hyperstructure

\tableofcontents

#Idea#

[[Nils Baas]] has been emphasizing for many years, in print and in private communication, the conviction that the usual  notions of [[n-category]], [[infinity-category]] [[omega-category]] in [[higher category theory]] are __not__ naturally suited for describing

* [[extended cobordisms]] such as appearing in

  * the [[generalized tangle hypothesis|tangle hypothesis]]

  * in [[FQFT|extended quantum field theory]]

* hierarchical systems such as appearing in

  * complex systems;

  * biology.

The point is essentially that the _directedness_ of morphisms and --- related to that --- the binary notion of [[source]] and [[target]] in categories and higher categories are notions alien to these contexts, which in applications have to and are essentially removed again in a second step by adding extra structure and requiring further properties, such as various monoidal structures and dualities, which allow to change the direction of morphisms, to collect objects together, etc.

In contrast to that, Baas pointed out that more naturally the above situations are thought of from the beginning in terms of hierarchies of what he calls _bonds_, where, quite generally, a bond is an object equipped with information of how a collection of sub-bonds sits inside it, _bound by the bond_.

A sketch of a generic such a situation of hierarchical bonds is a diagram

$$
  \array{
     \beta_2 &&=&& \beta_2 && \beta_4
     \\
     \downarrow && && \downarrow & \swarrow
     \\
     b_1 &\to& B &\leftarrow& b_2
     \\
     \uparrow && \uparrow  && \uparrow
     \\
     \beta_1 && b_3 && \beta_3
     \\
     && \uparrow
     \\
     && \beta_5
  }
$$

where a bond $B$ binds sub-bonds $b_1, b_2$ and $b_3$ which in turn bind sub-bonds $\beta_i$.

For instance $B$ might be an [[extended cobordism]] with three boundary components $b_1, b_2, b_3$ which in turn have pieces of boundary components $\beta_i$.

Nils Baas has made, in print and in private communication, suggestions for a formalization of such systems of hierarchical bonds and coined the term __hyperstructures__ for these. Some references are

* Nils A. Baas, _On Higher Order Structures_, [arXiv:1509.00403](http://arxiv.org/abs/1509.00403) (2015), 15 pp.

* Nils A. Baas, _Higher order architecture of collections of objects_, International Journal of General Systems vol. 44 (1), pp. 55-75, 2015. [arXiv:1409.0344](http://arxiv.org/abs/1409.0344)

* Nils A. Baas, N.C. Seeman and A. Stacey, _Synthesising topological links_, Journal of Mathematical Chemistry, vol. 53, pp. 183-199, 2015.

* Nils A. Baas, D.V. Fedorov, A.S. Jensen, K. Riisager, A.G. Volosniev and N.T. Zinner, _Higher-Order Brunnian Structures and Possible Physical Realizations_, Physics of Atomic Nuclei, vol. 77(3), pp. 336-343, 2014. [arXiv:1205.0746](http://arxiv.org/abs/1205.0746)

* Nils A. Baas, _New states of matter suggested by new topological structures_, International Journal of General Systems, vol. 42 (2), pp. 137-169, 2012. [arXiv:1012.2698](http://arxiv.org/abs/1012.2698)

* Nils A. Baas, _On structure and organization: an organizing principle_, International Journal of General Systems, vol. 42 (2), pp. 170-196, 2012. [arXiv:1201.6228](http://arxiv.org/abs/1201.6228)

* Nils A. Baas and N.C. Seeman, _On the chemical synthesis of new topological structures_, Journal of Mathematical Chemistry, vol. 50 (1), pp. 220-232, 2012.

* Nils A. Baas, _New structures in complex systems_, The European Physical Journal Special Topics, vol. 178 (1), pp. 25-44, 2010.

* Nils A. Baas, _Hyperstructures, Topology and Datasets_, Axiomathes, vol. 19 (3), pp. 281-295, 2009.

* Nils A. Baas, _Hyperstructures as Abstract Matter_, Advances in Complex Systems, vol. 9 (3), pp. 157-182, 2006.

* Nils A. Baas, A. Ehresmann and J.-P. Vanbremeersch, _Hyperstructures and Memory Evolutive Systems_, International Journal of General Systems, vol. 33 (5), pp. 553-568, 2004.

* Nils A. Baas, _Higher order cognitive structures and processes_. In _Toward a Science of Consciousness_, S.R. Hameroff, A.W.Kaszniak and A.C.Scott (editors), MIT Press, pp. 633-648, 1996.

* Nils A. Baas, _Hyper-structures as a tool in nanotechnology_, Nanobiology, vol. 3 (1), pp. 49-60, 1994.

#Remarks#

* The notion of replacing morphisms by _bonds_ is familiar and concretely realized at least in a hierarchy of depth 2 in the context of [[groupoidification]] and [[geometric function theory]], where morphisms are replaced by [[spans]]. Regarding these as [[cospans]] in the [[opposite category]] produces diagrams like the above sketch of a generic bond system.

* Accordingly, the notion of [[multispan]] and [[multi-cospan]] may come close to exhibiting some crucial aspects of the idea that motivated the concept of hyperstructures.

#Blog discussion#

* David Corfield, [Category Theory and Biology](http://golem.ph.utexas.edu/category/2007/11/category_theory_and_biology.html)

  * The birth of the idea of [[multispans]] as a formalization of hyperstructures is mentioned in [this comment](http://golem.ph.utexas.edu/category/2007/11/category_theory_and_biology.html#c013165).
Its application to the description of extended cobordisms and the [[generalized tangle hypothesis]] is topic of some thought chatted about at [this entry](http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html#c021486).

+--{.query}

[[Mike Shulman|Mike]]: Shouldn't we allow "oriented bonds" as well?  I am thinking of the case of, say, rings and modules, where a module $M$ that has right actions by rings $R$ and $S$ and a left action by $T$ should have $R,S,T$ as "sub-bonds" but with different "orientations."  And the "gluing" operation is tensor product, which only works if one module is a left module and the other is a right module.  This example also goes up in dimension, for instance modules over algebras over rings.  However, it seems that one would also needs this for cobordisms; aren't you only allowed to glue cobordisms along boundaries whose orientations match?

[[Urs Schreiber|Urs]]: right, the module example currently does not really fit yet. The reason seems to be that the would-be [[dagger category|dagger]] functor is not the identity on objects here, but sends objects to their opposites.

Concerning the cobordisms: in the cospan picture, every boundary component explicitly comes with its embedding into the cobordism. If two boundary components are equal as unembedded objects, then the pushout will automatically glue the corresponding cobordisms in the right way by identifying the components as embedded pieces. Do you see what I mean. Maybe I am mixed up about this.

[[Mike Shulman|Mike]]: Yes, the point is that a lot of these examples aren't really dagger at all, only [[compact closed category|compact closed]] (which I would prefer to call "autonomous").  My point about cobordisms has to do with orientations.  If $M$ and $N$ are two manifolds sharing a boundary component $B$, then I can glue them along it, but if they are _oriented_ then the gluing will only inherit an orientation if $M$'s boundary component is $B$ and $N$'s boundary component is $B^{op}$.  So it's just like rings; the category of oriented cobordisms is not dagger, only compact closed.  An oriented cobordism from $A$ to $B$ is not the same as an oriented cobordism from $B$ to $A$, but it is an oriented cobordism from $B^{op}$ to $A^{op}$.

[[Urs Schreiber|Urs]]: True. I am not sure about the best answer yet, but have thought about the following:

Suppose we have a span

$$
  A \leftarrow C \to B
$$

and want to think of its index objects $A$ and $B$ to carry an orientation which gets reversed when we flip the legs around to

$$
  B \leftarrow C \to A
  \,.
$$

We might be able to emulate this within the setting described below by supposing that there are two different dummy objects $\bullet$ and $\circ$ around which behave at least to some extent like initial objects and that the original span was secretly really the multispan:

$$
  \array{
     && \bullet
     \\
     & \swarrow && \searrow
     \\
     A &\leftarrow& C &\to& B
     \\
     & \nwarrow && \nearrow
     \\
     &&
     \circ
  }
  \,.
$$

Now, if in this multispan we want to exchange the original two legs, we could for instance rotate the entire multispan by a half turn to get

$$
  \array{
     && \circ
     \\
     & \swarrow && \searrow
     \\
     B &\leftarrow& C &\to& A
     \\
     & \nwarrow && \nearrow
     \\
     &&
     \bullet
  }
  \,.
$$

This would introduce the desired orientation on objects, as the second multispan behaves like:

$$
  B^{op} \leftarrow C \to A^{op}
  \,.
$$

For this to work out as desired, we would of course have to concentrate on planar multispans and disallow identification of multispans that can be turned into each other by reflections at lines in the plane.

_Mike_: It seems as though restricting to planar multispans would be quite a serious thing to do.

=--

***

#Laboratory section#

+--{.query}
[[Urs Schreiber|Urs]]: I am thinking about a way to formalize the idea of hyperstructure with a concrete application to extended quantum field theory and the Baez-Dolan hypothesis in mind. I would like to develop this in the following here on this page. If things work out as hoped for, this should eventually become an entry in its own right. If not, this should eventually be discarded.

[[Ronnie Brown|Ronnie]] In this connection I would like to draw attention here to

* D.W.Jones,  Polyhedral $T$-complexes, University of Wales PhD Thesis, 1984; published as  A general theory of polyhedral sets and their corresponding $T$-complexes, Diss. Math. 266, 1988.

The idea here was to give expression to the notion of: what is wrong with pentagons? Or rhombic dodecahedra? as part of the basic models. Two major problems were solved:

1. What should be the basic cells, and the polyhedral category?

1. How should one orient or more the basic cells?

The answer to the first question is given in terms of cells with polyhedral boundary and in which there is  a shellability condition. The answer to the second question is in terms of each cell has a marked face. This turns out to imply orientation, but is more of a homotopy condition than a homology condition, i.e. is stronger than the latter. There a relation to certain posets studied by Bjorner.

On the other hand this theory is probably more group than category oriented, there is a tendency to imply inverses.

The algebra comes in by defining _poly-$T$-complexes_, i.e. poly sets with [[thin element]]s.

=--

##Idea##

**Motivation**

A _bond system_ or _hyperstructure_ as intended in the following should be

* a conglomerate of _cells_ each of which is equipped with a prescribed collection of _sub-cells_, which in turn have their prescribed collection of subcells, etc;

* such that whenever two such cell complexes coincide on any one part of their sub-cell complex, there exists a cell which can be regarded as gluing the two cells along this sub-cell complex.

The idea is hence akin to the [[geometric definition of higher category]], but differs in two crucial aspects:

1. there is no fixed choice of [[geometric shapes for higher structures]] (such as [[globe]]s, [[simplex|simplices]], [[cube]]s, etc.) out of which the entire structure consists, instead _all_ possible sub-cell complexes are admitted;

2. there is no notion of directionality imposed on the cell complexes, and in particular _no ordering_ on the sub-cells of a given cell, neither binary into source and target cells as in [[globe category|globular]] higher categories, nor linearly into indexed faces as in [[simplex category|simplicial]] higher categories.

The idea is that a hyperstructure is like a **$\infty$-[[dagger category|dagger]] [[infinity-category]]** but capturing this notion more directly than an $\infty$-category equipped with an $\infty$-dagger-operation would (which amounts to first introducing directionality only to remove it in a second step). Notice that in this context that categories of ordinary [[span]]s are always dagger-categories, where the dagger operation is nothing but the re-labelling of the legs of the span into source and target.  Hyperstructures in particular generalize categories of spans (as described below) and their special nature is supposed to intrinsically realize the dagger-structure.

**Formalization**

To formalize this,  the idea is to observe that every _cell_ with its complex of sub-cells should naturally form a [[partial order|poset]] under inclusions of sub-cells. Since for any collection of sub-cells which share a common collection of sub-sub-cells the result of gluing these sub-cells along their common sub-sub-cells should again be a sub-cell, this poset should be closed under [[colimit]]s.

Therefore a _bond system_ or _hyperstructure_ should in particular assign to every poset with colimits a collection to be interpreted as the collection of cells whose sub-cell structure is of the form given by the poset. Such an assignment would naturally allow to restrict any cell to any of its sub-cells along an inclusion of posets to obtain another cell, and to extend any cell to a cell with richer but degenerate sub-cell structure along a surjection of posets. This clearly suggests that the _bond system_ or _hyperstructure_ is a [[presheaf]] on the [[category]] of [[partial order|poset]]s with [[colimit]]s.

Given this, the **gluing condition** or **sewing condition** that for any collection of cells which match on a part of their sub-cell complex there exists a cell obtained by gluing the cells along their common sub-cells can be formulated in a way analogous to the horn-filling condition in a [[Kan complex]] by requiring that the obvious map from the collection of cells which have the right shape of being glued cells to the collection of cells that admit gluing is an [[epimorphism]].

**Examples**

The archetypical example of a _bond system_ or _hyperstructure_ as intended in the following is supposed to be the collection $MultiCoSpans(C)$ of [[multispan|multi co-span]]s in a category $C$ with colimits. Since a [[multispan|multi co-span]] in $C$ is supposed to be nothing but the image of a poset in $C$, this means that the _bond structure_ of [[multispan|multi co-span]]s should be the presheaf on $Posets$ [[representable functor|represented]] by $C$. This does satisfy the _gluing condition_ in that for all multi-cospans which coincide in parts there exists in $C$ the colimit over their joint diagram. For ordinary [[cospan]]s this is supposed to reproduce the ordinary composition of [[cospan]]s by pushout over a joint leg.

**The hyperstructure of hyperstructures**

As there is a canonical notion of homo[[morphism]]s of
[[presheaf|presheaves]] we naturally obtain a [[category]] of _bond systems_ or _hyperstructures_. By the above example this in turn naturally induces a hyperstructure of hyperstructures consisting of the multi-cospans in the category of presheaves on posets.

This allows then finally to formalize one of the situations motivating the notion of hyperstructure in the first place, that of [[FQFT|extended quantum field theory]]: for instance for _topological_ QFT this is expected to be a morphism (or more general span) of hyperstructures
$$
  \array{
     ExtendedCobordisms :=
     MultiCoSpans(Top) &\to& MultiSpans(S)
     =:
     \infty Vect_S
  }
  \,,
$$
where multi-cospans in the category [[Top]] are thought of as modelling [[extended cobordism]]s as described for instance at [[Cospans in Algebraic Topology]], while multi-spans in some suitable category $S$ (multi co-spans in the [[opposite category]] $S^{op}$) are thought of as modelling morphisms between higher vector spaces as in [[groupoidification]].

##Definitions##

First recall some basics of [[partial order|poset]]s to fix our notation.

We write $2 = \{\bottom \to \top\}$ for the category with two objects and a single nontrivial morphism between them.
Recall that a [[partial order|poset]] is a [[enriched category|category enriched]] over $(2,\otimes, I = \top)$, where the canonical tensor product on $2$ is has tensor unit $\top$ and $\bottom \otimes \bottom = \bottom$.

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

Write $Posets$ for the (1-)category of small [[partial order|poset]]s
and $\overline{Posets} \subset Posets$ for the sub-category of small posets with all [[colimit]]s.

=--

Recall that for $D$ a poset the [[Yoneda embedding]]
$Y_D : D \to 2^{D^{op}}$ is the free cocompletion of $D$ and notice that for every poset $D$ also the [[presheaf]] category $2^{D^{op}}$ is again a poset. This yields a functor

$$
  \overline{(-)} := Y  : Posets \to \overline{Posets}
$$

which sends posets to their free cocompletion.

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

A **bond system** or **hyperstructure** $K$ is

* a [[Set]]-valued [[presheaf]] on $\overline{Posets}$
$$
  K : \overline{Posets}^{op} \to Set
$$

* which satisfies the following **gluing condition** or **sewing condition**: for every diagram
$$
  D_1 \leftarrow D_{glue} \to D_2
$$
in $\overline{Posets}$ the unique morphism $\phi$
$$
  \array{
     K(\overline{D_1 \sqcup_{D_{glue}} D_2} )
     &&&\stackrel{K(Y)}{\to}& K(D_1 \sqcup_{D_{glue}} D_2)
     \\
     & \searrow^{\phi} &&& \downarrow
     \\
     &&
     K(D_1) \times_{K(D_{glue})} D(D_2) &\to& K(D_2)
     \\
     \downarrow^{K(Y)}
     &&
     \downarrow && \downarrow
     \\
     K(D_1 \sqcup_{D_{glue}} D_2)
     &\to&
     K(D_1) &\to& K(D_{glue})
  }
$$
is an [[epimorphism]].

For $\Gamma$ a hyperstructure and $D \in \overline{Posets}$ we call $\Gamma(D)$ the collection
of **cell**s of $\Gamma$ of **shape** $D$.

=--

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

For $C$ a category with colimits let
$$
  MultiCoSpan(C) := Cat(-,C) : \overline{Posets} \to Set
$$
be the presheaf which assigns to any poset $D$ the set $Cat(D,C) = Funct(D,C)$ of [[functor]]s  from $D$ to $C$.

=--

+-- {: .un_lemma }
###### Lemma

For $C$ a category with colimits,
$MultiCoSpan(C)$ is a bond structure.

=--

+-- {: .proof}
###### Proof

We have to check the sewing condition. First notice that for $K := Cat(-,C)_\sim$ we have a natural bijection
$$
  K(D_1) \times_{K(D_{glue})} K(D_2)
  \simeq
  K(D_1 \sqcup_{D_{glue}} D_2)
  \,.
$$
Then recall from the discussion at [[Yoneda embedding]] that every functor $F \in K(D_1 \sqcup_{D_{glue}} D_2)$,
$F : D_1 \sqcup_{D_{glue}} D_2 \to C$ canonically extends to a functor $\hat F \in K(\overline{D_1 \sqcup_{D_{glue}} D_2})$ given by the [[end|coend]]
$$
  \hat F(-) = \int^{d \in D_1 \sqcup_{D_{glue}} D_2}
    hom(d,-)\cdot F(d)
$$
in that $\phi = K(Y_{D_1 \sqcup_{D_{glue}} D_2} ) :
\hat F \mapsto F$. This says that $\phi$ is epi.

=--

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

For $C$ a category with colimits we call
the cells of $MultiCoSpans(C)$ the **multi co-span**s in
$C$, call all of $MultiCoSpans(C)$ the
**hyperstructure of multi-cospans** in $C$. Dually,
for $C^{op}$ a category with colimits we
write
$$
  MultiSpan(C) := MultiCoSpan(C^{op})
$$
for the **hyperstructure of multi-spans** in $C$.
=--

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

We write
$$
  Hyperstructures := MultiCoSpans(Set^{\overline{Posets}^{op}})
$$
for the **hyperstructure of hyperstructures**.

=--

=--

##Examples##

### Composition of ordinary cospans ###

The above general definition in particular reproduces the ordinary composition of [[cospan]]s.

Let $C$ be a category with colimits.
The cells of the hyperstructure $MultiCoSpan(C)$
of shape the [[pullback]] poset
$$
  D =
  \left\{
  \array{
     && top
    \\
    & \nearrow && \nwarrow
    \\
    a &&&& b
  }
  \right\}
$$
are the ordinary [[cospan]]s $c \in MultiCoSpan(C)(D)$ in $C$. Let $D_{glue} = pt = \{\bullet\}$ and consider the functors
$\alpha : D_{glue} \to D : \bullet \mapsto a$ and
$\beta : D_{glue} \to D : \bullet \mapsto b$ in the diagram
$$
  D_1 := D \stackrel{\beta}{\leftarrow}
  D_{glue}
  \stackrel{\alpha}{\to}
  D =: D_2
$$
as above.

Then
$$
  D_1 \sqcup_{D_{glue}} D_2 =
  \left\{
    \array{
       && t_1 &&&& t_2
       \\
       & \nearrow && \nwarrow && \nearrow && \nwarrow
       \\
       a &&&& b &&&& c
    }
  \right\}
$$
and
$MultiCoSpans(C)(D) \times_{MultiCoSpans(C)(D_{glue})}
 MultiCoSpans(C)(D)
 \simeq
  MultiCoSpans(C)(D_1 \sqcup_{D_{glue}} D_2)
 $
is the collection of pullback-diagrams in $C$ which share one leg. The cocompletion of $D_1 \sqcup_{D_{glue}} D_2$ looks in parts like
$$
  \overline{D_1 \sqcup_{D_{glue}} D_2}
  =
  \left\{
    \array{
       &&&& \top = t_1 \sqcup t_2
       \\
       &&& \nearrow && \nwarrow
       \\
       && t_1 &&\cdots&& t_2
       \\
       & \nearrow &\uparrow& \nwarrow && \nearrow
       &\uparrow& \nwarrow
       \\
       a &\rightarrow& a \sqcup b& \leftarrow& b
       &\rightarrow& b \sqcup c& \leftarrow& c
       \\
       & \nwarrow &&& \uparrow &&& \nearrow
       \\
       &&&& \bottom
    }
  \right\}
  \,,
$$
where the important point is that it contains the terminal object $\top$. Given any two cospans in $C$ that coincide on one foot

$$
  F(D)
  =
  \left\{
  \array{
    && F(t_1)
    \\
    & \nearrow && \nwarrow
    \\
    F(a) &&&& F(b)=F'(b)
  }
  \right\}
  \;\;\;\;\;\;\;\;
  F'(D)
  =
  \left\{
  \array{
    && F'(t_2)
    \\
    & \nearrow && \nwarrow
    \\
    F(b)= F'(b) &&&& F'(c)
  }
  \right\}
$$

they induce the element

$$
  F \sqcup_{b} F'(D_1 \sqcup_{D_{glue}} D_2)
  =
  \left\{
  \array{
    && F(t_1) &&&& F'(t_2)
    \\
    & \nearrow && \nwarrow && \nearrow && \nwarrow
    \\
    F(a) &&&& F(b)=F'(b) &&&& F'(c)
  }
  \right\}
$$

whose lift to $\overline{D_1 \sqcup_{D_{glue}} D_2}$
is given on $\top$ by

$$
 \begin{aligned}
  \widehat{F \sqcup_b F'}(\top)
  &=
  \int^{d \in D_1 \sqcup_{D_{glue}} D_2}
  hom(d,\top)\cdot (F \sqcup_b F')(d)
  \\
  &=
  \int^{d \in D_1 \sqcup_{D_{glue}} D_2}
  (F \sqcup_b F')(d)
  \\
  &=
  colim_{d \in D_1 \sqcup_{D_{glue}} D_2}
  (F \sqcup_b F')(d)
 \end{aligned}
  \,.
$$

This is indeed the ordinary composite of the two [[cospan]]s $F$ and $F'$.

+--{.query}
[[Mike Shulman|Mike]]: It seems to me that this definition doesn't contain enough information.  Yes, the ordinary composite of the two cospans is _one_ lift to $\overline{D_1 \sqcup_{D_{glue}} D_2}$, but there are plenty of other lifts, and there doesn't seem to be anything in the presheaf you've described that can characterize that particular lift as the "correct" gluing (in particular, its universal property in $C$ seems to have been forgotten).

[[Urs Schreiber|Urs]]: I was thinking of next defining "non-flabby" $MultiCoSpans(C)$ as the presheaf of co-continuous functors from a given poset into $C$, so that the above lifts are the only ones (right?). Then I want to say that a morphism of hyperstructures from the bordism hyperstructure defined below to a non-flabby multicospan hyperstructure is fixed by its value on the point if it regards the interval as weakly equivalent to the point.

I am thinking of the difference between the above "flabby" version and the non-flabby version as the difference between general Kan complexes and those that are the nerve of an $n$-groupoid for which above $n$ the fillers are unique.

[[Mike Shulman|Mike]]: Using cocontinuous functors will make the lifts unique, at least up to isomorphism.  It doesn't seem that you've included any information about isomorphisms, but perhaps that can be remedied with higher spans or with presheaves of groupoids or categories.  But I don't see the "flabby" version as very much like a Kan complex; in a Kan complex the fillers of horns _are_ unique up to homotopy, because of the higher-dimensional horn-filling conditions, but I don't see anything like that going on here.

[[Urs Schreiber|Urs]]: But we have these higher filling conditions here automatically, too: if there are two different multispans which glue two given ones, they share these two given ones and can be glued along them. This should say that any two fillers are themselves connected by a filler.

_Mike_: It seems as though if you're considering two multispans to be equivalent if they can be connected by a filler, and your horn-filling conditions stipulate that all fillers exist, then your structure is fairly trivial: everything is equivalent to everything else.

=--

### Bordism hyperstructure ###

Let
$$
  I :=
  \left(
  \array{
    && [0,1]
    \\
    & {}^{in}\nearrow && \nwarrow^{out}
    \\
    pt &&&& pt
  }
  \right)
$$
be the standard [[interval object]] in [[Top]], regarded as a [[cospan]].

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

Let $ExtendedCobordisms$, the **hyperstructure of extended cobordisms**, be the smallest sub-hyperstructure of $MultiCospans(Top)$ which contains the interval object $I$ and is closed under cartesian product $\times$ of multi-cospans in $Top$.

=--

+--{.query}
[[Mike Shulman|Mike]]: In order for that to make sense, you need there to be a smallest sub-hyperstructure such that (blah).  But I don't see any reason for that to exist; the intersection of sub-hyperstructures need not be a hyperstructure.  Even in a simplicial set the intersection of sub-Kan-complexes need not be a Kan complex; if $X$ contains one filler for a given horn and $Y$ contains a different filler, then $X\cap Y$ may contain no filler at all.

[[Urs Schreiber|Urs]]: I agree that it is not a priori clear that some smallest hyperstructure with a given property exists. But I am not sure that I see what this has to do with intersections. Shouldn't we we be looking at _inclusions_?

But in the particular case at hand, it feels that this smallest hyperstructure should certainly exist, no? I should think about formalizing it, but heuristically it should be given by the colimit over inclusions of the following steps for building it:

* start with the interval cospans and take all ways to glue the interval with copies of itself.

1. then include for everey multi-cospan obtained so far also its product with the interval cospans;

2. then form all composites of multi-cospans obtained so far

3. then continue with 1. .

_Mike_: In general, there are two ways to construct the smallest sub-widget of $X$ such that (blah):

* Take the intersection of all sub-widgets of $X$ such that (blah).
* Start with (blah) and close it up, iteratively, under all the operations that define a sub-widget.

Usually, if one works the the other does too.  Occasionally one fails where the other succeeds due to set-theoretic technicalities.  But here I think the problem is intrinsic and will rear its head wearing a different mask in either case.

It definitely does not feel to me as though the smallest hyperstructure should exist.  The problem with your proposal is when you say "form all composites."  Since composites are not an _operation_, this can't mean "for any $x$ and $y$ already in our sub-hyperstructure, add in _the_ composite of $x$ and $y$."  If we just arbitrarily pick some particular composite of $x$ and $y$ to add, then there might be other sub-hyperstructure that chose a different composite, so what we end up with won't be the smallest (in the "minim<i>um</i>" sense) sub-hyperstructure such that (blah).  We might be able to construct a minim<i>al</i> sub-hyperstructure, but it wouldn't be uniquely determined, so we couldn't talk about [[generalized the|the]] hyperstructure of extended cobordisms.

_Toby_:  And then, you might have to rely on Zorn\'s Lemma (the [[axiom of choice]]) to prove that even such a minimal structure exists!
=--

This means that multi-cospans in $ExtendedCobordisms$ arise from iteratively tensoring with the interval and forming pushouts, which amounts to foming [[co-span co-trace]]s.

For instance with the interval $I$ there is also the circle $S^1$ in $ExtendedCobordisms$, arising as the composition of the interval with itself over both legs
$$
  \array{
    && I
    \\
    & \nearrow &\downarrow& \nwarrow
    \\
    pt &&S^1&& pt
    \\
    & \searrow &\uparrow& \swarrow
    \\
    && I
  }
  \,.
$$

With the circle there is then also the cylinder

$$
  \left(
  \array{
    && S^1
    \\
    & \nearrow && \nwarrow
    \\
    \emptyset &&&& \emptyset
  }
  \right)
  \times
  \left(
  \array{
    && I
    \\
    & {}^{in}\nearrow && \nwarrow^{out}
    \\
    pt &&&& pt
  }
  \right)
  =
  \left(
  \array{
    && S^1 \times I
    \\
    & {}^{in}\nearrow && \nwarrow^{out}
    \\
    S^1 &&&& S^1
  }
  \right)
  \,.
$$

With the cylinder there is also the torus

$$
  \array{
    && S^1 \times I
    \\
    & \nearrow &\downarrow& \nwarrow
    \\
    S^1 &&T^2&& S^1
    \\
    & \searrow &\uparrow& \swarrow
    \\
    && S^1 \times I
  }
  \,.
$$

Etc. Then there are the extended cobordisms proper, which have deeper layers of sub-cells. For instance the product of the interval with itself gives the disk regarded as an extended cobordisms

$$
  \left(
  \array{
    && I
    \\
    & {}^{in}\nearrow && \nwarrow^{out}
    \\
    pt &&&& pt
  }
  \right)
  \times
  \left(
  \array{
    && I
    \\
    & {}^{in}\nearrow && \nwarrow^{out}
    \\
    pt &&&& pt
  }
  \right)
  =
  \left(
  \array{
    pt &\to& I &\leftarrow& pt
    \\
    \downarrow && \downarrow && \downarrow
    \\
    I &\to&  I^2  &\leftarrow& I
    \\
    \uparrow && \uparrow && \uparrow
    \\
    pt &\to& I &\leftarrow& pt
  }
  \right)
$$

with four 1-dimensional and four 0-dimensional boundary components.

+--{.query}
[[Mike Shulman|Mike]]: I'm not sure if this is intentional or not, because I don't know what your goal is, but even if the definition makes sense, these "extended cobordisms" include things that aren't very manifold-like.  For instance, if you glue the interval-with-endpoints to another copy of the interval-with-endpoints along one of the endpoints, you get a cell like
$$
    \array{
       &&&& [0,2]
       \\
       &&& \nearrow && \nwarrow
       \\
       && [0,1] &&\cdots&& [1,2]
       \\
       & \nearrow &\uparrow& \nwarrow && \nearrow
       &\uparrow& \nwarrow
       \\
       \{0\} &\rightarrow& \{0,1\}& \leftarrow& \{1\}
       &\rightarrow& \{1,2\}& \leftarrow& \{2\}
       \\
       & \nwarrow &&& \uparrow &&& \nearrow
       \\
       &&&& \bottom
    }
$$
and now restricting along a suitable inclusion of posets, we get
$$ \array{ [0,2] \\ \uparrow \\ \{1\}}$$
and we can now glue this to another interval along the point $1$, obtaining a wedge of three copies of the interval.  I think the things you glue along should somehow be "canceled out" and no longer appear as sub-cells of the glued result (this is certainly what happens in the case of modules).

[[Urs Schreiber|Urs]]: true. I need to think about a nice way to formalize such a cancellation.

[[Z|Z]]: I'd like to know more about composition of bonds as described on p.8 of "Higher Order Architecture of Collections of Objects" (Nils Baas). Can someone please clarify the rules on this page?
=--

[[!redirects hyperstructural]]
[[!redirects hyperstructures]]


hypersurface

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Broadly speaking, a "hypersurface" is a higher-[[dimension|dimensional]] analogue of a _[[surface]]_.
Typically the term is used in generalization of [[curves]] and [[surfaces]] that are [[embedding of topological spaces|embedded]] into an ambient [[Cartesian space]]/[[Euclidean space]] -- such as is the case for _[[iso-hypersurfaces]]_ -- and used in this sense  the notion is essentially synonymous to that of a _[[submanifold]]_.

In algebraic geometry, a hypersurface is a codimension 1 subvariety.

## Related concepts

* [[iso-hypersurface]]
* [[hypersurface singularity]]
* [[Mather-Yau theorem]]

[[!redirects hypersurfaces]]


hypostatic abstraction

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Hypostatic_abstraction).

[[!redirects hypostasis]]
[[!redirects hypostases]]
[[!redirects hypostatic object]]
[[!redirects reification]]
[[!redirects subjectal abstraction]]


hypothesis

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

In [[formal logic]] (such as [[sequent calculus]] or [[natural deduction]])
a **hypothesis** or **premise** is the [[antecedent]] of a [[sequent]] (or part of one):

A general [[judgement]] asserts the [[proof]] of a [[proposition]] (or, [[propositions as types|more generally]], the [[term]] of some [[type]]) *assuming* the given hypothesis, which itself may or may not have a (known) proof.

In the practice of [[mathematics]] (or beyond), hypotheses that that are *expected* to have a [[proof]], even if currently unknown, are known as *[[conjectures]]*.

For example, the "[[standard conjectures]]" in [[algebraic geometry]] serve as hypotheses in a wealth of [[theorems]] which are all proven (only) "assuming the standard conjectures" (cf. e.g. [arXiv:9804123](https://arxiv.org/abs/math/9804123)).

## Related concepts

[[!include mathematical statements --- contents]]

[[!redirects hypothesis]]
[[!redirects hypotheses]]

[[!redirects premise]]
[[!redirects premises]]

[[!redirects assumption]]
[[!redirects assumptions]]


Hypothesis H

> This page is a non-maintained mirror of [ncatlab.org/schreiber/show/Hypothesis+H](https://ncatlab.org/schreiber/show/Hypothesis+H). See there for the up-to-date version.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

**Hypothesis H** is a hypothesis about the mathematical formulation of the theory of fundamental physics known as _[[M-theory]]_. The core statement of the hypothesis is that quantum [[charge quantization]] in [[M-theory]] happens in the [[non-abelian cohomology]] theory called *[[cohomotopy]]* (in more detail: *[[twisted cohomotopy]]*). This originates with an observation due to [Sati 2013](#Sati13) ([Sec. 2.5](https://arxiv.org/pdf/1310.1060.pdf#page=17)) and has become the theme of a research program developed by [[Hisham Sati]] and [[Urs Schreiber]], see the list of references [below](#References).

(In a play on how the "M" in "M-theory" has been [said](https://ncatlab.org/nlab/show/M-theory#MagicMysteryQuote) to be either for "Membrane" or for "Mystery", so the "H" in "Hypothesis H" may be read as being either for "Homotopy" or for "Hisham".)

## The open problem of formulating M-Theory
 {#TheOpenProblemOfFormulatingMTheory}

It is an open problem to mathematically formulate _[[M-theory]]_: Supposedly a single coherent [[theory (physics)|theory of physics]] whose various [[limit of a sequence|limiting]] cases reproduce the zoo of [[perturbative string theory|perturbative string theories]] ([[heterotic string theory|HET]]/[[type I string theory|I]]/[[type IIA string theory|IIA]]/[[type IIB string theory|IIB]]&[[F-theory|F]]/[[D=11 supergravity|11d SuGra]]) and the expected [[duality in string theory|dualities]] relating them; but which also makes mathematical sense of [[non-perturbative effect|non-perturbative]] [[D-brane|D-]]/[[M-brane]] physics, and hence solves, via [[intersecting D-brane models]] of [[confinement|confined]] [[quantization of Yang-Mills theory|quantum]] [[Yang-Mills theory]] ([[holographic QCD]]), the [[mass gap problem|last of the Millennium Problems]]:

In the [[physics]]-minded [[string theory]] community, parts of the literature tends to forget or downplay the [[conjecture|conjectural]] nature of many assumptions and leaps of faiths that are being made when it comes to discussion of [[D-brane]]/[[M-brane]] dynamics and generally of [[non-perturbative effects]] in [[string theory]]. More widely recognized is the fact that the theory of [[intersecting branes|coincident]] [[M5-branes]] is missing, together with the formulation of its expected decoupling limit to the [[D=6 N=(2,0) SCFT]] -- but of course these major open problems are but one aspect of full [[M-theory]].
For string-theoretic references that do highlight the open problem of formulating [[M-theory]] see at _[M-theory -- The open problem](https://ncatlab.org/nlab/show/M-theory#TheOpenProblem)_.

In [[mathematics]], it is rare but not unfamiliar that an open problem is not the proof of a theorem within an established theory, but is the establishing of a theory in the first place. A famous example of such a situation is the search for a theory of "absolute geometry" over the "[[field with one element]]". In this [[analogy]], the various [[perturbative string theories]] ([[heterotic string theory|HET]], [[type I string theory|I]], [[type IIA string theory|IIA]], [[type IIB string theory|IIB]] and their [[KK-compactification|KK-compactified]] [[perturbative string theory vacua]]) correspond to [[arithmetic geometry|arithmetic geometries]] over [[ground field|base]] [[prime fields]] $\mathbb{F}_p$ for $p \geq 2$, and the would-be [[M-theory]] corresponds to a  theory of a "[[field with one element]]" that unifies and completes all this, by describing it at a deeper level (literally: a deeper base).

Hence the task here is to conjure a mathematical theory $\mathcal{X}$, hypothesize that and explain how $\mathcal{X}$ _is_ the putative [[M-theory]], and then rigorously work out the mathematical implications of $\mathcal{X}$ in order to check to which extent they include the required design criterion "$\mathcal{X} \Rightarrow ST$". To the extent that $\mathcal{X}$ implies known or expected phenomena in [[perturbative string theory]] the hypothesis that $\mathcal{X}$ _is_ [[M-theory]] finds support, to the extent that it doesn't $\mathcal{X}$ needs to be modified to or be replaced by some $\mathcal{X}'$, and the process re-started.

## Hypothesis H

### Statement
 {#StatementOfHypothesisH}

In the large-[[scale]]/low-[[energy]] [[limit of a sequence|limit]] of [[M-theory]] given by [[D=11 supergravity]], the [[covariant phase space]] must consist of [[torsion constraints in supergravity|super-torsion free]] [[super Cartan geometry|super orbi]] $\mathbb{R}^{10,1\vert \mathbf{32}}$[[V-manifold|-folds]]  equipped with a suitable [[higher gauge field]]: the [[C-field]].

The first ingredient of a [[non-perturbative effect|non-perturbative]] [[quantum gravity|quantization]] of this [[phase space]] must be a choice of [[Dirac charge quantization]]-condition for the [[C-field]].

{#HypothesisH} **Hypothesis H:** ([FSS 19b](#FSS19b),[FSS 19c](#FSS19c)) _The [[C-field]] is [[Dirac charge quantization|charge quantized]] in [[J-homomorphism|J-]][[twisted cohomotopy|twisted]] [[cohomotopy cohomology theory]]_.

### Motivation from homotopy theoretic re-analysis of the brane-scan
 {#MotivationFromHomotopyTheoreticReanalysisOfBraneScan}

The [Hypothesis H](#HypothesisH) ([above](#StatementOfHypothesisH)) is motivated from re-analysis (based on [Sati 13, Sec. 2.5](#Sati13)  see [FSS 19a](#FSS19a) for comprehensive review) of [[geometry of physics -- fundamental super p-branes|super p-brane WZ terms]] in [[super homotopy theory]]. The resulting _[[schreiber:brane bouquet]]_ _proves_ that -- in the approximation of [[rational homotopy theory]] -- [[M-brane]] charge is in [[rational cohomotopy]] in exactly the same way that [[D-brane charge quantization in K-theory|D-brane charge is in twisted K-theory]].

Detailed lecture note on this point are at _[[geometry of physics -- fundamental super p-branes]]_, in the section _[M-Flux fields](https://ncatlab.org/nlab/show/geometry+of+physics+--+fundamental+super+p-branes#MFluxFields)_.

***

"

## Implications
 {#Implications}

Assuming [Hypothesis H](#HypothesisH), we have a well-defined [[covariant phase space]] and may rigorously derive properties of the resulting [[states]] and [[observables]]. These are the implications of [Hypothesis H](#HypothesisH). To the extent that these implications match known facts of [[perturbative string theory]] and to the extent that they match [[folklore]] about [[M-theory]], [Hypothesis H](#HypothesisH) finds support, to the extend that they do not [Hypothesis H](#HypothesisH) may need to be adjusted or abandoned.

### Anomaly cancellation

What physicists call _[[anomalies]]_ are what mathematician call _[[obstructions]]_ to the construction of a [[theory (physics)|physical theory]]. Hence [[anomaly cancellation|cancellation]] of all [[anomalies]] in every [[quantum field theory]] that [[folklore]] expects to be a [[limit of a sequence|limiting case]] of [[M-theory]] is strictly necessary for [[M-theory]] to exist: Any remaining anomaly is an [[obstruction]] against the existence of the theory that it is an anomaly of.

A long list of anomaly cancellation conditions in M-theory has been argued for in the literature, obtained from a variety of different arguments and under various assumptions, but supported by a web of consistency checks.

**Theorem** ([FSS 19b](#FSS19b), [FSS 19c](#FSS19c), [SS 19a](#SS19a))

_[Hypothesis H](##HypothesisH) implies the following list of [[anomaly cancellation]] consistency conditions expected in the [[M-theory]] [[folklore]]_:

#### On curved but non-singular 8-manifolds
 {#OnCurvedButNonSingularManifolds}

Here is a quick survey of the implications of _[Hypothesis H](#HypothesisH)_ on expected M-theory anomaly cancellation conditions for [[M-theory on 8-manifolds]]:

  **download:** [[schreiber:HypothesisHIntroductionAndSurvey191125.pdf:file]]

#### On flat but singular orbifolds:

[Hypothesis H](#HypothesisH) implies [[RR-field tadpole cancellation]] on [[flat orbifolds]] with [[ADE-singularities]], including the _absolute_ [[O-plane charge]] and hence the [[SemiSpin(32)]]-[[gauge group]] of [[type I' string theory]], [[duality between type I and heterotic string theory|dually]] of [[heterotic string theory]]. This is the heart of what came to be known as the "first string theory revolution" ([Schwarz 07](https://ncatlab.org/nlab/show/string+theory#Schwarz07)).

From [SS 19a](#SS19a):

### Microscopic brane physics

The core aspect of [[D-brane]] physics is the [[folklore]] (see [BMSS 19, p. 3](#BMSS19)) that, at low [[energy]], [[coincident branes|coincident]] [[D-branes]] carry [[Yang-Mills theory|Yang-Mills]] [[gauge theory]] on their [[worldvolumes]]. This is the starting point of [[geometric engineering of quantum field theory]] in [[intersecting D-brane models]] and without this phenomenon, string theory would be irrelevant for [[phenomenology]].

Moreover, [[folklore]] has it (but see the issues with the [[nonabelian DBI action]] [here](https://ncatlab.org/nlab/show/M-theory#OpenProblemNonabelianDBI)) that at small-[[scale]]/high-[[energy]] [[intersecting brane|intersecting]] [[branes]] show the deep quantum phenomena associated with [[monopoles in Yang-Mills theory]], which ultimately lead to the partial formulation of [[M-theory]] via the [[BFSS matrix model]] and the [[BMN matrix model]].

[Hypothesis H](#HypothesisH) implies that the [[states]] and [[observables]] of [[M-theory]] reflect the following expected phenomena for [[Dp-D(p+2)-brane intersections]]:

From [SS 19c](#SS19c):

## See also

* [[twisted cohomotopy]]
* [[equivariant cohomotopy]]
* [[differential cohomotopy]]

## FAQ
 {#FAQ}

* [MO-question:377071/381](https://mathoverflow.net/q/377071/381):

  > "In M-theory, what can hypothesis H tell us that quantization in ordinary cohomology cannot?"

  answer at: [MO-answer:377154/381](https://mathoverflow.net/a/377154/381)

### References for the [FAQ](#FAQ)

* {#DMW00} E. Diaconescu, G. Moore, E. Witten, _$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory_, Adv. Theor. Math. Phys. 6:1031-1134, 2003 ([arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090)), summarised in: _A Derivation of K-Theory from M-Theory_ ([arXiv:hep-th/0005091](http://arxiv.org/abs/hep-th/0005091))

* {#HS02} M Hopkins, I. Singer, *[[nLab:Quadratic Functions in Geometry, Topology, and M-Theory]]*_, J. Differential Geom. Volume 70, Number 3 (2005), 329-452 ([arXiv:math.AT/0211216](http://arxiv.org/abs/math.AT/0211216), [euclid:1143642908](https://projecteuclid.org/euclid.jdg/1143642908))

* {#DFM03} E. Diaconescu, D. Freed, G. Moore,  _The $M$-theory 3-form and $E_8$-gauge theory_, chapter in: _Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues_, Cambridge University Press 2007 ([arXiv:hep-th/0312069](http://arxiv.org/abs/hep-th/0312069), [doi:10.1017/CBO9780511721489](https:
//doi.org/10.1017/CBO9780511721489))

* {#MSa03} V. Mathai, H. Sati, _Some Relations between Twisted K-theory and E8 Gauge Theory_, JHEP 0403:016, 2004 ([arXiv:hep-th/0312033](https://arxiv.org/abs/hep-th/0312033), [doi:10.1088/1126-6708/2004/03/016](https://iopscience.iop.org/article/10.1088/1126-6708/2004/03/016))

* {#Sa13} H. Sati, _Framed M-branes, corners, and topological invariants_, J. Math. Phys. 59 (2018), 062304 ([arXiv:1310.1060](http://arxiv.org/abs/1310.1060))

* {#FSS15} D. Fiorenza, H. Sati, U. Schreiber,  _The WZW term of the M5-brane and differential cohomotopy_, J. Math. Phys. 56, 102301 (2015) ([arXiv:1506.07557](http://arxiv.org/abs/1506.07557), [doi:10.1063/1.4932618](http://scitation.aip.org/content/aip/journal/jmp/56/10/10.1063/1.4932618))

* {#FSS16} D. Fiorenza, H. Sati, U. Schreiber, _Rational sphere valued supercocycles in M-theory and type IIA string theory_, J. Geom. Phys., Vol 114 (2017) ([arXiv:1606.03206](http://arxiv.org/abs/1606.03206), [doi:10.1016/j.geomphys.2016.11.024](http://dx.doi.org/10.1016/j.geomphys.2016.11.024))

* {#BMSS18} V. Braunack-Mayer, H. Sati, U. Schreiber, _Gauge enhancement of super M-branes via parametrized stable homotopy theory_, Comm. Math. Phys. 371: 197 (2019) ([arXiv:1806.01115](https://arxiv.org/abs/1806.01115), [doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4))

* {#BSS18} S. Burton, H. Sati, U. Schreiber, _Lift of fractional D-brane charge to equivariant Cohomotopy theory_, J. Geom. Phys., 2020 (in print) ([arXiv:1812.09679](https://arxiv.org/abs/1812.09679))

* {#FSS19a} D. Fiorenza, H. Sati, U. Schreiber, _The rational higher structure of M-theory_, in: _Proceedings of Higher Structures in M-Theory 2018_, Fortsch. Phys. 2019 ([arXiv:1903.02834](https://arxiv.org/abs/1903.02834), [doi:10.1002/prop.201910017](https://doi.org/10.1002/prop.201910017))

* {#FSS19b} D. Fiorenza, H. Sati, U. Schreiber, _Twisted Cohomotopy implies M-Theory anomaly cancellation on 8-manifolds_, Comm. Math. Phys.  377(3), 1961-2025 (2020) ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207), [doi:10.1007/s00220-020-03707-2](https://doi.org/10.1007/s00220-020-03707-2))

* {#FSS19c} D. Fiorenza, H. Sati, U. Schreiber, _Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane_, Comm. Math. Phys. 2020 (in print) ([arXiv:1906.07417](https://arxiv.org/abs/1906.07417))

* {#FSS19d} D. Fiorenza, H. Sati, U. Schreiber, _Twistorial Cohomotopy Implies Green-Schwarz anomaly cancellation_ ([arXiv:2008.08544](https://arxiv.org/abs/2008.08544))

* {#SS19a} H. Sati, U. Schreiber, _Equivariant Cohomotopy implies orientifold tadpole cancellation_ J. Geom. Phys. Vol 156, 2020, 103775 ([arXiv:1909.12277](https://arxiv.org/abs/1909.12277), [doi:10.1016/j.geomphys.2020.103775](https://doi.org/10.1016/j.geomphys.2020.103775))

* {#SS19b} H. Sati, U. Schreiber, _Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams_ ([arXiv:1912.10425](https://arxiv.org/abs/1912.10425))

* {#SS20a} H. Sati, U. Schreiber, _Twisted Cohomotopy implies M5-brane anomaly cancellation_ ([arXiv:2002.07737](https://arxiv.org/abs/2002.07737))

* {#SS20b} H. Sati, U. Schreiber, _The character map in equivariant twistorial Cohomotopy implies the Green-Schwarz mechanism with heterotic M5-branes_ ([arXiv:2011.06533](https://arxiv.org/abs/2011.06533))

* {#Sa20} H. Sati, _M-theory and cohomotopy_, talk at _M-Theory and Mathematics_, NYUAD 2020 ([pdf](https://ncatlab.org/nlab/files/SatiMTheoryCohomotopy2020.pdf))

* {#Sc20} U. Schreiber, _Microscopic brane physics from Cohomotopy theory_, talk at _M-Theory and Mathematics_, NYUAD 2020 ([pdf](https://ncatlab.org/schreiber/files/Schreiber-MTheoryMathematics2020-v200126a.pdf))

## References
 {#References}

Introduction and survey:

* {#IntroductionToHypothsisH} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Introduction to Hypothesis H]]*.

More references:

* [Motivation by results in rational Cohomotopy](#ReferencesInRational)

* [Formulation in full Cohomotopy](#ReferencesFormulationInFullCohomotopyTheory)

* [Exposition and talk notes](#ExpositionAndTalkNotes)

### Motivation by results in rational Cohomotopy
 {#ReferencesInRational}

[Hypothesis H](#HypothesisH) is motivated by analysis of [[geometry of physics -- fundamental super p-branes|super p-brane WZ terms]] in [[super homotopy theory]], which _proves_ that -- in the approximation of [[rational homotopy theory]] -- [[M-brane]] charge is in [[rational Cohomotopy]] in exactly the same way that [[D-brane charge]] is in [[twisted K-theory]].

This is due to:

* {#Sati13} [[Hisham Sati]]:

  Section 2.5 of:

  **Framed M-branes, corners, and topological invariants**

  J. Math. Phys. **59** (2018) 062304

  [doi:10.1063/1.5007185](https://doi.org/10.1063/1.5007185)

  [arXiv:1310.1060](http://arxiv.org/abs/1310.1060)

and was further developed in:

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

  **[[schreiber:The WZW term of the M5-brane and differential cohomotopy]]**

  J. Math. Phys. **56** (2015) 102301

  [doi:10.1063/1.4932618](http://scitation.aip.org/content/aip/journal/jmp/56/10/10.1063/1.4932618)

  [arXiv:1506.07557](http://arxiv.org/abs/1506.07557)

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

  **[[schreiber:Rational sphere valued supercocycles in M-theory|Rational sphere valued supercocycles in M-theory and type IIA string theory]]**

  J. Geom. Phys. **114** (2017) 91-108

  [doi:10.1016/j.geomphys.2016.11.024](http://dx.doi.org/10.1016/j.geomphys.2016.11.024)

  [arXiv:1606.03206](http://arxiv.org/abs/1606.03206)

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

  **[[schreiber:Higher T-duality of super M-branes]]**

  Phys. Lett. B **781** (2018) 694-698

  [doi:10.1016/j.physletb.2018.04.058](https://doi.org/10.1016/j.physletb.2018.04.058)

  [arXiv:1805.00233](https://arxiv.org/abs/1805.00233)

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

  **[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]**

  Comm. Math. Phys. **371** (2019) 425–524

  [doi:10.1007/s00220-019-03442-3](https://doi.org/10.1007/s00220-019-03442-3)

  [arXiv:1805.05987](https://arxiv.org/abs/1805.05987)

* {#BMSS19} [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Gauge enhancement of Super M-Branes|Gauge enhancement of super M-branes via parametrized stable homotopy theory]]**

  Comm. Math. Phys. **371** (2019) 197–265

  [doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4)

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

Exposition and review of the rational situation is in:

* {#FSS19a} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:The rational higher structure of M-theory]]**

  Fort. Phys. (2019)

  _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/):_

  *[[Higher Structures in M-Theory 2018]]*

  [doi:10.1002/prop.201910017](https://doi.org/10.1002/prop.201910017)

  [arXiv:1903.02834](https://arxiv.org/abs/1903.02834)

Partial derivation of the [[M5-brane]]-model at this rational level:

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

  **[[schreiber:Super-exceptional embedding construction of the M5-brane]]**

  J. High Energy Phys. **2020** 107 (2020).

  doi:10.1007/JHEP02(2020)107

  [arxiv:1908.00042](https://arxiv.org/abs/1908.00042)

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

  **[[schreiber:Super-exceptional M5-brane model -- Emergence of SU(2)-flavor sector]]**

  J. Geom. Phys. **170**(2021) 104349

  [doi:10.1016/j.geomphys.2021.104349](https://doi.org/10.1016/j.geomphys.2021.104349)

  [arXiv:2006.00012](https://arxiv.org/abs/2006.00012)

On [[U-duality]] (and possibly [[mysterious duality]]) as [[automorphisms]] of [[iterated loop space|iterated]] ([[Sullivan model of loop space|rational]]) [[cyclic loop spaces]] of the ([[rational n-sphere|rational]]) [[4-sphere]]:

* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious triality* ([arXiv:2111.14810](https://arxiv.org/abs/2111.14810))

### Formulation in full Cohomotopy theory
 {#ReferencesFormulationInFullCohomotopyTheory}

The detailed form of the [Hypothesis H](#HypothesisH) (that the above rational situation should lift to full [[twisted Cohomotopy|twisted]] [[equivariant Cohomotopy|equivariant]] [[differential Cohomotopy|differential]] [[Cohomotopy theory]])
was formulated, and first consistency checks were made, in:

* {#FSS19b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds]]**

  Comm Math. Phys **377** 3 (2020) 1961-2025

  [doi:10.1007/s00220-020-03707-2](https://doi.org/10.1007/s00220-020-03707-2)

  [arXiv:1904.10207](https://arxiv.org/abs/1904.10207)

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]**

  Comm. Math. Phys. **384** 403–432 (2021)

  [doi:10.1007/s00220-021-03951-0](https://doi.org/10.1007/s00220-021-03951-0)

  [arXiv:1906.07417](https://arxiv.org/abs/1906.07417)

* {#GS20} [[Daniel Grady]], [[Hisham Sati]]:

  **Differential cohomotopy versus differential cohomology for M-theory**

  J. Geom. Phys. **165** (2021) 104203

  [https://doi.org/10.1016/j.geomphys.2021.104203](doi:10.1016/j.geomphys.2021.104203)

  [arXiv:2001.07640](https://arxiv.org/abs/2001.07640)

* {#SS19a} [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]**

  J. Geom. Phy. **156** (2020) 103775

  [doi:10.1016/j.geomphys.2020.103775](https://doi.org/10.1016/j.geomphys.2020.103775)

  [arXiv:1909.12277](https://arxiv.org/abs/1909.12277)

* {#SS19b} [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Lift of fractional D-brane charge to equivariant Cohomotopy theory]]**

  J. Geom. Phys. **161** (2021) 104034

  [doi:10.1016/j.geomphys.2020.104034](https://doi.org/10.1016/j.geomphys.2020.104034)

  [arXiv:1812.09679](https://arxiv.org/abs/1812.09679)

* {#SS20a} [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Twisted Cohomotopy implies M5-brane anomaly cancellation]]**

  Lett. Math. Phys. **111** 120 (2021)

  [doi:10.1007/s11005-021-01452-8](https://doi.org/10.1007/s11005-021-01452-8)

  [arXiv:2002.07737](https://arxiv.org/abs/2002.07737)

* {#FSS20a} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Twisted Cohomotopy implies twisted String structure on M5-branes]]**

  J. Math. Phys. **62** 042301 (2021)

  [doi:10.1063/5.0037786](https://doi.org/10.1063/5.0037786)

  [arXiv:2002.11093](https://arxiv.org/abs/2002.11093)

* {#SS19c} [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Differential Cohomotopy implies intersecting brane observables]]**

  Adv. Theor. Math. Physics **26** 4 (2022)

  [arXiv:1912.10425](https://arxiv.org/abs/1912.10425)

* {#CSS21a} [[David Corfield]], [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Fundamental weight systems are quantum states]]**

  [arXiv:2105.02871](https://arxiv.org/abs/2105.02871)

* {#SS20b} [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Proper Orbifold Cohomology]]**

  [arXiv:2008.01101](https://arxiv.org/abs/2008.01101)

* {#FSS20b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]:

  **[[schreiber:Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation]]**

  Rev. Math. Phys. **34** 05 (2022) 2250013

  [arXiv:2008.08544](https://arxiv.org/abs/2008.08544)

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

  **[[schreiber:The Character Map in Twisted Non-Abelian Cohomology]]**

  [arXiv:2009.11909](https://arxiv.org/abs/2009.11909)

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

  **[[schreiber:The Character Map in Equivariant Twistorial Cohomotopy]]**

  [arXiv:2011.06533](https://arxiv.org/abs/2011.06533)

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

  **[[schreiber:M-Theory as Mf-Theory|M/F-Theory as Mf-Theory]]**

  [arXiv:2103.01877](https://arxiv.org/abs/2103.01877)

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

  **[[schreiber:Anyonic defect branes in TED-K-theory]]**

See also:

* [[David Michael Roberts]]:

  **Topological sectors for heterotic M5-brane charges under Hypothesis H*

   J. High Energ. Phys. **2020** 52 (2020)

  doi:10.1007/JHEP06(2020)052

  [arXiv:2003.09832](https://arxiv.org/abs/2003.09832)

### Exposition and talk notes
 {#ExpositionAndTalkNotes}

* [[Luigi Alfonsi]], §4 in: *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

* *[[schreiber:Some Quantum States of M-Branes under Hypothesis H]]*

  talk at [Centre for Research in String Theory](https://strings.ph.qmul.ac.uk/welcome-crst)

  Queen Mary University London, 2021

* _[[schreiber:Proper Orbifold Cohomotopy for M-Theory]]_

  talk at [String and M-Theory: The New Geometry of the 21st Century – II](https://ims.nus.edu.sg/events/string-and-m-theory-the-new-geometry-of-the-21st-century-ii/)

  via NUS Singapore, 2021

* _[[schreiber:Microscopic Brane Physics from Cohomotopy]]_

  talk at [[M-Theory and Mathematics]]

  NYU Abu Dhabi, 2020

*  _[[schreiber:Strings2019|Twisted Cohomotopy implies M-theory anomaly cancellation]]_

   presentation at [Strings2019](https://sis-pc15.ulb.ac.be/event/2/)

   Brussels, 2019

* _[[schreiber:StringMath2017|Super p-Brane Theory emerging from Super Homotopy Theory]]_

  talk at [[String Math 2017]],

  Hamburg, 2017

* _[[schreiber:Equivariant Cohomotopy and Branes]]_

  talk at _[String and M-Theory: The New Geometry of the 21st Century](http://ims.nus.edu.sg/events/2018/wstring/index.php)_

  NUS Singapore, 2018

* _[[schreiber:The Higher Structure of 11d Supergravity]]_

   talk at _[Souriau 2019](http://souriau2019.fr/)_

   IHP Paris, 2019

* _[[schreiber:Equivariant Cohomotopy of toroidal orbifolds]]_

  talk at [[Sadok Kallel|Prof. Sadok Kallel]]‘s group seminar

  AUS Sharjah, 2019

* _[[schreiber:Cohomotopy Theory and Branes]]_

  talk at _Geometry, topology and Physics_

  NYU Abu Dhabi, 2020

* _[[schreiber:Equivariant Stable Cohomotopy and Branes]]_

  talk at _Geometry, Topology and Physics_

  NYU Abu Dhabi, 2018

* _[[schreiber:Equivariant cohomology of M2/M5-branes]]_

  talk at _Seminar on Higher Structures_

  MPI Bonn, 2016

* [[schreiber:ToposesCohomologyPhysics201002.pdf:file]]

* [[schreiber:CenturyOfChargeQuantization201012.pdf:file]]

See also:

* *[[schreiber:New Foundations for TDA -- Cohomotopy]]*


hypothèse inspiratrice

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _hypothèse inspiratrice_ or _inspiring assumption_  is an assumption formulated by [[Grothendieck]] in _[[Pursuing Stacks]]_, that the [[category]] of auto-[[equivalence of categories|equivalences]] of the [[classical homotopy category]] is [[equivalence of categories|equivalent]] to the [[final category]]. See below for Grothendieck's own formulation of it. It has not been proven in its original form.

## Proofs of lifts of the assumption

The _hypothèse inspiratrice_ can be proven relatively easily if one re-formulates the question for a lift of the homotopy category to [[derivator|derivators]], [[(∞,1)-category|(∞,1)-category theory]], or some other such setting for [[homotopy theory]]. One such [[proof]] is given in [Cisinski 08](#Cisinski08). A different proof is given in [Toën-Vezzosi 02](#ToenVezzoi02), as the case $X=1$ of Corollary 5.2.2.

## Proof of the analogous statement for the category of sets

\begin{prpn} The category of auto-equivalences of [[Set]] is equivalent to the final category. \end{prpn}

\begin{proof} Any equivalence of categories preserves [[colimit|colimits]]. Since [[Set]] is the [[free co-completion]] of the final category $1_{\mathsf{Cat}}$, and since $1_{\mathsf{Cat}}$ is sent to $1_{\mathsf{Set}}$ under the Yoneda embedding, any auto-equivalence $F$ of [[Set]] is determined uniquely, up to isomorphism, by $F(1_{\mathsf{Set}})$.

Now, equivalences of categories also preserve [[final object|final objects]]. Thus we have that $F(1_{\mathsf{Set}}) = 1_{\mathsf{Set}}$ for all $F$.

The proposition follows immediately from these two observations. \end{proof}

This argument is rather canonical. In principle, it 'should' lift to a proof of the hypothèse inspiratrice, simply by replacing set-theoretic notions by homotopy-theoretic ones: colimits by homotopy colimits, and so on. And this does work, but with the caveat that such a proof can only be formulated if one works inside 'homotopy category theory' rather than ordinary category theory.

## Undirected version

The original hypothèse inspiratrice is about the homotopy *1-category*, and its lifted versions are about the corresponding *$(\infty,1)$-category* of homotopy types ($\infty$-groupoids).  One might formulate an analogous conjecture about the *$\infty$-groupoid* of $\infty$-groupoids (or about its 0-truncation, the set of equivalence classes of $\infty$-groupoids).  But here the situation is rather different.

If we assume the law of [[excluded middle]], then there definitely exist nontrivial automorphisms of the $\infty$-groupoid of homotopy types.  For instance, if $X$ and $Y$ are two homotopy types that themselves have no nontrivial automorphisms (such as the [[empty set]] and the [[terminal set]]), then there is an automorphism of the $\infty$-groupoid of homotopy types that switches $X$ and $Y$ and leaves everything else fixed.  See for instance [[Vladimir Voevodsky|Voevodsky]]'s comment in [this thread](https://groups.google.com/d/msg/homotopytypetheory/Wyxmcvo5K2Y/SUm3nwYECQAJ) on the HoTT mailing list.  This requires excluded middle because we have to decide, for each homotopy type, whether it is equivalent to $X$ or to $Y$ or to neither.

It is not known whether any nontrivial automorphisms of the $\infty$-groupoid of homotopy types can be constructed without excluded middle.  (To talk about homotopy types without assuming excluded middle, it is best to work in constructive [[homotopy type theory]].)  But it is known that certain kinds of such automorphisms (e.g. those that move the empty set) do imply excluded middle or some other nonconstructive axiom; see [BELS](#BELS17).

## Grothendieck's formulation

The original formulation in _[[Pursuing Stacks]]_ (page 30 in the pagination of the original document, section 28, 8th of March 1983) is preceded by

> Now this maybe isnt [sic] so silly after all, in view of the following
>
> Assumption: The category of equivalences of $(Hot)$ with itself, and of natural isomorphisms (possibly even any morphisms) between such, is equivalent to the one point category.
>
>   This means 1) any equivalence $(Hot) \stackrel{\approx}{\to} (Hot)$ is isomorphic to the identity functor, and 2) any automorphism of the identity functor (possibly even any endomorphism ?) is the identity.
>
>   Maybe these are facts well-known to the experts, maybe not - it is not my business here anyhow to prove such kind of things. It looks pretty plausible , because if there was any non trivial autoequivalence of $(Hot)$, or automorphism of its identity functor, I guess I would have heard about it, or something of the sort would flip to my mind. It would not be so if we abelianized $(Hot)$ some way or other, as there would be the loop and suspension functor, and homotheties by $-1$ of $id_{(Hot)}$.

In most copies of Pursuing Stacks available, the assumption itself is blacked out. Tim Porter suggests that this is likely due to the text having been highlighted, probably in yellow, leading to the blacking out upon photocopying. Because of the blacking out, the assumption is also not available in circulating typed versions of Pursuing Stacks. Happily, Tim Porter has scanned _[[page 30.pdf|here:file]]_ the relevant page from his personal copy of Pursuing Stacks. The assumption is about mid-page.

Later on (page 71 in the pagination of the original document, section 41, around the 24th of March 1983), Grothendieck writes the following.

> One comment still, upon the role played in the theory I am developing of the assumption (p.30) that the category of autoequivalences of $(Hot)$ is equivalent to the final category. This assumption has been a crucial guide for putting the emphasis where it really belongs, namely upon the set $W \subset Fl(H)$ of weak equivalences within a category $H$ [letter is a bit illegible] which one would like to take in some sense as a category of models for homotopy types -  the functors $H \rightarrow (Hot)$ following along automatically. However, in no statement whatever I proved so far, was this assumption ever used. On the other hand, the notion of a _modelizer_ introduced in the wake of the 'assumption' (of p.31) [this seems like a typo, presumably it should be p.30] was tacitly changed during the reflection, ...

Slightly later again (page 73, section 42, 25th of March 1983), the following is written, introducing the phrase "inspiring assumption":

> This is indeed very much in the spirit of the "inspiring assumption" of p.30, that the category of autoequivalences of $(Hot)$ is equivalent to the unit category, which implies indeed that for two categories $H, H'$ equivalent to $(Hot)$, there is a unique isomorphism between them.

A further reference comes on page 80-81, section 44, 26th of March 1983.

> But it is so if we grant the "inspiring assumption", implying that any autoequivalence of $(Hot)$ is isomorphic to the identity functor

And again on page 86, still on the same day.

> If we admit the "inspiring assumption" that any autoequivalence of $(Hot)$ is uniquely isomorphic to the identity functor, it will follow that the autoequivalence $(Hot) \rightarrow (Hot)$ induced by (11) (where again $i$ is any weak test functor) is canonically isomorphic to the identity.

Next is a reference on pg.93, section 46, probably the 28th or 29th of March 1983.

> (If we admit the "inspiring assumption", there is no real choice, as a matter of fact - but we don't want to use this in a technical sense, but only as a guide and motivation.)

There is also a reference on pg.216-217, section 72, 10th of June 1983.

> This is of course very much in keeping with the "inspiring assumption" (section 28), which just means that up to unique isomorphism, there is indeed but _one_ equivalence from $Hot_B$ to $Hot_A$ (both categories being equivalent to $Hot(\underline{W})$. Here we are thinking of the extension of the "assumption" contemplated earlier, when usual weak equivalence is replaced by a basic localizer $\underline{W}$ as above. It seems plausible that the assumption holds true in all cases - anyhow we did'nt [sic] have at any moment to make explicit use of it (besides at a moment drawing inspiration from it...).

The final reference is on pg.273, section 83, probably the 27th or 28th of June 1983.

> Here also arises the question whether a functor $\phi: M \rightarrow Hot_{\underline{W}}$ stemming from a contractibility structure $M_{c}$ on $M$ can have non trivial automorphisms - a question closely connected to the "inspiring assumption" of section 28.

## References

* {#Cisinski08} [[Denis-Charles Cisinski]], _Propriétés universelles et extensions de Kan dérivées_, Theory Appl. Categ. 20, 605-649 (2008). [Link to article](http://www.tac.mta.ca/tac/volumes/20/17/20-17abs.html)

* {#ToenVezzoi02} [[Bertrand Toën]] and [[Gabriele Vezzosi]], _Segal topoi and Segal stacks over them_, [arXiv:math/0212330](https://arxiv.org/abs/math/0212330).

* {#BELS17} [[Auke Booij]], [[Martín Escardó]], [[Peter Lumsdaine]], [[Michael Shulman]].  _Parametricity, automorphisms of the universe, and excluded middle_, [arXiv:1701.05617](https://arxiv.org/abs/1701.05617)

[[!redirects inspiring assumption]]


Hyungrok Kim

## Selected writings

On [[quantum L-infinity algebras]] for the [[BV-BRST formalism]]:

* [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], *Perturbative Quantum Field Theory and Homotopy Algebras*, PoS (CORFU2019) 199 [[arXiv:2002.11168](https://arxiv.org/abs/2002.11168), [doi:10.22323/1.376.0199](https://doi.org/10.22323/1.376.0199)]

On the [[classical double copy]] ([[KLT relations]], relating [[gravity]] to [[Yang-Mills theory]]) via [[L-infinity algebras|$L_\infty$-algebras]]:

* [[Leron Borsten]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], _Double copy from homotopy algebras_, Fortschritte der Physik, **69** 8-9 (2021) 2100075 $[$[arXiv:2102.11390](https://arxiv.org/abs/2102.11390), [doi:10.1002/prop.202100075](https://doi.org/10.1002/prop.202100075)$]$

On the [[higher parallel transport]] for [[higher gauge theory]] with [[adjusted Weil algebras]]:

* {#KimSaemann20} [[Hyungrok Kim]], [[Christian Saemann]], *Adjusted Parallel Transport for Higher Gauge Theories*, J. Phys. A **52** (2020) 445206 [[arXiv:1911.06390](https://arxiv.org/abs/1911.06390), [doi:10.1088/1751-8121/ab8ef2](https://doi.org/10.1088/1751-8121/ab8ef2)]

and relation to [[EL-infinity algebras|$E L_\infty$-algebras]] and [[tensor hierarchies]]:

* [[Leron Borsten]], [[Hyungrok Kim]], [[Christian Saemann]].  *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies* [[arXiv:2106.00108](https://arxiv.org/abs/2106.00108)]

Proposal for further enhancement of the [[higher differential geometry]]-formulation of [[T-folds]]  (such as with [[connection on a 2-bundle|2-connections]] and further [[non-geometric backgrounds]]):

* [[Hyungrok Kim]], [[Christian Saemann]], *Non-Geometric T-Duality as Higher Groupoid Bundles with Connections* [[arXiv:2204.01783](https://arxiv.org/abs/2204.01783)]

On [[higher gauge theory]]:

* [[Leron Borsten]], [[Mehran Jalali Farahani]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Jiří Nárožný]], [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], *Higher Gauge Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.05275](https://arxiv.org/abs/2401.05275)]

category: people


HZR-theory

#Contents#
* table of contents
{:toc}

## Idea

The genuinely $\mathbb{Z}_2$-[[equivariant cohomology]] version of [[ordinary cohomology]], taking into account the $\mathbb{Z}_2$-[[action]] on the [[coefficients]]. A [[real-oriented cohomology theory]]. In degree 3 it serves as a [[twisted cohomology|twist]] for [[KR-theory]].

## Examples

The [[B-field]] over [[orientifold]] background of the [[bosonic string]]  is a cocycle in (twisted) HZR-theory. More generally for the [[type II superstring]] it is a genuinely $\mathbb{Z}_2$-[[equivariant cohomology|equivariant]] [[super line 2-bundle]]

## Related concepts

[[!include chromatic tower examples - table]]

## References

A detailed model ("Jandl gerbes") for [[ordinary differential cohomology|differential]] $HZR$-theory in degree 3 ("[[orientifold]] [[B-fields]]") is in

* {#SSW05} [[Urs Schreiber]], [[Christoph Schweigert]], [[Konrad Waldorf]], _Unoriented WZW models and Holonomy of Bundle Gerbes_, Communications in Mathematical Physics **274** 1 (2007) 31-64 [[arXiv:hep-th/0512283](http://arxiv.org/abs/hep-th/0512283), [doi:10.1007/s00220-007-0271-x](https://doi.org/10.1007/s00220-007-0271-x)]

* {#GawedzkiSuszekWaldorf08} [[Krzysztof Gawedzki]], Rafal R. Suszek,  [[Konrad Waldorf]], _Bundle Gerbes for Orientifold Sigma Models_, Adv. Theor. Math. Phys. 15(3), 621-688 (2011) ([arXiv:0809.5125](http://arxiv.org/abs/0809.5125))

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Real bundle gerbes, orientifolds and twisted KR-homology_, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number  ([arXiv:1608.06466](http://arxiv.org/abs/1608.06466), [doi:10.4310/ATMP.2019.v23.n8.a5](https://dx.doi.org/10.4310/ATMP.2019.v23.n8.a5))

[[!redirects HZR theory]]

[[!redirects Jandl gerbe]]
[[!redirects Jandl gerbes]]


Håkon Robbestad Gylterud

* [website](https://hakon.gylterud.net/)

## Selected writings

* [[Henrik Forssell]], [[Håkon Robbestad Gylterud]], [[David Spivak]], *Type theoretical databases*, Journal of Logic and Computation, Volume 30, Issue 1, January 2020, Pages 217–238, ([doi:10.1093/logcom/exaa009](https://doi.org/10.1093/logcom/exaa009), [arXiv:1406.6268](https://arxiv.org/abs/1406.6268))

On [[constructive set theory]] in [[homotopy type theory]]:

* [[Håkon Robbestad Gylterud]], [[Elisabeth Bonnevier]], *Non-wellfounded sets in HoTT* [[arXiv:2001.06696](https://arxiv.org/abs/2001.06696), talk video:[YT](https://www.youtube.com/watch?v=ALj5hOmwXps)]

## Talks

* [[Håkon Robbestad Gylterud]], *Defining and relating theories*, [[Homotopy Type Theory Electronic Seminar Talks]], 25 March 2021 ([video](https://www.youtube.com/watch?v=8aTDEn2D2pA), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Gylterud-2021-03-25-HoTTEST.pdf))

category: people


Hélène Esnault

* [Website](http://www.mi.fu-berlin.de/users/esnault/)
* [Wikipedia](http://en.wikipedia.org/wiki/H%C3%A9l%C3%A8ne_Esnault)
* [Mathematical genealogy](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=68523)
* [publication list](http://www.mi.fu-berlin.de/users/esnault/helene_publ.html)

## Selected writings

On [[Deligne cohomology]]:

* {#EsnaultViehweg88} [[Hélène Esnault]], [[Eckart Viehweg]], _Deligne-Beilinson cohomology_, in: [[Michael Rapoport]], [[Norbert Schappacher]], [[Peter Schneider]] (eds.), _[[Beilinson's Conjectures on Special Values of L-Functions]]_, Perspectives in Mathematics **4**, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, [[EsnaultViehweg-DeligneBeilinsonCohomology.pdf:file]]]

On [[secondary characteristic classes]]:

* {#BlochEsnault97} [[Spencer Bloch]], [[Hélène Esnault]], _Algebraic Chern-Simons theory_, American Journal of Mathematics, Volume 119, Number 4, August 1997, pp. 903-952 ([pdf](http://www.mi.fu-berlin.de/users/esnault/preprints/helene/39-algebraic-chern-simons.pdf))

* [[Hélène Esnault]],  _Algebraic Differential Characters of Flat Connections with Nilpotent Residues_,  In: Baas N., Friedlander E., Jahren B., Østvær P. (eds.) _Algebraic Topology_, Abel Symposia, vol 4. Springer 2009 ([doi:10.1007/978-3-642-01200-6_5](https://doi.org/10.1007/978-3-642-01200-6_5))

## Related $n$Lab entries

* [[Deligne cohomology]]

* [[Beilinson regulator]]

* [[function field analogy]]

* [[renormalization]]

category: people

[[!redirects Helene Esnault]]


Hông Vân Lê

Hông Vân Lê is a Vietnamese-Czech mathematician with research in [[differential geometry]] (especially [[contact geometry]], [[symplectic geometry]] and [[special holonomy]] manifolds), [[information geometry]] and mathematics of [[machine learning]].

* [webpage](https://users.math.cas.cz/~hvle) at the Institute of Mathematics of Czech Academy of Sciences in Prague

## Selected writings

* Hông Vân Lê, _Geometric structures associated with a simple Cartan 3-form_, J. of Geometry and Physics 70 (2013) 205--223 [arXiv:1103.1201](https://arxiv.org/abs/1103.1201)
* Nihat Ay, Jürgen Jost, [[Hông Vân Lê]], Lorenz Schwachhöfer, _Information geometry_, Ergeb. der Mathematik and ihrer Grenzgebiete 3. Folge, 64, Springer 2017

The following treatment of [[information geometry]] (and Fisher metric in particular) is using [[diffeological space]]s (motivated by singular statistical models, including from [[machine learning]])

* Hông Vân Lê, _Natural differentiable structures on statistical models and the Fisher metric_, Information Geometry (2022) [arXiv:2208.06539](https://arxiv.org/2208.06539) [doi](https://doi.org/10.1007/s41884-022-00090-w)

* Hông Vân Lê, _Diffeological statistical models and diffeological Hausdorff measures_, video [yt](https://youtu.be/lY1BmjwUmpQ), slides [pdf](https://users.math.cas.cz/~hvle/PHK/Lediffeological10032021.pdf)

* Hông Vân Lê, Alexey A. Tuzhilin, _Nonparametric estimations and the diffeological [[Fisher metric]]_, In: Barbaresco F., Nielsen F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning, p. 120--138, SPIGL 2020. Springer Proceedings in Mathematics & Statistics __361__, [doi](https://doi.org/10.1007/978-3-030-77957-3_7)

>  In this paper, first, we survey the concept of diffeological Fisher metric and its naturality, using functorial language of probability morphisms, and slightly extending Lê's theory in (Le2020) to include weakly $C^k$-diffeological statistical models. Then we introduce the resulting notions of the diffeological Fisher distance, the diffeological Hausdorff--Jeffrey measure and explain their role in classical and Bayesian nonparametric estimation problems in statistics.

* Hông Vân Lê, _Diffeological  statistical  models,the Fisher metric and probabilistic mappings_, Mathematics 2020, 8(2) 167 [arXiv:1912.02090](https://arxiv.org/abs/1912.02090)

category: people

[[!redirects Hong Van Le]]


Hölder's inequality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

*Hölder's inequality* is a basic [[inequality]] in [[analysis]], which can be interpreted as saying that the $\mathbf{C}$-graded \*-algebra of [[L^p-spaces]]
$$L^{1/p}(X,\mu)$$
(the unfortunate reciprocal in the grading is explained in the article [[Lebesgue space]])
is a $\mathbf{C}$-graded __normed__ \*-algebra.
That is, the canonical bilinear multiplication map
$$L^{1/p}(X,\mu)\otimes L^{1/q}(X,\mu)\to L^{1/(p+q)}(X,\mu)$$
is a [[contractive map]], i.e.,
if $f\in L^{1/p}(X,\mu)$ and $g\in L^{1/q}(X,\mu)$, then
$$f g \in L^{1/(p+q)}(X,\mu)$$
and
$$\|f g \|\le \|f\|\cdot\|g\|,$$
where the norms are [[p-norms]] for the corresponding values of $p$.

As a consequence, it implies that the canonical pairing
$$L^{1/p}(X,\mu)\otimes L^{1/(1-p)}(X,\mu)\to L^1(X,\mu)\to \mathbf{C}$$
exhibits $L^{1/(1-p)}(X,\mu)$ as the [[Banach space]] dual of the [[Banach space]] $L^{1/p}(X,\mu)$.

## Statements

Let $(X, \mu)$ be a [[measure space]], and for $p \in \mathbf{C}$ a complex number with a nonnegative real part let $L^{1/p}$ denote $L^{1/p}(X, \mu)$, the Banach space of complex-valued functions on $X$ with finite [[p-norm|1/p-norm]] modulo equality almost everywhere.

Suppose $p,q,r\in \mathbf{C}$ have nonnegative real parts and $p+q=r$. Then **Hölder's inequality** states that for any $f \in L^{1/p}$, $g \in L^{1/q}$ we have
$$f g \in L^{1/r}(X,\mu)$$
and
$$\|f g\|_{1/r}\le\|f\|_{1/p}\cdot \|g\|_{1/q}.$$

## Duality

In particular, if $r=1$ we have
$$\int_X \left| f g \right| \leq {\|f\|_{1/p}} {\|g\|_{1/q}}$$
(in particular, $f g$ is an $L^1$ function).

The "[[nPOV]]" meaning is this: in this situation there is a canonical pairing $\langle -, - \rangle$ between $L^{1/p}$ and $L^{1/q}$,
$$L^{1/p} \times L^{1/q} \to \mathbb{C}: (f, g) \mapsto \langle f, g \rangle \coloneqq \int_X f \cdot g,$$
which gives a bounded linear map
$$L^{1/p} \otimes L^{1/q} \to \mathbb{C}$$
between Banach spaces. The point of Hölder's inequality is that this pairing is a *short* map, i.e., a map of norm bounded above by $1$. In other words, this is morphism in the [[symmetric monoidal closed category]] [[Ban]] consisting of Banach spaces and short linear maps between them. Accordingly, the map
$$L^{1/p} \otimes L^{1/q} \to \mathbb{C}$$
induces (by [[currying]]) a map from $L^{1/p}$ to the Banach dual of $L^{1/(1-p)}$:
$$L^{1/p} \to (L^{1/q})^\ast \coloneqq [L^{1/q}, \mathbb{C}]$$
(again a short map of course), and reciprocally a map $L^{1/q} \to (L^{1/p})^\ast$.

It is a short step to prove that in fact the norm of the pairing $L^{1/p} \otimes L^{1/q} \to \mathbb{C}$ is *exactly* $1$, and even better that the maps $L^{1/p} \to (L^{1/q})^\ast$ and $L^{1/q} \to (L^{1/p})^\ast$ are in fact [[isometry|isometric]] embeddings.

If the real parts of $p$ and $q$ are nonzero, then
with a little more work (with the help of the [[Radon-Nikodym theorem]]; see for example [here](https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch7.pdf)), one sees these maps are [[surjective map|surjective]] and thus [[isomorphisms]] in $Ban$.

+-- {: .num_remark}
###### Remark
It is true also that $(L^1)^\ast \cong L^\infty$, *but* it is not true that $(L^\infty)^\ast$ is isomorphic to $L^1$. Or, it is at least not true in [[ZFC]], although it may be true in [[dream mathematics]].
=--

## The noncommutative case

Remarkably, [[L^p-spaces]] can be defined for arbitrary [[von Neumann algebras]] (Haagerup, 1979)
and the Hölder inequality continues to hold in this generality (Kosaki, 1984).

The $L^{1/p}$-spaces are now [[bimodules]] over the underlying [[von Neumann algebra]].

If the real part of $p$ is zero
and the [[von Neumann algebra]] is commutative,
than the space $L^{1/p}$ is (noncanonically) isomorphic to $L^\infty$.

In the noncommutative case this is no longer true,
which is the starting point of the [[Tomita–Takesaki theory]].

## Proof of Hölder's inequality in the commutative case

The proof is remarkably simple. First, if $p, q \gt 0$ and $\frac1{p} + \frac1{q} = 1$, then we have *[[Young's inequality]]*, viz. for $a, b \gt 0$
$$
  a b
  \;\leq\;
  \frac{a^p}{p}
    +
  \frac{b^q}{q}
  \,,
$$
with equality precisely when $a^p = b^q$. This is quickly derived from the (strict) convexity of the [[exponential function]], that $0 \leq t \leq 1$ implies
$$
  e^{t x + (1-t)y}
  \leq
  t e^x + (1-t)e^y
  \,,
$$
where equality holds iff $e^x = e^y$. All one has to do is put $t = \frac{1}{p}$ and arrange $x, y$ so that $e^x = a^p$ and $e^y = b^q$.

Then, to prove ${|\langle f, g \rangle|} \leq {\|f\|_p} {\|g\|_q}$, we may assume $f, g$ nonzero (so their norms are positive) and normalize them to unit vectors $u = f/{\|f\|_p}, v = g/{\|g\|_q}$, so that now the object is to prove

$$\int_X {|u|} \cdot {|v|} \leq 1.$$

But since we are dealing with unit vectors, we have $\int_X {|u|^p} = 1$ and $\int_X {|v|^q} = 1$, and now what we want follows straightaway from Young's inequality applied to integrands:

$$\int_X {|u|} \cdot {|v|} \leq \int_X \frac{|u|^p}{p} + \frac{|v|^q}{q} = \frac1{p} + \frac1{q} = 1$$

and so the proof of Hölder's inequality is complete.

To prove that the norm of the pairing $\langle -, - \rangle$ is exactly $1$ (is not less than $1$), it's enough to take any $u \in L^p$ of norm $1$, so $f = {|u|}$ is a nonnegative function of norm $1$, and then put $g = f^{p-1} = f^{p/q}$. We then have $f^p = g^q$ (almost) everywhere, where we then have $f g = \frac{f^p}{p} + \frac{g^q}{q}$, and now

$$\int_X f g = \int_X \frac{f^p}{p} + \frac{g^q}{q} = \frac1{p} + \frac1{q} = 1.$$

Actually these calculations do a little better: they show that upon [[currying]], the map

$$
  \begin{array}{ccc}
    L^p
    &\longrightarrow&
   [L^q, \mathbb{C}]
   \\
   f
   &\mapsto&
   \lambda g. \langle f, g \rangle
  \end{array}
$$

preserves the norm, so that $L^p$ [[isometry|isometrically]] embeds into $(L^q)^\ast$.

## Properties

### Relation to Minkowski's inequality

Recall that [[Minkowski's inequality]] is just the [[triangle inequality]] in the context of [[L-p space]]. There is a well-known trick, covered in just about every [[functional analysis]] text, that allows one to deduce Minkowski's inequality as a corollary of Hölder's inequality. You can look it up for instance in the English Wikipedia, [here](https://en.wikipedia.org/wiki/Minkowski_inequality).

What seemingly most such presentations lack is motivation for the trick, so let us try to say something about this.

First, [[Minkowski's inequality]] can be restated as asserting the convexity of the unit ball $B = \{f \in L^p: {\|f\|_p} \leq 1\}$ of $L^p$. If we place ourselves for a moment in the context of $L^p$ *real*-valued functions, then it suffices to show that $B$ is the intersection of a collection of [[affine space|affine]] half-spaces, say $H_\lambda = \{f \in L^p: \lambda(f) \leq 1\}$ where $\lambda: L^p \to \mathbb{R}$ is a (continuous) linear functional. But with hindsight into the meaning of Hölder's inequality, seen as paving the way to characterizing linear functionals on $L^p$ as those of the form $\lambda(g) = (f \mapsto \langle f, g \rangle)$ for some $g \in L^q$, it's only natural to see whether we can find a sufficiently large collection $B'$ of such $g$ such that $B = \bigcap_{g \in B'} H_{\lambda(g)}$, and in fact the intuition is that the unit ball $B'$ in $L^q$ ought to work.

Thus the idea is clear, and it's just a matter of technique from here. We let the relation ${|\langle f, g \rangle|} \leq 1$ on $L^p \times L^q$ set up a [[Galois connection]] between subsets of $L^p$ and subsets of $L^q$. The connection takes the unit ball $B'$ in $L^q$ to

$$
  (B')^\perp
  \coloneqq
  \big\{
    f \in L^p
     \colon
   (\forall_{g \in B'})
   \;
   {|\langle f, g \rangle|}
    \leq 1
  \big\}
  \,,
$$

which is clearly convex, being an intersection of convex sets $\{f: {|\langle f, g \rangle|} \leq 1\}$, one for each $g \in B'$. Hölder's inequality itself just asserts the containment $B \subseteq (B')^\perp$. If we show the other inclusion $(B')^\perp \subseteq B$, then $B = (B')^\perp$ is convex. So we want to show that if ${|\langle f, g \rangle|} \leq 1$ whenever ${\|g\|_q} \leq 1$, then ${\|f\|_p} \leq 1$. But we already did that calculation when we proved $L^p \hookrightarrow (L^q)^\ast$ is an isometry. Explicitly: take $h = {|f|^p}/f$ (with $h = 0$ where $f = 0$). Then ${|h|} = {|f|^{p-1}} = {|f|^{p/q}}$, so ${|h|^q} = {|f|^p}$ whence ${\|h\|_q^q} = {\|f\|_p^p}$. Put $g = \frac{h}{{\|h\|_q}}$; since ${\|g\|_q} \leq 1$, it follows by the hypothesis on $f$ that $1 \geq {|\langle f, g \rangle|}$. But this gives

$$1 \geq \frac1{{\|h\|_q}} \int_X f h = \frac1{{\|f\|_p^{p/q}}} \int_X {|f|^p} = {\|f\|_p^{p - p/q}} = {\|f\|_p}$$

as was to be shown.

The standard derivation of Minkowski's inequality from Hölder's inequality is nothing more than a very tidied-up rendering of this argument, but without the additional conceptual explanation given here.

### Relation to Log-convex functions

Let $D$ be a [[convex space]], e.g. an [[affine space]]. We say that a [[function]] $f: D \to (0, \infty)$ is *log-convex* if its [[logarithm]] $\log(f)$ is a [[convex function]].

Hölder's inequality is closely related to the notion of log-convexity. On the one hand, we saw that the inequality follows from the convexity of the [[exponential function]], which is the most basic log-convex function of all. On another hand, we have the following result which uses Hölder's inequality.

+-- {: .num_theorem}
###### Theorem
The collection of log-convex positive functions on a convex domain $D$ is closed under pointwise multiplication, pointwise addition, and pointwise sups.
=--

+-- {: .proof}
###### Proof
The statement for multiplication is clear since $\log(f \cdot g) = \log(f) + \log(g)$ and any sum of convex functions is convex.

Similarly, $\log: (0,\infty) \to \mathbb{R}$ is an isomorphism of [[partially ordered sets]] and so $\log (\sup\{f_i\}) = \sup\{\log(f_i)\}$. It thus suffices to show that if $f_i$ is a collection of convex functions on $D$, then so is $\sup\{f_i\}$. For $x, y \in D$ and $a, b \geq 0$ such that $a + b = 1$, we must show

$$\sup\{f_i\}(a x + b y) \leq a \sup\{f_i\}(x) + b\sup\{f_i\}(y);$$

letting $c$ denote the right side, this holds iff $f_i(a x + b y) \leq c$ for all $i$ (by definition of $\sup$). But

$$\array{
f_i(a x + b y) & \leq & a f_i(x) + b f_i(y) & since\; f_i\; is\; convex \\
 & \leq & a\sup\{f_i\}(x) + b\sup\{f_i\}(y) &
}.$$

Finally, for the sum $f + g$, in order to show $\log(f + g)$ is convex, it suffices to show that

$$\label{sum} (f + g)\left(\frac1{p}x + \frac1{q}y\right) \leq (f+g)(x)^{\frac1{p}} (f+g)(y)^{\frac1{q}}$$

for $p, q \gt 1$ such that $\frac1{p} + \frac1{q} = 1$. But setting

$$s = f(x)^{\frac1{p}}, \qquad t = g(x)^{\frac1{p}}, \qquad u = f(y)^{\frac1{q}}, \qquad v = g(y)^{\frac1{q}},$$

the right side of (eq:sum) is $(s^p + t^p)^{\frac1{p}} \cdot (u^q + v^q)^{\frac1{q}}$. By Hölder's inequality, this is greater than or equal to

$$\array{
s u + t v & = & f(x)^{\frac1{p}} f(y)^{\frac1{q}} + g(x)^{\frac1{p}} g(y)^{\frac1{q}} \\
 & \geq & f\left(\frac1{p} x + \frac1{q} y\right) + g\left(\frac1{p} x + \frac1{q} y\right)
}$$

where the last inequality is by log-convexity of $f$ and $g$.
=--

This last theorem enters into the account of [Artin (1931)](#Artin31) of the basic theory of the [[Gamma function]]:

+-- {: .num_cor}
###### Corollary
The [[Gamma function]]

$$\Gamma(x) = \int_0^\infty t^x e^{-t} \frac{d t}{t}$$

is log-convex over the domain $x \gt 0$.
=--

+-- {: .proof}
###### Proof
The function $x \mapsto t^{x-1}$ is log-linear, hence log-convex. Hence the integral defining $\Gamma(x)$ over $x \gt 0$ is a sup over suitable Riemann sums that are positive-linear combinations of the form

$$\sum_{i=1}^n t_i^{x-1} e^{-t_i} \Delta t_i$$
and these, and together with their sup, are log-convex by the Theorem.
=--

The main fact underlying Artin's 1931 discussion is the [Bohr-Haagerup theorem](Gamma+function#BohrMollerupTheorem) ([Artin (1931), Thm. 2.1](#Artin31)):

+-- {: .num_theorem #CharacterizationOgGamma}
###### Theorem
The [[Gamma function]] is characterized as the unique function $\Gamma: \{x \in \mathbb{R}|\; -x \notin \mathbb{N}\} \to \mathbb{R}$ satisfying the following conditions:

* $\Gamma(1) = 1$,

* $\Gamma(x+1) = x\Gamma(x)$,

* $\Gamma$ is log-convex over $(0, \infty)$.

=--

## References

* {#Artin31} [[Emil Artin]], *Einführung in die Theorie der Gammafunktion*, Hamburger Mathematische Einzelschriften
l. Heft, Verlag B. G. Teubner, Leipzig (1931)

  English translation by Michael Butler: *The Gamma Function*, Holt, Rinehart and Winston (1931) [[[Artin-TheGammaFunction.pdf:file]]]

[[!redirects Hölder inequality]]

[[!redirects Holder inequality]]
[[!redirects Holder's inequality]]

[[!redirects Hoelder inequality]]
[[!redirects Hoelder's inequality]]


Hörmander topology

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### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The "fundamental operations" on [[distributions]] ([[pullback of distributions]], [[product of distributions]]) are in general not [[continuous map|continuous]] in the [[subspace topology]], for subspaces where [[Hörmander's criterion]] is satisfied, of the canonical [[dual topological vector space]]-topology ("[[weak topology]]").

There is however a pseudo-topology which respect to which the fundamental operations are continuous. This is due to [Hörmander 71, p. 125](#Hoermander71) and hence called the _Hörmander topology_.

## References

Original articles include

* {#Hoermander71} [[Lars Hörmander]], _Fourier integral operators. I_ ,  Acta Math. 127 (1971), 79–183

* Yoann Dabrowski, [[Christian Brouder]], _Functional properties of Hörmander's space of distributions having a specified wavefront set_ ([arXiv:1308.1061](https://arxiv.org/abs/1308.1061))

* [[Christian Brouder]], Nguyen Viet Dang, [[Frédéric Hélein]], _Continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker)_, Studia Mathematica 232 (2016), 201-226  ([arXiv:org](https://arxiv.org/abs/1409.7662))

Review in the context of [[causal perturbation theory]]/[[perturbative QFT]] is in

* * {#FredenhagenRejzner12} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], around def. 5.1 in _Perturbative algebraic quantum field theory_, In _Mathematical Aspects of Quantum Field Theories_, Springer 2016 ([arXiv:1208.1428](https://arxiv.org/abs/1208.1428))

[[!redirects Hörmander toopologies]]

[[!redirects Hoermander topology]]
[[!redirects Hoermander topologies]]


Hörmander's condition

> This entry is about Hörmander's condition on [[tangent vector fields]]. This is different from [[Hörmander's criterion]] on [[wave front sets]].

## References

See also

* Wikipedia, _[Hörmander's condition](https://en.wikipedia.org/wiki/H%C3%B6rmander%27s_condition)_


Hörmander's criterion

> This entry is about Hörmander's criterion on [[wave front sets]]. This is different from "[[Hörmander's condition]]" on [[tangent vector fields]].

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Hörmander criterion_ ([Hörmander 90, theorem 8.2.10](#Hoermander90)) says that the [[product of distributions|product of two distributions]] $u_1 \cdot u_2$ (on some [[manifold]] $X$) is well-defined if their [[wave front sets]] $WF(u)$ are such that for $v \in T^\ast_x X$ a [[covector]] contained in one of the two wave front sets then the covector $-v \in T^\ast_x X$ with the opposite [[direction of a vector|direction]] in _not_ contained in the other wave front set, i.e. the [[intersection]] [[fiber product]]
inside the [[cotangent bundle]] $T^\ast X$ of the pointwise [[sum]] of wave fronts with the [[zero section]] is [[empty set|empty]]:

$$
  \left( WF(u_1) + WF(u_2) \right) \underset{T^\ast X}{\times} X
  \;=\;
  \emptyset
$$

i.e.

$$
  \array{
     && \emptyset
     \\
     & \swarrow && \searrow
     \\
     WF(u_1) + WF(u_2) && (pb) && X
     \\
     & \searrow && \swarrow_{\mathrlap{0}}
     \\
     && T^\ast X
  }
$$

See at _[[product of distributions]]_ for details.

## References

* {#Hoermander71} [[Lars Hörmander]], _Fourier integral operators. I_.  Acta Mathematica 127, 79–183 (1971) ([Euclid](https://projecteuclid.org/euclid.acta/1485889699))

* {#Hoermander90} [[Lars Hörmander]], section 8.1 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

[[!redirects Hoermander's criterion]]
[[!redirects Hormander's criterion]]

[[!redirects Hörmander criterion]]
[[!redirects Hoermander criterion]]
[[!redirects Hormander criterion]]


I-category

#Idea#
In his development of an '[[algebraic homotopy]]' theory,
Baues uses interacting structures, one of Quillen type (or rather of
K. Brown's version of half of Quillen's theory) and the other of cylinder
functor type.  The two structures are called [[cofibration category|cofibration categories]] and
_$I$-categories_.

# Definition

An **$I$-category** has various data specified: $(C, \mathit{cof}, I, \emptyset)$.

 Here $C$ is a category, $\mathit{cof}$ is a class of 'cofibrations', $\emptyset$ is an [[initial object]] of $C$, and $I$ is a [[cylinder functor]] (written as a functor, so $I(X)$ is the cylinder on object $X$).

These are required to satisfy:

I 1)  $I$ is a cylinder functor;

I 2)  Pushout axiom (almost as in the first part of C2 of [[cofibration category]], but $I$ is also
to preserve pushouts, and $I(\emptyset) = \emptyset$ so in fact $I$ preserves all finite colimits);

I 3) Cofibration axiom:

* $\iso \subset \mathit{cof}$ (where $\iso$ is the class of [[isomorphism]]s in $C$);
* $\emptyset \rightarrow X$ is always in $\mathit{cof}$;
* a composition of cofibrations is a cofibration and all morphisms in $\mathit{cof}$ satisfy the [[homotopy
extension property]].

I 4)  Relative cylinder axiom:

If $i : B \rightarrow A$ is a cofibration and one forms the pushout
$A \cup_B I(B)\cup_B A$,
then the natural map
$$A \cup_B I(B)\cup_B A \rightarrow A\times I$$
is a cofibration;

I 5) The 'interchange' axiom.

For all objects $X$, there is a map
$$T : I^2(X) \rightarrow I^2(X)$$
interchanging the two copies of $I$,
i.e.
$$T \circ e_i(I(X)) = I(e_i(X)), \quad T \circ I(e_i(X)) = e_i(I(X))$$
for $i = 0,1$.

(This corresponds to exchanging the first and second $I$-coordinates of
$X\times \mathbf{I} \times \mathbf{I}$ (where $I(X)$ is thought of as $X \times \mathbf{I}$), that is
$$(x,s,t) \rightarrow (x,t,s).$$