horizontal categorification

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### Context
#### Category theory
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[[!include category theory - contents]]
=--
#### Algebra
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[[!include higher algebra - contents]]
=--
#### Categorification
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[[!include categorification - contents]]
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=--
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#Contents#
* table of contents
{:toc}

## Idea

_Horizontal categorification_ or _Oidification_
describes the process by which

 1. a concept is realized to be equivalent to a certain type of [[category]] or [[magmoid]] with a _single [[object]]_;

 2. and then this concept is generalized -- or oidified -- by passing to instances of such types of categories with more than one object.

## Remarks

* This is to be contrasted with [[vertical categorification]].

* It can be argued that the term 'categorification' should be reserved for [[vertical categorification]],  since we can use 'oidification' for the horizontal concept.

* It has rightly been remarked that [[groupoid]]s are more fundamental than [[groups]], [[algebroids]] are more fundamental than algebras, etc. Hence in a better world, the suffix would be characterizing the one-object special cases, not the general concepts.

## Examples

* The horizontal categorification of [[groups]] are [[groupoids]]: [[category|categories]] in which every morphism is invertible.

* A horizontal categorification of [[algebras]] are [[algebroids]]: [[enriched category|enriched categories]] in the category of vector spaces.

* A horizontal categorification of rings are [[ringoid]]s: [[enriched category|enriched categories]] over the category of abelian groups. ([blog](http://golem.ph.utexas.edu/category/2006/09/ringoids.html))

* A horizontal categorification of $C^*$-algebras hence ought to be known as _$C^*$--algebroids_  but is usually known as [[C-star-category|C*-categories]].

* Since, by the [[Gelfand-Naimark theorem]], [[C-star algebra]]s are dual to [[topological space]]s, Paolo Bertozzini et. al proposed to define [[spaceoid]]s to be entities dual to $C^*$-categories ([blog](http://golem.ph.utexas.edu/category/2008/01/spaceoids.html)).

* {#Monoidoid} And an **exception to the rule**: a many-object [[monoid]] is _not_ called a _[[monoidoid]]_ -- but is called a [[category]]!

  On the other hand, if the category is $R$-linear ([[enriched category|enriched]] in [[R Mod]]) then it may again be referred to as an *[[algebroid]]* over $R$.

* A horizontal categorification of the notion of *[[monads]]* is that of *[[categories enriched in a bicategory]]*.

* A horizontal categorification of a single-sorted [[Lawvere theory]] is a multisorted Lawvere theory. Analogously an [[operad]] categorifies to a many-colored operad.

* {#InTypeTheory} In [[type theory]]/[[programming language theory]], horizontal categorification is analogous to *introducing a type distinction*: an untyped language is a special case of a typed language where there is exactly one [[type]]. This fits with the [[categorical semantics]]: untyped [[lambda calculus]] has models in [[Lawvere theories]], whereas (simply) [[typed lambda calculus]] has models in Cartesian [[multicategories]].

[[!include oidification - table]]

## Further discussion

Related $n$-Café-discussion is in

* [[John Baez]], [_What is categorification?_](http://golem.ph.utexas.edu/category/2008/10/what_is_categorification.html)

* [[John Baez]], [_Ringoids_](http://golem.ph.utexas.edu/category/2006/09/ringoids.html)

[[!redirects Oidification]]
[[!redirects oidification]]
[[!redirects oidifications]]
[[!redirects horizontal categorifications]]


horizontal chord diagram

> For discussion of standard round chord diagrams see at _[[chord diagram]]_.

***

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### Context
#### Graph theory
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[[!include graph theory - contents]]
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#### Knot theory
+-- {: .hide}
[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _horizontal chord diagram_ on $n$ strands is a finite undirected [[graph]] that is obtained from a trivalent graph with $n$ numbered embedded disjoint [[circles]] by cutting the circles open (to give the _strands_), such that the result has all edges not inside the circles (the _chords_) be vertically  ordered (i.e. along the strands) and going between distinct strands.

Here is an example of a horizontal chord diagram on 5 strands:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Defintions

### Monoid of horizontal chord diagrams
 {#MonoidOfHorizontalChordDiagrams}

For $n \in \mathbb{N}$ (the *number of strands*), the *monoid of horizontal chord diagrams* is the [[free monoid]]

\[
  \label{HorizontalChordDiagramsAsFreeMonoidOnChords}
  \mathcal{D}^{pb}_n
  \;\coloneqq\;
  FreeMonoid
  \Big(
    \big\{
      (i j)
      \,\vert\,
      1 \leq i \lt j \leq n
    \big\}
  \Big)
\]

on the [[set]] of pairs of distinct elements of $\{1, \cdots, n\}$, i.e. of pairs of strands, called the *chords* (the traditional superscript $pb$ is for _[[pure braids]]_).

Hence a horizontal chord diagram is equivalently a finite [[list]] of chords

$$
  D
  \;=\;
  (i_d j_d)
    \cdot
    \cdots
    \cdot
  (i_2 j_2)
    \cdot
  (i_1 j_1)
$$

and the product of chord diagrams is the concatenation of these list, with the [[empty list]] being the [[neutral element]].

The [[function]] that sends the chord $(i j)$ to that [[permutation]] of $n$ elements (strands) which is given by the [[transposition]] $t_{i j}$ of the $i$th with the $j$th strand extends to a unique [[monoid]] [[homomorphism]] from the monoid of horizontal chord diagrams (eq:HorizontalChordDiagramsAsFreeMonoidOnChords) to the [[symmetric group]] on $n$ elements:

$$
  \array{
    \mathcal{D}^{pn}_n
    &\overset{perm}{\longrightarrow}&
    Sym(n)
    \\
    (i_d j_d)
      \cdot \cdots \cdot
    (i_1 j_1)
    &\mapsto&
    t_{i_d j_d}
      \circ
    \cdots
      \circ
    t_{i_1 j_1}
    \,.
  }
$$

### Closure to round chord diagrams
 {#TraceToRoundChordDiagrams}

Given a [[horizontal chord diagram]] on $n$ strands and given any choice of [[cyclic permutation]] of $n$ elements, the [[trace of horizontal to round chord diagrams]] is the [[round chord diagram]] obtained by gluing the ends of the strands according to the cyclic permutation, and retaining the chords in the evident way.

The following shows an example of the trace operation for [[cyclic permutation]] of strands one step to the left:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

The following are the analogous traces of the four types of horizontal chord diagrams appearing in the [[4T relation]]:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

This defines a [[function]]

$$
  tr \;\colon\; \mathcal{C}^{pb} \longrightarrow \mathcal{C}^c
$$

from the [[set]] of [[horizontal chord diagrams]] to the set of [[round chord diagrams]].

### Algebra of horizontal chord diagrams
 {#AlgebraOfHorizontalChordDiagrams}

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

For $n \in \mathbb{N}$ and for , the [[linear span]] $Span\big( \mathcal{D}_n^{pb}\big)$ on the set of horizontal chord diagrams on $n$ strands becomes an [[graded algebra|graded]] [[associative algebra]]

$$
  \big( Span\big( \mathcal{D}_n^{pb}\big), \circ \big)
$$

under [[concatenation]] of strands.

=--

For example:

### The 2T- and 4T-relations
 {#The2TAnd4TRelations}

+-- {: .num_defn #2TAnd4TRelations}
###### Definition

On the $R$-[[module]] $R\langle \mathcal{D}^{pb}\rangle$ of [[horizontal chord diagrams]] consider the following [[relations]]:

The [[2T relations]] -- for pairwise distinct indices $i,j,k,l$ the $(i,j)$-chord commutes with the $(k,l)$-chord, for instance:

and the [[4T relations]] -- for pairwise distinct indices $i,j,k$ we have:

=--

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

In terms of the [[commutator]] [[Lie algebra]] of the [above](#AlgebraOfHorizontalChordDiagrams) algebra $\big( R\langle \mathcal{D}^{pb}\rangle, \circ \big)$ of horizontal chord diagrams, these are the [[infinitesimal braid relations]].

One writes

\[
  \label{QuotientModule}
  \mathcal{A}^{{}^{pb}}
  \;\coloneqq\;
  R\langle \mathcal{D}^{pb}\rangle/(2T,4T)
\]

for the [[quotient algebra]] of horizontal chord diagrams by these relations.

### Universal enveloping of infinitesimal braid relations

+-- {: .num_prop #UniversalEnvelopingOfInfinitesimalBraidsIsHorizontalChords}
###### Proposition
**([[universal enveloping algebra]] of [[infinitesimal braid Lie algebra]] is [[horizontal chord diagrams]] [[quotient vector space|modulo]] [[2T relations|2T]]&[[4T relations|4T]])

The [[associative algebra]] (eq:QuotientModule)

$$
  \Big(
  \mathcal{A}_n^{{}^{pb}}
  \;\coloneqq\;
  Span
  \big(
    \mathcal{D}_n^{pb}
  \big)/(2T, 4T)
  ,
  \circ
  \Big)
$$

of [[horizontal chord diagrams]] on $n$ strands with product given by [[concatenation]] of strands (Def. \ref{AlgebraHorizontalChordDiagrams}) [[quotient vector space|modulo]] the [[2T relations]] and [[4T relations]] (Def. \ref{2TAnd4TRelations}) is [[isomorphism|isomorphic]] to the [[universal enveloping algebra]] of the [[infinitesimal braid Lie algebra]] ([this Def.](infinitesimal+braid+relation#InfinitesimalBraidLieAlgebra)):

$$
  \big(\mathcal{A}_n^{pb}, \circ\big)
  \;\simeq\;
  \mathcal{U}(\mathcal{L}_n(D))
  \,.
$$

=--

### Horizontal weight systems

An $R$-[[linear map]] from the [[quotient module]] (eq:QuotientModule) of [[horizontal chord diagrams]] to $R$

\[
  \label{AHorizontalWeightSystem}
  w
  \;\colon\;
  \mathcal{A}_n^{{}^{pb}}
  \longrightarrow
  R
\]

is called a _[[weight system]]_ on horizontal chord diagrams (of $n$ strands), or maybe a _[[horizontal weight systems]]_.

Hence for $R = k$ a [[field]], the [[vector space]] of all horizontal weight systems is the degreewise [[dual vector space]]

$$
  \mathcal{W}_{pb}
  \;\coloneqq\;
  \big(
    \mathcal{A}^{pb}
  \big)^\ast
$$

([Bar-Natan 96, p. 3](#BarNatan96))

### Star-algebra structure
 {#StarAlgebraStructure}

Over a [[ground ring]] $R$ that is itself equipped with the [[mathematical structure|structure]] of a [[star-algebra]] $\mathbb{F} \overset{(-)^\ast}{\to} \mathbb{F}$ (such as the [[real numbers]], trivially, or the [[complex numbers]] via [[complex conjugation]]), we have that
also the [[associative algebra]] (eq:QuotientModule)

$$
  \Big(
  \mathcal{A}_n^{{}^{pb}}
  \;\coloneqq\;
  Span
  \big(
    \mathcal{D}_n^{pb}
  \big)/(2T, 4T)
  ,
  \circ
  \Big)
$$

of [[horizontal chord diagrams]] on $n$ strands with product given by [[concatenation]] of strands (Def. \ref{AlgebraHorizontalChordDiagrams}) [[quotient vector space|modulo]] the [[2T relations]] and [[4T relations]] (Def. \ref{2TAnd4TRelations})

becomes a [[star-algebra]]  with star-operation

$$
  (-)^\ast
  \;:\;
  \mathcal{A}_n^{{}^{pb}}
  \longrightarrow
  \mathcal{A}_n^{{}^{pb}}
$$

given by reversing the [[orientation]] of strands:

Since [[horizontal chord diagrams are the homology of the loop space of configuration space]] and the [[homology of a loop space]] is an [[involutive Hopf algebra]], this is a special case of the general fact that involutive Hopf algebras are star-algebras ([here](star-algebra#InvolutiveHopfAlgebrasAreStarAlgebras)).

With respect to this [[star-algebra]]-[[mathematical structure|structure]] one may ask (setting $R \coloneqq \mathbb{C}$ for definiteness) whether a given [[weight system]] (eq:AHorizontalWeightSystem)

$$
  w
  \;\colon\;
  \mathcal{A}_n^{{}^{pb}}
  \longrightarrow
  \mathcal{C}
$$

is a [[state on a star-algebra]] in that for any $D \in \mathcal{A}_n^{{}^{pb}}$ we have that the value of $w$ on the corresponding [[normal operator]] $D \cdot D^\ast$ is a non-[[negative number|negative]] [[real number]]:

$$
  \Big(
    w \in \mathcal{W}_n^{{}^{pb}}
    \;
    \text{is a state}
  \Big)
  \phantom{AAA}
  \Leftrightarrow
  \phantom{AAA}
  \left(
    \begin{aligned}
      \text{1.}\;\;\;
      &
      w(1) = 1
      \\
      \text{2.}\;\;\;
      &
      \underset{
        \mathclap{
          D \in \mathcal{A}_n^{{}^{pb}}
        }
      }{
        \forall
      }
      \;\;\;\;
      \Big(
        w(D \cdot D^\ast)
        \;
          \geq 0
        \;
        \in
        \mathbb{R}
        \subset
        \mathbb{C}
      \Big)
    \end{aligned}
  \right)
  \,.
$$

The [[weight systems]] which are [[states on a star-algebra]] with respect to this star-involution are discussed in [CSS 21](#CSS21).

### Closure to Sullivan chord diagrams
 {#ClosureToSullivanChordDiagrams}

More generally, one obtains [[Sullivan chord diagrams]] with $p$ disjoint embedded circles from [[horizontal chord diagrams]] by closing up strands after acting with a [[permutation]] with $p$ cycles ($p$ [[orbits]])

> from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

## Applications

### Knizhnik-Zamolodchicov connection

[[!include Knizhnik-Zamolodchikov-Kontsevich construction -- definition]]

[[!include chord diagrams as multi-trace observables in BMN matrix model -- example]]

## Related concepts

[[!include chord diagrams and weight systems -- table]]

## References

Original articles:

* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

* {#Kohno02} [[Toshitake Kohno]], _Loop spaces of configuration spaces and finite type invariants_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arXiv:math/0211056](https://arxiv.org/abs/math/0211056))

* [[Adrien Brochier]], _Cyclotomic associators and finite type invariants for tangles in the solid torus_, Algebr. Geom. Topol. 13 (2013) 3365-3409 ([arXiv:1209.0417](https://arxiv.org/abs/1209.0417))

Textbook accounts:

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 5.11 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

Discussion of the [[star-algebra]]-structure and associated [[state on a star-algebra|states]] on horizontal chord diagrams:

* {#CSS21} [[David Corfield]], [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Fundamental weight systems are quantum states]]* ([arXiv:2105.02871](https://arxiv.org/abs/2105.02871))

[[!redirects horizontal chord diagrams]]


horizontal composition

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### Context
#### 2-Category theory
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[[!include 2-category theory - contents]]
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#### Higher category theory
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[[!include higher category theory - contents]]
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# Contents
* table of contents
{:toc}

## Definition

In a [[2-category]], the composition of [[2-morphisms]] along [[object]]s is called _horizontal composition_ .

+-- {: style="text-align:center"}
[[!include horizontal composition - SVG]]
=--

This is in contrast to the [[vertical composition]] of 2-morphisms, which is their composition along 1-[[morphism]]s.

## Properties

Horizontal and vertical composition are subject to the compatibility condition called the [[interchange law]], which in this case and the higher categorical ones means that if we fix three objects $A$, $B$, and $C$ then the composition of 1-cells between them defines a functor.

## Examples

* In [[Cat]], horizontal composition is the [[Godement product]] of [[natural transformation]]s.

## Related topics

* [[whiskering]]

## References

Horizontal composition of [[natural transformations]] is first describes in

* {#Godement58} [[Roger Godement]], Appendix (pp. 269) of: *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]

for textbook accounts see most of those listed at *[[category]]*, such as

* [[Saunders MacLane]], §II.5, p. 43 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]


horizontal composition - SVG

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       <mn>1</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">G_2\circ G_1</annotation>
    </semantics>
   </math>
  </foreignObject>
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  </marker>
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  </marker>
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category: svg


horizontal differential form

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

#Contents#
* table of contents
{:toc}

## Definition

There are two different meanings of _horizontal differential form_

Given a [[bundle]] $\pi \colon P \to X$ of [[smooth manifolds]],

* then a [[differential form]] on the total space $P$ is _horizontal_ if it vanishes on [[vertical vector fields]]:

* in the context of [[variational calculus]]:

  a differential form on the the total space $J^\infty_X(E)$ of the [[jet bundle]] of $E$ is horizontal if it is in the horizontal component of the [[variational bicomplex]] of $J^\infty(E)$.

## Related concepts

* [[vertical differential form]]

* [[basic differential form]]

* [[variational bicomplex]]

[[!redirects horizontal differential forms]]

[[!redirects horizontal form]]
[[!redirects horizontal forms]]


horn

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

#Contents#
* table of contents
{:toc}

## Idea ##

In [[simplicial homotopy theory]] a *horn* of a [[simplex]] is a simplicial incarnation of the notion of a *boundary segment* (as opposed to the full boundary) of a [[cell]].

Specifically, the $k$th horn $\Lambda_k[n] \hookrightarrow \Delta[n]$ of the $n$-dimensional [[simplex]]  (defined for $n \geq 1$, see Rem. \ref{ZeroSimplexHasNoHorns}) is the [[simplicial set]] obtained from the [[boundary of a simplex|boundary of the n-simplex]] $\partial \Delta^n$ of the standard simplicial [[n-simplex]] $\Delta^n$ by discarding its $k$th face.

The property of a [[morphism]] of [[simplicial set]] to have the [[right lifting property]] against the inclusions of horns into their simplices (have "horn fillers") characterizes the basic types of [[fibrations]] and [[fibrant objects]] in [[simplicial homotopy theory]]:

* [[Kan fibrations]] (hence [[Kan complexes]], when over the point) are characterized as having the [[right lifting property]] against *all horns*;

* [[left/right Kan fibrations]] are characterized as having the [[right lifting property]] against all horns *except* possibly the first/last horn inclusion in each dimension, respectively;

* [[quasi-categories]] are characterized as having the [[right lifting property]] against all horns *except* possibly the first and last horn inclusion in each dimension, respectively.

Specifically, [[Kan fibrations]] are the simplicial analog of [[Serre fibrations]] (in a precise sense so under the [[Quillen equivalence]] between the [[classical model structure on simplicial sets]] and [[classical model structure on topological spaces|on topological spaces]]) and under this correspondence the horn inclusions into the $n$th simplex correspond to the inclusion of the $(n-1)$-dimensional [[topological space|topological]] [[disk]] into its [[cylinder]]:

$$
  \Lambda_k[n] \hookrightarrow \Delta[n]
  \;\;\;\;
  \overset{\left\vert-\right\vert}{\mapsto}
  \;\;\;\;
  D^{n-1} \overset{(id,0)}{\hookrightarrow} D^{n-1} \times [0,1]
  \,,
$$

as both are the [[generating cofibrations]] in their respective [[model categories]] ([[classical model structure on simplicial sets|on sSet]], [[classical model structure on topological spaces|on Top]], respectively).

## Definition

Let

$$
  \Delta[n] = \mathbf{\Delta}( -, [n]) \in  Simp Set
$$

denote the standard simplicial $n$-[[simplex]] in [[SimpSet]].

+-- {: .num_defn #Horn}
###### Definition
**(horns)**

For all $n, k \,\in\, \mathbb{N}$ with $n\geq 1$ and $0\leq k \leq n$, the **$(n,k)$-horn** or **$(n,k)$-box**  is the [[subobject|sub]]-[[simplicial set]]

$$
  \Lambda_k[n]
  \hookrightarrow
  \Delta[n]
$$

which is the [[union]] of all faces _except_ the  $k$th one.

{#InnerHorns} This is called an **outer horn** if $k = 0$ or $k = n$, otherwise it is an **inner horn**.

=--

\begin{remark}
Since [[sSet]]  is a [[presheaf topos]], [[unions]] of [[subobjects]] make sense and they are calculated objectwise, thus in this case dimensionwise.  This way it becomes clear what the structure of a horn as a functor $\Lambda^k[n]: \Delta^{op} \to Set$ must therefore be: it takes $[m]$ to the collection of ordinal maps $f \colon [m] \to [n]$ which factor through some coface map $[n-1] \to [n]$ which is _not_ the $k$th one.
\end{remark}

\begin{remark}\label{ZeroSimplexHasNoHorns}
**(the 0-simplex has no horns)**\linebreak
Beware that any would-be horn of the 0-simplex (whose [[boundary of a simplex|boundary]] is the [[empty simplicial set]]) is *ex*-cluded in Def. \ref{Horn}: The 0-simplex has *no* horn.

This is not a matter of convention if one sticks to the usual definition of [[Kan fibration]] as having [[right lifting property|right lifting]] against *all horns*:  In particular one must *not* declare a would-be horn of the 0-simplex to be itself the [[empty simplicial set]], as that would make the [[nerve of a category|nerve]] of the [[empty groupoid]] fail to be a [[Kan complex]] (Def. \ref{KanComplex}).

Conversely, the fact that every horn is [[inhabited set|inhabited]] means (see also [this Example](Kan+fibration#KanFibrationNeedNotBeSurjective)) that any morphism out of the [[empty simplicial set]] (an [[empty bundle]] $\varnothing \to X$) is a [[Kan fibration]] (Def. \ref{KanFibration}).

This is in contrast to the situation for [[acyclic Kan fibrations]], see [there](acyclic+Kan+fibration#AcyclicKanFibrationsAreSurjective).
\end{remark}

## Examples ##

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

The [inner horn](#InnerHorns) of the 2-simplex

$$\Delta^2 =
  \left\{
      \array{
        && 1
        \\
        & \nearrow &\Downarrow& \searrow
        \\
        0 &&\to&& 2
      }
     \right\}
$$

with [[boundary]]

$$\partial \Delta^2 =   \left\{
      \array{
        && 1
        \\
        & \nearrow && \searrow
        \\
        0 &&\to&& 2
      }
     \right\}
$$

looks like

$$
\Lambda^2_1
  =
   \left\{
      \array{
        && 1
        \\
        & \nearrow && \searrow
        \\
        0 &&&& 2
      }
     \right\}
 \,.
$$

The two outer horns look like

$$\Lambda^2_0 =
\left\{
      \array{
        && 1
        \\
        & \nearrow &&
        \\
        0 &&\to&& 2
      }
     \right\}
$$

and

$$\Lambda^2_2 =
\left\{
      \array{
        && 1
        \\
        & && \searrow
        \\
        0 &&\to&& 2
      }
     \right\}
$$

respectively.

\[\label{The2Horns}\]
\begin{imagefromfile}
    "file_name": "The2Horns.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -20,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

=--

## Lifting against horn inclusion

\begin{defn}\label{KanFibration}
**([[Kan fibration]])**\linebreak
A [[Kan fibration]] is a [[morphism]] of [[simplicial sets]] which has the [[right lifting property]] with respect to all horn inclusions $\Lambda^k[n] \hookrightarrow \Delta^n$.

\end{defn}

\begin{defn}\label{KanComplex}
**([[Kan complex]])**\linebreak
A [[Kan complex]] is a simplicial set in which "*all horns have fillers*": a simplicial set for which the morphism to the point is a Kan fibration (Def. \ref{KanFibration}).
\end{defn}

\begin{defn}\label{QuasiCategory}
**([[quasi-category]])**\linebreak
A [[quasi-category]] is a [[simplicial set]] in which *all [inner horns](#InnerHorns) have fillers*, hence such that the morphism to the [[point]] is an [[inner fibration]].
\end{defn}

## Related concepts

* The [[boundary of a simplex]] is the union of its faces.

* The [[spine]] of a [[simplex]] is the [[union]] of all its generating 1-cells.

## References

See the references at *[[simplicial homotopy theory]]*.

[[!redirects horns]]
[[!redirects inner horn]]
[[!redirects inner horns]]

[[!redirects horn inclusion]]
[[!redirects horn inclusions]]

[[!redirects simplicial horn]]
[[!redirects simplicial horns]]


Horn theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Type theory
+--{: .hide}
[[!include type theory - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **Horn theory** is a [[theory]] in which every [[axiom]] has a certain special form.

## Definition

Let $\Sigma$ be a [[signature (in logic)|signature]]. A **[[term]]** of $\Sigma$ is an expression built out of [[variables]] and [[function symbols]]. (For example, $x y^{-1} z$ is a term in the language of [[groups]].) An **atomic formula** relative to $\Sigma$ is a formula that consists of a [[relation symbol]] of $\Sigma$ (including [[equality]]) applied to terms.

A **Horn clause** is a logical expression of the form

$$\phi_1 \wedge \ldots \wedge \phi_n \vdash \phi$$

where each $\phi_i$ and $\phi$ is atomic. A (universal) **Horn theory** is a [[theory]] in which every [[axiom]] is a Horn clause.

## Related concepts

* [[modular lattice]]

* [[pseudolattice ordered ring]]

## References

* [[Michael Barr]], *Models of Horn theories*, in: *Categories in Computer Science and Logic*, Contemporary Math. **92**, Amer. Math. Soc. (1989) 1–7 [[pdf](https://www.math.mcgill.ca/barr/papers/horn.pdf), [[Barr-ModelsOfHornTheories.pdf:file]]]

[[!redirects Horn theories]]

[[!redirects Horn clause]]
[[!redirects Horn clauses]]

[[!redirects Horn logic]]


horocycle correspondence

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

#Contents#
* table of contents
{:toc}

## Idea

In [[geometric representation theory]] the _horocycle correspondence_ for $G$ a complex [[reductive group]] and $B \subset G$ a [[Borel subgroup]] is the [[correspondence]] of [[group]] [[quotients]]  given by

$$
  G//_{ad} G \stackrel{p}{\longleftarrow} G_{ad}//B \stackrel{\delta}{\longrightarrow} B \backslash  G / B
 \,.
$$

This generalizes the [[Grothendieck-Springer correspondence]]. ([Ben-Zvi & Nadler 09, example 1.2](#BenZviNadler09))

The pull-push [[integral transform]] of [[D-modules]] from right to left through this correspondence is the _Harish-Chandra transform_

$$
  p_\ast \delta^! \;\colon\; \mathcal{D}(B\backslash G / B) \longrightarrow \mathcal{D}(G/_{ad}G)
$$

from the [[Hecke category]] on the left.

A (unipotent) [[character sheaf]] ([Lusztig 85](#Lusztig85)) for $G$ is a [[simple object]]-[[direct summand]] of a [[D-module]] that is in the image of a [[simple object]] under this transform. ([Ginzburg 89](#Ginzburg89))

## Related concepts

* [[twistor correspondence]], [[Penrose transform]]

* [Schubert calculus -- Correspndences](Schubert+calculus#Correspondences)

## References

Original articles include

* {#Lusztig85} [[George Lusztig]], _Character sheaves I_. Adv. Math 56 (1985) no. 3, 193-237.

* {#Ginzburg89} [[Victor Ginzburg]], _Admissible modules on a symmetric space. Orbites unipotentes et représentations, III. Astérisque No. 173-174 (1989), 9–10, 199–255.

Interpretation in the context of [[extended TQFT]] is in

* {#BenZviNadler09} [[David Ben-Zvi]], [[David Nadler]], _The Character Theory of a Complex Group_ ([arXiv:0904.1247](http://arxiv.org/abs/0904.1247))

[[!redirects horocycle correspondences]]
[[!redirects Harish Chandra transform]]
[[!redirects Harish Chandra transforms]]
[[!redirects Harish-Chandra transform]]
[[!redirects Harish-Chandra transforms]]

[[!redirects Grothendieck-Springer correspondence]]
[[!redirects Grothendieck-Springer correspondences]]


horseshoe lemma

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

#Contents#
* table of contents
{:toc}

## Idea

A basic [[lemma]] in [[homological algebra]].

## Related concepts

* [[salamander lemma]]

* [[3x3 lemma]], [[snake lemma]], [[5-lemma]]

## References

The horseshoe lemma appears as lemma 2.2.8 in

* [[Charles Weibel]], _[[An introduction to homological algebra]]_

[[!redirects hoerseshoe lemma]]


Horst Herrlich

Horst Herrlich (b. September 11, 1937; d. March 13, 2015) was a German mathematician, known primarily for work in [[general topology]] using methods of [[category theory]]. He is the author of a book on the [[axiom of choice]].

## Selected writings

On [[category theory]] with emphasis on [[concrete categories]]:

* [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]], *[[Abstract and Concrete Categories -- The Joy of Cats]]*, John Wiley and Sons, New York (1990), reprinted in: Reprints in Theory and Applications of Categories **17** (2006) 1-507 ([tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html), [webpage](http://katmat.math.uni-bremen.de/acc/), [pdf](http://katmat.math.uni-bremen.de/acc/acc.pdf))

On [[category theory|category theoretic]] aspects of [[general topology]]:

* [[Horst Herrlich]], [[George Strecker]], *Categorical topology -- Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971* ([pdf](https://link.springer.com/content/pdf/10.1007%2F978-94-017-0468-7_15.pdf)), pages 255-341 in: C. E. Aull, R Lowen (eds.), *Handbook of the History  of General Topology. Vol. 1* , Kluwer 1997 ([doi:10.1007/978-94-017-0468-7](https://link.springer.com/book/10.1007/978-94-017-0468-7))

category: people


Horst Schubert

* [Wikipedia entry](https://en.wikipedia.org/wiki/Horst_Schubert)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=21623)

## Selected writings

On [[category theory]]:

* [[Horst Schubert]], *Categories*, Springer (1972) [[doi:10.1007/978-3-642-65364-3](https://doi.org/10.1007/978-3-642-65364-3)]

category: people


hosohedron

[[!include ADE -- table]]

## References

See also

* Wikipedia, _[Hosohedron](https://en.wikipedia.org/wiki/Hosohedron)_

[[!redirects hosohedra]]


HoTT at DMV2015

This page records talks that were given at the meeting:

* **[Homotopy Type Theory and Univalent Foundations](https://sites.google.com/site/dmv2015hott/)**

  at [DMV 2015](https://www.math.uni-hamburg.de/dmv2015/)

  Hamburg, 2015

on [[homotopy type theory]] and [[univalent foundations of mathematics]].

* [[Thomas Streicher]], _Various ways of splitting and equality of objects_ ([slides](http://www.mathematik.tu-darmstadt.de/~streicher/FIBR/SpFi2015.pdf))

* [[Martin Hofmann]], _The groupoid interpretation of type theory, a personal retrospective_ ([[HofmannDMV.pdf|slides:file]])

* [[Nicola Gambino]], _Aspects of univalence_ ([[GambinoDMV.pdf|slides:file]])

* [[Urs Schreiber]], _Some thoughts on the future of modal homotopy type theory_ ([[schreiber:Some thoughts on the future of modal homotopy type theory|Notes]])

* [[Bas Spitters]], _Cubical sets as a classifying topos_ ([[SpittersDMV.pdf|slides:file]])

* [[Benno van den Berg]], _Weak universes and homotopy exact completion_

* [[Steve Awodey]], _On the cubical model of HoTT_ ([[AwodeyDMVrev.pdf|slides:file]])

* [[Simon Huber]], _A Cubical Type Theory_ ([slides](http://www.cse.chalmers.se/~simonhu/slides/hamburg2015.pdf))

* [[Thorsten Altenkirch]], _The coherence problem in HoTT_ ([[Altenkirch.pdf|slides:file]])

* [[Rasmus Møgelberg]], _Towards guarded recursion in HoTT_ ([[Mogelberg.pdf|slides:file]])

* [[Tamara von Glehn]], _Constructing models of type theory_ ([[vonGlehnDMV2015.pdf|slides:file]])

* [[Peter LeFanu Lumsdaine]], _Formalising the categorical semantics of type theory, in type theory_ ([slides](http://peterlefanulumsdaine.com/research/Lumsdaine-2015-DMV-Hamburg-slides-Formalising-semantics-of-DTT-in-DTT.pdf))

## See also

* [[homotopy type theory events]]

category: reference

[[!redirects DMV2015]]


HoTT book real number

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

#Contents#
* table of contents
{:toc}

## Idea ##

A construction ([UFP13, Sec. 11.3](#UFP13)) of the [[type]] of [[real numbers]] in [[homotopy type theory]] as a [[higher inductive-inductive type]].

As opposed to the "naive" [[quotient type]] of [[regular sequence|regular]] [[Cauchy sequences]] of [[rational numbers]] (i.e. the quotient type of the [[setoid]]/[[Bishop set]] by which the [[Cauchy real numbers]] are traditionally modeled in [[constructive analysis]]) this [[higher inductive-inductive type]] is [[sequentially modulated Cauchy complete]].

Notice that this crucial property of the [[real number]] fails for other constructions:

* The [[quotient type]] of [[regular sequence|regular]] [[Cauchy sequences]] of [[rational numbers]] is Cauchy complete only assuming that any sequence of its [[terms]] has a [[lift]] to a sequence of representing [[Cauchy sequences]], which requires an extra [[axiom of countable choice]] (and is the reason why traditional [[Errett Bishop|Bishop]]-[[constructive analysis]] does not pass to the quotient but stays with the [[setoid]]/[[Bishop set]]).

* The [[modulated Cauchy real numbers]] are not sequentially modulated Cauchy complete either, in [[constructive mathematics]].

## Definition ##

### As a sequentially modulated Cauchy complete Archimedean ordered field ###

Let $K$ be an [[Archimedean ordered field]]. $K$ is *[[sequentially modulated Cauchy complete]]* if every [[Cauchy sequence]] in $K$ with a [[modulus of convergence]] [[converges]]. The set of __HoTT book real numbers__ $\mathbb{R}_H$ is the [[initial object]] in the [[category]] of sequentially modulated Cauchy complete Archimedean fields.

More abstractly, let $\mathrm{ArchiFields}$ be the [[category]] of [[Archimedean ordered fields]] and Archimedean ordered field homomorphisms, and let $F:\mathrm{ArchiFields} \to \mathrm{ArchiFields}$ be the [[endofunction]] which takes an Archimedean ordered field $K$ to the Archimedean ordered field $F(K)$ which is the [[quotient set]] of the set of [[Cauchy sequences]] in $K$ with a [[modulus of convergence]]. $K$ is [[sequentially modulated Cauchy complete]] if there is an isomorphism $K \cong F(K)$, and the **HoTT book real numbers** $\mathbb{R}_H$ is the initial Archimedean ordered field with an isomorphism $\mathbb{R}_H \cong F(\mathbb{R}_H)$.

### As a Cauchy structure ###

The set of __HoTT book real numbers__ $\mathbb{R}_H$ is the initial object in the category of [[Cauchy structures]] and [[Cauchy structure homomorphisms]].

### As a higher inductive-inductive type ###

As a [[higher inductive-inductive type]]:

Let $\mathbb{Q}$ be the [[rational numbers]] and let

$$\mathbb{Q}_{+} \coloneqq \sum_{x:\mathbb{Q}} 0 \lt x$$

be the [[positive number|positive]] rational numbers. The __HoTT book real numbers__ $\mathbb{R}_H$ are inductively generated by the following:

* a function $inj: \mathbb{Q} \to \mathbb{R}_H$

* a function $lim: C(\mathbb{R}_H) \to \mathbb{R}_H$, where $C(\mathbb{R}_H)$ is the type of [[Cauchy approximation]]s in $\mathbb{R}_H$.

* a dependent family of terms

$$a:\mathbb{R}_H, b:\mathbb{R}_H \vdash eq_{\mathbb{R}_H}(a, b): \left( \prod_{\epsilon:\mathbb{Q}_{+}} (a \sim_{\epsilon} b) \right) \to (a =_{\mathbb{R}_H} b)$$

and the [[premetric]] type family $\sim$ is simultaneously inductively generated by the following:

* a dependent family of terms
$$a:\mathbb{Q}, b:\mathbb{Q}, \epsilon:\mathbb{Q}_{+} \vdash \mu_{\mathbb{Q}, \mathbb{Q}}(a, b, \epsilon): (\vert a - b \vert \lt \epsilon) \to (inj(a) \sim_{\epsilon} inj(b))$$

* a dependent family of terms
$$a:\mathbb{Q}, b:C(\mathbb{R}_H), \epsilon:\mathbb{Q}_{+}, \eta:\mathbb{Q}_{+} \vdash \mu_{\mathbb{Q}, C(\mathbb{R}_H)}(a, b, \epsilon, \eta): (inj(a) \sim_{\epsilon} b_{\eta}) \to (inj(a) \sim_{\epsilon + \eta} lim(b))$$

* a dependent family of terms
$$a:C(\mathbb{R}_H), b:\mathbb{Q}, \delta:\mathbb{Q}_{+}, \epsilon:\mathbb{Q}_{+} \vdash \mu_{C(\mathbb{R}_H), \mathbb{Q}}(a, b, \delta, \epsilon): (a_{\delta} \sim_{\epsilon} inj(b) ) \to (lim(a) \sim_{\delta + \epsilon} inj(b))$$

* a dependent family of terms
$$a:C(\mathbb{R}_H), b:C(\mathbb{R}_H), \delta:\mathbb{Q}_{+}, \epsilon:\mathbb{Q}_{+}, \eta:\mathbb{Q}_{+} \vdash \mu_{C(\mathbb{R}_H), C(\mathbb{R}_H)}(a, b, \delta, \epsilon, \eta): (a_{\delta} \sim_{\epsilon} b_{\eta} ) \to (lim(a) \sim_{\delta + \epsilon + \eta} lim(b))$$

* a dependent family of terms
$$a:\mathbb{R}_H, b:\mathbb{R}_H, \epsilon:\mathbb{Q}_{+} \vdash \pi(a, b, \epsilon): isProp(a \sim_{\epsilon} b)$$

## Properties

The [[analytic functions]], such as the [[exponential function]], the [[sine function]], and the [[cosine function]] are well defined in the HoTT book real numbers, since all modulated Cauchy sequences converge in the HoTT book real numbers.

## See also ##

* [[higher inductive-inductive type]]

* [[Cauchy real numbers]]

* [[generalized Cauchy real numbers]]

* [[premetric space]]

* [[Escardo-Simpson real numbers]]

## References ##

* {#UFP13} [[Univalent Foundations Project]], Section 11.3 of: *[[Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

Survey and review:

* [[Andrej Bauer]], *The real numbers in homotopy type theory*, talk at *[Computability and Complexity in Analysis](http://cca-net.de/cca2016/)*, Faro (2016) [[pdf](https://math.andrej.com/wp-content/uploads/2016/06/hott-reals-cca2016.pdf), [[Bauer-RealsInHoTT.pdf:file]]]

* {#Booij20} [[Auke Booij]], *Analysis in Univalent Type Theory* (2020) [[etheses:10411](http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf), [[Booij-AnalysisInUF.pdf:file]]]

[[!redirects HoTT book real numbers]]


HoTT in Bonn2018

[[!redirects Bonn2018]]

This entry records tites and abstracts of the meeting

* **[Types, Homotopy Type Theory, and Verification](https://www.him.uni-bonn.de/programs/past-programs/past-trimester-programs/types-sets-constructions/workshop-types-homotopy-type-theory-and-verification/)**

  June 4-8, 2018

  at the [Hausdorff Institute for Mathematics](https://www.him.uni-bonn.de/), Bonn

  Organizers: [[Steve Awodey]], [[Thierry Coquand]], [[Maria Emilia Maietti]]

on [[homotopy type theory]].

This was part of the:

* **Trimester Program "[Types, Sets and Constructions](https://www.him.uni-bonn.de/programs/past-programs/past-trimester-programs/types-sets-constructions/description/)"**

  May 2 - August 24, 2018

  at the [Hausdorff Institute for Mathematics](https://www.him.uni-bonn.de/),

  Organizers: Douglas S. Bridges, Michael Rathjen, Peter Schuster, Helmut Schwichtenberg

#Contents#
* table of contents
{:toc}

## Schedule of the Workshop ##

Monday, June 4

11:00 - 12:00	Tutorial 1: Dan Licata: A Fibrational Framework for Modal Simple Type Theories
([slides](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/him-tutorial.pdf))

14:00 - 15:00	Christian Sattler: Do cubical models of type theory also model homotopy types?

15:00 - 16:00	Floris van Doorn: Spectral Sequences in Homotopy Type Theory ([[vanDoorn.pdf:file]])

16:30 - 17:30	Thorsten Altenkirch: Towards higher models and syntax of type theory
([slides](http://www.cs.nott.ac.uk/~psztxa/talks/bonn18.pdf))

Tuesday, June 5

09:30 - 10:30	Klaus Mainzer: Algorithms of Type and Proof Checking

11:00 - 12:00	Tutorial 2: Felix Wellen: The Shape Modality in Real-cohesive HoTT and Covering Spaces

15:00 - 16:00	Kuen-Bang Hou (Favonia): Cartesian cubical computational type theory ([slides](http://favonia.org/files/ccctt-bonn2018-slides.pdf))

16:30 - 17:30	Benno van den Berg: Univalent polymorphism  ([[vdB.pdf:file]])

Wednesday, June 6

09:30 - 10:30	Tutorial 3: Dan Licata: Discrete and Codiscrete Modalities in Cohesive HoTT, I ([slides](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/him-tutorial.pdf))

11:00 - 12:00	Tutorial 4: Felix Wellen: Discrete and Codiscrete Modalities in Cohesive HoTT, II

15:00 - 16:00	Anders Mortberg: Yet Another Cartesian Cubical Type Theory (yacctt)  ([[MortbergBonn.pdf:file]])

16:30 - 17:30	Jacopo Emmenegger: W-types in the setoid model

Thursday, June 7

09:30 - 10:30	Tutorial 5: Dan Licata: A Fibrational Framework for Modal Dependent Type Theories ([slides](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/him-tutorial.pdf))

11:00 - 12:00	Bas Spitters: Modal Dependent Type Theory and the Cubical Model ([PDF](http://www.cs.au.dk/~spitters/bonn-slides.pdf))

14:00 - 15:00	Pino Rosolini: Embedding the effective topos into cubical assemblies

15:00 - 16:00	[[Fabio Pasquali]]: Assemblies as an elementary quotient completion

16:30 - 17:30	Hugo Herbelin: Investigations into cubical type theory

Friday, June 8

09:30 - 10:30	Tutorial 6: Felix Wellen: Differential Cohesive HoTT

11:00 - 12:00	Andrej Bauer: Toward an initiality theorem for general type theories

14:00 - 15:00	Nicolai Kraus: The Challenge of Free Groups

15:00 - 16:00	Paul-André Melliès: Refinement type systems and Martin-Lof type theory

##Abstracts##

### Thorsten Altenkirch: Towards higher models and syntax of type theory

We (Ambrus Kaposi and myself) have defined the intrinsic (no preterms) syntax of type theory in type theory using quotient inductive inductive types, that is set truncated higher inductive inductive types. Set truncation is necessary because we want to have decidability which we established using normalisation by evaluation. However, this stops us from defining any interesting semantic models, e.g. interpreting contexts as sets, types as families etc, because sets do not from a set (not for size reason but dimension). I suggest that we can overcome this problem by defining higher models of type theory and a higher syntax. I envisage that the higher syntax does form a set without being explicitly truncated, hence decidability should still hold. In my talk I will describe the first steps towards this programme.

### Andrej Bauer: Toward an initiality theorem for general type theories

I will report on a project of defining carefully and formally what a general type theory is. This is kind of step 1 to a generally applicable initiality theorem. It's not really a big surprise: it's what one would expect, but there's a number of things one has to organize and think through so that the whole thing holds water. We've got large chunks of it formalized. Work in progress with Philipp Haselwarter and Peter Lumsdaine.

### Benno van den Berg: Univalent polymorphism

We show that Martin Hyland's effective topos can be exhibited as the homotopy category of a path category EFF. Path categories are categories of fibrant objects in the sense of Brown satisfying two additional properties and as such provide a context in which one can interpret many notions from homotopy theory and Homotopy Type Theory. Within the path category EFF one can identify a class of discrete fibrations which is closed under push forward along arbitrary fibrations (in other words, this class is polymorphic or closed under impredicative quantification) and satisfies propositional resizing. This class does not have a univalent representation, but if one restricts to those discrete fibrations whose fibres are propositions in the sense of Homotopy Type Theory, then it does. This means that, modulo the usual coherence problems, it can be seen as a model of the Calculus of Constructions with a univalent type of propositions. We will also build a more complicated path category in which the class of discrete fibrations whose fibres are sets in the sense of Homotopy Type Theory has a univalent representation, which means that this will be a model of the Calculus of Constructions with a univalent type of sets.

### Floris van Doorn: Spectral sequences in Homotopy Type Theory

In algebraic topology, spectral sequences form a powerful tool which can be used to compute homotopy and (co)homology groups of spaces in a wide variety of cases. We have constructed two spectral sequences in homotopy type theory, the Atiyah-Hirzebruch and Serre spectral sequences for cohomology. In this talk, I will introduce the concept of spectral sequences, talk about the spectral sequences we constructed and explore some of the applications and future work. (This is joint work with Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke and Mike Shulman.)

### Jacopo Emmenegger: W-types in the setoid model

Current arguments to obtain initial algebras for polynomial endofunctors in categories of equivalence relations rely on assumptions like UIP or on constructions that involve recursion into a universe. We present a new argument, verified in Coq, that instead uses a generalisation of W-types in the underlying type theory. This in turn allows us to both give a direct construction of the initial algebra and to use induction of the underlying W-type to prove initiality.

### Hugo Herbelin: Investigations into cubical type theory

We report on a work in progress in building a variant of cubical type theory based on the following principles:

- equality is primitively attached to types;
- in particular, as in Altenkirch-Kaposi and Polonsky, equality of types is defined to be equivalence (so that univalence holds by construction); equality of proofs of equality of types is defined to be commutative squares of equivalences, etc.;
- type coercion (and thus transport) is direct by projecting the first component of a proof of equivalence;
- on top of type-based equality, a generic transverse level of dependent abstraction and application over a formal interval is provided (as in Section 9 of Cohen-Coquand-Huber-Mörtberg), together with connections and inverses, emphasizing that equality gives to types the structure of a symmetric (Cartesian) cubical structure with connections and reversals;
- a primitive notion of strictly associative composition of equality proofs is added which allows to interpret a function from the formal interval to the universe as a proof of equivalence;
- enforcing that transport computes on reflexivity proofs does not seem to pose problems;
- the theory is intended to be directly justified by an iterated parametricity translation to type theory with identity type, as explored e.g. by Altenkirch-Kaposi.

### Kuen-Bang Hou (Favonia): Cartesian cubical computational type theory

Homotopy type theory (HoTT) equips Martin-Löf type theory with additional features: univalence and higher inductive types. This has led to a fruitful development of synthetic homotopy theory. Unfortunately, those new features stated as axioms disrupt the computational content of type theory and affect the practicality of using HoTT to mechanize proofs. A well-known example is the fourth homotopy group of the 3-sphere, which was shown by Brunerie to be isomorphic to Z/nZ for some number n in HoTT, and the number n could have been automatically evaluated to the number two if the disruption did not happen. Instead, it took Brunerie several more years to prove that the number is indeed two. There have been many attempts to restore computational content while retaining all the additional features. To date, the most promising approach is through cubical structure. Recently, Cohen et al. constructed a type theory based on De Morgan cubes with univalence and higher inductive types. Inspired by their work, we also built a type-theoretic semantics with all the features but instead based on Cartesian cubes. In addition to using a different cubical structure, our semantics is built from computational content directly, following a computation-first methodology which itself is interesting. In this talk, I will compare two different cubical type theories and discuss how the computation-first methodology works.

(Joint work with Carlo Angiuli, Evan Cavallo, Robert Harper and Jonathan Sterling)

### Klaus Mainzer: Algorithms of Type and Proof Checking, Towards Univalent Foundations of Mathematics and Computer Science

Homotopy Type Theory (HoTT) is inspired by the idea of a proof-checking software to guarantee trust and security in mathematics. Since the Curry-Howard interpretation, basic intuitionistic type theory was extended by more advanced type formers (e.g., Martin-Löf). With higher inductive types in HoTT, we get direct, logical descriptions and constructions of advanced mathematical objects (e.g. in homotopy theory) as well as convenient machine implementations. These are not only theoretical insights connecting mathematics and computer science. They are deeply involved in great challenges of our digitalized world: Current AI-technology of machine learning is based on inductive principles of learning algorithms from data. But, although these are data-efficient methods, they need more explainability, verification, and provability of their foundations, in order to improve trust, reliability and security in society. HoTT should serve as a practical aid to working mathematicians as well as to computer scientists.

### Paul-André Melliès: Refinement type systems and Martin-Lof type theory

In this talk, I will review my recent work with Noam Zeilberger on refinement type systems [1,2]. I will then discuss how to equip refinement type systems with various notions of comprehension, in such a way as to connect them with Martin-Lof type theory.

[1] Paul-André Melliès and Noam Zeilberger. Functors are Type Refinement Systems.
[2] Paul-André Melliès and Noam Zeilberger. An Isbell Duality Theorem for Type Refinement Systems. Math. Struct. in Comp. Science (2018), vol. 28, pp. 736-774.
[3] Paul-André Melliès and Noam Zeilberger. A bifibrational reconstruction of Lawvere's presheaf hyperdoctrine. Proceedings of LICS 2016: 555-564

### Anders Mörtberg: Yet Another Cartesian Cubical Type Theory (yacctt)

I will discuss recent work on developing a Cartesian cubical type theory inspired by the computational semantics of Computational Higher Type Theory of Angiuli et. al. This system differs from cubical type theory (à la Coquand et. al.) in that it has an interval with less structure (no reversals or connections) and more general Kan composition operations. Another major difference is that there are no general Glue types, instead there is a special case called V-types which allows an equivalence to be transformed into a line between types. This specialization enables us to give a more direct proof that these "univalence types" are fibrant which in turn should lead to more efficient computations with univalence. If time permits I will sketch two proofs of the univalence theorem in this system.

This is joint work with Carlo Angiuli and we have implemented a primitive proof assistant based on this type theory: https://github.com/mortberg/yacctt.

### Fabio Pasquali: Assemblies as an elementary quotient completion

We characterize the category of assemblies as an elementary quotient completion, a construction introduced by Maietti and Rosolini to axiomatize the category of setoids used to build the Maietti-Sambin's Minimalist Foundation. As an application we link this characterization to the known fact that the effective topos, and related subcategories of it, satisfies Formal Church's Thesis. (This is a joint work with M.E. Maietti and G. Rosolini.)

### Pino Rosolini: Embedding the effective topos into cubical assemblies

The effective topos, introduced by J.M.E. Hyland in a paper in the Proceedings of the L.E.J.Brouwer Centenary Symposium, has proved to be a very useful environment where to test constructive aspects of mathematics, and many different presentations for the topos have been provided since its original appearance. In this line of work, we exhibit the effective topos as a subcategory of the cubical sets in the category of assemblies.
(Joint work with Steve Awodey and Jonas Frey)

### Christian Sattler: Do cubical models of type theory also model homotopy types?

I will give an alternative exposition of Kapulkin and Voevodsky's result about simplicial sets forming a subtopos of certain cubical sets: https://arxiv.org/abs/1805.04126. I will also give a brief summary of my current state of knowledge on which cubical models do or do not model homotopy types.

### Bas Spitters: Modal Dependent Type Theory, and its application to the cubical model

In recent years we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory, and spatial and cohesive type theory. In this paper we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K-axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any finite limit category with an adjunction of endofunctors gives rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal lambda-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.
[Ranald Clouston, Bassel Mannaa, Rasmus Ejlers Møgelberg, Andrew M. Pitts, Bas Spitters, Modal Dependent Type Theory and Dependent Right Adjoints. https://arxiv.org/abs/1804.05236]
I will also present the related work on the application of modal dependent type theory to the construction of a universe in cubical sets in the internal logic.
[Daniel R. Licata, Ian Orton, Andrew M. Pitts, Bas Spitters, Internal Universes in Models of Homotopy Type Theory FSCD18 https://arxiv.org/abs/1801.07664]

### Tutorial Daniel R. Licata and Felix Wellen: Synthetic Mathematics in Modal Dependent Type Theories

Several [[modal homotopy type theory|modal extensions]] of [[homotopy type theory]] have been or are being developed, with applications to [[synthetic mathematics|synthetic]] formalizations of aspects of topology, differential geometry, and spectra, as well as internal language presentations of cubical models of HoTT. In this tutorial, we will describe some recent work on these type theories, the frameworks we use to design them, and their applications in real-cohesive and differential-cohesive HoTT.

The preliminary lecture schedule is:

A Fibrational Framework for Modal Simple Type Theories
The Shape Modality in Real-cohesive HoTT and Covering Spaces
Discrete and Codiscrete Modalities in Cohesive HoTT, I
Discrete and Codiscrete Modalities in Cohesive HoTT, II
A Fibrational Framework for Modal Dependent Type Theories
Differential Cohesive HoTT
These talks will describe joint work with and work by Jacob Gross, Max S. New, Ian Orton, Jennifer Paykin, Andrew M. Pitts, Egbert Rijke, Mitchell Riley, Urs Schreiber, Michael Shulman, and Bas Spitters.

* {#LicataWellen18} [[Dan Licata]], [[Felix Wellen]], _Synthetic Mathematics in Modal Dependent Type Theories_, tutorial at *[[HoTT in Bonn2018|Types, Homotopy Theory and Verification]]*, 2018

  Tutorial 1, Dan Licata:  _A Fibrational Framework for Modal Simple Type Theories_
([recording](https://www.youtube.com/watch?v=uLLbSAYd3yI))

  Tutorial 2, Felix Wellen: _The Shape Modality in Real cohesive HoTT and Covering Spaces_ ([recording](https://www.youtube.com/watch?v=ACGjJDarEc4))

  Tutorial 3, Dan Licata: _Discrete and Codiscrete Modalities in Cohesive HoTT_ ([recording](https://www.youtube.com/watch?v=OHN9vEVPBN8))

  Tutorial 4, Felix Wellen, _Discrete and Codiscrete Modalities in Cohesive HoTT, II_ ([recording](https://www.youtube.com/watch?v=KXx9UDQdWPw))

  Tutorial 5, Dan Licata: _A Fibrational Framework for Modal Dependent Type Theories_ ([recording](https://www.youtube.com/watch?v=R4SsxUnIcng))

  Tutorial 6, Felix Wellen: _Differential Cohesive HoTT_, ([recording](https://www.youtube.com/watch?v=uEZXHPdwvJU&t=226s))

* {#Licata18} [[Dan Licata]], _Synthetic Mathematics in Modal Dependent Type Theories_, notes for tutorial at _[Types, Homotopy Theory and Verification](https://www.him.uni-bonn.de/programs/past-programs/past-trimester-programs/types-sets-constructions/workshop-types-homotopy-type-theory-and-verification/)_, 2018 ([pdf](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/him-tutorial.pdf))

## See also

* [[homotopy type theory events]]

category: reference

[[!redirects Bonn18]]


HoTT methods for homotopy theorists

{:bluebox: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

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

This page on aspects of [[homotopy type theory]] is meant for readers who are interested in [[homotopy theory]] but not (necessarily) in [[formal logic]] and [[formal proof]]. This page is meant to help answer the question:

> I am a homotopy theorist; what can homotopy type theory do for me?

See also at _[[homotopy type theory FAQ]]_ the section _[Why should I care? -- For homotopy theorists](homotopy+type+theory+FAQ#WhyShouldICareForHomotopyTheorists)_.

Since homotopy theory takes place in [[model categories]] and similar categorical structures, the input from homotopy type theory is via its [[categorical semantics]]. In this sense the question which this page is meant to help to answer is

> I am a homotopy theorist; which methods can I learn from the [[categorical semantics of homotopy type theory]]?

For the moment this page will mostly list pointers with brief comments to other entries where details are given. You should maybe read it like an instructional exercise list and follow the pointers for help.

#Contents#
* table of contents
{:toc}

## Basic dictionary for the internal language
 {#BasicDictionary}

Starting from ([[(infinity,1)-category theory|homotopy]]) [[category theory]],
the corresponding ([[homotopy type theory|homotopy]]) [[type theory]] is simply a formal language for denoting familiar constructions, as indicated in the following table.

|  ([[(infinity,1)-category theory|homotopy]]) [[category theory]] | ([[homotopy type theory|homotopy]]) [[type theory]] |
|----------------------|------------------|
|  [[semantics]]       | [[syntax]] |
| [[object]] $X$ | [[type]] $x : X$ |
| [[fibration]]([[display map]]) $\array{A \\ \downarrow^{\mathrlap{p}} \\ X}$  | [[dependent type]]  $x : X \vdash A(x) : Type$ |
| [[section]] $\array{ X &&\stackrel{t}{\to}&& A \\ & {}_{\mathllap{id}}\searrow && \swarrow_{p} \\ && X}$ | [[term]] $x : X \vdash t(x) : A(x)$ |
|  [[pullback]] $\array{ f^* A &\to& A \\ \downarrow && \downarrow^{\mathrlap{p}} \\ Y &\stackrel{f}{\to} & X }$ | [[substitution]] $y : Y \vdash A(f(y)) : Type$ |
| [[direct image]] $\array{ A && f_* A \\ {}^{\mathllap{p}}\downarrow && \downarrow ^{f_* p}\\ X &\stackrel{f}{\to}& Y}$ | [[dependent product]] $ y : Y \vdash \underset{x : X(y)}{\prod } A(x) : Type$ |
| [internal hom in slice](locally%20cartesian%20closed%20category#EquivalentCharacterizations) $\array{ X \times f^* A && f_* f^* A & = [X,A]_Y \\ {}^{\mathllap{}}\downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y}$ | [[function type]] $ y : Y \vdash X(y) \to A(y) : Type$ |
| postcomposition $\array{ A &=& f_! A \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y}$ | [[dependent sum]] $y : Y \vdash \underset{x : X(y)}{\sum} A(x) : Type$ |
| [[fiber product|fiber]][[product]] $\array{ X \times f^* A &=& f_! f^* A & = X \times_Y A\\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y}$ | [[product type]] $y : Y \vdash X(y) \times A(y) : Type$ |
| [[Beck-Chevalley condition]] of [[codomain fibration]] | [[substitution]] commutes with [[dependent sum]] |
| [[path space object]] $\array{A^I \\ \downarrow^{\mathrlap{(d_1,d_0)}} \\ A \times A}$ | [[identity type]] $a,b : A \vdash (a = b)$ |
| [[truncated object in an (infinity,1)-category|(-2)-truncated morphism]]/[[equivalence in an (infinity,1)-category|equivalence]] $\array{Y \\ \downarrow^{\mathrlap{\simeq}} \\ X}$ | [[true]]/[[unit type]] $x : X \vdash 1 : Type$ |
| [[truncated object in an (infinity,1)-category|(-1)-truncated morphism]]/[[monomorphism in an (infinity,1)-category|monomorphism]] $\array{\phi \\ \downarrow \\ X}$ | [[proposition]] $x : X \vdash \phi(x) : Type$ |
| [[direct image]] of [[truncated object in an (infinity,1)-category|(-1)-truncated morphism]] | [[universal quantifier]] $y : Y \vdash \underset{x \in X(y)}{\forall} \phi(x) : Type$ |
| [[truncated object of an (infinity,1)-category|(-1)-truncation]] of postcomposition of [[truncated object in an (infinity,1)-category|(-1)-truncated morphism]] | [[existential quantifier]] $y : Y \vdash \underset{x \in X(y)}{\exists} \phi(x) : Type$ |

The symbols in the right column may be formally manipulated according to the rules of _[[type theory]]_.
For the case of ordinary categories, this table defines a [[functor]]

$$
  Lang : LocallyCartesianClosedCategories \to DependentTypeTheories
$$

from _[[locally cartesian closed categories]]_ to _[[dependent type theories]]_ that sends each category to its _[[internal language]]_.

The important fact is that

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

The functor $Lang$ is an [[equivalence of categories]].

=--

This is discussed at _[[relation between type theory and category theory]]_. So the (dependent) type theory is just an equivalent way of talking about a (locally cartesian closed category).

For the case of [[(∞,1)-categories]]/[[homotopy theories]] that we are interested in here, there remain some things to be fully worked out, but it is clear that we get an analogous construction

$$
  Lang : LocallyCartesianClosed(\infty,1)Categories \to HomotopyTypeTheory
$$

from [[locally cartesian closed (∞,1)-categories]] to [[homotopy type theory]] which is expected to be an [[equivalence of (∞,1)-categories]].

As the above table shows, from the perspective of [[dependent type theory]] [[categories]] $\mathcal{C}$ are regarded systematically via the collection of their [[slice categories]] (their associated "[[hyperdoctrines]]"). If $\mathcal{C}$ is a [[locally cartesian closed category]] then every [[morphism]] $f : X \to Y$ in $\mathcal{C}$ induces an [[adjoint triple]] of functors between the corresponding slice categories

+-- {: bluebox}
######

([[dependent sum]] $\dashv$ [[base change]] $\dashv$ [[dependent product]]) = $(\sum_f \dashv f^* \dashv \prod_f) : \mathcal{C}_{/X} \to \mathcal{C}_{/Y}$.

=--

Many familiar constructions are usefully expressed entirely in terms of these [[adjoint triples]]. For instance the [[internal hom]] in a slice category.

While this is in principle clear/well known, the systematic use of the base change adjoint triple enforced by type theory turns out to lead to various elegant constructions that have not found much attention before, and which can be useful in applications.

## Homotopy pullbacks

The yoga of [[homotopy pullbacks]], [[homotopy fibers]], [[loop space objects]], [[fiber sequences]] etc. is basic to [[homotopy theory]], and of course is also fairly elementary. Homotopy type theory can hardly add a previously unknown fact here. Nevertheless, it is noteworthy that many of these constructions, elementary as they are, look _even simpler_ when formulated in homotopy type theory.

|  [[category theory]] | [[type theory]] |
|---|---|
| [[homotopy pullback]] $\array{A \times_C^h B &\to& B \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{g}} \\ A &\stackrel{f}{\to}& X}$ | $\underset{a : A, b : B}{\sum} (f(a) = g(b))$ /  $\{ a : A, b : B ; (f(a) = g(b)) \}$  |
| [[homotopy fiber]] $\array{hfib(f) &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{}} \\ A &\stackrel{f}{\to}& X}$ | $\sum_{a : A} (f(a) = *)$ / $\{ a : A ; (f(a) = *) \}$  |
| [[mapping cocylinder]] $\array{ CoCyl(f) &\to& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ Y^I &\stackrel{d_0}{\to} & Y \\ \downarrow^{\mathrlap{d_1}} \\ Y } $ | $y : Y \vdash \underset{x : X}{\sum} (f(x) = y)$ |
| [[free loop space object]] $\array{\mathcal{L}X &\to& X \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ X &\to& X \times X}$ | $\underset{x,y : X}{\sum} (x = y) and (x = y)$ |

* [[pasting law]] (...)

## Detecting $n$-truncation

The central insight (due to [[Vladimir Voevodsky]]) that boosts dependent type theory with identity types to genuine homotopy type theory is that in terms of [[identity types]] there are simple natural expressions for [[n-truncated|n-truncation]] and detection of $n$-truncation of objects and morphisms. Translated via [[categorical semantics]] to [[homotopy theory]], these formulas turn out to refomulate some basic yoga of model category computation in a new way that hasn't received attention before in homotopy theory, emphasizing the base change adjoint triple.

| [[category theory]] | [[type theory]] |
|----|----|
| object $X$ is [[(-2)-truncated]]/[[contractible]] | [[isContr]]$(X) = \sum_{x : X} \prod_{y : X} (x = y)$ |
| morphism $X$ is [[(-2)-truncated]]/[[equivalence in an (infinity,1)-category|equivalence]] | [[isEquiv]]$(f) \coloneqq \prod_{x : {X}} isContr(hfiber(f,x) $ |

+-- {: .num_remark }
###### Remark

In the [[(∞,1)-category]] [[∞Grpd]] it is true that a morphism $f : X \to Y$ is an equivalence precisely if for all global points $y : * \to Y$ the corresponding homotopy fiber of $f$ is contractible. The same simple statement is not available _in the "external" logic_ for a general [[(∞,1)-category]], simply because there an object $Y$ may not even have enough global points (it may be non-trivial and have no global point).

The above is useful in that it says that _in the [[internal logic]]_ of the $(\infty,1)$-category, the simple statement familiar form [[∞Grpd]] is true generally, after all.

=--

## Homotopy algebras over homotopy monads
 {#Induction}

We have seen that homtopy type theory naturally describes
[[homotopy pullbacks]] and some other finite [[(∞,1)-limits]] in terms of [[identity types]] and the [[base change]] [[adjoint triple]]. The [[duality|dual]] notion, _[[homotopy colimits]]_, is conceived of in homotopy type theory as a (vast) generalization of the basic notion of _[[induction]]_ and _[[recursion]]_, subsumed type-theoretically in the notion of _[[inductive type]]_, roughly a kind of [[type]] that is given by [[generators and relations]].

Traditionally inductive types are in category theory [[categorical semantics|interpreted]] as _[[algebras over an endofunctor]]_. While useful, this is a concept somewhat alien to usual constructions in category theory or homotopy theory. The natural notion is instead that of an [[algebra over a monad]]. While an [[algebra over an endofunctor]] may be thought of as a monad-algebra over a [[free monad]], from the point of view of homotopy theory it still seems unnatural to restrict attention to free monads.

However, after generalization to _[[homotopy type theory]]_ this is rectified: here the homotopy-theoretic notion of [[induction]] turns out to be about [[∞-algebras over an (∞,1)-monad]], as one would hope, and a _[[higher inductive type]]_ is an [[initial algebras of a presentable (∞,1)-monad]].

(...)

## Detecting $n$-connectedness

| [[category theory]] | [[type theory]] |
|--|--|
| [[n-connected object of an (infinity,1)-topos|(-1)-connected object]]/ [[inhabited object]] $X$ | $isInhab(X) \coloneqq ...$ |
| [[n-connected object of an (infinity,1)-topos|(-1)-connected morphism]]/ [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] $f : X \to Y$ | $\prod_{y : Y} isInhab(\sum_{x : X} (f(x) = y))$ |

## Homotopy pushouts

| [[category theory]] | [[type theory]] |
|---|---|
| [[homotopy pushout]] $\array{ C &\stackrel{g}{\to}& B \\ {}^{\mathllap{f}}\downarrow &\swArrow_{\simeq}& \downarrow \\ A &\to&  A \coprod_C^h B}$ | $hpushout (A\,B\,C : Type) (f : C \to A) (g : C \to B) : Type \coloneqq \left\{ \array{inl : B \to hpushout(f,g) \\ inr : A \to hpushout(f,g) \\ glue \prod_{c : C} (inl(f(c)) = inr(g(c)))} \right.$  |

(...)

## Constructing $n$-truncation

| [[category theory]] | [[type theory]] |
|---|---|
| [[truncated object in an (infinity,1)-category|(-1)-truncation]] $\tau_{-1}(-)$ | $supp(X : Type) : Type \coloneqq \left\{ \array{ proj : X \to supp X \\ \underset{x, y : supp X}{\prod} (x = y) }\right.  $ |

## Specific HoTT proofs in homotopy theory

We list in the following theorems in homotopy theory together with such proofs of them in terms of homotopy type theory language that are either new (to the best of our knowledge), or else that are at least considerably simpler than earlier proofs with traditional homotopy theory tools.

### Blakers-Massey theorem

see at _[[Blakers-Massey theorem]]_

### Essential uniqueness of group units

(...)

### Equivalence of notions of stabilizer $\infty$-groups

(...) [...](http://nforum.mathforge.org/discussion/3454/stabilizer-subgroup/?Focus=28287#Comment_28287)

(...)


HoTT Mini-Course 2016

[[!redirects HoTT Mini-Course]]
#Contents#
* table of contents
{:toc}

## HoTT Mini-Course at CMU ##
### Marc Bezem, Spring Semester 2016 ###

####Place and Time####
* Baker Hall 152
* Monday and Wednesday 13:30 - 16:30
* Exercises 13:30 - 15:00
* Lectures 15:00 -16:30
* We continue during the Spring break
* No lectures 23,28,30 March

####Resources####
* [The HoTT book](http://www.homotopytypetheory.org/book)
* Slides [[DTUA.pdf:file]]
* Slides [[HoTT.pdf:file]]

####Recordings and Homework####
* 2 March [video](https://www.youtube.com/watch?v=G4quix5Z51g)
  1. Prove that Eq_A on slide 2 of DTUA.pdf is an equivalence relation for all types A.
  1. Untyped $\lambda$-calculus: maximally reduce (+ 1 2) and (* 3 0).
  1. Maximally reduce (+ 0 x) and (+ x 0). Is there a difference?
* 7 March [video](https://www.youtube.com/watch?v=DY5DzmIfVDo)
  1. Let = be Eq_A (slide 2 of DTUA.pdf). Try to prove the groupoid laws on slide 9 of DTUA.pdf (not all are provable, which shows that Leibniz equality is too weak to give types groupoid structure).
  1. Same task as previous, but now with Eq_A from slide 8 of DTUA.pdf (now they are
all provable).
  1. Read the Introduction of the HoTT book.
* 9 March [video](https://www.youtube.com/watch?v=Xhu3VVyIdH8)
  1. Find the term _ac_ on slide 21 of HoTT.pdf.
  1. Do Exercise 1.1 and 1.2 of the HoTT book.
  1. Let _Bool_ be the inductive type with two nullary constructors _false: Bool_ and _true: Bool_. Formulate formation, introduction, elimination and computation rules for Bool.
  1. Find a term _t: U ->  U -> Bool -> U_ such that _t A B true = A_ and _t A B false = B_, for all _A,B: U_.
* 14 March [video](https://www.youtube.com/watch?v=Jzb8mXXz7xI)
  1. Do as many exercises from Chapter 1 of the HoTT book as you can. You can find some solutions in the [HoTT library](https://github.com/HoTT/HoTT/blob/master/contrib/HoTTBookExercises.v)
and by running `make exercise_solutions.pdf` after forking the [HoTT book](https://github.com/HoTT/book) on Github.
* 16 March (no video due to connectivity problems)
  1. As to Exercises 1.7 and 1.14, give the $\Sigma$-type $S$ and prove for all $a:A$ that $(a,refl_a) =_S (x,p)$ for all $x:A, p: a =_A x$.
  1. Do Exercises 2.1 - 2.4 in the HoTT book.
* 21 March [video](https://www.youtube.com/watch?v=Cd8rHVnSCwg)
  1. Do Exercises 2.5 - 2.10 in the HoTT book.
* 4 April
  1. Do as many of the remaining exercises from Chapter 2 of the HoTT book as you can.
* 6 April
  1. Investigate the action on paths of the function $f : \Sigma A B \to \Sigma A' B'$ given by $g: A\to A'$ and $h: \Pi x:A. B x \to B' (g x)$ in the following canonical way: $f(a,b) = (g a,h a b)$. (Hint: this is a dependent version of a simpler result for Cartesian products).
  2. Prove that for all $x,y : *$ we have $(x=_*y) \simeq \mathbf{1}$.

## See also

* [[CMU HoTT Research Group's local activities]]

category: people


HoTT2019 Summer School open problems list

Here are the open problems stated at the [HoTT 2019 Summer School](https://hott.github.io/HoTT-2019//summer-school/) (on [[homotopy type theory]]),
listed by the instructor who posed the problem:

#### Mathieu Anel: Logos Theory####

- Notion of higher theory? (syntactic approach? categorical approach à la Lawvere?)
	(e.g. CAT^lex, CAT_cc, LOGOS, …)

- Explicit distributivity formula between finite limits and colimit in a logos?
	colim_{something} lim_{something} —> lim_{something} colim_{something}
This should lead to a polynomial calculus for logoi (in the sense of polynomial functors)

- Explicit description of the symmetric logo functor? Sym: CAT_cc —> LOGOI

- Develop internal cat theory in a logos

- Define a class of logoi suited for the purpose of logic (and containing all presentable logoi)

####Egbert Rijke: Synthetic Homotopy Theory####

- deloop the 3-sphere: find a pointed connected type X s.th Omega(X) = S^3

- define the Grassmanians (& other interesting CW-complexes) using HITS.

- prove the BoTT periodicity theorem

####Jonas Frey: The Coherence Problem####

- define internal operads, semi-simplicial types, (oo,1)-categories in HoTT.

####Anders Mortberg: Cubical Type Theory####

- Efficient evaluation/ computation of cubical programs

- have a cubical TT where transport ref x == x.

####Guillaume Brunerie: Computation in Cubical Type Theory####

- find other interesting examples of cubical terms to compute (e.g. cohomology cup products)

####Kristina Sojakova:####

- conservativity of cubical TT over Book HoTT

- (suggested by Nicolai Kraus): Does adding a path between fixed points in a type preserve truncation level? Namely, if X is an n-type for n > 0 and a, b : X, is the HIT H generated by [-] : X —> H , p : [a] = [b] , still an n-type?

####Steve Awodey:####

- crossed modules classify 2-types

- define a notion of cubical quasi-category

- Cubical Homotopy Hypothesis:
 sSet (w/Kan QMS) ~ cSet (Dedekind cubes w/Sattler QMS)

Return to [[open problems in homotopy type theory]].


Houman Zolfaghari

* [ResearchGate page](https://www.researchgate.net/profile/Houman-Zolfaghari)

## Selected writings

An introduction to [[topos theory]] with emphasis on [[presheaf toposes]]:

* [[Marie La Palme Reyes]], [[Gonzalo E. Reyes]], [[Houman Zolfaghari]], _Generic figures and their glueings.  A constructive approach to functor categories_, Polimetrica (2008) [[pdf](https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf), ISBN:8876990046]

category: people


Houssam Abdul-Rahman

* [personal  page](https://wp.nyu.edu/houssama/)

* [Institute page at NYU Abu Dhabi](https://nyuad.nyu.edu/en/academics/divisions/science/faculty/houssam-abdul-rahman.html)

## Selected writings

On [[topological phases]] (i.e. [[mass gap|gapped]] [[ground states]]) of [[spin chains]]:

* [[Houssam Abdul-Rahman]], Marius Lemm, Angelo Lucia, [[Bruno Nachtergaele]], Amanda Young,  *A class of two-dimensional AKLT models with a gap*, in: *Analytic Trends in Mathematical Physics*,  Contemporary Mathematics **741**,  AMS (2020) [[arXiv:1901.09297](https://arxiv.org/abs/1901.09297), [doi:10.1090/conm/741/14917](https://doi.org/10.1090/conm/741/14917)]

category: people


How to get started

see *[[HowTo]]*


Howard Georgi

* [webpage](http://www.people.fas.harvard.edu/~hgeorgi/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Howard_Georgi)

## Selected writings

Introducing [[grand unified theory]]:

* {#GeorgiGlashow74} [[Howard Georgi]], [[Sheldon Glashow]], _Unity of All Elementary-Particle Forces_, Phys. Rev. Lett. **32** (1974) 438 ([doi:10.1103/PhysRevLett.32.438](https://doi.org/10.1103/PhysRevLett.32.438))

On [[supersymmetry breaking]] in [[grand unified theory]]:

* [[Savas Dimopoulos]], [[Howard Georgi]], _Softly broken supersymmetry and $SU(5)$_, Nuclear Physics B **193** 1 (1981) 21  (doi:10.1016/0550-3213(81)90522-8)

On [[hidden local symmetry]] in [[chiral perturbation theory]]/[[quantum hadrodynamics]]:

* [[Howard Georgi]], _Vector realization of chiral symmetry_, Nuclear Physics B **331** 2 (1990) 311-330 (doi:10.1016/0550-3213(90)90210-5)

On [[waves]] ([[wave equations]], [[Fourier analysis]], etc.):

* [[Howard Georgi]], *The Physics of Waves*, Prentice Hall (1993) $[$[web](https://www.people.fas.harvard.edu/~hgeorgi/new.htm), [pdf](https://www.people.fas.harvard.edu/~hgeorgi/onenew.pdf)$]$

Review of ([[Lie group|Lie]]-)[[group theory]] ([[Lie algebras]]) and [[representation theory]] with application to the ([[standard model of particle physics|standard model of]]) [[particle physics]] and [[grand unified theory]]:

* [[Howard Georgi]], *Lie Algebras In Particle Physics -- From Isospin to Unified Theories*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

## Related entries

* [[grand unified theory]]

* [[gauge coupling unification]]

* [[R-parity]]

category: people


Howard Marcum

* [webpage](https://math.osu.edu/people/marcum.1)

## Selected writings

On the [[Toda bracket]] understood via [[homotopy coherent diagram|homotopy-coherent]] [[pasting diagrams]] in the [[homotopy 2-category]] of [[pointed topological spaces]]:

* [[Keith Hardie]], [[Howard Marcum]], [[Nobuyuki Oda]], _Bracket operations in the homotopy theory of a 2-category_, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001) ([rendiconti:33/02](https://rendiconti.dmi.units.it/volumi/33/02.pdf))

* {#HardieKampsMarcum02} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Howard Marcum]], _The Toda bracket in the homotopy category of a track bicategory_, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)

* [[Howard Marcum]], [[Nobuyuki Oda]], _Long Box Bracket Operations in Homotopy Theory_, Appl Categor Struct 19, 137–173 ([doi:10.1007/s10485-009-9186-3](https://doi.org/10.1007/s10485-009-9186-3))

category: people