* [[Vladislav Kupriyanov]]

  **$L_\infty$-Bootstrap Approach to Non‐Commutative Gauge Theories**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [doi:1903.02867](https://arxiv.org/abs/1903.02867) [doi:10.1002/prop.201910010](https://doi.org/10.1002/prop.201910010)

## Higher Structures in Algebraic Quantum Field Theory:

*  [[Marco Benini]], [[Alexander Schenkel]]

  **Higher Structures in Algebraic Quantum Field Theory**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02878](https://arxiv.org/abs/1903.02878), [doi:10.1002/prop.201910015](https://doi.org/10.1002/prop.201910015)

on [[homotopical algebraic quantum field theory]]

## Coloring Operads for Algebraic Field Theory

* [[Simen Bruinsma]]:

  **Coloring Operads for Algebraic Field Theory**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02863](https://arxiv.org/abs/1903.02863) [doi:10.1002/prop.201910004](https://doi.org/10.1002/prop.201910004)

on [[homotopical algebraic quantum field theory]]

## Full Logarithmic Conformal Field theory

* [[Jürgen Fuchs]], [[Christoph Schweigert]],

  **Full Logarithmic Conformal Field theory --an Attempt at a Status Report**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02838](https://arxiv.org/abs/1903.02838), [doi:10.1002/prop.201910018](https://doi.org/10.1002/prop.201910018)

on [[logarithmic conformal field theory]] in view of the [[FRS-theorem on rational 2d CFT]]

## Categorical Equivalence and the Renormalization Group

* [[Eric Sharpe]]:

  **Categorical Equivalence and the Renormalization Group**

  [arXiv:1903.02880](https://arxiv.org/abs/1903.02880), [doi:10.1002/prop.201910019](https://doi.org/10.1002/prop.201910019)

## Quantization of Magnetic Poisson Structures

* [[Richard Szabo]]:

  **Quantization of Magnetic Poisson Structures**

  in: Proceedings of _[[Higher Structures in M-Theory 2018]]_

  Fortschritte der Physik, Special Issue Volume 67, Issue 8-9

  [arXiv:1903.02845](https://arxiv.org/abs/1903.02845), [doi:10.1002/prop.201910022](https://doi.org/10.1002/prop.201910022)

\linebreak

\linebreak

category: reference


higher supergeometry

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###Context###
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Higher supergeometry_ is the [[higher differential geometry]] modeled on [[supergeometry]].

* [[super ∞-groupoid]]

* [[smooth super ∞-groupoid]]

* [[Super Gerbes]]


higher symplectic geometry

+-- {: .rightHandSide}
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###Context###
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Higher symplectic geometry_ is the generalization of [[symplectic geometry]] to the context of [[higher geometry]].

It involves two kinds of generalizations:

1. the [[symplectic form]] generalizes from a 2-form to a form of arbitrary arity. This aspect is called [[multisymplectic geometry]].

1. the base [[manifold]] is generalized to a [[smooth ∞-groupoid]] or [[∞-Lie algebroid]]. For binary symplectic forms this is called a [[symplectic Lie n-algebroid]].

In the full higher symplectic geometry both of these aspects are unified: a _multisymplectic $\infty$-groupoid_ is a [[smooth ∞-groupoid]] equipped with a [[smooth infinity-groupoid -- structures|differential n-form on smooth ∞-groupoids]] satisfying some condition.

Examples of higher symplectic geometries arise naturally as the [[covariant phase space]]s "over the point" or "in top codimension" (in the sense of [[extended topological quantum field theory]]) in systems of [[schreiber:∞-Chern-Simons theory]]: their $\infty$-multisymplectic form is the [[invariant polynomial]] that defines the theory.

## Definition

+-- {: .num_defn}
###### Definition

Let $\mathfrak{a}$ be an [[L-∞ algebroid]]. For $n \in \mathbb{N}$, an **[[n-plectic form]]** or **[[multisymplectic form]]** of $n$ arguments on $\mathfrak{a}$ is

* an [[invariant polynomial]] $\omega$ on $\mathfrak{a}$ which is $n$-linear (takes $n$ arguments):

  $$
    \omega \in  W^n(\mathfrak{a}) = \Omega^n(\mathfrak{a})
    \,;
  $$

* such that the contraction morphism

  $$
    \iota_{(-)}\omega : T \mathfrak{a} \to W^{n-1}(\mathfrak{a}) = \Omega^{n-1}(\mathfrak{a})
  $$

in injective.

=--

+-- {: .num_example}
###### Example

If $\mathfrak{a}$ is a Lie 0-algebroid (over a smooth manifold) then it is simply that [[smooth manifold]], $\mathfrak{a} = X$. In this case $W(\mathfrak{a}) = \Omega^{\bullet}(X)$ is the ordinary [[de Rham complex]] and an [[invariant polynomial]] is a closed [[differential form]] of positive degree.

In this case an $n$-plectic form on $\mathfrak{a}$ is a closed $n$-form $\omega(-,-,.\dots, -)$ on $X$ such that for every [[vector field]] $v \in \Gamma(T X)$ we have

$$
  (\omega(v,-,\cdots,-) = 0) \;\; \Rightarrow \;\;
  (v = 0).
$$

=--

## Applications

### Higher Chern-Simons theory

Every [[invariant polynomial]] $\omega$ induces an [[schreiber:∞-Chern-Simons theory]] [[action functional]]

$$
  S_{CS} : \Omega(\Sigma,\mathfrak{a}) \to \mathbb{R}
  \,.
$$

The [[variational calculus|variation]] of that functional is

$$
  \delta S_{CS} : A \mapsto \int_\Sigma \omega(\delta A, F_A, \cdots, F_A)
  \,.
$$

Therefore the condition that the invariant polynomial is $n$-plectic amounts to saying that $S_{CS}$ has no spurious global symmetries.

(...)

## Related concepts

* [[symplectic Lie n-algebroid]]

* [[symplectic ∞-groupoid]]

[[!include Isbell duality - table]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

Discussion of what here we call "higher symplectic geometry over Lie 0-algebroids" ([[multisymplectic geometry]]) is in

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068)).

For more references see [[multisymplectic geometry]].


Higher Topos Theory

+-- {: .rightHandSide}
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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

This entry collects links related to the book

* [[Jacob Lurie]],

  _Higher Topos Theory_,

  Annals of Mathematics Studies 170,

  Princeton University Press 2009

  ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

which discusses the [[higher category theory]] of [[(∞,1)-categories]] in general and that of [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-categories of (∞,1)-sheaves]] (i.e. of [[∞-stack | ∞-stacks]]) -- called (Grothendieck-Rezk-Lurie) [[(∞,1)-topos | (∞,1)-toposes]] -- in particular;

following an earlier sketch in

* {#Lurie03} [[Jacob Lurie]], *On $\infty$-Topoi* ([arXiv:math/0306109](https://arxiv.org/abs/math/0306109))

The book is available online from the arXiv and also from Lurie's web site:

* [PDF of published version](https://www.math.ias.edu/~lurie/papers/highertopoi.pdf) from Lurie's web site

* [arXiv:math.CT/0608040](http://arxiv.org/abs/math.CT/0608040) -- this has been updated since the publication of the print version, including addition of some new material!

* [updated version from Lurie's web site](https://www.math.ias.edu/~lurie/papers/HTT.pdf) -- more recent even than the arXiv version, as of 2019

An online textbook of a similar content is developing at:

* [[Jacob Lurie]], _[[Kerodon]]_

#Contents#
* table of contents
{:toc}

#Related entries#

For general information on higher category and higher topos theory see also

* [[higher category theory]]

* [[higher topos theory]].

If you need basics, see

* [[category theory]]

* [[sheaf and topos theory]].

If you need more motivation see

* [[motivation for sheaves, cohomology and higher stacks]].

If you need to see applications see for instance

* [[cohomology]]

* [[structured (∞,1)-topos]]

* [[A Survey of Elliptic Cohomology]]

#Summary#

## General idea ##

Recall the following familiar 1-categorical statement:

* Working in the 1-[[category]] [[Set]] of [[0-category|0-categories]] is the same as  doing [[set theory]]. The point of [[Categories and Sheaves|categories and sheaves]] is to pass to _parameterized_ [[0-category|0-categories]], namely [[presheaf]] categories: these [[topos|topoi]] behave much like the category [[Set]] but their objects are generalized [[space and quantity|spaces]] that may carry more structure, for instance they may be [[generalized smooth space | generalized smooth spaces]] if one considers (pre)[[sheaf|sheaves]] on [[Diff]].

One can think of Lurie's book as a comprehensive study of the generalization of the above statement  from $1$ to $(\infty,1)$  (recall the notion of [[(n,r)-category]]):

* Working in the $(\infty,1)$-[[category]] [[∞Grpd]] of [[infinity-groupoid|(∞,0)-categories]] is the same as  doing [[topology]]. The point of  [[∞-stacks]] is to pass to _parameterized_ [[infinity-groupoid|(∞,0)-categories]], namely [[(∞,1)-presheaf]] categories: these [[(infinity,1)-topos|(∞,1)-topoi]] behave much like the $(\infty,1)$category [[∞Grpd]] but their objects are generalized [[space and quantity|spaces]] with higher [[homotopy|homotopies]] that may carry more structure, for instance they may be $\infty$-[[differentiable stack | differentiable stack]] if one considers [[∞-stacks]] on [[Diff]].

## First part, sections 1-4

Based on work by André Joyal on the [[quasi-category]] model for [[(∞,1)-categories]], Lurie presents a comprehensive account of the theory of [[(∞,1)-categories]] including the definitions and properties of all the standard items familiar from category theory (limits, fibrations, etc.)

## Second part, sections 5-7 ##

Given the $(\infty,1)$-categorical machinery from the first part there are natural $(\infty,1)$-categorical versions of $(\infty,1)$-[[presheaf]] and $(\infty,1)$-[[sheaf]] categories (i.e. $(\infty,1)$-categories of [[∞-stacks]]): the "$\infty$-topoi" that give the book its title (more descriptively, these would be called "Grothendieck $(\infty,1)$-topoi"). Lurie investigates their properties in great detail and thereby in particular puts the work by Brown, Joyal, Jardine, Toën on the [[model structure on simplicial presheaves]] into a coherent $(\infty,1)$-categorical context by showing that, indeed, these are [[models for ∞-stack (∞,1)-toposes]].

#How to read the book#

## 1-categorical background ##

The book _Higher topos theory_ together with Lurie's work on [[Stable ∞-Categories]] is close to an $(\infty,1)$-categorical analog of the 1-categorical material as presented for instance in

* Kashiwara and Shapira, [[Categories and Sheaves]].

## Sections with crucial concepts ##

The book discusses crucial concepts and works out plenty of detailed properties. On first reading it may be helpful to skip over all the technical parts and pick out just the central conceptual ideas. These are the following:

* [section 1.1](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=13) : the concept of [[(∞,1)-category]]

* [section 5.1](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=256): the concept of [[(∞,1)-presheaves]]

* [section 6.1](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=426): the concept of [[(infinity,1)-topos|(∞,1)-topoi]]

* [section 6.2](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=459)
[section 6.5](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=521)
and : relation to the Brown-Joyal-Jardine-Toën
theory of [[models for ∞-stack (∞,1)-toposes]] in terms of the [[model structure on simplicial presheaves]].

#Content#

##1 An overview of higher category theory

* [[higher category theory]]

  * [[∞-category]]

  * [[(n,r)-category]]

  * [[(∞,0)-category]]

    * [[∞-groupoid]]

    * [[Kan complex]]

    * [[topological space]]

  * [[(∞,1)-category]]

    * [[quasi-category]]

      * [[equivalence of quasi-categories]]

    * [[simplicially enriched category]]

      * [[homotopy coherent nerve]]

    * [[relation between quasi-categories and simplicial categories]]

    * [[complete Segal space]]

    * [[Segal category]]

  * [[(∞,n)-category]]

* constructions in quasi-categories

  * [[homotopy category of an (∞,1)-category]]

  * [[join of quasi-categories]]

    * [[join of simplicial sets]]

  * [[over quasi-category]]

  * [[terminal object in a quasi-category]]

  * [[limit in quasi-categories]]

  * [[sub-quasi-category]]

  * [[opposite (∞,1)-category]]/[[opposite quasi-category]]

* [[(∞,1)-functor]]

  * [[full and faithful (∞,1)-functor]]

  * [[(∞,1)-presheaf]]

  * [[(∞,1)-category of (∞,1)-functors]]

* [[adjoint functor|adjoint (∞,1)-functor]]

##2 Fibrations of Simplicial Sets ##

* [[model structure for quasi-categories]]

* [[fibrations of quasi-categories]]

  * [[Kan fibration]]

  * [[inner Kan fibration]]

  * [[left Kan fibration]]

  * [[right Kan fibration]]

  * [[Cartesian fibration]]

    * [[Cartesian morphism]]

  * [[anodyne morphism]]

### 2.1 Left fibrations

* [[left fibration]]

* [[model structure for left fibrations]]

### 2.2 Simplicial categories and $\infty$-categories

* [[relation between quasi-categories and simplicial categories]]

* [[model structure for quasi-categories]]

### 2.3 Inner fibrations

* [[inner fibration]]

#### 2.3.1 Correspondences

* [[correspondence]]

* [[graph of a functor]], [[cograph of a functor]]

#### 2.3.2 Stability properties of inner fibrations

(...)

#### 2.3.3 Minimal fibrations

* [[minimal inner fibration]]

#### 2.3.4 $n$-Categories

* [[(n,1)-category]]

* [[coskeleton]]

### 2.4 Cartesian fibrations

#### 2.4.1 Cartesian morphisms

* [[Cartesian morphism]]

#### 2.4.2 Cartesian fibrations

* [[Cartesian fibration]]

* [[model structure on marked simplicial over-sets]]

* [[(∞,1)-Grothendieck construction]]

#### 2.4.3 Stability properties of Cartesian fibrations

#### 2.4.4 Mapping spaces and Cartesian fibrations

#### 2.4.5 Application: Invariance of Undercategories

#### 2.4.6 Application: Categorical fibrations over a point

#### 2.4.7 Bifibrations

* [[codomain fibration]]

##3 The $\infty$-Category of $\infty$-Categories

* the [[(∞,2)-category]] [[(∞,1)Cat]]

* the [[(∞,1)-category of (∞,1)-categories]]

* [[(∞,1)-Grothendieck construction]]

  * [[cartesian fibration]]

    * [[cartesian morphism]]

  * [[model structure on marked simplicial over-sets]]

    * [[marked simplicial set]]

  * [[universal fibration of (∞,1)-categories]]

* [[limit in a quasi-category|limits of (∞,1)-categories]]

##4 Limits and Colimits {#Limits}

* [[limit in quasi-categories]]

  * [[fibration sequence]]

* [[idempotent]]

  * [[split idempotent]]

  * [[Karoubi envelope]]

### 4.1 Cofinality

* [[cofinal (∞,1)-functor]]

### 4.2 Techniques for computing colimits

* [[homotopy limit]]

### 4.3 Kan extensions

#### 4.3.1 Relative colimits

* [[relative (infinity,1)-colimit]]

...

### 4.4 Examples of colimits

...

##5 Presentable and Accessible $\infty$-Categories##

### 5.1 $(\infty,1)$-Categories of presheaves ###

* [[(∞,1)-category of (∞,1)-presheaves]]

* [[Yoneda lemma for (∞,1)-categories]]

### 5.2 Adjoint $(\infty,1)$-functors ###

* [[adjoint (∞,1)-functor]]

  * [[graph of a functor]]

  * [[cograph of a functor]]

  * [[adjoint functor theorem]]

  * [[Quillen adjunction]] / [[derived functor]]

#### 5.2.8 Factorization systems

* [[orthogonal factorization system in an (∞,1)-category]]

### 5.3 $(\infty,1)$-Categories of inductive limits ###

#### 5.3.1 Filtered $\infty$-categories

* [[cardinal number]]

* [[filtered (∞,1)-category]]

#### 5.3.2 Right exactness

* [[exact (∞,1)-functor]]

#### 5.3.3 Filtered colimits

#### 5.3.4 Compact objects

* [[compact object in an (∞,1)-category]]

#### 5.3.5 Ind-objects

* [[ind-object in an (∞,1)-category]]

* [[pro-object in an (∞,1)-category]]

#### 5.3.6 Adjoining colimits to $\infty$-categories

### 5.4 Accessible $(\infty,1)$-categories ###

#### 5.4.1 Locally small $\infty$-categories

* [[essentially small (∞,1)-category]]

* [[locally small (∞,1)-category]]

#### 5.4.2 Accessible $(\infty,1)$-categories

* [[accessible (∞,1)-category]]

* [[accessible (∞,1)-functor]]

#### 5.4.3 Accessible and idempotent-complete $(\infty,1)$-categories

* [[idempotent-complete (∞,1)-category]]

### 5.5 Presentable $(\infty,1)$-categories ###

#### 5.5.1 Presentability

* [[locally presentable (∞,1)-category]]

#### 5.5.2 Representable functors and the adjoint functor theorem

* [[adjoint (∞,1)-functor theorem]]

#### 5.5.3 Limits and colimits of presentable $\infty$-categories

* [[symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]]

#### 5.5.4 Local objects

* [[localization of an (∞,1)-category]]

  * [[reflective sub-(∞,1)-category]]

  * [[local object]]

  * [[local equivalence]]

#### 5.5.5 Factorization systems on presentable $\infty$-categories

#### 5.5.6 Truncated objects

* [[n-truncated object of an (∞,1)-category]]

  * [[monomorphism in an (∞,1)-category]]

  * [[subobject in an (∞,1)-category]]

* [[Postnikov tower in an (∞,1)-category]]

#### 5.5.7 Compactly generated $\infty$-categories

* [[compactly generated (∞,1)-category]]

#### 5.5.8 Nonabelian Derived Categories

* [[sifted (∞,1)-category]]

* [[(∞,1)-algebraic theory]]

#### 5.5.9 Quillen's model for $\mathcal{P}_\Sigma(C)$

* [[(∞,1)-algebraic theory|model structure for (∞,1)-algebras over (∞,1,)-algebraic theories]]

##6 $\infty$-Topoi##

### 6.1 Definitions and characterizations ###

#### 6.1.1 Giraud's Axioms in the $\infty$-Categorical setting

* [[∞-topos]]

  * [[(∞,1)-topos]]

* [[Giraud's axioms]]

  * [[groupoid object in an (∞,1)-category]]

  * [[universal colimits]]

* [[object classifier]]

#### 6.1.2 Groupoid objects

* [[simplicial object in an (∞,1)-category]]

* [[groupoid object in an (∞,1)-category]]

  * [[?ech nerve]]

  * [[effective epimorphism]]

  * [[simplicial resolution]]

  * [[delooping]]

    * [[quotient object]]

#### 6.1.3 $\infty$-Topoi and descent

* [[descent]]

* [[(∞,2)-sheaf]]

#### 6.1.4 Free Groupoids

(...)

#### 6.1.5 Giraud's theorem for $\infty$-Topoi

* [[Giraud's theorem]]

#### 6.1.6 $\infty$-Topoi and classifying objects
 {#ClassifyingObjects}

* [[object classifier in an (infinity,1)-topos]]

### 6.2 Constructions of $(\infty,1)$-toposes ###

#### 6.2.1 Left exact localization

* [[exact functor]]

* [[reflective sub-(∞,1)-category|exact reflective sub-(∞,1)-category]]

#### 6.2.2 Grothendieck topologies and sheaves in higher category theory

* [[(∞,1)-category of (∞,1)-sheaves]]

  * [[(∞,1)-site]]

  * [[(∞,1)-sheaf]]

  * [[topological localization]]

  * [[hypercompletion]]

  * [[local object]]

  * [[descent]]

  * [[sheaf]]

  * [[stack]]

  * [[∞-stack]]

  * [[derived stack]]

  * [[∞-stackification]]

* [[models for ∞-stack (∞,1)-toposes]]

* [[model structure on simplicial presheaves]]

  * [[simplicial presheaf]]

  * [[global model structure on simplicial presheaves]]

  * [[local model structure on simplicial presheaves]]

  * [[local model structure on simplicial sheaves]]

  * [[descent for simplicial presheaves]]

#### 6.2.3 Effective epimorphisms

* [[(∞,1)-semitopos]]

* [[effective epimorphism in an (∞,1)-category]]

### 6.3 The $\infty$-Category of $\infty$-Topoi

#### 6.3.1 Geometric morphisms

* [[(∞,1)-geometric morphism]]

* [[(∞,1)Toposes]]

#### 6.3.2 Colimits of $\infty$-topoi

#### 6.3.3 Filtered limits of $\infty$-topoi

#### 6.3.4 General limits of $\infty$-topoi

#### 6.3.5 Etale Morphisms of $\infty$-topoi

* [[étale geometric morphism]]

### 6.4 $n$-Topoi ###

* [[(n,1)-topos]]

* [[n-localic (∞,1)-topos]]

### 6.5 Homotopy theory in an $(\infty,1)$-topos

#### 6.5.1 Homotopy groups

* [[homotopy groups in an (∞,1)-topos]]

  * [[n-truncated object of an (∞,1)-topos]]

  * [[n-connected object of an (∞,1)-topos]]

  * [[Eilenberg-MacLane object]]

* [[cohomology]]

#### 6.5.2 $\infty$-Connectedness

#### 6.5.3 Hypercovering

* [[(∞,1)-sheafification]]/[[∞-stackification]]

* [[hypercover]]

* [[hypercomplete (∞,1)-topos]]

* [[Whitehead theorem]]

#### 6.5.4 Descent versus Hyperdescent

* [[topological localization]]

* [[hypercompletion]]

##7 Higher Topos Theory in Topology

### 7.1 Paracompact spaces

* [[non-abelian cohomology]]

  of [[paracompact topological spaces]]

* [[shape of an (∞,1)-topos]]

* [[shape theory]]

### 7.2 Dimension theory

* [[homotopy]]

* [[cohomology]]

  * [[cocycle]]

  * [[Eilenberg-MacLane object]]

  * [[gerbe]], [[∞-gerbe]]

* [[dimension]]

  * [[homotopy dimension]]

  * [[cohomological dimension]]

  * [[covering dimension]]

  * [[Heyting dimension]]

##Appendix##

### A.1 Category theory

* [[category theory]]

* [[presentable category]]

* [[accessible category]]

* [[weak factorization system]]

* [[transfinite composition]]

* [[small object argument]]

* [[enriched category theory]]

  * [[monoidal category]]

### A.2 Model categories {#ModelCategories}

* [[homotopy theory]]

* [[category with weak equivalences]]

* [[homotopical category]]

  * [[enriched homotopical category]]

  * [[category of fibrant objects]]

  * [[Waldhausen category]]

  * [[simplicial localization]]

  * [[model category]]

    * [[Quillen adjunction]]

      * [[Quillen equivalence]]

    * [[Quillen bifunctor]]

    * [[monoidal model category]]

    * [[enriched model category]]

    * [[localization of a simplicial model category]]

      * [[Bousfield localization]]

  * [[Reedy model structure]]

### A.3 Simplicial categories

  * [[presentable (∞,1)-category]]

#### A.3.1 Enriched and monoidal model categoires

* [[enriched model category]]

* [[monoidal model category]]

* [[Quillen bifunctor]]

* [[simplicial model category]]

#### A.3.2 The model structure on $\mathbf{S}$-enriched categories

* [[simplicially enriched category]]

* [[combinatorial simplicial model category]]

* [[excellent model category]]

* [[model structure on sSet-categories]]

#### A.3.3 Model structures on diagram categories

* [[global model structure on functors]]

* [[Reedy model structure]]

* [[homotopy Kan extension]]

  * [[homotopy limit]]

  * [[homotopy colimit]]

#### A.3.4 Path spaces in $\mathbf{S}$-enriched categories

* [[(∞,1)-category of (∞,1)-functors]]

#### A.3.5 Homotopy colimits of $\mathbf{S}$-enriched categories

...

#### A.3.5 Exponentiation in model categories

...

#### A.3.7 Localizations of simplicial model categories

* [[Bousfield localization of model categories]]

* [[localization of a simplicial model category]]

* [[Combinatorial model categories have presentations]]

category: reference

[[!redirects HTT]]


higher topos theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Higher topos theory is the generalisation to [[higher category theory]] of [[topos theory]]. It is partly motivated by [[Grothendieck]]'s program in _[[Pursuing Stacks]]_.

More generally, the concept _$(n,r)$-topos_ is to [[topos]] as [[(n,r)-category]] is to [[category]].

Rather little is known about the very general notion of higher topos theory. A rich theory however exists in the context of [[(∞,1)-category|(∞,1)-categories]], see at *[[(∞,1)-topos theory]]*

## Examples

### Flavors of higher toposes

[[!include flavors of higher toposes -- list]]

### Archetypical higher toposes

Just as the archetypical example of an ordinary [[topos]] (i.e. a [[(1,1)-topos]]) is [[Set]] -- the category of [[0-categories]] -- so the $\infty$-category of [[(n,r)-categories]] should form the archetypical example of an $(n+1,r+1)$-topos:

\begin{example}\label{ExamplesOfArchetypicalHigherToposes}
**(examples of archetypical higher toposes)** \linebreak

* [[Grpd]] -- the [[(2,1)-category]] of [[groupoids]], hence of [[(1,0)-categories]] -- should be the archetypical [[(2,1)-topos]].

* [[Cat]] -- the [[2-category]] of [[categories]], hence of [[(1,1)-categories]] -- should be the archetypical [[2-topos]] (i.e. [[(2,2)-topos]]).

* [[Pos]] -- the category of [[posets]] and monotone maps, hence of [[(0,1)-categories]] -- should be the archetypal [[(1,2)-topos]].

* [[∞Grpd]] -- the [[(∞,1)-category]] of [[∞-groupoids]], hence of [[(∞,0)-categories]] -- is the archetypical [[(∞,1)-topos]];

* [[(∞,1)Cat]] -- the [[(∞,2)-category]] of [[(∞,1)-categories]] -- should be the archetypical [[(∞,2)-topos]].

\end{example}

## Related concepts

* [[higher structures]]

* [[topos theory]]

* [[2-topos theory]]

* [[(∞,1)-topos theory]]

## References

* [[Michael Shulman]], _[[michaelshulman:n-topos for large n]]_ .

[[!redirects higher topos]]
[[!redirects higher toposes]]
[[!redirects higher topoi]]


Higher toposes of laws of motion

> In ([Lawvere 67](#Lawvere67), [Lawvere 86](#Lawvere86), [Lawvere 97](#Lawvere79)) there was proposed a notion of _[[toposes of laws of motion]]_ meant to formalize [[classical mechanics|classical]] [[continuum mechanics]] in [[synthetic differential geometry]]/in [[topos theory]]. This page here gives an introductory survey of the refinements possible when lifting this from [[topos theory]] to [[(infinity,1)-topos theory|higher topos theory]] and of the applications of the resulting formalism to [[quantum field theory]].

> This text originates in a [talk](http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos#TalkAt8thScottishCategorySeminar) at the _[Eighth Scottish Category Theory Seminar](http://homepages.inf.ed.ac.uk/als/SCT/sct131129.html)_. Accordingly, these notes amplify aspects of [[category theory]] and [[topos theory]] and generally stick to a [[William Lawvere|Lawverian]] perspective.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Introduction and motivation

Urban legend [has it](http://en.wikipedia.org/wiki/History_of_general_relativity#Sir_Arthur_Eddington) that there was a time when only three people understood [[Einstein]]'s [[theory (physics)|theory]] of [[classical field theory|classical]] [[gravity]] -- "[[general relativity]]".

Whether true or not, one of the three was [[David Hilbert]]. He made sure every beginning student today can understand [[general relativity]], he did so by giving it a clear and precise (= rigorous) formalization in [[mathematics]]:

classical Einstein gravity is simply the study of the [[critical points]] of the [[integral]] of the [[scalar curvature]] [[density]] [[action functional|functional]] on the [[moduli space]] of [[pseudo-Riemannian metrics]] on [[spacetime]].

By trusting that a fundamental theory of physics should have a fundamental formulation in mathematics, [[Hilbert]] was able to essentially scoop Einstein (see [here](Einstein-Hilbert+action#History) for the history), that's why this functional is now called the _[[Einstein-Hilbert action functional]]_.

Hilbert had promoted this general idea before as part of the famous eponymous _[[Hilbert's problems]] in mathematics, from 1900. Here [[Hilbert's 6th problem]] asks [[mathematics|mathematicians]] generally to find [[axioms]] for [[theory (physics)|theories]] in [[physics]].

Since then a list of such axiomatizations has been found, for instance

| [[physics]] | [[mathematics]] |
|--|--|
| [[mechanics]] | [[symplectic geometry]] |
| [[gravity]] | [[Riemannian geometry]] |
| [[gauge theory]] | [[Chern-Weil theory]] |
| [[quantum mechanics]] | [[operator algebra]] |
| [[TFT|topological]] [[local quantum field theory]] | [[monoidal (∞,n)-category]] theory |
| $\vdots$ | $\vdots$ |

Two aspects of this list are noteworthy: on the one hand, it contains crown jewels of mathematics, on the other the items appear unrelated and piecemeal.

As a student, [[William Lawvere]] was exposed to the proposal to axiomatize [[thermodynamics]] as what was called "[[rational thermodynamics]]". He realized that a fundamental [[foundation]] of such [[continuum physics]] first of all requires a good foundation of [[differential geometry]] itself.
Looking over his life publication record (see [[William Lawvere|here]]) one sees that he pursued the following grand plan.

**Plan.**

1. lay the [[foundations]] of [[mathematics]] in [[topos theory]] ("[[ETCS]]")

1. lay the foundations of [[geometry]] in [[topos theory]] ([[synthetic differential geometry]], [[cohesion]])

1. lay the foundations of [[classical mechanics|classical]] [[continuum physics]] in [[synthetic differential geometry]] ([[toposes of laws of motion]]).

Lawvere became famous for his groundbreaking contributions to the first two items ([[categorical logic]], [[topos theory|elementary topos theory]], [[algebraic theories]], [[synthetic differential geometry|SDG]]). For some reason the motivation of all this by the third item is not as widely recognized, even thought Lawvere continuously emphasized this third point, see the [list of quotations here](William%20Lawvere#MotivationFromFoundationsOfPhysics).

Grandiose as this plan is, we have to note that in the above form it falls short in each item, by modern standards:

1. modern [[mathematics]] is naturally founded not in [[topos theory]]/[[type theory]], but in [[(infinity,1)-topos theory|higher topos theory]]/[[homotopy type theory]].

1. modern [[geometry]] is not just about "variable sets" ([[sheaves]]) but is [[higher geometry]] about "variable [[homotopy types]]",  "[[geometric homotopy types]]", "[[infinity-stack|higher stacks]]";

1. modern [[physics]] goes beyond [[classical mechanics|classical]] [[continuum physics]]; at high [[energy]] (small distance) classical physics is refined by [[quantum physics]], specifically by [[quantum field theory]].

Therefore what is needed is a foundation of  [[high energy physics]]  in [[higher differential geometry]] formulated in [[higher topos theory]].

In the following we illustrate three aspects of such a refined theory, following ([dcct](#dcct), [sythQFT](#syntheticQFT)). We close by indicating how this theory serves to solve subtle open problems in modern [[quantum field theory]] and [[string theory]].

## **I)** Mapping spaces in gauge theory and General covariance
 {#HigherMappingSpaces}

One basic point emphasized in ([Lawvere 67](#Lawvere67)) is that central to the formulation of [[physics]] is the existence of _[[mapping spaces]]_ which satisfy the [[exponential law]].

(notice: [[mapping space]] ... space of [[trajectories]] ... [[path integral]])

Indeed, generically a [[physical system]] is specified by

* a [[space]] $\Sigma$ called _[[spacetime]]_ or _[[worldvolume]]_;

* a [[space]] $X$ called _[[target space]]_ or _[[field bundle]]_ or _[[moduli space]] of [[field (physics)|fields]]_.

such that a _[[trajectory]]_ or _history_ of [[physical field|field configurations]] is a [[map]]

$$
  \Sigma \longrightarrow X
  \,.
$$

For instance for $\Sigma = \mathbb{R}$ the abstract [[worldline]] and $X$ [[spacetime]], then $\Sigma \to X$ may be taken to be the [[trajectory]] of a [[particle]] in [[spacetime]].

(Notice that after [[second quantization]] the roles change. First the domain $\Sigma$ is [[worldvolume]] and the codomain $X$ is [[spacetime]] ("[[sigma-model]])", then after second quantization spacetime $X$ becomes the domain (hence becomes $\Sigma$ in the above).)

Hence the [[mapping space]] $[\Sigma,X]$ is the space of all [[trajectories]] (the _[[path space]]_, famous as the [[domain]] of the infamous _[[path integral]]_).

Lawvere observed that this not only needs to exist as a decent "space", it also needs to satisfy the axiom of an [[cartesian closed category|cartesian]] [[internal hom]]. Because if we consider a split

$$
  \Sigma \coloneqq \mathbb{R} \times \Sigma_{d-1}
$$

into [[time]] and [[space]], then we want that spacetime field configurations

$$
  \mathbb{R} \times \Sigma_{d-1} \longrightarrow X
$$

are equivalently trajectories of fields on space

$$
  \mathbb{R}  \longrightarrow [\Sigma_{d-1}, X]
$$

and also equivalently a collection of field trajectories for each point of space

$$
  \Sigma_{d-1}  \longrightarrow [\mathbb{R}, X]
  \,.
$$

This led Lawvere to recognize that physics ([[prequantum field theory|prequantum physics]], to be precise) is to be formulated in a [[cartesian closed category]], such as a [[topos]].

The [[category]] $SmthMfd$ of [[smooth manifolds]] is too small to accomplish this. But the [[category of sheaves]] $Sh(SmthMdf)$ on the [[site]] of smooth manifolds is the canonical improvement. Objects in here include [[smooth manifolds]], also [[diffeological spaces]] and general [[smooth spaces]].

Better still, there is the [[category of sheaves]] $Sh(FSmthMfd)$ on the site of  [[formal smooth manifolds]] -- known as the _[[Cahiers topos]]_ . This also contains [[infinitesimal objects]] and indeed interprets the [[axioms]] of [[synthetic differential geometry]].

But actually in modern physics one needs a bit more than this. Physics is fundamentally governed by [[gauge equivalence]], which means that there is no sense in asking if [[field (physics)|field]] configurations are equal, we must ask if they are [[equivalence|equivalent]].

A fundamental example of this is [[Einstein]]'s notion of _[[general covariance]]_. This says that if

$$
  s \;\colon\; U \hookrightarrow \Sigma
$$

is a region in [[spacetime]], and $\phi \colon \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma$ is a [[diffeomorphism]] acting on spacetime, then the "translated region"

$$
  \phi^\ast s \;\colon\; U \hookrightarrow \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma
$$

is "the same", for all physical purposes. Stated this way in ordinary [[topos theory]] this is confusing, and historically it was confusing: this confusion is essentially what is known as the "[[hole paradox]]".

This apparent [[paradox]] is resolved in [[higher topos theory]]. Here a [[space]] $S$ has internal symmetries, it is a _[[groupoid]]_, a _[[homotopy type]]_. This means that it is not sensible to ask if two maps into it are equal, but between any two maps there is a space of equivalences between them.

For [[general covariance]] this means by the above that spacetime is not actually the spacetime [[manifold]] $\Sigma$, but is the [[action groupoid]]/[[quotient stack]]

$$
  \Sigma//Diff(\Sigma)
$$

of $\Sigma$ by its [[diffeomorphism group]] (regarded as a [[diffeological group]]). By the very definition of "[[stack]]", this $\Sigma//Diff(\Sigma)$ is the thing which is such that maps $U \longrightarrow \Sigma//Diff(\Sigma)$ into it behave just as they should as demanded by general covariance.

This is the formalization of "[[general covariance]]" for regions _inside_ spacetime. The other thing now is general covariance for [[field (physics)|fields]] _on_ spacetime. It turns out that the formalism automatically handles these now:

For notice that the above means now that we also need to consider the [[mapping spaces]] refined to  higher topos theory. A fundamental fact is that for $G \in Grp(\mathbf{H})$ [[group object in an (infinity,1)-category|group object]] then the [[slice (infinity,1)-topos|higher slice topos]] over its [[delooping]] is equivalently the collection of [[infinity-action|G-actions]]

$$
  \mathbf{H}_{/\mathbf{B}G} \simeq G Act(\mathbf{H})
  \,.
$$

Under this equivalence a [[infinity-action|higher action]] is identified with the universal [[associated infinity-bundle|associated bundle]] which it induces

$$
  \left(
  \array{
    \Sigma &\longrightarrow& \Sigma//Diff(\Sigma)
    \\
    && \downarrow
    \\
    && \mathbf{B}Diff(\Sigma)
  }
  \right)
  \;\;
  \in
  \;\;
  \mathbf{H}_{/\mathbf{B}Diff(\Sigma)}
  \simeq Diff(\Sigma)Act(\mathbf{H})
  \,.
$$

Lawvere also introduced [[categorical logic]] and understood [[dependent product]] $\prod$ and [[dependent sum]] $\sum$ as [[base change]]. In [[(infinity,1)-topos theory|higher topos theory]] this becomes [[representation theory]] as follows:

[[!include homotopy type representation theory -- table]]

Using this we have the mapping space of [[general covariance|covariant]] [[field (physics)|fields]] formed in the [[context]] of $Diff(\Sigma)$-covariance/[[equivariance]]

$$
  \mathbf{B}Diff(\Sigma)
  \;\;
   \vdash
  \;\;
   \left[
     \Sigma//Diff(\Sigma)
     \,,\;
      X
   \right]
$$

**Theorem.** Down in the absolute [[context]], under [[dependent sum]], this is

$$
   \vdash
  \;\;
   \underset{\mathbf{B}Diff(\Sigma)}{\sum}
   \left[
     \Sigma//Diff(\Sigma)
     \,,\;
      X
   \right]
   \;\;
   \simeq
   \;\;
   [\Sigma, X]//Diff(\Sigma)
  \,.
$$

Here $[\Sigma, X]$ is the space of fields on spacetime as one would find it in ordinary topos theory, and $[\Sigma,X]//Diff(\Sigma)$ is its [[homotopy quotient]] by the [[action]] of the [[diffeomorphism group]] by pullback of fields. This is precisely the group of [[gauge equivalences]] on fields in [[general relativity]].

So we obtain a formalization of the famous insight of [[Einstein]], derived by lifting [[Lawvere]]'s argument about [[mapping spaces]] in [[physics]] from [[topos theory]] to [[higher topos theory]].

In a slogan we may conclude, in view of the above table, that in the formalization of [[physics]] in [[(infinity,1)-topos theory|higher topos theory]] we have:

* _[[general covariance]] is homotopy [[coinvariants]]_

in [[mapping spaces]].

## **II)** Toposes of laws of motion and Hamilton-Jacobi-Lagrange mechanics

In ([Lawvere 97](#Lawvere97)) it was observed that [[equations of motion]] in [[physics]] can (almost, see below) be formalized in [[synthetic differential geometry]] as follows.

Let $\mathbf{H}$ be an ambient [[smooth topos|synthetic differential]] [[topos]] (such as the [[Cahiers topos]] of [[smooth spaces]] and [[formal smooth manifolds]]).

The canonical [[line object]] $\mathbb{A}^1 = \mathbb{R}$ of this models the [[continuum]] line, the abstract [[worldline]]. Let

$$
  D \hookrightarrow \mathbb{R}
$$

be the inclusion of the first order [[infinitesimally thickened point|infinitesimal]] neighbourhood of the origin of $\mathbb{R}$ -- in the [[internal logic]] this is $D = \{x \in \mathbb{R}| x^2 = 0\}$, externally it is the [[spectrum of a commutative ring|spectrum]] of the [[ring of dual numbers]] over $\mathbb{R}$.

Then consider $X \in \mathbf{H}$ any [[object]] which we are going to think of as a [[configuration space]] of a [[physical system]]. For instance if the system is a [[particle]] propagating on a [[spacetime]], then $X$ is that spacetime. Or $X$ may be the [[phase space]] of the system.

Accordingly the [[mapping space]] $[\mathbb{R}, X] \in \mathbf{H}$ is the [[smooth space|smooth]] [[path space]] of $X$. This is the space of potential [[trajectories]] of the [[physical system]].

If $X$ is thought of as [[phase space]], then every point in there determines a unique [[trajectory]] starting at that point. This means that time evolution is then an [[action]] of $\mathbb{R}$ on $X$. As $X$ here might be any space, we have the collection

$$
  \mathbb{R}Act(\mathbf{H}) \in Topos
$$

of all $\mathbb{R}$-[[actions]] on objects in $\mathbf{H}$. This is again a [[topos]], and hence this is a first version of what one might call a _[[topos of laws of motion]]_.

On the other hand, if we think of $X$ as [[configuration space]], then it is (in the simplest but common case of [[physical systems]]) a [[tangent vector]] in $X$ that determines a [[trajectory]], hence a point in $[D,X]$. There is the canonical projection $[\mathbb{R},X] \longrightarrow [D,X]$ from the smooth [[path space]] to the [[tangent bundle]], which sends each path to its [[tangent vector]]/[[derivative]] at $0 \in \mathbb{R}$. A [[section]] of this map is hence an assignment that sends each tangent vector to a [[trajectory]] which starts out with this tangent. Specifying such a section is hence part of what it means to have [[equations of motion]] in [[physics]]. Accordingly in _[[Toposes of laws of motion]]_ [[Lawvere]] called the collection of such data a Galilean [[topos of laws of motion]].

Of course this is not quite yet what is actually used and needed in physics.  On p. 9 of ([Lawvere 97](#Lawvere97)) this problem is briefly mentioned:

> But what about actual dynamical systems in the spirit of [[Galileo]], for example, second-order ODE's? (Of course, the [[symplectic geometry|symplectic]] or [[Hamiltonian mechanics|Hamiltonian systems]] that are also much studied do address this question of states of [[becoming|Becoming]] versus locations of [[being|Being]], but in a special way which it may not be possible to construe as a topos;

We observe now (following _[[schreiber:Classical field theory via Cohesive homotopy types]]_) that it does exist as a "[[higher topos theory|higher topos]]".

First notice that in [[physics]] a [[phase space]] is not any space $X$, but is a space $X$ equipped with a closed [[differential 2-form]], a "[[presymplectic form]]". Speaking in terms of mapping spaces as above let $\mathbf{\Omega}^2_{cl}$ be the [[moduli space]] of closed 2-forms, then this means that a phase space is really a map

$$
  \array{
    X
    \\
    & \searrow
    \\
    && \mathbf{\Omega}^2_{cl}
  }
$$

In fact this is still not quite the accurate statement. Rather a phase space is a "[[prequantization]]" of such data. This means the following.

The [[circle group]] $S^1$ naturally [[action|acts]] on the space of [[differential 1-forms]] $\mathbf{\Omega}^1$ by

$$
 \mathbf{\Omega^1} \times S^1 \longrightarrow \mathbf{\Omega}^1
$$

$$
  (A, g) \mapsto A + \mathbf{d}log(g)
  \,,
$$

where $\mathbf{d}$ is the [[de Rham differential]]. The resulting [[quotient stack]] we write

$$
  \mathbf{B}S^1_{conn} = \mathbf{\Omega}^1//S^1
  \,.
$$

The [[de Rham differential]]  $\mathbf{d} \;\colon\; \mathbf{\Omega}^1 \longrightarrow \mathbf{\Omega}^2_{cl}$ descends to this quotient to yield a map

$$
  F_{(-)} \;\colon\; \mathbf{B}S^1_{conn} \longrightarrow \mathbf{\Omega}^2_{cl}
  \,.
$$

A [[prequantization]] of a [[presymplectic form]] is a lift $\nabla$ through this map

$$
  \array{
    X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}U(1)_{conn}
    \\
    & {}_{\mathllap{\omega}}\searrow & \downarrow^{\mathrlap{F_{(-)}}}
    \\
    && \mathbf{\Omega}^2
  }
  \,.
$$

Now suppose $\omega$ is actually a [[symplectic form]]. Then:

**Theorem.** Concrete actions of $\mathbb{R}$ on $(X, \nabla) \in \mathbf{H}_{/\mathbf{B}S^1_{conn}}$ are equivalent to "[[Hamiltonians]]" $H \in [X,\mathbb{R}]$, where under the equivalence an element $t \in \mathbb{R}$ is sent to the slice automorphism

$$
  \array{
    X && \stackrel{\exp(t \{H,-\})}{\longrightarrow} && X
    \\
    & {}_{\mathllap{\theta}} \searrow
    & \swArrow_{\exp( i S_t  )} & \swarrow_{\mathrlap{\theta}}
    \\
    && \mathbf{B}U(1)_{conn}
  }
  \,,
$$

where $\exp(t \{H,-\})$ denotes the [[flow]] of [[Hamilton's equations of motion]] induced by $H$ and where $S_t = \int_0^t L \, d t$ is the [[Hamilton-Jacobi action]] given by the [[integration|integral]] of the [[Lagrangian]] $L$ (the [[Legendre transform]] of $H$).

This statement subsumes the core ingredients of [[classical mechanics]]. See at _[[prequantized Lagrangian correspondence]]_ for details.

In conclusion we find that $\mathbb{R}$-[[actions]] in the higher slice topos $\mathbf{H}_{/\mathbf{B}S^1_{conn}}$ over the [[moduli stack]] of [[circle group]]-[[principal connections]] are equivalent to actual [[laws of motion]] in [[classical mechanics]]

$$
  LawsOfMotion(\mathbf{H})
  \simeq
  \mathbb{R}Act\left(
    \mathbf{H}_{/\mathbf{B}S^1_{conn}}
  \right)
  \,.
$$

More precisely, this applies to [[laws of motion]] in [[mechanics]]. One obtains more generally the [[Hamilton-de Donder-Weyl equations of motion]] of $n$-dimensional [[local field theory|local]] [[classical field theory]] by replacing $\mathbf{B} S^1_{conn}$ here with $\mathbf{B}^n S^1_{conn}$ ([Schreiber 13](#dcct), [Schreiber 13b](#Schreiber)).

## **III)** Extensive/intensive duality and Cohomological quantization

In [[physics]] and especially in [[continuum mechanics]] and [[thermodynamics]], a physical quantity associated with a [[physical system]] extended in [[space]] is called

* _[[intensive]]_ if it is a [[function]] on (the physical system extended in) space;

* _[[extensive]]_ if it is a [[density]] or [[linear distribution]] on (the physical system extended in) space.

For instance for a [[solid body]] its [[temperature]] is intensive, but its [[mass]] is extensive: there is a temperature assigned to every point of the body (in the idealization of [[classical mechanics|classical]] [[continuum mechanics]] anyway) but a mass is assigned only to every little "extended" piece of the body, not to a single point.

This terminology in [[physics]] apparently originates with [[Richard Tolman]] in 1917.

In ([Lawvere 86](#Lawvere86)) it is amplified that this [[duality]] is generally a fundamental one also in [[mathematics]]: given a [[topos]] $\mathbf{H}$ with a [[commutative ring|commutative]] [[ring object]] $R \in CRing(\mathbf{H})$, then

* the space of _intensive quantities_ on an [[object]] $X \in \mathbf{H}$ is the [[mapping space]] $[X,R]_{\mathbf{H}} \in CRing(\mathbf{H})$ formed in $\mathbf{H}$;

* the space of _extensive quantities_ on $X$ is the $R$-linear dual, namely the mapping space $[X,R]^\ast \coloneqq [[X,R], R]_{R Mod}$ formed in $R$-[[modules]] in $\mathbf{H}$.

* the [[integration]] map is the canonical [[evaluation]] pairing

  $$
    \int_X \;\colon\; [X,R] \times [X,R]^\ast  \longrightarrow R
    \,.
  $$

Viewed this way, this naturally generalizes to the case where $\mathbf{H}$ is in fact an [[(∞,1)-topos]] and $R \in CRing(\mathbf{H})$ an [[E-∞ ring]]. In this case $[X,R]$ is called the  $R$-[[cohomology]] [[spectrum]] of $X$ and $[X,R]^\ast$ is the corresponding [[generalized homology]] spectrum. In this form intensive and extensive properties appear in [[physics]] in the context of [[motivic quantization]] of [[local prequantum field theory]].

More generally, for $\chi$ an $R$-[[(∞,1)-line bundle]] over $X$ then the corresponding extensive object is the $\chi$-twisted [[Thom spectrum]] $R_{\bullet + \chi}(X)$ and the intensive object is the $\chi$-[[twisted cohomology]] [[spectrum]] $R^{\bullet + \chi}(X) = [R_{\bullet+ \chi}(X),R]_{R Mod}$. See at _[[motivic quantization]]_ for how this appears in [[physics]].

In particular ordinary [[quantum mechanics]] is recovered by settin $R = $ [[KU]], the [[complex K-theory]] [[spectrum]] ([Nuiten 13](#Nuiten13)).

The [[monoidal (infinity,1)-category]] $KU Mod$ is the refined ambient home for $Hilb = \mathbb{C} Mod$ (used for [[finite quantum mechanics in terms of dagger-compact categories]]).

## Application: Cohesion and Superstring anomaly cancellation

On first sight, formalization of [[physics]] in ([[higher topos theory|higher]]) [[topos theory]] might seem like a fruitless exercise. But on the contrary, it is hardly possible to understand the deep structure of [[quantum field theory]] without such ([[geometric homotopy theory|geometric]]) [[homotopy theory]].

We close here by briefly indicating one example problem of recent interest, concerned with  the fine-structure of [[quantum anomaly cancellation]] in 2d [[QFT]].

One reason why the need for geometric homotopy theory in QFT is not mentioned  in the bulk of the QFT literature is that traditionally the bulk of the discussion of [[quantum field theory]] is in [[perturbation theory]] (perturbative both in [[Planck's constant]] and in terms of the [[coupling constant]]). This perspective tends to hide the rich nature of what QFT fundamentally is, as [[non-perturbative quantum field theory]].

Phenomena that arise from the global structure of a [[moduli]] of [[field (physics)|field]] configurations in [[physics]] are alien to [[perturbation theory]], and hence are _[[quantum anomaly|anomalies]]_. Such an [anomalous action functional](quantum+anomaly#AnomalousActionFunctional) is something that ought to be a [[function]] $[\Sigma,X] \longrightarrow S^1$ on configuration space, but possibly comes out just as a [[section]] of a [[bundle]] over configuration space (examples include [[gravitational anomaly|gravitational anomalies]], the [[conformal anomaly]], the [[Freed-Witten-Kapustin anomaly]], the [[Green-Schwarz anomaly]], the [[Diaconescu-Moore-Witten anomaly]].)

The anomaly bundles on $[\Sigma,X]$ typically arise as the [[transgression]] of [[principal infinity-bundles|higher bundles]] on the [[moduli space]] of [[field (physics)|fields]] $X$ itself (see at [[twisted smooth cohomology in string theory]] for more on this).  So these are phenomena which are intrinsically phenomena in [[geometric homotopy theory]]/[[(infinity,1)-topos theory]].

We consider now specifically a general aspect of what is called the _[[Freed-Witten-Kapustin anomaly]]_. It is usually read out as follows:

Just as above we saw that the basic example of a quantum field theory on $\Sigma = \mathbb{R}$ descibes the [[dynamics]] of a [[particle]], so the basic example of a quantum field theory on a 2-dimensional $\Sigma$ describes the dynamics of a [[string]]. This naturally feels [[forces]] excerted in particular by two [[background gauge fields]] called the _[[B-field]]_ and the _[[RR-field]]_.

The global nature of these fields is more subtle than for, say, the [[electromagnetic field]], since they are [[higher gauge fields]]. To a first approximation one finds that the [[RR-field]] is a [[cocycle]] in [[twisted K-theory]], where the [[twisted cohomology|twist]] is the [[B-field]] which in turn is a  [[cocycle]] in [[ordinary cohomology]].

But this is not the full story, in the full story these fields are cocycles in [[differential cohomology]]. The [[RR-field]] is a [[cocycle]] in [[twisted cohomology|twisted]] [[differential K-theory]] twisted by the
[[B-field]] which is a  [[cocycle]] in [[ordinary differential cohomology]].

In ([DFM 09](#DFM09)) is indicated the rich subtleties in the quantum anomaly consistency conditions on these [[background fields]], assuming that [[twisted differential K-theory]] exists with some properties, but without having constructed it.

The infamous "[[landscape of string theory vacua]]" is essentially
the [[moduli space]] of certain 2d field theories satisfying consistency  conditions like this.

The central problem in showing the existence of a [[differential cohomology]] theory is to show that this cohomology theory sits inside a double square diagram called the "differential cohomology diagram".

Now a miracle happens. After developing [[synthetic differential geometry]], Lawvere explored a more fundamental axiomatization of [[differential geometry]], which he called _[[cohesion]]_ ([Lawvere 94](#Lawvere94), [Lawvere 07](#Lawvere07) ) (Earlier: "being and becoming" ([Lawvere 91](#Lawvere91))). Slightly paraphrased, cohesion means that the ambient [[type theory]] is equipped with an [[adjoint triple]] of (co-)[[modalities]]

$$
  \int \;\dashv\; \flat \;\dashv\; \sharp
$$

called: [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]].

This has an immediate extension to [[homotopy type theory]] ([[cohesive homotopy type theory]]). But there it has more dramatic consequences. In
([Bunke-Nikolaus-Völkl 13](#BunkeNikolausVoelkl13)) it was observed that on [[stable homotopy types]] $A$ cohesion implies that the canonical diagram formed from modality units and counits

$$
  \array{
    &&  \int_{dR} \Omega A && \longrightarrow && \flat_{dR}\Sigma A
    \\
    & \nearrow & & \searrow & & \nearrow_{\mathrlap{\theta_A}} && \searrow
    \\
    \flat \int_{dR} \Omega A  && && A && && \int \flat_{dR}\Sigma A
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow_{\mathrlap{\int \theta_A}}
    \\
    && \flat A && \longrightarrow && \int A
  }
  \,,
$$

is guaranteed to consist of [[homotopy pullback]] squares, by the nature of [[adjoint triples]] of [[modalities]] (see at _[[tangent cohesion]]_ for more on this).

In ([Bunke-Nikolaus-Völkl 13](#BunkeNikolausVoelkl13)) it is shown that this is universally the "differential cohomology diagram" which hence exhibits _every_ [[stable homotopy type]] $A$ in [[cohesive homotopy type theory]] as a [[differential cohomology theory]], hence as the [[moduli stack]] for abelian [[higher gauge fields]] in [[quantum field theory]].  Hence [[cohesive homotopy type theory]] is a universal ambient context for [[differential cohomology]] and hence for [[higher gauge fields]] appearing in [[quantum field theory]] -- whence the title "[[schreiber:differential cohomology in a cohesive topos]]".

Using this and the [[twisted cohomology]] available in [[tangent cohesion]] ([Bunke-Nikolaus](#BunkeNikolaus)) show the existence of [[twisted differential K-theory]], the way it needs to exist for 2d QFT to be consistent.

## References

* {#Lawvere67} [[William Lawvere]], _[[Categorical dynamics]]_, 1967 Chicago lectures ([pdf](http://www.mat.uc.pt/~ct2011/abstracts/lawvere_w.pdf))

* {#Lawvere86} [[William Lawvere]], Introduction to _[[Categories in Continuum Physics]], Lectures given at a Workshop held at SUNY, Buffalo 1982. Lecture Notes in Mathematics 1174. 1986

* {#Lawvere91} [[William Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_ in A. Carboni, M. Pedicchio, G. Rosolini, _Category Theory_  , [[Como|Proceedings of the International Conference held in Como]], Lecture Notes in Mathematics 1488, Springer (1991)

* {#Lawvere94} [[William Lawvere]] _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_ Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. ([[LawvereCohesiveToposes.pdf:file]])

* {#Lawvere97} [[William Lawvere]], _[[Toposes of laws of motion]]_, 1997

* {#Lawvere07} [[William Lawvere]], _Axiomatic cohesion_, Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

* {#dcct} [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

* {#Schreiber13b} [[Urs Schreiber]], _[[schreiber:Classical field theory via Cohesive homotopy types]]_ ([arXiv:1311.1172](http://arxiv.org/abs/1311.1172))

* {#Schreiber14} [[Urs Schreiber]], _[[schreiber:Quantization via Linear homotopy types]]_ ([arXiv:1402.7041](http://arxiv.org/abs/1402.7041))

* {#Nuiten13} [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local boundary prequantum field theory]]_, 2013

* {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

* {#BunkeNikolaus} [[Ulrich Bunke]], [[Thomas Nikolaus]], _Twisted differential cohomology_, in preparation


higher trace

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For morphisms between [[dualizable objects]] in a [[symmetric monoidal category]] there is a notion of _[[trace]]_. More generally, for a [[fully dualizable object]] $V$ in a [[symmetric monoidal (∞,n)-category]] $\mathcal{C}^\otimes$ there is a notion of $k$-dimensional trace for each $k \leq n$ and indeed for each diffeomorphism type of a [[closed manifold]].

The [[cobordism theorem]] says that a [[monoidal (∞,n)-functor]] $Z \colon Bord_n \to \mathcal{C}^\otimes$ from the [[(∞,n)-category of cobordisms]] picks a [[fully dualizable object]] $V \coloneqq Z(\ast) \in \mathcal{C}$ and then sends the $k$-[[sphere]] $S^k$ to the $k$-dimensional higher trace of the identity on $X$:

$$
  tr_{S^k}(id_{id_{ \cdots id_V}}) \in \Omega^k \mathcal{C}
  \,.
$$

This are the "round traces". More generally the [[cobordism theorem]] gives a higher dimensional trace of the "shape" of any [[closed manifold]] $\Sigma$ of [[dimension]] $k$ on any [[fully dualizable object]] $X$

$$
  tr_{\Sigma}(id_V) \in \Omega^k \mathcal{C}
$$

and for every $k$-manifold with [[boundary]] $\partial \Sigma$ the relative trace is a morphism

$$
  (1 \stackrel{tr_{\Sigma}(V)}{\longrightarrow} tr_{\partial \Sigma}(id_V))
  \in
  Hom_{\Omega^{k-1}\mathcal{C}}(1,tr_{\partial \Sigma}(id_V))
  \,.
$$

## Examples

Higher dimensional traces in a an [[(∞,n)-category of correspondences]] are give by higher [[span traces]]. Those of the shape of [[Riemann surfaces]] are spelled out for instance in ([lpqft](#lpqft)).

## Related concepts

* [[trace]]

* [[bicategorical trace]]

* [[span trace]]

* [[opetopic type theory]]

## References

* [[David Ben-Zvi]], [[David Nadler]], _Secondary Traces_ ([arXiv:1305.7177](http://arxiv.org/abs/1305.7177))

* [[David Ben-Zvi]], [[David Nadler]], _Nonlinear traces_ ([arXiv:1305.7175](http://arxiv.org/abs/1305.7175))

* {#lpqft} _[[schreiber:Local prequantum field theory]]_

[[!redirects higher traces]]

[[!redirects higher dimensional trace]]
[[!redirects higher dimensional traces]]


higher U(1)-gauge theory

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[higher gauge theory]] analog of [[electromagnetism]], including in degree 2 the [[B-field]], in degree 3 the [[supergravity C-field|C-field]], and so on.

## Definition

Over a [[spacetime]] $X$, a field configuration of order $n$ $U(1)$-gauge theory is a [[circle n-bundle with connection]] $\hat F : X \to \mathbf{B}^n U(1)_{conn}$.

The [[action functional]] of the bare theory is given by

$$
  \exp(i S(-))
  :
  \hat F \mapsto \exp(i \int_X   F\wedge \ast F)
  \,,
$$

where $F \in \Omega^{n+1}_{cl}(X)$ is the [[field strength]]/[[curvature]] of $\hat F$, and where $\ast$ denotes the [[Hodge star operator]].

The presence of background [[electric charge]] on $X$ is modeled by a fixed [[circle n-bundle with connection|circle (d-n-1)-bundle with connection]]

$$
  \hat j_{el} : X \to \mathbf{B}^{d-n-1} U(1)_{conn}
  \,,
$$

where $d$ is the [[dimension]] of $X$, and adding to the action the [[higher electric background charge coupling]] term

$$
  \exp(i S_{el}(-))
  :
  \hat F
    \mapsto
  \exp(i \int_X \hat F \cup \hat j)
  \,,
$$

given by the [[Beilinson-Deligne cup product]] of the higher electromagnetic field with the background electric current, followed by [[fiber integration in ordinary differential cohomology]].

The presence of background [[magnetic charge]], on the other hand, is modeled by changing the configurations from circle $n$-bundles with connection to _twisted_ circle $n$-bundles with connection (...)

## Examples

* [[electromagnetic field]]

* [[B2-field]]

* [[C3-field]]

* [[B6-field]]

* [[C6-field]]

## Related concepts

* [[self-dual higher gauge theory]]

## References

* [[Dan Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_

* [[Dan Freed]], [[Greg Moore]], [[Graeme Segal]], _The uncertainty of fluxes_ 	Commun.Math.Phys.271:247-274 (2007) ([arXiv:hep-th/0605198](http://arxiv.org/abs/hep-th/0605198))

* [[Dan Freed]], [[Greg Moore]], [[Graeme Segal]], _Heisenberg Groups and Noncommutative Fluxes_ 	AnnalsPhys.322:236-285 (2007) ([arXiv:hep-th/0605200](http://arxiv.org/abs/hep-th/0605200))

[[!redirects higher U(1)-gauge field]]
[[!redirects higher U(1)-gauge fields]]


higher van Kampen theorem

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _higher homotopy van Kampen theorem_ is a theorem that asserts that the [[homotopy type]], or certain invariants thereof, of a [[topological space]] can be 'computed', in some sense, by a suitable [[colimit]] or [[homotopy colimit]] over the homotopy types / invariants of its pieces.

These all in some sense generalize the [[van Kampen theorem]], (SvKT), which only deals with the underlying 1-type (the fundamental groupoid). The actual statements of the results depend on the models of the homotopy invariants being considered.

## For topological spaces {#ForTopSpaces} (The homotopy small simplex theorem.)

The general form that this takes is similar to the classical homology small simplex theorem which can be found in many books on homology. This states that given an open cover, $\mathcal{U}$, of a space, $X$, the inclusion of the chain complex of $\mathcal{U}$-small singular simplices into the singular complex of $X$ induces isomorphisms on all homology groups. It is closely related to results on [[excision]] and on the [[Mayer-Vietoris sequence|Mayer-Vietoris theorem]].  Various homotopy analogues of this result have been given with one of the most recent being that given by Luire (see below). He explicitly relates it to the van Kampen theorem, and it is a similar 'local-to-global' result.

The experience from the uses of the homology small simplex theorem suggests that it may be most useful when used in conjunction with specific invariants of homotopy types. (Note excision fails for homotopy groups and so one has to be more subtle.) The precise relationship to the other forms of generalised van Kampen theorem has yet to be determined.

### General statement (Small simplex theorem aka generalised SvKT)

+-- {: .un_theorem}
###### Theorem

Let $X$ be a [[topological space]], write $Op(X)$ for its [[category of open subsets]] and let

$$
  \chi : C \to Op(X)
$$

be a [[functor]] out of a [[small category]] $C$ such that

* for each point $x\in X$ the [[full subcategory]] $C_x$ of objects $c$ such that $\chi(c)$ contains $x$ has a [[weak homotopy equivalence|weakly]] [[contractible]] [[nerve]].

Then:

the canonical morphism in [[sSet]] out of the [[colimit]]

$$
   {\lim_\to} Sing \circ \chi \to Sing(X)
$$

into the [[singular simplicial complex]] of $X$ exhibits $Sing(X)$ as the [[homotopy colimit]]  $hocolim Sing \circ \chi$.

=--

This is theorem A.1.1 in ([Lurie](#Lurie)).

A related theorem is given by ([Dugger-Isaksen 2004](#DuggerIsaksen)) and earlier versions were given by [[G. Segal]], (amongst others).

This form of the general type of result gives a relationship between the homotopy type of the space and the homotopy types of parts of that space, telling one how they fit together.  To get explicit more 'algebraic' calculations out it has so far proved necessary to consider spaces with extra structure, such as a [[filtered topological space|filtration]] (Brown-Higgins). The explicit algebraic models do not, in general, give a complete classifying invariant of the homotopy types concerned, but that allows much more precise and calculable invariants  to be found. In the first 'strict' version, the invariant, namely the  [[fundamental ∞-groupoid]],  is a [[strict ∞-groupoid]] and could be substituted by the corresponding [[crossed complex]] (according to your taste and needs!)

## Filtered space / strict  ∞-groupoid generalisation of the SvKT (Brown-Higgins)

Suppose $X_*$ is a [[filtered space]] and $X$ is the union of the interiors of sets $U^i$, $i \in I$. Let $U^i_*$ be the filtered space given by the intersections $U^i \cap X_n$ for $n \geq 0$. If $d=(i,j) \in I^2$ we write $U^d$ for $U^i \cap U^j$. We then have a coequaliser diagram of filtered spaces

$$\bigsqcup_{d \in I^2} U^d_* \rightrightarrows ^a_b \bigsqcup _{i \in I} U^i_* \to ^c X_*.$$

+-- {: .un_theorem}
###### Strict ∞-groupoid Higher van Kampen Theorem for filtered spaces.

If the filtered spaces $U^f_*$ are [[connected filtered space]]s for all finite intersections $U^f_*$ of the filtered spaces $U^i_*$, then

1. (Conn) The filtered space $X_*$ is connected; and

1. (Iso) The fundamental [[crossed complex]] functor $\Pi$ takes the above coequaliser diagram of filtered spaces to a coequaliser diagram of crossed complexes.

=--

**Remarks**

* Note that because $\Pi$ uses groupoids, it obviously takes disjoint unions $\bigsqcup$ of filtered spaces into disjoint unions (= coproducts) $\bigsqcup$ of crossed complexes.

* The proof of the theorem is not direct but goes via the fundamental cubical $\omega$-groupoid with connections of the filtered spaces, as that context allows the notions of _algebraic inverse to subdivision_ and of _commutative cube_. However the proof is a direct generalisation of a proof for the [[van Kampen theorem]] for the [[fundamental groupoid]].

* Applications of this theorem include many basic facts in algebraic topology, such as the Relative Hurewicz Theorem, the Brouwer degree theorem, and new nonabelian results on 2nd relative homotopy groups, not of course obtainable by the traditional wholly abelian methods. No use is made of _singular homology theory_ or of _simplicial approximation_.

* This result includes the crossed module version of the  generalised SvKT. This was the first extension in this direction. It handles relative homotopy groups. That version is also a special case of the next of this type of result given by the Brown-Loday theory.

##Generalised SvKT for models of $n$-types (Brown-Loday)

After [[Loday]] had proved that [[cat-n-group]]s modelled all connected [[homotopy n-type|homotopy (n+1)-type]]s, Brown and Loday investigated generalised forms of the SvKT, in terms of these new invariants. (to be continued)

## Van Kampen spectral sequence (Artin-Mazur)

Artin and Mazur proved the existence of a spectral sequence determined by the [[Cech complex]] of an open cover. This converges to the homotopy groups of the big space. There does nnot seem to have been any study done of the relationship of their results with the others mentioned above.  Their methods are very similar to those used for the Dugger-Isaksen small simplex theorem, however.

## For objects in a cohesive $(\infty,1)$-topos

In a [[cohesive (∞,1)-topos]] higher van Kampen theorems hold in great generality.

See the section cohesive (∞,1)-topos -- van Kampen theorem.

In particular for the cohesive $(\infty,1)$-topos [[?TopGrpd]] of [[topological ∞-groupoid]]s this reproduces the [topological higher van Kampen theorem](#ForTopSpaces) discussed above.

## Examples

Here is one application in dimension 2 not easily obtainable by traditional [[algebraic topology]].

Let $0 \to P \to Q \to R \to 0$ be an [[exact sequence]] of [[abelian groups]]. Let $X$  be the [[mapping cone]] of the induced map $K(P,1) \to K(Q,1)$ of [[Eilenberg-Mac Lane space]]s. Then a [[crossed module]] representing the [[homotopy 2-type]] of $X$ is $\mu: C \to Q$ where $C$ is abelian and is the direct sum $\oplus_{r \in R} P^r$  of copies of $P$ one for each $r \in R$ and the action of $Q$ is via $R$ and permutes the copies by $(p,r)^s=(p,r+s)$. Similar examples for $P,Q,R$ nonabelian are do-able,   more complicated, and certainly **not** obtainable by traditional methods.

## References
 {#References}

Various versions exist, generalising the classical theorem.

The homology small simplex theorem is to be found, for instance, in [[Spanier]]'s book on algebraic topology.

The homotopy version for topological spaces and the [[fundamental infinity-groupoid]] functor is discussed in  [Appendix A.1](http://www.math.harvard.edu/~lurie/papers/DAG-VI.pdf#page=134) of

* [[Jacob Lurie]], _[[Ek-Algebras]]_ (now part of _[[Higher Algebra]]_)

A very similar result occurs before as prop. A.5 in

* [[Graeme Segal]], _Classifying spaces related to foliations_, Topology, Vol 17, (1978) 367-382
 {#Segal}

An important  central lemma (the _small simplices theorem_) also appears as corollary 3.5 of

* [[Daniel Dugger]], [[Daniel Isaksen]], _Topological hypercovers and $\mathbb{A}^1$-realizations_, Mathematische Zeitschrift 246 (2004)
 {#DuggerIsaksen}
where many similar results are discussed.

The version for filtered topological spaces and the strict homotopy $\infty$-groupoid functor was one of the aims of the long research programme of [[Ronnie Brown]] and [[Philip Higgins]]. For the original work see the publication lists of Brown and Higgins.  A complete view of this approach is discussed in

* [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], _[[Nonabelian Algebraic Topology]]_

The Brown-Loday results can be found in

* R. Brown and [[J.-L. Loday]], _Homotopical excision, and Hurewicz theorems for n-cubes of spaces_, Proc. London Math. Soc., (3)54, (1987),
176 – 192.

and

*  R. Brown and [[J.-L. Loday]], _Van Kampen Theorems for diagrams of
spaces_, Topology, 26, (1987), 311 – 337.

The paper of Artin and Mazur is

* [[M. Artin]] and [[B. Mazur]], _On the van Kampen theorem_, [Topology, vol. 5 (1966), pp. 179 - 189](http://www.maths.ed.ac.uk/~aar/topology/Topology%20Vol%2005/Artin_On-the-van-Kampen-theorem_1966.pdf).

[[!redirects Higher Homotopy van Kampen Theorem]]
[[!redirects higher homotopy van Kampen Theorem]]

[[!redirects higher van Kampen theorem]]

[[!redirects higher homotopy van Kampen theorem]]

[[!redirects higher van Kampen theorems]]
[[!redirects Higher van Kampen Theorem]]
[[!redirects Higher van Kampen Theorems]]
[[!redirects higher homotopy van Kampen theorems]]


higher van Kampen theorem > history

< [[higher van Kampen theorem]]


_Higher-Dimensional Types in the Mechanization of Homotopy Theory_, [[Kuen-Bang Hou (Favonia)]], Ph. D. Thesis 2017, [pdf](https://favonia.org/files/thesis.pdf)

## Abstract
Mechanized reasoning has proved effective in avoiding serious mistakes
in software and hardware, and yet remains unpopular in the practice of mathematics. My thesis is aimed at making mechanization easier so that more
mathematicians can benefit from this technology. Particularly, I experimented
with higher-dimensional types, an extension of ordinary types with a hierarchy of stacked relations, and managed to mechanize many important results
from classical homotopy theory in the proof assistant Agda. My work thus
suggests higher-dimensional types may help mechanize mathematical concepts.

## See also
* [[homotopy type theory - references]]
* [[synthetic homotopy theory]]
* [[covering space]]

category: reference


higher-level foundations

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# Higher-level foundations
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## Definition

By **higher-level foundations**, or simply **higher foundations**, we mean any [[foundation of mathematics]] in which there are, or at least can be, basic objects of [[h-level]] higher than [[h-sets]].  Thus, these objects behave like [[groupoids]] or [[infinity-groupoids]] (or even [[categories]] or [[infinity-categories]]).  This is in contrast to [[set-level foundations]] such as [[set theory]] and [[set-level type theory]] in which all objects behave like sets.

Some higher-level foundations include:

* [[homotopy type theory]] / [[univalent foundations]] of all sorts:
  * [[Book HoTT]]
  * [[cubical type theory]], including CCHM cubical type theory, Cubical Agda, and Cartesian cubical type theory
  * [[higher observational type theory]]
* More generally, [[intensional type theory]], in which no types of higher h-level can be shown to exist, but neither can all types be shown to be [[h-sets]].
* [[formal category theory]]
* [[synthetic (infinity,1)-category theory]], such as [[simplicial type theory]] and bicubical type theory
* Hypothetical elementary theories of the [[bicategory]] of [[categories]] and the [[(2,1)-category]] of [[groupoids]], expressed in [[first-order logic]] with dependent sorts in the same way that [[ETCS]] is.  One extant theory along these lines is [[ETCC]], although it hasn't been successfully made into a full [[foundation of mathematics]].

[[!redirects higher foundations]]


higher-order abstract syntax

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## Idea
{#Idea}

The term **higher-order abstract syntax** has been used to refer to several distinct, but related, concepts. In chronological order, and from least to most generality:

1. A [[logical framework]] proposed by [[Pfenning & Elliot (1988)](#PfenningElliot88)] based on the [[simply-typed lambda-calculus]] with [[polymorphism]].
2. The [[logical framework]] technique of representing [[abstract syntax]] for [[theories]] with [[variable-binding operators]] using the [[function types]] of another theory viewed as a meta-theory, popularised by [[LF]] and Pfenning–Elliot's logical framework described in (1).
3. [[abstract syntax|Abstract syntax]] for [[theories]] with variable-binding operators, metavariable-binding operators, and so on. The logical framework technique in (2) is a particular representation of such abstract syntax.

Usage (1) does not appear to be in common usage. Usage (2) tends to be the convention in literature on [[logical frameworks]]; while usage (3) is the convention in literature on algebraic type theory, though note that authors often restrict to [[second-order abstract syntax]], which is sufficient for most [[type theories]].

Examples of variable-binding operators are [[lambda-abstraction]] and [[quantifiers]] (e.g. in [[higher-order logic]], whence the name).

## Higher-order abstract syntax in logical frameworks

In logical frameworks, one formalizes [[languages]] supporting forms of [[name binding]] (such as function parameters or [[quantifiers]]) by using the meta-language [[function space]] rather than a more syntactic notion of [[bound variable]].

## Higher-order abstract syntax in algebraic type theory

Traditional approaches to [[abstract syntax]], e.g. for [[algebraic theories]], typically involve [[abstract syntax trees]]. However, it is not possibly directly to capture [[variable-binding operators]] with such structures. Therefore, alternative structures are necessary to capture the syntax of [[theories]] with such operators, e.g. [[type theories]].

One possible approach is via the [[logical framework]] approach described above.

## Related Concepts

* [[abstract syntax tree]]

* HOAS is used extensively in [[logical frameworks]]

* [[synthetic Tait computability]]

## References

* {#PfenningElliot88} [[Frank Pfenning]], Conal Elliot, *Higher-order abstract syntax*, ACM SIGPLAN Notices **23** 7 (1988) 199–208 [[doi;10.1145/960116.54010](https://doi.org/10.1145/960116.54010), [pdf](https://www.cs.cmu.edu/~fp/papers/pldi88.pdf)]

* [[Martin Hofmann]], _Semantical analysis of higher-order abstract syntax_, LICS 1999 [[ieee:782616](https://ieeexplore.ieee.org/document/782616), [doi:10.1109/LICS.1999.782616](https://doi.ieeecomputersociety.org/10.1109/LICS.1999.782616)]

* [[Marcelo Fiore]], [[Gordon Plotkin]], [[Daniele Turi]]. _Abstract syntax and variable binding_, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158). IEEE (1999) [[doi:10.1109/LICS.1999.782615](https://doi.org/10.1109/LICS.1999.782615), [pdf](https://homepages.inf.ed.ac.uk/gdp/publications/Abstract_Syn.pdf), [webpage](https://www.dcs.ed.ac.uk/home/dt/abstractsyn.html)]

* [[Marcelo Fiore]], [[Chung-Kil Hur]]. _Second-order equational logic_, International Workshop on Computer Science Logic. Springer, Lecture Notes in Computer Science **6247** (2010) [[doi:10.1007/978-3-642-15205-4_26](https://doi.org/10.1007/978-3-642-15205-4_26), [pdf](https://www.cl.cam.ac.uk/~mpf23/papers/Types/soeqlog.pdf)]

See also:

* Wikipedia, *Higher-order abstract syntax*

[[!redirects HOAS]]


higher-order logic

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## Idea

A _higher-order logic_ is any [[logic]] which features _higher-order predicates_, which are [[predicates]] of predicates or of operations.  If we think of a predicate as a [[function]] to [[truth values]], then a higher-order predicate is a function on a [[power set]] or a [[function set]].

[[type theory|Typed]] higher-order logic may be called higher-order type theory. Typed higher-order [[intuitionistic logic]] is often identified with the [[internal logic]] of a [[topos]].

There are also examples of [[dependent type theory|dependently typed]] higher-order logics. One example of a dependently typed higher-order logic is a [[dependent type theory]] with a separate [[type]] [[judgment]], [[identity types]], [[dependent sum types]], [[dependent product types]], [[weak function extensionality]], a [[type of all propositions]], and a [[natural numbers type]]. This uses the [[propositions as some types]] interpretation of [[dependent type theory]].

## Related concepts

* [[logic]]

  * [[propositional logic]] (0th order)

  * [[predicate logic]] (1st order)

  * [[second-order logic]]

  * **higher order logic**

    * [[HOL]], [[Isabelle]]
    * Girard's System Fω (also called λω) and Coquand-Huet's [[calculus of constructions]] are the higher-order subsystems of [[pure type system#lambda_cube|Barendregt's lambda-cube]]

* [[higher-order abstract syntax]]

* [[dependent type theory]]

* [[tripos]]

## References

* [[Colin Rothgang]], [[Florian Rabe]], [[Christoph Benzmüller]], *Theorem Proving in Dependently-Typed Higher-Order Logic -- Extended Preprint*, The 29th International Conference on Automated Deduction (CADE-29), July 1-5, 2023. ([arXiv:2305.15382](https://arxiv.org/abs/2305.15382))

[[!redirects Higher-order logic]]
[[!redirects higher order logic]]

[[!redirects higher-order type theory]]
[[!redirects higher order type theory]]

[[!redirects higher-order proposition]]
[[!redirects higher-order propositions]]
[[!redirects higher order proposition]]
[[!redirects higher order propositions]]
[[!redirects higher-order predicate]]
[[!redirects higher-order predicate]]
[[!redirects higher order predicate]]
[[!redirects higher order predicate]]


highly structured spectrum

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## Idea

The term _highly structured spectrum_ refers to models for [[spectra]] by [[model categories]] which carry more [[structure]] than [[sequential spectra]], such as to support a [[symmetric monoidal smash product of spectra]] (see there for more background). This includes [[excisive functors]], [[orthogonal spectra]], [[symmetric spectra]] and [[S-modules]]. For details see _[[Introduction to Stable homotopy theory]]_, _[[Introduction to Stable homotopy theory -- 1-2|Part 1-2 -- Structured spectra]]_.

Similarly a _highly structured ring spectrum_ is a [[monoid]] in this context (a model for an [[A-infinity algebra]]/[[E-infinity algebra]]).

See at the following entries:

[[model structure on spectra]], [[symmetric monoidal smash product of spectra]]

* [[symmetric spectrum]], [[model structure on symmetric spectra]]

* [[orthogonal spectrum]], [[model structure on orthogonal spectra]]

* [[S-module]], [[model structure on S-modules]]

## Related concepts

* [[brave new algebra]]

## References

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], _Modern foundations for stable homotopy theory_ ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May|P. May]], *Rings, modules and algebras in stable homotopy theory* (aka "EKMM")

* {#MMSS00} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society Volume 82, Issue 2, 2000  ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf), [doi:10.1112/S0024611501012692]( https://doi.org/10.1112/S0024611501012692))

* {#Schwede12} [[Stefan Schwede]], p.214-216 of _[[Symmetric spectra]]_ (2012)

[[!redirects highly structured spectra]]

[[!redirects highly structured ring spectrum]]
[[!redirects highly structured ring spectra]]

[[!redirects structured spectrum]]
[[!redirects structured spectra]]


Higman's embedding theorem

[[!redirects Higman's Theorem]]
[[!redirects Higman's theorem]]
[[!redirects Higman embedding theorem]]
[[!redirects Higman's Embedding Theorem]]

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## Idea
In 1961 G. Higman proved an important result in [[group theory]] that brings together in an intriguing way concepts from [[computability theory]] and [[finitely presentable object|finite presentability]].

## The result

**Higman's embedding theorem.** _A finitely generated group can be embedded in a finitely presented group iff it has a presentation (with a finite generating set) for which the set of defining relations is a recursively enumerable set of words._

**Theorem.** _There is a finitely presented group which contains an isomorphic copy of every finitely presented group._

## Remark

It clearly makes sense to ask the same question for other algebraic theories. In fact, it has been conjectured by the group theorist W. Boone that a similar result holds more generally for single-sorted [[Lawvere theory|algebraic theories]]. While in fact the conjecture is not true for all single-sorted algebraic theories (see [[Boone conjecture]] for further details), as was known by Soviet mathematicians, the general question "In which algebraic categories does (an analogue of) Higman's embedding theorem hold?" is still interesting. The importance of this question has been stressed and the parallel to results by Craig and Vaught in [[first-order logic]] has been pointed out by [[F. W. Lawvere]] ([2002](#Lawvere02)) (see at [[Boone conjecture]] for further details).

## Related pages

* [[combinatorial group theory]]

* [[Boone conjecture]]

## Link

* [Wikipedia entry](https://en.wikipedia.org/wiki/Higman%27s_embedding_theorem)

## References

* G. Higman, _Subgroups of finitely presented groups_ , Proc. Royal. Soc. London Ser. A (1961) pp.455-475.

* L. Dediu , _Higman's Embedding theorem - An Elementary Proof_ , Report CDMCTS-010 (1995).  ([pdf](https://www.cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/010luminita.pdf))

* {#Lawvere02}[[William Lawvere|W. F. Lawvere]], _On the [[effective topos]] - message to catlist_, January 2002. ([link](http://facultypages.ecc.edu/alsani/ct02%281-2%29/msg00016.html))


Hikaru Kawai

* [webpage](http://www.kyoto-u.ac.jp/static/en/research/forefronts/archives/kawai.html)

## Selected writings

On the [[large N limit]] and first hints of [[holographic QCD]] and [[matrix models]]:

* {#EguchiKawai82} [[Tohru Eguchi]], [[Hikaru Kawai]], _Reduction of Dynamical Degrees of Freedom in the Large-$N$ Gauge Theory_, Phys. Rev. Lett. 48, 1063 (1982) ([spire:176459](http://inspirehep.net/record/176459), [doi:10.1103/PhysRevLett.48.1063](https://doi.org/10.1103/PhysRevLett.48.1063))

Introducing the [[KLT relations]]:

* {#KawaiLewellenTye86} [[Hikaru Kawai]], D. C. Lewellen, S. H. H. Tye, _A Relation Between Tree Amplitudes of Closed and Open Strings_, Nucl. Phys. B269 (1986) 1-23 (doi:10.1016/0550-3213(86)90362-7)

Introducing the [[IKKT matrix model]]:

* {#IKKT96} [[Noboyuki Ishibashi]], [[Hikaru Kawai]], Y. Kitazawa, A. Tsuchiya, _A Large-$N$ Reduced Model as Superstring_, Nucl. Phys. B498 (1997) 467-491 ([arXiv:hep-th/9612115](http://arxiv.org/abs/hep-th/9612115), doi:10.1016/S0550-3213(97)00290-3)

* H. Aoki, S. Iso, [[Hikaru Kawai]], Y. Kitazawa, T. Tada, A. Tsuchiya, _IIB Matrix Model_, Prog.Theor.Phys.Suppl.134:47-83,1999 ([arXiv:hep-th/9908038](http://arxiv.org/abs/hep-th/9908038))

and relating it to [[Yang-Mills theory]]:

* H. Aoki, [[Noboyuki Ishibashi]], S. Iso, [[Hikaru Kawai]], Y. Kitazawa, T. Tada, _Noncommutative Yang-Mills in IIB Matrix Model_, Nucl.Phys. B565 (2000) 176-192 ([arXiv:hep-th/9908141](http://arxiv.org/abs/hep-th/9908141))

and to [[closed string field theory]]:

* M. Fukuma, [[Hikaru Kawai]], Y. Kitazawa, A. Tsuchiya, _String Field Theory from IIB Matrix Model_, Nucl.Phys.B510:158-174,1998 ([arXiv:hep-th/9705128](http://arxiv.org/abs/hep-th/9705128))

category: people