cubical set

\tableofcontents

\section{Definition}

\begin{notn} We make use of the notation of [[category of cubes]]. \end{notn}

\begin{defn} The _category of cubical sets_ is the [[free co-completion]] of $\square$, the category of cubes. \end{defn}

\begin{rmk} The free co-completion of a small category can be constructed as the category of [[presheaf | presheaves]] of sets on this category. Thus we can also think of the category of cubical sets as the category of presheaves of sets on $\square$. \end{rmk}

\begin{notn} We denote the category of cubical sets by $\mathsf{Set}^{\square^{op}}$. \end{notn}

\begin{defn} A _cubical set_ is an object of $\mathsf{Set}^{\square^{op}}$. \end{defn}

\begin{rmk} When we think of the category of cubical sets as the category of presheaves of sets on $\square$, we consequently think of a cubical set as a presheaf of sets on $\square$. \end{rmk}

\begin{defn} A _morphism_ of cubical sets is an arrow of $\mathsf{Set}^{\square^{op}}$. \end{defn}

## Monoidal structure

The strict monoidal structure of $\square$ gives rise to a (non-strict) monoidal structure on $\mathsf{Set}^{\square^{op}}$, by [[Day convolution]]. The unit of the monoidal structure is $\square^{0}$, in the notation of Notation \ref{NotationFreeStandingNCube}. Whenever we use the symbol $\otimes$ when working with cubical sets or morphisms of cubical sets, we shall always refer to the functor $- \otimes -$ of this monoidal structure.

## Notation

### Free standing $n$-cube, and an $n$-cube of a cubical set

+-- {: .num_defn #NotationFreeStandingNCube}
###### Notation

Let $y : \square \rightarrow \mathsf{Set}^{\square^{op}}$ denote the [[Yoneda embedding]] functor. Let $n \geq 0$ be an integer. We denote the cubical set $y(I^{n})$ by $\square^{n}$.

=--

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

We refer to $\square^{n}$ as the _free-standing $n$-cube_.

=--

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

Let $X$ be a cubical set. Let $n \geq 0$ be an integer. By an _$n$-cube_ of $X$, we shall mean a morphism of cubical sets $\square^{n} \rightarrow X$.

=--

+-- {: .num_defn #NotationOneCubePictorial}
###### Notation

Let $f$ be a 1-cube of $X$. We shall often depict $f$ as $f : x_{0} \rightarrow x_{1}$ or as follows.

$$
   \array{x_{0} & \overset{f}{\rightarrow} & x_{1}}
$$

In this case, $x_{0}$ is to be understood to be the $0$-cube $f \circ y(i_{0})$ of $X$, and $x_{1}$ is to be understood to be the $1$-cube $f \circ y(i_{1})$ of $X$.

=--

+-- {: .num_defn #NotationTwoCubePictorial}
###### Notation

Let $\sigma$ be a 2-cube of $X$. We shall often depict $\sigma$ as follows.

$$
   \array{
      x_{0}            & \overset{f_{0}}{\rightarrow}  & x_{1} \\
      f_{2} \downarrow & \sigma                & \downarrow f_{1} \\
      x_{2}            & \underset{f_{3}}{\rightarrow} & x_{3}
   }
$$

In this case, $f_{0}$ is to be understood to be the $1$-cube $\sigma \circ y(i_{0} \otimes I^{1})$ of $X$, $f_{1}$ is to be understood to be the $1$-cube $\sigma \circ y(I^{1} \otimes i_{1})$ of $X$, $f_{2}$ is to be understood to be the $1$-cube $\sigma \circ y(I^{1} \otimes i_{0})$ of $X$, and $f_{3}$ is to be understood to be the 1-cube $\sigma \circ y(i_{1} \otimes I^{1})$ of $X$.

It can be checked that this notation is consistent with Notation \ref{NotationOneCubePictorial}.

=--

### Boundary of the free standing $n$-cube, and of an $n$-cube of a cubical set

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

Let $n \geq 1$ be an integer. We denote by $\partial : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square^{op}}$ the functor given by defined by $sk_{n-1} \circ tr_{n-1}$, where $tr_{n-1}$ is the $(n-1)$-[[cubical truncation, skeleton, and co-skeleton | truncation]] functor for cubical sets, and $sk_{n-1}$ is the $(n-1)$-[[cubical truncation, skeleton, and co-skeleton | skeleton]] functor for cubical sets.

=--

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

Let $n \geq 0$ be an integer. We refer to $\partial \square^{n}$ as the _boundary_ of $\square^{n}$.

=--

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

We also denote by $\partial \square^{0}$ (recalling that $\mathsf{Set}^{\square^{op}}$ is, by construction, co-complete) the initial object of $\mathsf{Set}^{\square^{op}}$.

=--

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

Let $n \geq 1$ be an integer. We denote by $i_{n} : \partial \square^{n} \rightarrow \square^{n}$ the morphism of cubical sets corresponding, under the adjunction between $sk_{n-1}$ and $tr_{n-1}$ described at [[cubical truncation, skeleton, and co-skeleton]], to the identity arrow $tr_{n}(\square^{n}) \rightarrow tr_{n}(\square^{n})$ in $\mathsf{Set}^{\square_{n-1}^{op}}$.

=--

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

Let $n \geq 0$ be an integer, and let $\sigma : \square^{n} \rightarrow X$ be an $n$-cube of a cubical set $X$. We refer to the morphism of cubical sets

$$
   \array{\partial \square^{n} & \overset{i_{n}}{\rightarrow} & \square^{n} & \overset{\sigma}{\rightarrow} & X }
$$

as the _boundary_ of $\sigma$.

=--

+-- {: .num_defn #NotationTwoCubeBoundaryPictorial}
###### Notation

Let $\sigma$ be a 2-cube of $X$ as follows.

$$
   \array{
      x_{0}            & \overset{f_{0}}{\to}  & x_{1} \\
      f_{2} \downarrow & \sigma                & \downarrow f_{1} \\
      x_{2}            & \underset{f_{3}}{\to} & x_{3}
   }
$$

We shall often depict the boundary of $\sigma$ as follows.

$$
   \array{
      x_{0}            & \overset{f_{0}}{\to}  & x_{1} \\
      f_{2} \downarrow &                     & \downarrow f_{1} \\
      x_{2}            & \underset{f_{3}}{\to} & x_{3}
   }
$$

=--

### Horns of the free-standing $n$-cube

+-- {: .num_defn #NotationHorn}
###### Notation

We define inductively, for any integer $n \geq 1$, any integer $1 \leq i \leq n$, and any integer $0 \leq \epsilon \leq 1$, a cubical set $\sqcap^{n,i,\epsilon}$ and a morphism of cubical sets $i_{i, \epsilon} : \sqcap^{n,i,\epsilon} \rightarrow \square^{n}$.

When $n=1$, we define both $\sqcap^{1,1,0}$ and $\sqcap^{1,1,1}$ to be $\square^{0}$. We define $i_{1,0} : \square^{0} \rightarrow \square^{1}$ to be $y(i_{0})$, and define $i_{1,1} : \square^{0} \rightarrow \square^{1}$ to be $y(i_{1})$.

Suppose that, for some integer $n \geq 1$, we have defined $\sqcap^{n,i,\epsilon}$ and a morphism of cubical sets $i_{i, \epsilon} : \sqcap^{n,i,\epsilon} \rightarrow \square^{n}$ for all integers $1 \leq i \leq n$, and all integers $0 \leq \epsilon \leq 1$. For $1 \leq i \leq n$, we define (recalling that $\mathsf{Set}^{\square^{op}}$ is co-complete by construction) $\sqcap^{n+1,i, \epsilon}$ to be a cubical set fitting into a [[pushout | co-cartesian square]] in $\mathsf{Set}^{\square^{op}}$ as follows.

$$
   \array{
      \sqcap^{n,i,\epsilon} \sqcup \sqcap^{n,i,\epsilon} & \overset{\big( \sqcap^{n,i,\epsilon} \otimes y(i_{0}) \big) \sqcup \big( \sqcap^{n,i,\epsilon} \otimes y(i_{1}) \big)}{\rightarrow}  & \sqcap^{n,i,\epsilon} \otimes \square^{1} \\
       \mathllap{i_{i,\epsilon} \sqcup i_{i,\epsilon}} \downarrow    &                                    & \downarrow \mathrlap{r_{0}} \\
      \square^{n} \sqcup \square^{n}  & \underset{r_{1 }}{\rightarrow} & \sqcap^{n+1,i,\epsilon}
   }
$$

We denote by $i_{i,\epsilon} : \sqcap^{n+1,i,\epsilon} \rightarrow \square^{n+1}$ the canonical arrow determined, by means of the universal property of $\sqcap^{n+1,i,\epsilon}$, by the following commutative square in $\mathsf{Set}^{\square^{op}}$.

$$
   \array{
      \sqcap^{n,i,\epsilon} \sqcup \sqcap^{n,i,\epsilon} & \overset{\big( \sqcap^{n,i,\epsilon} \otimes y(i_{0}) \big) \sqcup \big( \sqcap^{n,i,\epsilon} \otimes y(i_{1}) \big)}{\rightarrow}  & \sqcap^{n,i,\epsilon} \otimes \square^{1} \\
       \mathllap{i_{i,\epsilon} \sqcup i_{i,\epsilon}} \downarrow    &                                    & \downarrow \mathrlap{i_{i,\epsilon} \otimes \square^{1}} \\
      \square^{n} \sqcup \square^{n}  & \underset{\big( \square^{n} \otimes y(i_{0}) \big) \sqcup \big( \square^{n} \otimes y(i_{1}) \big)}{\rightarrow} & \square^{n+1}
   }
$$

We define $\sqcap^{n+1, n+1, \epsilon}$ to be a cubical set fitting into a [[pushout | co-cartesian square]] in $\mathsf{Set}^{\square^{op}}$ as follows.

$$
   \array{
      \sqcap^{n,n,\epsilon} \sqcup \sqcap^{n,n,\epsilon} & \overset{\big( y(i_{0}) \otimes \sqcap^{n,n,\epsilon} \big) \sqcup \big( y(i_{1}) \otimes \sqcap^{n,n,\epsilon} \big)}{\rightarrow}  & \square^{1} \otimes \sqcap^{n,n,\epsilon} \\
       \mathllap{i_{n,\epsilon} \sqcup i_{n,\epsilon}} \downarrow    &                                    & \downarrow \mathrlap{r_{0}} \\
      \square^{n} \sqcup \square^{n}  & \underset{r_{1}}{\rightarrow} & \sqcap^{n+1,n+1,\epsilon}
   }
$$

We denote by $i_{n+1,\epsilon} : \sqcap^{n+1,n+1,\epsilon} \rightarrow \square^{n+1}$ the canonical arrow determined, by means of the universal property of $\sqcap^{n+1,n+1,\epsilon}$, by the following commutative square in $\mathsf{Set}^{\square^{op}}$.

$$
   \array{
      \sqcap^{n,n,\epsilon} \sqcup \sqcap^{n,n,\epsilon} & \overset{\big( y(i_{0}) \otimes \sqcap^{n,n,\epsilon} \big) \sqcup \big( y(i_{1}) \otimes \sqcap^{n,n,\epsilon} \big)}{\rightarrow}  & \square^{1} \otimes \sqcap^{n,n,\epsilon} \\
       \mathllap{i_{n,\epsilon} \sqcup i_{n,\epsilon}} \downarrow    &                                    & \downarrow \mathrlap{\square^{1} \otimes i_{n,\epsilon}} \\
      \square^{n} \sqcup \square^{n}  & \underset{\big( y(i_{0}) \otimes \square^{n} \big) \sqcup \big( y(i_{1}) \otimes \square^{n} \big)}{\rightarrow} & \square^{n+1}
   }
$$

=--

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

We refer to $\sqcap^{n, i, \epsilon}$ together with the morphism $i_{i,\epsilon}$ as a _horn_ of $\square^{n}$.

=--

### Morphism from $\square^{n}$ to $\square^{0}$

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

We denote by $p : \square^{n} \rightarrow \square^{0}$ the arrow $y(\underbrace{p \otimes p \otimes \cdots p}_{n})$ of $\mathsf{Set}^{\square^{op}}$, making use of the fact that $\underbrace{\square^{0} \otimes \square^{0} \otimes \cdots \square^{0}}_{n}$ is $\square^{0}$, since $\square^{0}$ is the unit of the monoidal structure of $\mathsf{Set}^{\square^{op}}$.

=--

## Model structure

The category of cubical sets admits a [[Cisinski model structure]], which admits a [[Quillen equivalence]] to the [[Kan–Quillen model structure]] on [[simplicial sets]].
See the article [[model structure on cubical sets]] for more information.

The category of cubical sets also admits a Joyal-type model structure, which admits a [[Quillen equivalence]] to the [[Joyal model structure]] on [[simplicial sets]].
See the article [[model structure for cubical quasicategories]] for more information.

## Expository material

For Expository and other material, see [[cubical set - exposition]].

## Related entries

### General

* [[bicubical sets]]

### Applications of cubical sets

* [[cubical type theory]]

#### In higher category theory

* [[fundamental groupoid of a cubical set and the cubical nerve of a groupoid]]

* [[homotopy hypothesis for 1-types]]

### Theory of cubical sets

* [[category of cubes]]

* [[cubical geometric realisation]]

* [[cubulation]]

* [[cubical truncation, skeleton, and co-skeleton]]

* [[homotopy groups of a cubical Kan complex]]

* [[model structure on cubical sets]]

## References

The original reference for cubical sets (based on the 1950 paper by [[Samuel Eilenberg]] and [[J. A. Zilber]] on [[simplicial sets]]) is

* [[Daniel M. Kan]], _Abstract homotopy.  I_, Proceedings of the National Academy of Sciences 41:12 (1955), 1092–1096.  [doi](http://dx.doi.org/10.1073/pnas.41.12.1092).

Kan switched to [[simplicial sets]] in Part III of the series.

[[!redirects Cubical set]]
[[!redirects Cubical Set]]
[[!redirects Cubical set]]
[[!redirects cubical set]]
[[!redirects category of cubical sets]]


cubical set - exposition

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

#Contents#
* table of contents
{:toc}

## Idea

The definition is to be understood from the point of view of [[space and quantity]]: a _cubical set_ is a [[space]] characterized by the fact that, and the ways in which, it may be _probed_ by mapping standard cellular [[cubes]] into it: the set $S_n$ assigned by a cubical set to the standard $n$-[[cube]] $[n]$ is the set of $n$-[[cubes]] in this space, hence the way of mapping a standard $n$-[[cube]] into this spaces.

Being a [[functor]] $S : \Box^{op} \to Set$, a cubical set $S$ also assigns maps between its sets $S_n$ of $n$-cubes which determine in which way smaller cubes sit inside larger cubes.

The **face maps**  go from sets $S_{n+1}$ of $(n+1)$-dimensional cubes to the corresponding set $S_{n}$ of $n$-dimensional cubes and can be thought of as sending each cube in the cubical set to one of its faces, for instance for $n=1$ the set $S_2$ of 2-cubes  would be sent in four different ways by four different face maps to the set of $1$-cubes, for instance one of the face maps would send

$$
  \left(
  \array{
     a &\to& b
     \\
     \downarrow &\Downarrow^F& \downarrow
     \\
     c &\to& d
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \array{
     a &\to& b
  }
  \right)
$$

another one would send

$$
  \left(
  \array{
     a &\to& b
     \\
     \downarrow &\Downarrow^F& \downarrow
     \\
     c &\to& d
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \array{
     a
     \\
     \downarrow
     \\
     c
  }
  \right)
  \,.
$$

On the other hand, the **degeneracy maps** go the other way round and send sets $S_n$ of $n$-cubes to sets $S_{n+1}$ of $(n+1)$-cubes by regarding an $n$-cube as a degenerate or "thin" $(n+1)$-cube in the various different ways that this is possible. For instance again for $n=1$ a degeneracy map may act by sending

$$
  \left(
  \array{
     a
     &\stackrel{f}{\to}&
     b
  }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
  \array{
     a &\stackrel{f}{\to}& b
     \\
     \downarrow^{Id} &\Downarrow^{Id}& \downarrow^{Id}
     \\
     a &\stackrel{f}{\to}& b
  }
  \right)
  \,.
$$

Notice the $Id$-labels, which indicate that the edges and faces labeled by them are "thin" in much the same way as an [[identity morphism]] is thin (notice however that a cubical set by itself is not equipped with a notion of composition of cubes. If it were, we'd call it a [[cubical ∞-category]]).

In an ordinary cubical set all degeneracy maps act in the kind of way depicted above. One might also want to require a cubical set to contain "thin" cells between equal _adjacent_ faces. These **extra degeneracy maps** act by sending 1-cells to degenerate 2-cells of the form

$$\left(\array{
a&\stackrel{f}{\to}&b
}\right)
\;\;
\mapsto
\;\;
\left(\array{
a & \stackrel{f}{\to} & b \\
\downarrow^{f} & \Downarrow & \downarrow^{Id} \\
b & \stackrel{Id}{\to} & b
}\right)
\,.
$$

If the cubical set has this additional property, one calls it a [[connection on a cubical set|cubical set with connection]].

## Cubical sets in homotopy theory
 {#InHomotopyTheory}

[[Daniel Kan]]'s early work on [[homotopy theory]] used cubical sets instead of [[simplicial set]]s. But then a bit later it was found that plain cubical sets suffer from three disadvantages when it comes to modelling [[homotopy type|homotopy n-type]]s:

1. [[Moore complex|normalization of chains]] is essential

1. [[cubical group]]s are not automatically fibrant

1. the [[geometric realization]] of [[cartesian product]]s of cubical sets (see [geometric realization](#GeometricRealization) below) tends to have the wrong [[homotopy n-type|homotopy type]]:

   for instance the geometric realization of the cubical set $I \times I$ has non-trivial [[homotopy group]]s and hence does not model the [[topological space]] given by the standard square, which is [[contractible]].

It was then realized that none of these problems is shared by [[simplicial sets]]:

1. Eilenberg and Mac Lane proved a normalisation theorem which may be found in Mac Lane's book 'Homology'.

1. every [[simplicial group]] is necessarily a [[Kan complex]] and therefore fibrant in the standard [[model structure on simplicial sets]]

1. [[geometric realization]] of [[simplicial set]]s is well behaved and in fact constitutes a [[cartesian monoidal category|cartesian monoidal]] [[Quillen equivalence]]

   $$
      |-| : SSet \stackrel{\leftarrow}{\to} Top : S(-)
   $$

   between [[SSet]] and [[Top]]. For more on this see [[homotopy hypothesis]].

These observations led to a widespread use of simplicial methods, while cubical methods coexisted only with a kind of underground existence, to this date.

There is, however, a proof of parts of the [[homotopy hypothesis]] also for cubical sets: their [[homotopy category]] is equivalent, after all, to the standard homotopy catgeory [[Top]]. This is described at [[model structure on cubical sets]].

Also, it turns out that the second and third of the above disadvantages of [[cubical set]]s over [[simplicial sets]] in [[homotopy theory]] can be dealt with to some extent.

1. The first problem can't be avoided. See

   Rosa Antolini and Bert Wiest, _The singular cubical set of a topological space_ , Mathematical Proceedings of the Cambridge Philosophical Society, 126 (1),  (1999)

1. The second problem is resolved, at least when restricting to [[strict omega-groupoid]]s by using [[connection on a cubical set|cubical sets with connection]]. See

   A. P. Tonks, _Cubical groups which are Kan_, J. Pure Appl. Algebra, 81 (1),  (1992)

1. the third problem is due to the fact that the [[cube category]] is a [[test category]] but not a [[strict test category]]. However, the category of cubes _[[connection on a cubical set|with connection]]_ is a strict test category, as shown by [[Georges Maltsiniotis]], based on work by [[Denis-Charles Cisinski]]. See _[[connection on a cubical set]]_ for details.

### Geometric realization
 {#GeometricRealization}

If $X$ is a cubical set, the [[geometric realization]] $|X|$ may be defined as the [[weighted colimit]] (a [[coend]]) in [[Top]]

$$X \otimes_{\Box} I^{\bullet} = \int^{n \in \Box} X(n) \cdot I^n$$

where $I^{\bullet}: (\Box, \otimes, I) \to (\Top, \times, 1)$ is the unique (up to isomorphism) monoidal functor mapping the generating object $int$ to $[0, 1]$, and for $k = 0, 1$, mapping $i_k$ to the inclusion $\{k\} \hookrightarrow [0, 1]$. This is parallel to one way of defining the [[geometric realization]] of a [[simplicial set]].
Geometric realization is a functor [[adjoint functor|left adjoint]] to the functor
$$cub: Top \to Set^{\Box^{op}}$$
which takes a space $S$ to the functor $\hom_{Top}(I^{\bullet}-, S)$.

### Subdivision and fibrant replacement

A cubical subdivision functor $sd$ is discussed in [Jardine 0, Section 5](#Jardine02). It is an obvious subdivision of an $n$-cube, which is just a product of barycentric subdivisions of intervals. The (functorial) subdivision $sd X$ of a cubical set $X$ is constructed from this naive subdivision of the $n$-cube in the end.
See [Jardine's lecture notes](#Jardine02) for details. There is a natural sequence of maps of cubical sets

$$
\cdots  \to sd^n X \to \cdots  \to sd X  \to X
$$

defined similar to its [[Kan fibrant replacement|simplicial counterparts]]. Let its [[right adjoint]] be
denoted as usual by $Ex X$. So $n$-cubes of $Ex X$ are cubical maps from subdivision of the $n$-cube to $X$ (similar to the definition of [[Kan fibrant replacement|simplicial
Ex-functor]]). We therefore get maps

$$
 X  \to Ex X \to Ex^2 X  \to \cdots
$$

Let $Ex^\infty X$ be the union of the latter maps (similar to simplicial $Ex^\infty$).

**question**: Is $Ex^\infty X$  a fibrant cubical set for any cubical set $X$?.

Recall that a cubical set is fibrant if any
cubical horn has a filler (similar to [[simplicial set]]: any [[Kan complex|Kan fibrant]] simplicial set has [[horn]] fillers). See also ([Cisinski 2006](#Cisinski06)) or [Jardine's lectures on cubical sets](#JardineLecture) for definitions.

The first question is probably not true in general, but if we consider cubical sets with [[connection on a cubical set|connections]] in the sense of Brown-Higgins (we
add some degeneracy maps to cubical sets), see e.g. [Maltsiniotis paper](http://people.math.jussieu.fr/~maltsin/ps/cubique.ps) then the cubical subdivision remains the same and the $Ex^\infty X$ is
defined similarly. The question is whether $Ex^\infty X$ with $X$ a cubical set with connections is fibrant. **Is it true?**

### Model category structure and homotopy theory
 {#ModelCategoryStructureAndHomotopyTheory}

There is a [[model structure on cubical sets]] with the same [[homotopy theory]] as the standard [[model structure on simplicial sets]] ([Jardine 02](#Jardine02)), which models [[homotopy types]]/[[infinity-groupoids]].

In fact ([Jardine 02, theorem 29, theorem 30](#Jardine02)) gives an [[adjunction]] between the [[model categories]] which, while not quite a [[Quillen adjunction]], does have [[unit of an adjunction|unit]] and [[counit of an adjunction|counit]] being [[weak equivalences]]. Hence by the discussion at _[[adjoint (∞,1)-functor]]_  it should indeed follow that the [[derived functors]] of the adjunction exhibit the [[simplicial localizations]] of cubical sets equivalent to that of simplicial sets, hence make their [[(∞,1)-categories]] [[equivalence of (∞,1)-categories|equivalent]] (hence equivalent to [[∞Grpd]]).

This generalizes to a local [[model structure on cubical presheaves]] over any [[site]], which has at least the same [[homotopy category]] as the corresponding local [[model structure on simplicial presheaves]].

## Background

The notion of cubical set and of homology theory and homotopy theory based on singular cubes goes way back in the literature. Serre's work on spectral sequences and fibre spaces was based on cubes. Kan's early work on combinatorial homotopy was based on cubes. However the category of cubical sets was found to have a major disadvantage compared with simplicial sets, in that the [[cartesian product]] in this category failed to have the correct [[homotopy type]]. This is in striking contrast to the cartesian product on [[simplicial set]]s.

Nonetheless cubical sets continued to have a kind of underground existence.

Brown and Higgins introduced the extra structure of [[connection on a cubical set|connections]] $\Gamma^\pm_i$ on cubical sets, and included this structure into their cubical  [[strict ∞-groupoids]]. All this structure was essential for the equivalence with [[crossed complexes]] and for the applications to homotopy theory. For example these $\omega$-groupoids have a canonical structure of [[thin elements]], defined as any composition of elements of the form $\pm \epsilon_j x, \pm \Gamma^\pm_i$. Such elements have "commuting boundary".

The geometric realisation of cubical sets with connections, and the relation with cartesian products, has been analysed by Maltsiniotis in the paper referred to below.

Nonetheless, the advantages of cubes are:

1. Easy notions of multiple compositions (compared with the globular pasting schemes); we refer to [[compositions in cubical sets]].

1. Good notions of tensor product, because of the rule $I^m \times I^n \cong I^{m+n}$, and hence easy conceptual handling of homotopies.  This is exploited in the paper by Brown and Higgins on tensor products,  and also in the following paper

* Al-Agl, F. A., [[Ronnie Brown]],  and Steiner, R. Multiple categories: the equivalence of a globular and a   cubical approach. _Adv. Math._ 170~(1) (2002) 71--118.

which gives a monoidal closed structure on cubical $\omega$-categories with connections, allowing the transfer of this to globular $\omega$-categories. The tensor product here generalises the [[Gray  tensor product]] of 2-categories. This is also convenient in the [[homotopical structure on C*-algebras]].

Cubical methods are a key feature in using higher homotopy groupoids to prove homotopy classification results.

## Related concepts

* [[cubical object]]

* [[model structure on cubical presheaves]]

* [[simplicial set]], [[globular set]], [[cellular set]], [[dendroidal set]]

* [[simplicial object]]

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

* [[semi-simplicial object]]

  * [[semisimplicial set]]

## References

Early references include

* [[Jean-Pierre Serre]], _Homologie singulière des espaces fibrés_ , Ann.
Math. **54** no.3 (1951), pp.425-505. ([pdf](http://www.college-de-france.fr/media/jean-pierre-serre/UPL7235285843586540944_Serre_The_se.pdf))

* [[Dan Kan]], _Abstract homotopy I_ , Proc. Nat. Acad. Sci. U.S.A. **41** (1955) pp.1092–1096. ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528202/pdf/pnas00727-0082.pdf))
 {#Kan}

General introductions of the cube category and of cubical sets are in

* {#JardineLecture} [[Rick Jardine]], _Cubical sets_, Lecture 12 in [Fields Lectures on simplicial presheaves](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf) (unpublished).

* [[Sjoerd Crans]], section 2 of _Pasting schemes for the monoidal biclosed structure on $\omega$-$\mathbf{Cat}$_ ([web](http://crans.fol.nl/papers/thten.html), [ps](http://crans.fol.nl/papers/thten.ps.gz), [[thten.pdf|pdf:file]])

The **cubical identities** satisfied by a cubical set are given there in proposition 2.8 on p. 9.

Cubical [[singular homology]] is discussed in

* Massey, W. S.,   _Singular homology theory_, _Graduate Texts in
  Mathematics_, Volume~70. Springer-Verlag, New York (1980).

An axiomatization of cubical sets in [[constructive set theory]]/[[type theory]] (with the aim of building models of [[homotopy type theory]]) is in

* {#BezemCoquandHuber13} Marc Bezem, [[Thierry Coquand]], [[Simon Huber]], _A model of type theory in cubical sets_, 2013  ([web](http://drops.dagstuhl.de/opus/volltexte/2014/4628/), [pdf](http://drops.dagstuhl.de/opus/volltexte/2014/4628/pdf/7.pdf))

* {#KaposiAltenkirch14} Ambrus Kaposi, [[Thorsten Altenkirch]], _A syntax for cubical type theory_ ([pdf](http://mazzo.li/dump/aim-kaposi-pres.pdf))

* {#Docherty14} Simon Docherty, _A model of type theory in cubical sets with connection_, 2014 ([pdf](http://www.illc.uva.nl/Research/Publications/Reports/MoL-2014-12.text.pdf))

See also

* [[Dan Licata]], [[Guillaume Brunerie]], _A cubical approach to synthetic homotopy theory_ ([pdf](http://dlicata.web.wesleyan.edu/pubs/lb15cubicalsynth/lb15cubicalsynth.pdf))

For more on this see at _[[relation between category theory and type theory]]_.

The [[homotopy theory]] / [[model category]] structure on cubical sets is discussed in

* {#Jardine02} [[Rick Jardine]], _Cubical homotopy theory: a beginning_ (2002) ([pdf](http://hopf.math.purdue.edu/Jardine/cubical2.pdf))

The fact that the [[exponential object]] of two fibrant cubical sets is again fibrant follows from remark 8.4.33 in

* {#Cisinski06} [[Denis-Charles Cisinski]], _[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]_, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages ([pdf](http://www.mathematik.uni-regensburg.de/cisinski/ast.pdf))

in the context of [[Cisinski model structures]].

Finally cubical sets as [[categorical semantics]] for [[homotopy type theory]] with [[univalence]] is discussed in

* {#Cisinski14} [[Denis-Charles Cisinski]], _Univalent universes for elegant models of homotopy types_ ([arXiv:1406.0058](http://arxiv.org/abs/1406.0058))

The [[strict test category]] nature of cubical sets with connection is discussed in

* [[Georges Maltsiniotis]], _La catégorie cubique avec connexions est une catégorie test stricte_. Homology, Homotopy Appl. 11~(2) (2009) 309--326.

There is also the old work

* Victor Gugenheim, _On supercomplexes_ Trans. Amer. Math. Soc. 85 (1957), 35–51 [PDF](https://www.ams.org/journals/tran/1957-085-01/S0002-9947-1957-0086299-1/S0002-9947-1957-0086299-1.pdf)

in which "supercomplexes" are discussed, that combine simplicial sets and cubical sets (def 5). There are functors from simplicial sets to supercomplexes (after Defn 5) and, implicitly, from supercomplexes to cubical sets (in Appendix II). This was written in 1956, long before people were thinking as formally as nowadays and long before Quillen model theory, but a comparison of the homotopy categories might be in there.

A discussion of cubical sets and normal forms in several cases is in

* [[Marco Grandis]], and Mauri, L. Cubical sets and their site, _Theory Applic. Categories_ {11} (2003) 185--201.

Cubical sets as models for [[strict ∞-groupoids]] are discussed in

* [[Ronnie Brown]], P. Higgins, _The equivalence of $\omega$-groupoids and cubical $T$-complexes_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 22 no. 4 (1981), p. 349-370  ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1981__22_4/CTGDC_1981__22_4_349_0/CTGDC_1981__22_4_349_0.pdf)).

Their use for monoidal closed structures  and homotopy classification is given in

* [[Ronnie Brown]] and P. Higgins,  Tensor products and homotopies for $\omega$-groupoids and crossed complexes.  _J. Pure Appl. Algebra_ 47 (1987) 1--33.

and are essential in

* [[Ronnie Brown]], Philip J. Higgins and Rafael Sivera, _[[Nonabelian Algebraic Topology]]_ EMS Tracts in Math vol 15, 703 pages, August, 2011. ([web](http://www.bangor.ac.uk/~mas010/nonab-a-t.html)).

Essential subtoposes of $Set^{\Box^{op}}$ are discussed in the context of Lawvere's dialectical theory of dimension in

* {#KRRZ10} C. Kennett, [[Emily Riehl|E. Riehl]], M. Roy, M. Zaks, _Levels in the toposes of simplicial sets and cubical sets_ , JPAA **215** no.5 (2011) pp.949-961. ([preprint](http://arxiv.org/abs/1003.5944))

[[!redirects cubical sets]]


cubical structure

In the sense of higher order analogs of [[quadratic refinements]]:

* [[cubical structure on a line bundle]]

* [[cubical structure in M-theory]]

In the sense of geometric/cellular [[cubes]]:

* [[cubical set]]

category: disambiguation

[[!redirects cubical structures]]


cubical structure in M-theory

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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

It is well known that when the [[higher dimensional Chern-Simons theory|higher Chern-Simons term]] in [[11-dimensional supergravity]] is [[KK-compactification|compactified]] on a [[4-sphere]] to yield the [[7-dimensional Chern-Simons theory]] which inside [AdS7/CFT6](AdS-CFT#AdS7CFT6) is [[duality in string theory|dual]] to the [[M5-brane]] [[6d (2,0)-superconformal QFT]], the [[cup product]] square in [[ordinary differential cohomology]] that enters its definition is to receive a [[quadratic refinement]]. This was originally argued in ([Witten 97](http://ncatlab.org/nlab/show/7d+Chern-Simons+theory#Witten97)) and then formalized and proven in ([[Quadratic Functions in Geometry, Topology, and M-Theory|Hopkins-Singer 02]]).

What though is the situation up in 11 dimensions before compactifying to 7-dimensions?

In ([DFM 03, section 9](supergravity+C-field#DiaconescuFreedMoore03)) it is claimed that the full 11-dimensional Chern-Simons term evaluated on the [[supergravity C-field]] (with its flux quantization correction, see there) indeed carries a _cubic refinement_.

More precisely, and slightly paraphrasing, the [[fiber integration in ordinary differential cohomology|transgression]] $\int_X CS_{11}(\hat C)$ of the [[higher dimensional Chern-Simons theory|11-dimensional Chern-Simons term]] of [[11-dimensional supergravity|11d SuGra]]  to 10d spacetime $X$ is a [[complex line bundle]] on the [[moduli space]] $CField(X)$ of [[supergravity C-field|supergravity C-fields]] $\hat C$ is claimed to be such that its "cubical line" $\Theta(\int_X CS_{11}(\hat C))$ (in the notation at _[[cubical structure on a line bundle]]_) is the  line bundle on the space of triples of C-field configurations which is given by the transgression of the three-fold [[cup product in ordinary differential cohomology]],

$$
  \Theta\left(\int_X CS_{11}\left(-\right)\right)
  \simeq
  \int_X (-)_1 \cup (-)_2 \cup (-)_3
  \,.
$$

## Relation to F-theory and the topological Witten genus

In the context of "[[F-theory]] compactifications" of [[M-theory]], one considers [[supergravity C-field|C-fields]] on an [[elliptic fibration]] which are "factorizable fluxes", in that their underlying [[cocycle]] $\hat C$ in [[ordinary differential cohomology]] is the [[cup product in ordinary differential cohomology|cup product]] of a cocycle $\hat C_{fib}$ on the fiber with one $\hat C_b$ on the base

$$
  \hat C \coloneqq \hat C_{b} \cup \hat C_{fib}
 \,.
$$

In approaches like ([GKP 12 (around p. 19)](http://ncatlab.org/nlab/show/F-theory#GKP12), [KMW 12](http://ncatlab.org/nlab/show/F-theory#KMW12)) the [[supergravity C-field|C-field]] is factored as a [[cup product in ordinary differential cohomology|cup product]] of a degree-2 cocycle on the elliptic fiber with a degree-2 class in the [[Calabi-Yau space|Calabi-Yau]]-base. This makes the component of the C-field on the elliptic fiber a [[complex line bundle]] (with [[connection on a bundle|connection]]). Notice that the space of complex line bundles on an elliptic curve is dual to the elliptic curve itself.

On the other hand in  e.g. ([DFM 03, p. 38](supergravity+C-field#DiaconescuFreedMoore03)) the factorization is  taken to be that of two degree-3 cocycles in the base (which are then identified with the combined degree-3 [[RR-field]]/[[B-field]] flux coupled to the [[(p,q)-string]]) with, respectively, the two canonical degree-1 cocycles $\hat t_i$ on the elliptic fiber which are given by the two canonical coordinate functions $t_i$ (speaking of a [[framed elliptic curve]]). In this case the fiber-component of the [[supergravity C-field]] "is" the [[elliptic curve]]-fiber,

$$
  \hat C \coloneqq \hat B_{NS} \cup \hat t_1 + \hat B_{RR} \cup \hat t_2
$$

or equivalently each point in the moduli space of $H$-flux in 10d induces an identification of the $G$-flux with the elliptic curve this way.

This is maybe noteworthy in that when the [[supergravity C-field|C-field]] is identified with the compactification  [[elliptic curve]] in this way, then the formula for $\Theta\left(\int_X CS_{11}(\hat C)\right)$ as [above](#Idea) is exactly that appearing in the definition of a [[cubical structure on a line bundle]] over an [[elliptic curve]]. But a "cubical" trivialization of $\Theta(\mathcal{O}(-\{0\}))$ over a given elliptic curve is what in ([Hopkins 02](http://ncatlab.org/nlab/show/string+orientation+of+tmf#Hopkins02), [AHS01](http://ncatlab.org/nlab/show/string+orientation+of+tmf#AndoHopkinsStrickland01)) is used to induce the [[sigma-orientation]] of the corresponding [[elliptic cohomology theory]] and in totality the [[string-orientation of tmf]]. But that is the refinement of the [[Witten genus]], hence of the [[partition function]] of the [[heterotic string]].

Now, by the above fact that $\Theta\left(CS_{11}(-)\right) \simeq \int_X (-)_1 \cup (-)_2 \cup (-)_3$, a cubical trivialization of $\Theta(L)$ is also given by a trivialization of the topological class of the C-field. This is one way (or is at least closely related) to the trivialization of the anomaly line bundle which "sets the quantum integrand" of M-theory.

So there is a curious coincidence of concepts here, which might want to become a precise identification:

on the one hand there is naturally a [[cubical structure on a line bundle]] on the Chern-Simons line bundle over the moduli space of [[supergravity C-fields]] which for [[F-theory]] compactifications and factorizable flux configurations induces in particular a cubical structure on a line bundle over the compactification elliptic curve. On the other hand, the latter are the structures that enter the refined construction of the [[Witten genus]] via the [[string orientation of tmf]].

## Relation to S-duality and 3-form flux
 {#RelationToSDualityAnd3FormFlux}

The refined perspective on [[perturbative string theory|perturbative]] [[type II string theory]] is that (see also at _[[orientifold]]_) the [[B-field]] is a [[cocycle]] in ([[twisted cohomology|twisted]]) [[ordinary differential cohomology]], while the [[RR-field]] is a [[cocycle]] in [[differential K-theory]] (in fact [[KR-theory]]). This is however not compatible with [[non-perturbative effect|non-perturbative]] [[S-duality]], which mixes the degree- components here.

In ([DFM 03, section 9.3](supergravity+C-field#DiaconescuFreedMoore03)) it was argued that the cubical structure on the 11d CS term alleviates this problem, even though at face value it does not really solve it. But see at _[S-duality -- Cohomological nature of type II fields](S-duality#CohomologicalNatureOfTypeIIFieldsUnderSDuality)_ for more on this.


cubical structure on a line bundle

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#### Bundles
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[[!include bundles - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _cubical structure_ on a [[complex line bundle]] over an [[abelian group]] is a certain trivialization of a certain induced line bundle on the 3-fold Cartesian product ("cube") of the group which is constructed in a kind of cubical generalization of the [[polarization identity]] formula for [[quadratic forms]].

Over [[formal groups]] associated with [[complex oriented cohomology theories]] cubical structures encode [[orientation in generalized cohomology]].

## Definition

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

Given a [[circle group]]-[[principal bundle]]/[[complex line bundle]] $\mathcal{L}$ on an [[abelian group]] $A$, write $\Theta(\mathcal{L})$ for the line bundle on $G^3$ which is given by the formula

$$
  \Theta(\mathcal{L})_{x,y,z}
  =
  \mathcal{L}_{x+y+z} \otimes \mathcal{L}_{x+y}^{-1} \otimes \mathcal{L}_{x+z}^{-1} \otimes \mathcal{L}_{y+z}^{-1} \otimes \mathcal{L}_x \otimes \mathcal{L}_y \otimes \mathcal{L}_z \otimes \mathcal{L}_0^{-1}
  \,.
$$

=--

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

A _cubical structure_ on $\mathcal{L}$ is a trivializing [[section]] $s$ of $\Theta(\mathcal{L})$ such that

1. $s(0,0,0) = 1$

1. $s(x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}) = s(x_1, x_2, x_3)$

1. $s(w+x,y,z) s(w,x,z) = s(w,x + y, z) s(x,y,z)$

for all elements of $A$ as indicated, and for all [[permutations]] $\sigma$ of three elements. Here the equalities are equalities of sections after applying the canonical [[isomorphisms]] of [[complex lines]] on both sides.

=--

([Breen 83](#Breen83), [Hopkins 94, section 4](#Hopkins94), [Ando-Hopkins-Strickland 01, def. 2.40](#AndoHopkinsStrickland01))

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

The canonical isomorphsms hidden in def. \ref{CubicalStructure} are:

1. $\mathcal{L}_0^{\otimes 3} \otimes (\mathcal{L}_0^{-1})^{\otimes 3} \to 1$ the canonical map exhibiting $\mathcal{L}_0^{-1}$ as the inverse ([[dual object]]) of $\mathcal{L}_0$:

1. etc.

=--

There is the following further refinement.

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

In the situation of def. \ref{CubicalStructure}, if the line bundle $\mathcal{L}$ is equipped with a natural "symmetry"

$$
  t \colon \mathcal{L}_x \stackrel{\simeq}{\longrightarrow} \mathcal{L}_{-x}
$$

then a _$\Sigma$-structure_ on $\mathcal{L}$ is a cubical structure, def. \ref{CubicalStructure}, such that in addition

$$
  s(x,y , -x-y) = 1
 \,.
$$

=--

## Examples

### Relation to orientations in complex-oriented cohomology theory

For $E$ a [[multiplicative cohomology theory|multiplicative]] [[weakly periodic cohomology theory|weakly periodic]] [[complex orientable cohomology theory]], we have that $Spec E^0(B U\langle 6\rangle)$ is naturally equivalent to the space of cubical structures on the trivial line bundle over the [[formal group]] of $E$.

In particular, [[homotopy classes]] of [[morphisms]] of [[E-infinity ring]] spectra $MU\langle 6\rangle \to E$ from the [[Thom spectrum]] to $E$, and hence universal $MU\langle 6\rangle$-[[orientation in generalized cohomology|orientations]] (see there) of $E$ are in natural bijection with these cubical structures.

([Hopkins 94, theorem 6.1, 6.2](#Hopkins94), [Ando-Hopkins-Strickland 01, corollary 2.50](#AndoHopkinsStrickland01))

This way for instance the [[string orientation of tmf]] has been constructed. See there for more on this.

### On the 11-dimensional Chern-Simons term

The [[higher dimensional Chern-Simons theory|11-dimensional Chern-Simons]] [[action functional]] in [[11-dimensional supergravity]] gives a line bundle $L$ on the space of [[supergravity C-fields]] whose $\Theta^3(L)$ is the transgression of the [[cup product in ordinary differential cohomology]] of three factors. It seems that each trivialization of the class of the [[supergravity C-field]] induces a "cubical" trivialization of $\Theta^3(L)$ as above, and hence a cubical structure on $L$. See at _[[cubical structure in M-theory]]_ for more on this.

## Related concept

* [[theta characteristic]]

## References

An early reference discussing the relation with [[theta functions]] is

* {#Breen83} [[Lawrence Breen]], _Fonctions thêta et théorème du cube_, Springer Lecture Notes in Mathematics **980** (1983). ([MR0823233](http://www.ams.org/mathscinet-getitem?mr=823233)).

In relation to [[orientation in generalized cohomology]] cubical structures have been prominently discussed in

* {#Hopkins94} [[Michael Hopkins]], _Topological modular forms, the Witten genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, 554–565. MR 97i:11043 ([pdf](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf))

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. Math. 146 (2001) 595–687 MR1869850 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf))

[[!redirects cubical structures on a line bundle]]
[[!redirects cubical structures on line bundles]]

[[!redirects cubical structure on a complex line bundle]]
[[!redirects cubical structures on a complex line bundle]]


cubical T-complex

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

#Contents#
* table of contents
{:toc}

##Definition

A _cubical $T$-complex_ is a [[cubical set]] $K$ which is also a special kind of [[Kan complex]], in that it has a family $T_n \subseteq K_n$ (for $n \geq 1$) of _[[thin element]]s_ with the properties, (due first to Keith Dakin, for the [[simplicial T-complex|simplicial case]], in his  doctoral thesis):

* T1) Degenerate elements are thin.

* T2) Every box in $K$ has a unique thin filler.

* T2) If all but one face of a thin element are thin, then so also is the remaining face.

Cubical T-complexes are equivalent to [[crossed complex]]es, and also to [[cubical omega-groupoid]]s with connections.

A modification of axiom T2) to be more like an axiom for a [[quasi-category]] is suggested in the paper by Steiner referenced below.

## Related concepts

* [[T-complex]]

  * [[simplicial T-complex]]

  * **cubical $T$-complex**

* [[algebraic Kan complex]]

## References##

[[T-complex]]es first appeared in:

* M.K. Dakin, _Kan complexes and multiple groupoid structures_ , PhD Thesis, University of Wales, Bangor, (1976).

A main application of the idea of thin elements in the cubical case was to define the notion of commutative cube. In the case of cubical $\omega$-groupoids with connection, the boundary of a cube is commutative if and only if the boundary has a thin filler. One consequence is that _any well defined composition of thin elements is thin_. This is a key element of the proof of the Higher Homotopy van Kampen theorem in the _colimits_ paper, which thus nicely generalises the proof of the 1-dimensional van Kampen theorem for the fundamental groupoid on a set of base points.

Cubical T-complexes are thus a key concept in the long series of articles by  Brown and Higgins on [[strict omega-groupoid|strict omega-groupoids]]. For instance

* [[Ronnie Brown]], P.J. Higgins, _The equivalence of $\omega$-groupoids and cubical $T$-complexes_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 22 no. 4 (1981), p. 349-370  ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1981__22_4/CTGDC_1981__22_4_349_0/CTGDC_1981__22_4_349_0.pdf)).

*  [[Ronnie Brown]], P.J. Higgins, _On the algebra of cubes_,  J. Pure Appl.  Algebra 21 (1981) 233-260.

* R. Brown, P.J. Higgins, _Colimit theorems for relative homotopy groups_, J. Pure Appl. Algebra 22 (1981) 11-41.

* R. Brown, P.J. Higgins, _Tensor products and homotopies for
$\omega$-groupoids and crossed complexes_, J. Pure Appl.
Alg. 47 (1987) 1-33.

*  R. Brown, R. Street, _Covering morphisms  of  crossed
complexes and of cubical omega-groupoids are closed under   tensor
product_, Cah. Top. G\'eom. Diff. Cat., 52 (2011) 188-208.

and generally

* Ronald Brown, Philip J. Higgins and Rafael Sivera, _Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids_, EMS Tracts in Mathematics Vol 15 (2011).  ([web](http://www.bangor.ac.uk/~mas010/nonab-a-t.html)).

For more on 'thinness' in a cubical context, see:

* P.J. Higgins, _Thin elements and commutative shells in
cubical  $\omega$-categories_, Theory Appl. Categ.,
14 (2005) 60--74.

* R. Steiner, _Thin fillers in the cubical nerves of
omega-categories_, Theory Appl. Categ. 16 (2006), 144--173.


cubical truncation, skeleton, and co-skeleton

# Contents
* table of contents
{: toc}

## Truncation

We make use of the notation introduced in [[category of cubes]] and [[cubical set]].

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

Let $n \geq 0$ be an integer. We denote by $\square_{n}$ the full sub-category of $\square$ whose objects are $I^{0}$, $I^{1}$, $\ldots$, $I^{n}$.

=--

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

We refer to $\square_{n}$ as the _$n$-truncated category of cubes_.

=--

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

Let $n \geq 0$ be an integer. The inclusion functor $i_{n} : \square_{n} \rightarrow \square$ canonically determines a functor $i_{n}^{*} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square_{n}^{op}}$. We shall denote this functor by $tr_{n}$.

=--

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

We refer to $tr_{n}$ as the _$n$-truncation_ functor.

=--

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

Let $n \geq 0$ be an integer. The _category of $n$-truncated cubical sets is the free co-completion of $\square_{n}$.

=--

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

The free co-completion of a small category can be constructed as the category of [[presheaf | presheaves]] presheaves of sets on this category. Thus we can also think of the category of $n$-truncated cubical sets as the category of presheaves of sets on $\square_{n}$.

=--

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

We denote the category of $n$-truncated cubical sets by $\mathsf{Set}^{\square_{n}^{op}}$.

=--

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

An _$n$-truncated cubical set_ is an object of $\mathsf{Set}^{\square_{n}^{op}}$.

=--

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

When we think of the category of $n$-truncated cubical sets as the category of presheaves of sets on $\square_{n}$, we consequently think of an $n$-truncated cubical set as a presheaf of sets on $\square_{n}$.

=--

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

A _morphism of $n$-truncated cubical sets_ is an arrow of $\mathsf{Set}^{\square_{n}^{op}}$.

=--

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

Let $X$ be an $n$-truncated cubical set. Let $0 \leq m \geq n$ be an integer. By an _$m$-cube_ of $X$, we shall mean an $m$-cube of $sk_{n}(X)$, where $sk_{n}$ is $n$-skeleton functor defined in Notation \ref{NotationNSkeleton}.

=--

## Skeleton

+-- {: .num_defn #NotationNSkeleton}
###### Notation

Let $n \geq 0$ be an integer. By [[Kan extension | left Kan extension]], the functor $tr_{n} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square_{n}^{op}}$ admits a [[adjoint functor | left adjoint]] $i_{!} : \mathsf{Set}^{\square_{n}^{op}} \rightarrow \mathsf{Set}^{\square^{op}}$. We shall denote this functor by $sk_{n}$.

=--

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

We refer to $sk_{n}$ as the _$n$-skeleton_ functor.

=--

## Co-skeleton

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

Let $n \geq 0$ be an integer. By [[Kan extension | right Kan extension]], the functor $tr_{n} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Set}^{\square_{n}^{op}}$ admits a [[adjoint functor | right adjoint]] $i_{*} : \mathsf{Set}^{\square_{n}^{op}} \rightarrow \mathsf{Set}^{\square^{op}}$. We shall denote this functor by $cosk_{n}$.

=--

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

We refer to $cosk_{n}$ as the _$n$-skeleton_ functor.

=--


cubical type theory

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### Context
#### Type theory
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[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Cubical type theory_ is a flavor of [[dependent type theory]] in which maps out of an interval primitive is used to define [[cubical path types]], rather than the [[inductive family]] of Martin-Löf [[identity types]] as in [[Martin-Löf type theory]]. Cubical type theory additionally differs from [[Martin-Löf type theory]] in that [[function extensionality]] is a [[theorem]] in cubical type theory, rather than an axiom as is the case in [[Martin-Löf type theory]].

Similarly to [[Martin-Löf type theory]], cubical type theory comes in [[extensional type theory|extensional]] and [[intensional type theory|intensional]] flavours as well. One could also add an [[equality reflection]] rule for path types to make the cubical type theory definitionally extensional.

{#UnivalentCubicalTypeTheory} [[univalence|Univalent]] cubical type theory models the [[infinity-groupoid|$\infty$-groupoid]]-structure implied by [[Martin-Löf type theory|Martin-Löf]] [[identity types]] on [[constructive set theory|constructive]] [[cubical sets]] (whence the name) thus making it a form of [[homotopy type theory]].

## Syntax

### The interval and face formulas

We begin with a [[dependent type theory]] which has rules for [[dependent product types]], [[dependent sum types]], the [[empty type]], [[unit type]], [[booleans type]], and [[natural numbers type]].

Cubical type theory adds to dependent type theory an interval primitive $I$, whose elements are called "dimensions" as well as face formulas $F$, which behave like propositions in a [[propositional logic]]. Both come with the following judgments for dimension variables and face formulae:

$$\frac{\Gamma \; ctx}{\Gamma, i:I \; ctx} \qquad \frac{\Gamma \; ctx \quad \Gamma \vdash \phi:F}{\Gamma, \phi \; ctx}$$

Similarly to types, dimension variables and face formulas come with their own [[substitution rules]], [[weakening rules]], and [[hypothesis rules]].

In addition, there are rules for forming the endpoints of the interval:

$$\frac{\Gamma}{\Gamma \vdash 0:I} \qquad \frac{\Gamma}{\Gamma \vdash 1:I}$$

Equality of dimensions in the interval is a face formula:

$$\frac{\Gamma \vdash r:I \quad \Gamma \vdash s:I}{\Gamma \vdash r = s:I}$$

and face formulas are closed under disjunction

$$\frac{\Gamma \vdash \phi:F \quad \Gamma \vdash \psi:F}{\Gamma \vdash \phi \vee \psi:F}$$

Since face formulae behave like propositions, it should be possible to judge them to be true. Thus, we have the following rules:

$$\frac{\Gamma \vdash \phi \; true}{\Gamma \vdash \phi \vee \psi \; true} \qquad \frac{\Gamma \vdash \psi \; true}{\Gamma \vdash \phi \vee \psi \; true} \qquad \frac{\Gamma \vdash \phi \vee \psi \; true \quad \Gamma, \phi \vdash \chi \; true \quad \Gamma, \psi \vdash \chi \; true}{\Gamma \vdash \chi \; true}$$

The interval primitive $I$ has more points than $0$ and $1$, so it is not the case that the sequent
$$r:I \vdash r = 0 \vee r = 1 \;\mathrm{true}$$
holds.  Thus, there is a **boundary formula**
$$\delta(r) \coloneqq r = 0 \vee r = 1$$

## Canonicity in cubical type theory

In contrast to [[Martin-Löf type theory]], there exist cubical type theories, such as [[XTT]], in which [[UIP]] is not just an [[axiom]] but a [[theorem]]. Similarily, there exist cubical type theories in which [[univalence]] is not just an [[axiom]] but a [[theorem]]. As a result, in those cubical type theories, [[canonicity]] still holds. This is useful for computational purposes, and for the use of [[cubical type theory]] in [[proof assistants]].

However, it is equally valid to add an [[axiom]] to a cubical type theory that says all [[type universe]] satisfy [[UIP]] or [[univalence]], so canonicity is not required to hold in all cubical type theories.

## Univalent cubical type theories

The first constructive account of the univalence axiom was given in ([Coquand 13](#Coquand13), [Bezem-Coquand-Huber 17](#BezemCoquandHuber17)), called the "BCH-model".

The BCH model, unfortunately, has some problems that appear to make it unsuitable for general [[HoTT]].  In particular, it is not known how to model [[higher inductive types]]. The problem is that the BCH model is based on [[presheaves]] on the '[[symmetric monoidal category|symmetric monoidal]] [[cube category]]', which is basically the [[free construction|free]] [[PROP]] generated by an [[interval]].
In particular, the base category's maps are generated by face maps and permutative renamings of dimension variables (this is where the 'symmetric monoidal' part comes in). For somewhat technical reasons, this doesn't work out when you want to define the [[elimination rules]] for [[higher inductive types]] (like for the [[circle]]).

To account for [[HIT|HITs]], you seem to need [[diagonals]] in the base category.  There is more than one choice in addition as to what other morphisms to add.

* If you add only diagonals, then you have the '[[cartesian monoidal category|cartesian]] cube category'.  This was used as a basis for a cubical type theory by [Angiuli, Brunerie, Coquand, Favonia, Harper, and Licata](#ABCFHL).

* The first cubical type theory, due to [Cohen-Coquand-Huber-Moertberg 17](#CohenCoquandHuberMoertberg17), used an even richer [[cube category]], the free [[De Morgan algebra]] generated by an [[interval]]. In addition to diagonals, this includes what are called 'reversals' and 'connections'.

Both models validate [[univalence]] (like the BCH model) and can be used to model a variety of [[HIT|HITs]] as well as supporting a syntactic type theory based on 'dimension variables'.  However, while it is possible to interpret the Martin-Löf [[identity type]] in all of these models, it is only equivalent to, not definitionally isomorphic to, the native cubical path-types in the model.  Thus, the latter support the Martin-Lof eliminator, but only with a typal computation rule.

## Models

Cubical type theory can be modeled in a number of varieties of [[cubical sets]], for example in a [[type-theoretic model structure]].

## Related concepts

* [[synthetic Tait computability]]
* [[simplicial type theory]]
* [[type theory with shapes]]
* [[two-level type theory]]

## References

Introductory lecture notes:

* {#Mortberg19} [[Anders Mörtberg]], _Cubical methods in HoTT/UF_, 2019 ([pdf](http://staff.math.su.se/anders.mortberg/papers/cubicalmethods.pdf))

Exposition in view of [[synthetic homotopy theory]]:

* {#MortbergPujet20} [[Anders Mörtberg]], Loïc Pujet, _Cubical synthetic homotopy theory_,  CPP 2020: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs January 2020, pp. 158–171, [doi:10.1145/3372885.3373825](https://doi.org/10.1145/3372885.3373825), ([pdf](https://staff.math.su.se/anders.mortberg/papers/cubicalsynthetic.pdf))

Original articles on the BCH model:

* {#Coquand13} [[Thierry Coquand]] (with [[Marc Bezem]] and [[Simon Huber]]), _Computational content of the Axiom of Univalence_, September 2013 ([pdf](http://www.humboldt-kolleg.iam.unibe.ch/talks/Coquand.pdf))

* {#BezemCoquandHuber17} [[Marc Bezem]], [[Thierry Coquand]], [[Simon Huber]], _The univalence axiom in cubical sets_ ([arXiv:1710.10941](https://arxiv.org/abs/1710.10941))

On the CCHM model and type theory:

* {#CohenCoquandHuberMoertberg17} [[Cyril Cohen]], [[Thierry Coquand]], [[Simon Huber]], [[Anders Mörtberg]], _Cubical Type Theory: a constructive interpretation of the univalence axiom_ ([pdf](https://arxiv.org/pdf/1611.02108.pdf))

* {#CubicalCanonicity} [[Simon Huber]], _Canonicity for Cubical Type Theory_ ([arXiv:1607.04156](https://arxiv.org/abs/1607.04156))

* {#CubicalHIT} [[Thierry Coquand]], [[Simon Huber]], [[Anders Mörtberg]], _On Higher Inductive Types in Cubical Type Theory_ ([arXiv:1802.01170](https://arxiv.org/abs/1802.01170))

* [[homotopytypetheory:Evan Cavallo]], [[Robert Harper]], _Internal parametricity for cubical type theory_ ([PDF](http://www.cs.cmu.edu/~ecavallo/works/parametric-cubical.pdf))

On the cartesian cubical model and type theory:

* {#ABCFHL} Carlo Angiuli, Guillaume Brunerie, Thierry Coquand, Kuen-Bang Hou (Favonia), Robert Harper, and Daniel R. Licata, _Cartesian cubical type theory_, <https://www.cs.cmu.edu/~rwh/papers/uniform/uniform.pdf>

* Carlo Angiuli, _Computational Semantics of Cartesian Cubical Type Theory_, Ph.D. Thesis, <https://www.cs.cmu.edu/~cangiuli/thesis/thesis.pdf>

On comparing the models:

* [[homotopytypetheory:Evan Cavallo]], [[Anders Mörtberg]], Andrew W Swan, _Unifying Cubical Models of Univalent Type Theory_, 2019, [PDF](https://github.com/mortberg/gen-cart/blob/master/conference-paper.pdf)

* [[Thierry Coquand]], [[Simon Huber]], [[Christian Sattler]] _Homotopy canonicity for cubical type theory_, 2019 [PDF](https://arxiv.org/pdf/1902.06572)

* {#Bentzen2019} [[Bruno Bentzen]], _Naive cubical type theory_, 2019 ([pdf](https://arxiv.org/pdf/1911.05844.pdf))

Discussion of implementation in the [[proof assistant]] [[Cubical Agda]]:

* {#VMA19} [[homotopytypetheory:Andrea Vezzosi]], [[Anders Mörtberg]] and Andreas Abel, _Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types_, 2019 ([pdf](https://staff.math.su.se/anders.mortberg/papers/cubicalagda.pdf))

On [[normal form|normalization]] for cubical type theory:

* [[Jonathan Sterling]], Carlo Angiuli, _Normalization for Cubical Type Theory_, ([arXiv:2101.11479](https://arxiv.org/abs/2101.11479))
* [[Jonathan Sterling]], _First Steps in Synthetic Tait Computability: The Objective Metatheory of Cubical Type Theory_, ([pdf](https://www.jonmsterling.com/papers/sterling-2021-thesis.pdf))

On [[XTT]]:

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _Cubical syntax for reflection-free extensional equality_. In Herman Geuvers, editor, _4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)_, volume 131 of _Leibniz International Proceedings in Informatics (LIPIcs)_, pages 31:1-31:25. ([arXiv:1904.08562](https://arxiv.org/abs/1904.08562), [doi:10.4230/LIPIcs.FCSD.2019.31](https://doi.org/10.4230/LIPIcs.FSCD.2019.31))

* [[Jonathan Sterling]], [[Carlo Angiuli]], [[Daniel Gratzer]], _A Cubical Language for Bishop Sets_, Logical Methods in Computer Science, 18 (1), 2022. ([arXiv:2003.01491](https://arxiv.org/abs/2003.01491)).

[[!redirects cubical type theories]]
[[!redirects boundary in cubical type theory]]
[[!redirects boundary (cubical type theory)]]
[[!redirects boundary formula]]


cubical-type model category

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#### Type theory
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#### Model category theory
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# Cubical-type model categories

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{: toc}

## General

These exist a class of [[model category]] [[mathematical structure|structures]] on certain [[categories]] of [[cubical sets]], due to [[Christian Sattler]], motivated by [[cubical type theory]].

### Assumptions

Let $\mathbb{C}$ be a [[small category]] with [[finite products]].
Assume there is a (cartesian) [[interval object]] $\mathbb{I}$, i.e.,
an [[object]] $\mathbb{I}$ with two [[parallel morphisms]] $0,1  \colon e \to \mathbb{I}$, where $e$ is the [[terminal object]] in $\mathbb{C}$.

Assume further that we have [[connection on a cubical set|connection
maps]] $\vee,\wedge : \mathbb{I}\times\mathbb{I} \to \mathbb{I}$ such
that $x \vee 0 = 0 \vee x = x$ and $x \wedge 1 = 1 \wedge x = x$.

We also need a face lattice $\mathbb{F}$, that is, a [[subobject|sub]]-[[lattice]] of the [[subobject classifier]] $\Omega$ in the [[presheaf category]] $cSet = \mathbb{C}^{op} \to Set$ containing the endpoints of each [[cylinder object]] $J^+ = J \times \mathbb{I}$ for each object $J$ of $\mathbb{C}$, and an operation $\forall \colon \mathbb{F}^\mathbb{I} \to \mathbb{F}$ [[right adjoint]] to the [[projection]] in the sense that $\psi \le \forall \delta$ iff $\psi p \le \delta$, where $p : \mathbb{F} \times \mathbb{I} \to \mathbb{F}$.

If we want the model to be effective, we also require that each [[proposition]] $\psi = 1$ with $\psi \in \mathbb{F}(I)$ is [[decidability|decidable]]. (This rules out taking $\mathbb{F} = \Omega$.)

### Instances of the assumptions

A simple and central example is the [[full subcategory]] of the category
of [[posets]] $Pos$ on powers of $\mathbb{I} = (0 \lt 1)$. This is
also the [[Lawvere theory]] of [[distributive lattices]]. The
[[idempotent completion]] is the full subcategory of $Pos$ on finite
lattices.

Other prominent examples are the Lawvere theories of [[de Morgan algebra|de
Morgan]], [[Kleene algebra|Kleene]], and [[boolean algebra|boolean]] algebras.

For computational purposes we can take $\mathbb{F}$ to be the smallest
lattice containing the cylinder endpoints (the faces of cubes). To get
[[Cisinski model structures]] we can take $\mathbb{F} = \Omega$.

## Equivalence with simplicial sets

One may wonder whether these models structures are equivalent to the model in [[simplicial sets]]. This is not the case for Cartesian cubes; see [mailing-list](https://groups.google.com/d/msg/homotopytypetheory/RQkLWZ_83kQ/tAyb3zYTBQAJ). However, this model does form a Grothendieck [[(infinity,1)-topos]], because it satisfies the Giraud axioms. For deMorgan cubical sets the geometric realization is not a Quillen equivalence (because of the reversal map); the counterexample is the unit interval quotient by the symmetry); see [Sattler](#sattlerHIM). Whether they are equivalent by another map is not yet excluded.
The question whether the geometric realization for the cubical sets based on distributive lattices is an equivalence is still open, cf. also [Hackney and Rovelli](#hrInduced) and [Streicher and Weinberger](#swCuSi).

## References

* {#mod2} [[Thierry Coquand]], *A model structure on some presheaf categories*, [pdf](http://www.cse.chalmers.se/~coquand/mod2.pdf)

* {#cis3} [[Thierry Coquand]], *Some examples of complete Cisinski model structures*, [pdf](http://www.cse.chalmers.se/~coquand/cis3.pdf)

* {#sattlerHIM} [[Christian Sattler]], *Do cubical models of type theory also model homotopy types*, lecture at Hausdorff Trimester Program: Types, Sets and Constructions, [youtube](https://www.youtube.com/watch?v=wkPDyIGmEoA)

* {#hrInduced} [[Philip Hackney]] and [[Martina Rovelli]], *Induced model structures for ∞-categories and ∞-groupoids*, $[$[arXiv:2102.01104](https://arxiv.org/abs/2102.01104)$]$, [Proc. Amer. Math. Soc., June 10, 2022](https://www.ams.org/journals/proc/0000-000-00/S0002-9939-2022-15982-6/home.html)

* {#swCuSi} [[Thomas Streicher]] and [[Jonathan Weinberger]], *Simplicial sets inside cubical sets* $[$[arXiv:1911.09594](https://arxiv.org/abs/1911.0959)$]$, [Theory and Applications of Categories, Vol. 37, 2021, No. 10, pp 276–286](http://www.tac.mta.ca/tac/volumes/37/10/37-10abs.html)

[[!redirects type-theoretic model structures]]
[[!redirects type theoretic model structures]]
[[!redirects type theoretic model structure]]
[[!redirects type-theoretic model structure]]
[[!redirects cubical-type model categories]]
[[!redirects cubical-type model structure]]
[[!redirects cubical-type model structures]]


cubulation

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#### Topology
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#Contents#
* table of contents
{:toc}

## Idea

Where a *[[triangulation]]* of a [[topological space]] is its [[homeomorphism|homeomorphic]] identification with the [[topological realization]] of a [[simplicial set]] ([[simplicial complex]]), the analogous construction with [[cubical sets]] may be referred to as "cubulation".

## The category of cubes

The [[category of cubes]], $Cube$, may be defined universally to be a [[walking structure|walking]] [[interval]]: it is [[initial object|initial]] among [[monoidal categories]] that are equipped with an object $I$, two maps $i_0, i_1: 1 \to I$ (where $1$ is the monoidal unit) and a map $p: I \to 1$ such that $p \circ i_0 = id_1 = p \circ i_1$. The monoidal unit $1$ in $Cube$ is [[terminal object|terminal]], hence there is a unique map $!: X \to 1$ for any object $X$. The interval $I$ of $Cube$ monoidally generates $Cube$ in the sense of [[PROP|PROS]].

It may be shown that if $m \leq n$, there are $\binom{n}{m}2^{n-m}$ injections $I^{\otimes m} \to I^{\otimes n}$, the same as the number of $m$-dimensional faces of the geometric $n$-cube. There are no diagonal maps in the category of cubes as defined here.

* A different possibility is to consider the [[Lawvere theory]] of two constants, which gives a different category of cubes with diagonal maps.

## Standard geometric cube functor

From the [[universal property]] of $Cube$, it follows that if [[Top]] is considered as a [[cartesian monoidal category]] equipped with $I = [0, 1]$ in this sense of interval, we get an induced monoidal functor

$$\Box: Cube \to Top$$

The monoidal product on $Cube$ induces a monoidal product $\otimes$ on $Set^{Cube^{op}}$ by [[Day convolution]]. The [[topological realization of cubical sets|cubical realization functor]] $R_{cub}: Set^{Cube^{op}} \to Top$ is, up to isomorphism, the unique [[cocontinuous functor|cocontinuous]] [[monoidal functor]] which extends the monoidal functor $\Box$ along the Yoneda embedding; therefore $R_{cub}$ takes $\otimes$-products of [[cubical sets]] to the corresponding [[product topological space|cartesian products of spaces]].

In terms of an explicit formula, the [[topological cubical realization of a cubical set]] $C: \Cube^{op} \to Set$ is given by the [[coend]] formula

$$R_{cub} C = \int^{m \in Ob(Cube)} C(m) \times \Box(m)$$

## Definition

A **cubulation** of a [[topological space]] $Y$ is a [[cubical set]] $C: Cube^{op} \to Set$ together with a [[homeomorphism]] $h: R_{cub}C \to Y$.

## Relation between triangulation and cubulation

In rough terms, a space can be [[triangulation|triangulated]] if and only if it can be cubulated. This can be shown by simple conceptual arguments, as follows.

### Cubulating a triangulated space

In this section, $Top$ may be taken to be the category of topological spaces, or otherwise any sufficiently [[convenient category of topological spaces]] (completeness and cocompleteness are baseline assumptions).

#### Simplices as cubical sets

We define a functor

$$\Sigma: \Delta \to Set^{Cube^{op}}$$

The functor $\Sigma$ effectively regards an [[n-simplex|$n$-simplex]] as an iterated [[join of simplicial sets]] and then produces the analogous join in the category of [[cubical sets]]. This for instance regards the 2-simplex as a square with one degenerate edge.

To define $\Sigma: \Delta \to Set^{Cube^{op}}$, we mimic the [second definition](http://ncatlab.org/nlab/show/triangulation#second_definition_5) of the affine simplex functor given at [[triangulation]], replacing $Top$ by cubical sets and the topological simplicial join by a suitable "cubical simplicial join". Formally, we define a monoidal structure on cubical sets by taking $X \star Y$ to be the pushout of the diagram

$$X \stackrel{\pi_1}{\leftarrow} X \otimes Y \stackrel{1_X \otimes i_0 \otimes 1_Y}{\to} X \otimes I \otimes Y \stackrel{1_X \otimes i_1 \otimes 1_Y}{\leftarrow} X \otimes Y \stackrel{\pi_2}{\to} Y$$

where the projection maps $\pi_1$, $\pi_2$ are defined by taking advantage of the fact that the monoidal unit of $\otimes$ is terminal:

$$\pi_1 = (X \otimes Y \stackrel{1_X \otimes !}{\to} X \otimes 1 \cong X)$$

$$\pi_2 = (X \otimes Y \stackrel{! \otimes 1_Y}{\to} 1 \otimes Y \cong Y)$$

The terminal cubical set is of course a monoid with respect to this monoidal product, so by the walking monoid property we obtain a monoidal functor

$$\Sigma: \Delta \to Set^{Cube^{op}}$$

which plays a role analogous to the affine simplex functor into $Top$.

#### Cubulating geometric simplices

Observe that geometric realization $R_{cub}: Set^{Cube^{op}} \to Top$ takes cubical simplicial joins to topological simplicial joins, because $R_{cub}$ sends $\otimes$-products to cartesian products, and [[preserved colimit|preserves]] [[pushouts]] because it is [[cocontinuous functor|cocontinuous]]. We conclude that both $\sigma: \Delta \to Top$ and $R_{cub} \circ \Sigma: \Delta \to Top$ take monoidal products in $\Delta$ to topological simplicial joins, and both take the walking monoid of $\Delta$ to the one-point space. By the universal property of $\Delta$, it follows that there is a natural isomorphism

$$\sigma \cong R_{cub} \circ \Sigma$$

(as monoidal functors), giving the canonical cubulation of affine simplices. In terms of an explicit formula, we have

$$\sigma(n) \cong \int^m \Sigma_n(m) \cdot \Box(m)$$

#### Standard cubulation of a triangulated space

Given a [[triangulation]] $(X, h: R X \to Y)$ of a space $Y$, we have [[isomorphisms]]

$$\array{
Y & \cong & \int^n X(n) \cdot \sigma(n) & & \\
 & \cong & \int^n X(n) \cdot (\int^m \Sigma_n(m) \cdot \Box(m)) & & cubulation of \sigma(n) \\
 & \cong & \int^m (\int^n X(n) \cdot \Sigma_n(m)) \cdot \Box(m) & & interchange of coends
}$$

where in the last line we used the [coend Fubini theorem](end#Fubini). Thus, defining the cubical set $C$ by

$$C(m) = \int^n X(n) \cdot \Sigma_n(m)$$

we have a homeomorphism $Y \cong \int^m C(m) \cdot \Box(m) = R_{cub} C$, i.e., we obtain a cubulation of $Y$.

### Triangulating a cubulated space

In this section we assume $Top$ is a [[convenient category of spaces]], so that [[geometric realization]] of simplicial sets is product-preserving (see [there](geometric+realization#GeometricRealizationIsLeftExact)).

#### Cubes as simplicial sets

Define a monoidal functor $\Box_{\delta}: Cube \to Set^{\Delta^{op}}$ as follows: regard the category of simplicial sets as a cartesian monoidal category equipped with the representable $hom(-, [1])$ as an interval (with two face maps from and a projection to the terminal object $hom(-, [0])$). By the walking interval property of $Cube$, there is an induced functor

$$\Box_{\delta}: Cube \to Set^{\Delta^{op}}$$

#### Triangulating geometric cubes

Next, because $R: Set^{\Delta^{op}} \to Top$ is preserves cartesian products and preserves the interval objects, we have an isomorphism

$$(\Box: Cube \to Top) \cong (Cube \stackrel{\Box_\delta}{\to} Set^{\Delta^{op}} \stackrel{R}{\to} Top) \qquad (2)$$

by the universal property of $Cube$. In terms of an explicit formula, we have

$$\Box(m) \cong \int^n \Box_{\delta m}(n) \cdot \sigma(n)$$

#### Standard triangulation of a cubulated space

Given a cubulation $(C, h: R_{cub} X \to Y)$ of a space $Y$, we have [[isomorphism]]s

$$\array{
Y & \cong & \int^m C(m) \cdot \Box(m) & & \\
 & \cong & \int^m C(m) \cdot (\int^n \Box_{\delta m}(n) \cdot \sigma(n)) & & triangulation of \Box(m) \\
 & \cong & \int^n (\int^m C(m) \cdot \Box_{\delta m}(n)) \cdot \sigma(n) & & interchange of coends
}$$

where in the last line we used the [coend Fubini theorem](end#Fubini). Thus, defining the simplicial set $X$ by

$$X(n) = \int^m C(m) \cdot \Box_{\delta m}(n)$$

we have a homeomorphism $Y \cong \int^n X(n) \cdot \sigma(n) = R X$, i.e., we obtain a triangulation of $Y$.

## Related concepts

* [[triangulation]]

[[!redirects cubulation]]


Cumrun Vafa

Cumrun Vafa is a [[string theory|string theorist]] at Harvard.

* [personal webpage](https://www.cumrunvafa.org/)

* [institute website](http://physics.harvard.edu/~vafa/)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Cumrun_Vafa)

## Selected writings
 {#SelectedWritings}

On the [[quantum cohomology]] induced by [[supersymmetry|supersymmetric]] [[CP^N sigma-model|$\mathbb{C}P^N$ sigma-models]]:

* [[Sergio Cecotti]], [[Cumrun Vafa]], *Exact Results for Supersymmetric Sigma Models*, Phys. Rev. Lett. **68** (1992) 903-906 [[arXiv:hep-th/9111016](https://arxiv.org/abs/hep-th/9111016), [doi:10.1103/PhysRevLett.68.903](https://doi.org/10.1103/PhysRevLett.68.903)]

* {#CecottiVafa93} [[Sergio Cecotti]], [[Cumrun Vafa]], *On Classification of $N=2$ Supersymmetric Theories*, Comm. Math. Phys. **158** (1993) 569-644 [[arXiv:hep-th/9211097](https://arxiv.org/abs/hep-th/9211097), [doi:10.1007/BF02096804](https://doi.org/10.1007/BF02096804)]

On how the [[second quantization]] of the [[B-model]] yields [[Kodeira-Spencer gravity]]/[[BCOV theory]]:

* {#BCOV93} [[Michael Bershadsky]], [[Sergio Cecotti]], [[Hirosi Ooguri]], [[Cumrun Vafa]], *Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes*, Commun. Math. Phys. **165** (1994) 311-428 [[arXiv:hep-th/9309140](http://arxiv.org/abs/hep-th/9309140), [doi:10.1007/BF02099774](https://doi.org/10.1007/BF02099774)]

Introducing the idea of [[F-theory]]:

* {#Vafa96} [[Cumrun Vafa]], _Evidence for F-theory_, Nucl. Phys. B469:403-418, 1996, ([arXiv:hep-th/9602022](http://arxiv.org/abs/hep-th/9602022))

On [[F-theory]] [[KK-compactification|KK-compactified]] on [[Calabi-Yau manifolds]] and [[duality between F-theory and heterotic string theory]]:

* {#MorrisonVafa96a} [[David Morrison]], [[Cumrun Vafa]], _Compactifications of F-Theory on Calabi--Yau Threefolds -- I_, Nucl. Phys. B473 (1996) 74-92 ([arXiv:hep-th/9602114](http://arxiv.org/abs/hep-th/9602114))

* {#MorrisonVafa96b} [[David Morrison]], [[Cumrun Vafa]], _Compactifications of F-Theory on Calabi--Yau Threefolds -- II_, Nucl. Phys. B476:437-469, 1996 ([arXiv:hep-th/9603161](http://arxiv.org/abs/hep-th/9603161))

Introducing the idea of [[quantum cohomology]]:

* [[Cumrun Vafa]], *Topological mirrors and quantum rings*, in [[Shing-Tung Yau]] (ed.) *Essays on mirror manifolds*, International Press (1992), republished in *Mirror Symmetry I*, AMS/IP Studies in Advanced Mathematics **9** (1998) [[arXiv:hep-th/9111017](https://arxiv.org/abs/hep-th/9111017), [doi:10.1090/amsip/009](https://doi.org/10.1090/amsip/009)]

On [[Chern-Simons theory as a topological string theory]] ([[open/closed string duality]] in interpreting [['t Hooft double line notation]] in the [[large N limit]] and in [[AdS/CFT duality]]);

* {#GopakumarVafa98} [[Rajesh Gopakumar]], [[Cumrun Vafa]], _On the Gauge Theory/Geometry Correspondence_, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 ([arXiv:hep-th/9811131](https://arxiv.org/abs/hep-th/9811131))

Including [[Wilson loop observables]] in [[Chern-Simons theory as a topological string theory]]:

* [[Hirosi Ooguri]], [[Cumrun Vafa]], _Knot Invariants and Topological Strings_, Nucl. Phys. B 577:419-438, 2000 ([arXiv:hep-th/9912123](https://arxiv.org/abs/hep-th/9912123))

Introducing [[Gopakumar-Vafa duality]]:

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _Topological Gravity as Large N Topological Gauge Theory_, Adv. Theor. Math. Phys. 2:413-442, 1998 ([arXiv:hep-th/9802016](https://arxiv.org/abs/hep-th/9802016))

* [[Rajesh Gopakumar]], [[Cumrun Vafa]], _On the Gauge Theory/Geometry Correspondence_, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 ([arXiv:hep-th/9811131](https://arxiv.org/abs/hep-th/9811131))

On [[geometric engineering of quantum field theory]]:

* {#KatzKlemmVafa97} [[Sheldon Katz]], [[Albrecht Klemm]], [[Cumrun Vafa]], _Geometric Engineering of Quantum Field Theories_, Nucl.Phys. B497 (1997) 173-195 ([arXiv:hep-th/9609239](http://arxiv.org/abs/hep-th/9609239))

* {#KatzKlemm96} [[Sheldon Katz]], [[Cumrun Vafa]], _Geometric Engineering of $N=1$ Quantum Field Theories_, Nucl.Phys. B497 (1997) 196-204 ([arXiv:hep-th/9611090](http://arxiv.org/abs/hep-th/9611090))

On [[geometric engineering of QFT]] on [[D4-D6 brane intersections]] subject to the [[s-rule]]:

* {#OoguriVafa97} [[Hirosi Ooguri]], [[Cumrun Vafa]], _Geometry of $N = 1$ Dualities in Four Dimensions_, Nucl. Phys. B 500:62-74, 1997 ([arXiv:hep-th/9702180](https://arxiv.org/abs/hep-th/9702180))

On microscopic explanation of [[Bekenstein-Hawking entropy]] via [[geometric engineering of QFT|geometric engineering]] of [[black holes in string theory]] as [[bound states of D-branes]]:

* {#StromingerVafa96} [[Andrew Strominger]], [[Cumrun Vafa]], _Microscopic Origin of the Bekenstein-Hawking Entropy_, Phys. Lett. B379: 99-104, 1996 ([arXiv:hep-th/9601029](http://arxiv.org/abs/hep-th/9601029))

On [[topological string theory]]:

* {#NeitzkeVafa04} [[Andrew Neitzke]], [[Cumrun Vafa]], _Topological strings and their physical applications_, talk at [Simons Workshop in Mathematics and Physics 2004](http://insti.physics.sunysb.edu/conf/simonsworkII/) ([hep-th/0410178](http://xxx.lanl.gov/abs/hep-th/0410178))

On the [[large N limit]]:

* M. Bershadsky, Z. Kakushadze, [[Cumrun Vafa]], _String expansion as large $N$ expansion of gauge theories_, Nucl.Phys. B523 (1998) 59-72 ([hep-th/9803076](http://arxiv.org/abs/hep-th/9803076), doi)

* {#OoguriVafa02} [[Hirosi Ooguri]], [[Cumrun Vafa]], _Worldsheet Derivation of a Large $N$ Duality_, Nucl. Phys. B641:3-34, 2002 ([arXiv:hep-th/0205297](https://arxiv.org/abs/hep-th/0205297))

* S. Sinha, [[Cumrun Vafa]], _$SO$ and $Sp$ Chern-Simons at Large $N$_ ([arXiv:hep-th/0012136](https://arxiv.org/abs/hep-th/0012136))

  (for [[Chern-Simons theory]] and [[topological string theory]])

On [[Landau-Ginzburg models]]:

* {#VafaWarner89} [[Cumrun Vafa]], [[Nicholas Warner]], _Catastrophes and the Classification of Conformal Theories_, Phys.Lett. B218 (1989) 51 (doi:10.1016/0370-2693(89)90473-5)

On [[D0-D4-brane bound states]]:

* [[Cumrun Vafa]], _Instantons on D-branes_, Nucl. Phys. B463 (1996) 435-442 ([arXiv:hep-th/9512078](https://arxiv.org/abs/hep-th/9512078))

with emphasis on the resulting [[configuration spaces of points]]:

* [[Cumrun Vafa]], [[Edward Witten]], Section 4.1 of: _A Strong Coupling Test of S-Duality_, Nucl. Phys. B431:3-77, 1994 ([arXiv:hep-th/9408074](https://arxiv.org/abs/hep-th/9408074))

On [[electric-magnetic duality]]:

* [[Cumrun Vafa]], _Geometric Origin of Montonen-Olive Duality_, Adv. Theor. Math. Phys. 1:158-166, 1998 ([arXiv:hep-th/9707131](https://arxiv.org/abs/hep-th/9707131))

On a [[mysterious duality]]:

* {#INV01} [[Amer Iqbal]], [[Andrew Neitzke]], [[Cumrun Vafa]], _A mysterious duality_, ([arXiv:hep-th/0111068](https://arxiv.org/abs/hep-th/0111068))

On [[topological M-theory]]:

* [[Robbert Dijkgraaf]], [[Sergei Gukov]], [[Andrew Neitzke]], [[Cumrun Vafa]], *Topological M-theory as Unification of Form Theories of Gravity*, Adv. Theor. Math. Phys. **9** (2005) 603-665 [[arXiv:hep-th/0411073](http://arxiv.org/abs/hep-th/0411073), [doi:10.4310/ATMP.2005.v9.n4.a5](https://dx.doi.org/10.4310/ATMP.2005.v9.n4.a5)]

On [[topological string theory]] and [[integrable systems]]:

* [[Mina Aganagic]], [[Robbert Dijkgraaf]], [[Albrecht Klemm]], [[Marcos Marino]], [[Cumrun Vafa]],  *Topological Strings and Integrable Hierarchies*, Commun. Math. Phys. **261** (2006) 451-516 [[arXiv:hep-th/0312085](https://arxiv.org/abs/hep-th/0312085), [doi:10.1007/s00220-005-1448-9](https://doi.org/10.1007/s00220-005-1448-9)]

* [[Mina Aganagic]], [[Miranda Cheng]], [[Robbert Dijkgraaf]], Daniel Krefl, [[Cumrun Vafa]], *Quantum Geometry of Refined Topological Strings*, J. High Energ. Phys. **2012** 19 (2012) [[arXiv:1105.0630](https://arxiv.org/abs/1105.0630), doi:10.1007/JHEP11(2012)019]

On [[moduli stabilization]] in [[non-geometric string vacua]]:

* [[Katrin Becker]], [[Melanie Becker]], [[Cumrun Vafa]], [[Johannes Walcher]], _Moduli Stabilization in Non-Geometric Backgrounds_,  Nucl. Phys. B **770** (2007) 1-46 [[arXiv:hep-th/0611001](https://arxiv.org/abs/hep-th/0611001)]

On [[M-theory on S1/G_HW times H/G_ADE]] and [[D=6 N=(1,0) SCFT]]:

* {#DHTV14} [[Michele Del Zotto]], [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], _6d Conformal Matter_, JHEP02(2015) 054 ([arXiv:1407.6359](https://arxiv.org/abs/1407.6359))

On the [[KK-compactification]] of [[D=6 N=(1,0) SCFT]] (on [[M5-branes]]) to [[D=4 N=1 super Yang-Mills]]:

* Ibrahima Bah, Christopher Beem, Nikolay Bobev, Brian Wecht, _Four-Dimensional SCFTs from M5-Branes_ ([arXiv:1203.0303](http://arxiv.org/abs/1203.0303))

* Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _$4d$ $\mathcal{N} = 1$ from $6d (1,0)$_, J. High Energ. Phys. (2017) 2017: 64 ([arXiv:1610.09178](https://arxiv.org/abs/1610.09178))

* Ibrahima Bah, [[Amihay Hanany]], Kazunobu Maruyoshi, Shlomo S. Razamat, Yuji Tachikawa, Gabi Zafrir, _$4d$ $\mathcal{N}=1$ from $6d$ $\mathcal{N}=(1,0)$ on a torus with fluxes_ ([arXiv:1702.04740](https://arxiv.org/abs/1702.04740))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _E-String Theory on Riemann Surfaces_, Fortsch. Phys. ([arXiv:1709.02496](https://arxiv.org/abs/1709.02496))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _D-type Conformal Matter and SU/USp Quivers_, JHEP06(2018)058 ([arXiv:1802.00620](https://arxiv.org/abs/1802.00620))

* Hee-Cheol Kim, Shlomo S. Razamat, [[Cumrun Vafa]], Gabi Zafrir, _Compactifications of ADE conformal matter on a torus_, JHEP09(2018)110 ([arXiv:1806.07620](https://arxiv.org/abs/1806.07620))

On the [[KK-compactification]] of [[D=6 N=(1,0) SCFT]] (on [[M5-branes]]) to [[D=5 super Yang-Mills theory]]:

* Lakshya Bhardwaj, Patrick Jefferson, Hee-Cheol Kim, Houri-Christina Tarazi, [[Cumrun Vafa]], _Twisted Circle Compactification of 6d SCFTs_ ([arXiv:1909.11666](https://arxiv.org/abs/1909.11666))

On [[M-strings]]:

* [[Babak Haghighat]], [[Amer Iqbal]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _M-Strings_, Commun. Math. Phys. 334, 779–842 (2015) ([arXiv:1305.6322](https://arxiv.org/abs/1305.6322), [doi:10.1007/s00220-014-2139-1](https://doi.org/10.1007/s00220-014-2139-1))

* [[Babak Haghighat]], [[Can Kozcaz]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _On orbifolds of M-Strings_, Physical Review D 89.4 (2014): 046003 ([arXiv:1310.1185](https://arxiv.org/abs/1310.1185))

On [[E-strings]]:

* J. A. Minahan, D. Nemeschansky, [[Cumrun Vafa]], [[Nicholas Warner]], _E-Strings and $N=4$ Topological Yang-Mills Theories_, Nucl. Phys. B527 (1998) 581-623 ([arXiv:hep-th/9802168](https://arxiv.org/abs/hep-th/9802168))

* {#HLV14} [[Babak Haghighat]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _Fusing E-string to heterotic string: $E + E \to H$_, Phys. Rev. D 90, 126012 (2014) ([arXiv:1406.0850](https://arxiv.org/abs/1406.0850))

* {#KKLPV14} [[Joonho Kim]], [[Seok Kim]], Kimyeong Lee, [[Jaemo Park]], [[Cumrun Vafa]], _Elliptic Genus of E-strings_, JHEP 1709 (2017) 098 ([arXiv:1411.2324](https://arxiv.org/abs/1411.2324))

and emergence of [[SU(2)]]  [[flavor (particle physics)|flavor]]-[[chiral symmetry|symmetry]] on [[M5-branes]] in [[heterotic M-theory]] (in the [[D=6 N=(1,0) SCFT]] on [[small instantons]] in [[heterotic string theory]]):

* {#GHKKLV15} [[Abhijit Gadde]], [[Babak Haghighat]], [[Joonho Kim]], [[Seok Kim]], [[Guglielmo Lockhart]], [[Cumrun Vafa]], _6d String Chains_, JHEP **1802** (2018) 143 [[arXiv:1504.04614](https://arxiv.org/abs/1504.04614)]

On [[swampland conjectures]]:

* {#Vafa05} [[Cumrun Vafa]], _The String Landscape and the Swampland_ ([arXiv:hepth/0509212](http://arxiv.org/abs/hepth/0509212))

* {#OoguriVafa06} [[Hirosi Ooguri]], [[Cumrun Vafa]], _On the Geometry of the String Landscape and the Swampland_, Nucl.Phys.B766:21-33, 2007 ([arXiv:hep-th/0605264](https://arxiv.org/abs/hep-th/0605264))

* {#BrennanCartaVafa17} T. Daniel Brennan, Federico Carta, [[Cumrun Vafa]], _The String Landscape, the Swampland, and the Missing Corner_ ([arXiv:1711.00864](https://arxiv.org/abs/1711.00864))

* Hee-Cheol Kim, Houri-Christina Tarazi, [[Cumrun Vafa]], _Four Dimensional $\mathcal{N}=4$ SYM and the Swampland_ ([arXiv:1912.06144](https://arxiv.org/abs/1912.06144))

On the [[M5-brane elliptic genus]]:

* {#GukovPeiPutrovVafa18} [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*,  J. High Energ. Phys. **2021** 84 (2021) [[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), doi:10.1007/JHEP05(2021)084, [spire:1704312](https://inspirehep.net/literature/1704312)]

On [[Chern-Simons theory]] potentially applicable to a form of [[Berry curvature]] on the [[Brillouin torus]] of 3d [[topological semimetals]], with nodal lines playing the role of [[Wilson lines]]:

* [[Biao Lian]], [[Cumrun Vafa]], [[Farzan Vafa]], [[Shou-Cheng Zhang]], *Chern-Simons theory and Wilson loops in the Brillouin zone*, Phys. Rev. B **95** (2017) 094512 [[doi:10.1103/PhysRevB.95.094512](https://doi.org/10.1103/PhysRevB.95.094512)]

Introducing the [[swampland cobordism conjecture]]:

* {#McNamaVafa17} [[Jacob McNamara]], [[Cumrun Vafa]], _Cobordism Classes and the Swampland_ ([arXiv:1909.10355](https://arxiv.org/abs/1909.10355), [spire:1755441](https://inspirehep.net/literature/1755441))

* [[Miguel Montero]], [[Cumrun Vafa]], _Cobordism Conjecture, Anomalies, and the String Lamppost Principle_, J. High Energ. Phys. 2021, 63 (2021) ([arXiv:2008.11729](https://arxiv.org/abs/2008.11729))

## Selected talks

On [[mirror symmetry]], [[topological string theory]] and a [[mysterious duality]]:

* {#Vafa00} [[Cumrun Vafa]], _Mirror symmetry_, Talk at [String Theory at the Millennium](http://theory.caltech.edu/conf2000/), Caltech, January 2000 ([slides html](http://www.theory.caltech.edu/conf2000/Vafa/fs1.html))

* _[Strings, and the magic of extra dimensions](http://www.icts.res.in/lecture/1/details/1632/)_, public talk at [ICTS](http://www.icts.res.in/home/), Bangalore, June 2013 ([pdf slides](http://www.icts.res.in/media/uploads/Lecture/Slides/Cumrun%20Vafa.pdf), [video](http://www.icts.res.in/media/uploads/Lecture/Slides/Cumrun%20Vafa.pdf))

* {#Vafa18} _F-theory, Black holes and topological strings_, talk at [ift 2018](https://www.ift.uam-csic.es/en/content/f-theory-black-holes-and-topological-strings), ([pdf](https://workshops.ift.uam-csic.es/files/221/Vafa.pdf), [recording](https://www.youtube.com/watch?v=Fmri3ke8Q-g))

On [[Ray-Singer torsion]] in relation to the  [[topological string]] and [[black hole entropy]]:

* [[Cumrun Vafa]], *Ray-Singer Torsion, Topological Strings and Black Holes* [[arXiv:2401.12816](https://arxiv.org/abs/2401.12816)]

## Related $n$Lab entries

* [[string theory]]

* [[F-theory]],

  * [[F'-theory]]

  * [[duality between M/F-theory and heterotic string theory]]

  * [[F-theory on CY4-folds]]

  * [[F-theory on Spin(7)-manifolds]]

  * [[Witten's Dark Fantasy]]

* [[E-string]]

* [[Gopakumar-Vafa duality]]

* [[Brandenberger-Vafa mechanism]]

* [[topological string]]

  * [[A-model]], [[B-model]]

* [[black holes in string theory]]

* [[geometric engineering of quantum field theory]]

* [[mirror symmetry]]

* [[6d (2,0)-superconformal QFT]]

* [[wall crossing]]

* [[landscape of string theory vacua]]

* [[mysterious duality]]

* [[weak gravity conjecture]]

* [[grand unified theory]]

* [[topological M-theory]]

* [[discrete torsion]]

* [[Landau-Ginzburg model]]

* [[bosonization]]

* [[orbifold]]

* [[flux compactification]]

* [[moduli stabilization]]

* [[E-string]]

* [[cosmic censorship hypothesis]]

* [[little string theory]]

* [[fractional D-brane]]

* [[conifold transition]]

* [[string phenomenology]]

* [[Riemann theta function]]

* [[axion]]

* [[enhanced gauge symmetry]]

* [[quiver gauge theory]]

* [[Kaluza-Klein monopole]]

* [[universal exceptionalism]]

* [[M-theory on G2-manifolds]]

* [[Hitchin functional]]

* [[self-dual higher gauge theory]]

* [[duality in physics]]

* [[Khovanov homology]]

* [[zeta function of an elliptic differential operator]]

* [[ADE classification]]

* [[theta function]]

* [[7-dimensional supergravity]]

* [[NS5-brane]]

* [[N=2 D=4 super Yang-Mills theory]]

* [[topologically twisted D=4 super Yang-Mills theory]]

* [[U-duality]]

* [[TCFT]]

* [[confinement]]

category: people


cumulant

(...)

## References

A discussion of cumulants as [[A-infinity algebra]] [[homomorphisms]] is in

* Gabriel C. Drummond-Cole, Jae-Suk Park, John Terilla, _Homotopy Probability Theory I_ ([arXiv:1302.3684](http://arxiv.org/abs/1302.3684))

[[!redirects cumulants]]


cumulative hierarchy

# Contents #
* table of contents
{:toc}

## Idea

In [[material set theory]], especially versions of set theory that accept the [[axiom of foundation]] that the membership relation $\in$ is an extensional (but not necessarily [[transitive relation|transitive]]) [[well-founded relation]], there is a picture of sets as forming a _cumulative hierarchy_, hierarchically ordered by a rank function valued in the ordinals. The idea is that on Day 0 the empty set is born; on Day $n+1$ is born the power set of Day $n$. On limit Days all the sets born earlier are collected together into a single set which is their union. The cumulative hierarchy picture amounts to the assertion that every set belongs to some set produced by this iterative procedure.

## In ZFC

Suppose $V$ is a [[model]] of [[ZFC]]. Recall that in the theory ZFC, a set $x$ is defined to be *transitive* if $b \in x$ implies $b \subset x$; this is the same as saying that if $a \in b \in x$ then $a \in x$. Note this doesn't mean that the relation $\in$ on the set consisting of $x$ and the elements of its transitive closure is itself [[transitive relation|transitive]]: the condition concerns only chains $a \in b \in c$ where $c = x$. However, if that relation *is* transitive, then it is transitive, and extensional (by the extensionality axiom), and well-founded (by the axiom of foundation), and hence a well-order according to the argument [here](/nlab/show/well-order#wellorders_are_linear). In this case, $x$ is by definition an [[ordinal number]] (in the sense of von Neumann).

Letting $On(V)$ be the class of ordinals (= ordinal numbers), one may define with the help of the replacement axiom a function

$$R: On(V) \to V$$

by transfinite induction as follows: $R(0) = 0$, while

* $R(\alpha + 1) = P(R(\alpha))$ (where $P$ denotes power set),

* $R(\beta) = \bigcup_{\alpha \lt \beta} R(\alpha)$ when $\beta$ is a limit ordinal.

+-- {: .num_prop}
###### Proposition
Each of the $R(\alpha)$ is transitive, and $\alpha \leq \beta$ implies $R(\alpha) \subseteq R(\beta)$.
=--

+-- {: .proof}
###### Proof
First, if $X$ is a transitive set, then so is $P(X)$. For suppose $A \in P(X)$. Then $A \subseteq X$, and so if $x \in A$, we have $x \in X$ and thus $x \subset X$ since $X$ is transitive, so that $x \in P(X)$. We have thus shown $A \subset P(X)$.

That each $R(\alpha)$ is transitive now follows by an easy induction.

And so $R(\alpha) \subset R(\alpha + 1) = P(R(\alpha))$ since $R(\alpha) \in P(R(\alpha))$, and now $\alpha \leq \beta \Rightarrow R(\alpha) \subseteq R(\beta)$ follows by an easy induction.
=--

If $x \in R(\gamma)$, then there is a least $\beta$ such that $x \in R(\beta)$, and this $\beta$ must be a successor ordinal, $\beta = \alpha + 1$. We define the _rank_ of $x$ to be that $\alpha$. Thus

* $\emptyset$ has rank $0$,

* $1 \coloneqq \{\emptyset\}$ has rank $1$,

* $\{1\}$ and $2 \coloneqq \{0, 1\}$ have rank $2$,

and so on. Each ordinal $\alpha$ has rank $\alpha$.

+-- {: .num_theorem}
###### Theorem
For every element $x$ of $V$, there is some $\alpha \in On(V)$ such that $x \in R(\alpha)$. Thus every set $x$ appears as an element somewhere within the cumulative hierarchy

$$R(0) \subset R(1) \subset \ldots \subset R(\omega) \subset R(\omega + 1) \subset \ldots \subset R(\omega + \omega) \subset \ldots$$
=--

The proof is essentially that $\bigcup_{\alpha \in On(V)} R(\alpha)$ is an $\in$-[[well-founded relation|inductive set]] of $V$, and so must be all of $V$ since $(V, \in)$ is well-founded (by the axiom of foundation). Details may be found in any reasonable text on ZFC set theory, for example [Kunen](#Kunen).

+-- {: .num_remark}
###### Remark
The notation $V$ so widely seen in set theory texts and articles is a kind of visual pun that refers to the cumulative hierarchy: one imagines the V as outlining an angle, with a horizontal cross-section of the space inside the angle at height $\alpha$ suggesting a set $R(\alpha)$, which expands as $\alpha$ increases; the ordinals $\alpha$ themselves may be pictured as vertebrae of a spine or line therein. All sets in the cumulative hierarchy lie somewhere within the V.
=--

## In algebraic set theory

The idea of the cumulative hierarchy is realized in [[algebraic set theory]] via the construction of an initial "ZF-algebra". In broad-brush terms, there is a general connection between well-foundedness and initial algebras, as in Paul Taylor's theory of recursion where an initial $T$-algebra (for a [[taut functor]] $T$) is seen as a [[well-founded coalgebra]] $(X, \theta: X \to T X)$ for which $\theta$ is an isomorphism (i.e., a maximal well-founded coalgebra). Algebraic set theory can be seen as exploiting this connection and working out the details in cases specific to operations on "small sets", eventually enabling one to get at the cumulative hierarchy *per se*, i.e., the universe of *well-founded sets* (as well as the universe of ordinals, etc.).

This deserves to be discussed at length, but let us try to give a few hints for now. One starts with a [[pretopos]] $\mathcal{C}$ (whose objects are regarded as "classes") equipped with a suitable notion of "smallness": to say a map $f: E \to X$ in the pretopos is "small" means intuitively that all its [[fibers]] are "small" (i.e., sets). Thus one assumes some reasonable axioms on the class of small maps in $\mathcal{C}$, including the existence of a universal small map "$el$": $E \to U$, with the elements $u$ of $U$ naming small sets and the fiber over $u$ the actual (small) set of its elements. The smallness axioms allow one to construct a small-[[power set]] functor $P_s: \mathcal{C} \to \mathcal{C}$; intuitively this sends a class $C$ to the class of small subsets of $C$. This carries a [[monad]] structure whose algebras $(X, \sup: P_s X \to X)$ are "small-complete" posets in $\mathcal{C}$.

To get at the actual cumulative hierarchy (with attendant global membership relation $\in$), one defines a *ZF-algebra* to be a small-complete poset $(V, \leq)$ in $\mathcal{C}$, equipped with a function $s: V \to V$ satisfying suitable conditions; here one is to think of $\leq$ as "inclusion" and $s(x)$ as a singleton $\{x\}$. The relation $x \in y$ can then be interpreted as $s(x) \leq y$. The initial object in the category of ZF-algebras then captures the desired universe of well-founded sets.

+-- {: .num_remark}
###### Remark
The details of the construction of the initial ZF-algebra should be examined with attention to connections with Taylor's theory of well-founded coalgebras; for example, systematic use is made of bisimulations which has a general meaning in coalgebra theory.
=--

This program, initiated by [[André Joyal]] and [[Ieke Moerdijk]], permits a fine-grained analysis of *intuitionistic* ZF-set theory and intuitionistic ordinals. The reader is referred to their [monograph](#JM) for details, and to the [Algebraic Set Theory](http://www.phil.cmu.edu/projects/ast/) page for further pointers to the literature.

## Related entries

* [[constructible universe]]

* [[reflection principle]]

* [[algebraic set theory]]

* [[set-theoretic multiverse]]

## References

The cumulative hierarchy made its first appearance in

* D. Mirimanoff, _Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles_ , L'enseignement Mathématique **19** (1917) pp.37-52. ([pdf](http://retro.seals.ch/cntmng?pid=ensmat-001:1917:19::9); 19,4MB)

Another historically important contribution is

* Dana Scott, _Axiomatizing Set Theory_ , pp.207-214 in _Axiomatic Set Theory_ - Proc. Symp. Pure Math. **13**, AMS Providence 1974.

A modern textbook account can be found e.g. in

* Kenneth Kunen, _Set Theory: An Introduction to Independence Proofs_, Studies in Logic and the Foundations of Mathematics Vol. 102 (2006), Elsevier.
 {#Kunen}

For algebraic set theory consult the following monograph

* André Joyal and Ieke Moerdijk, _Algebraic Set Theory_, London Math. Soc. Lecture Series Notes 220 (1995), Cambridge University Press.
 {#JM}


Cuntz algebra

A type of [[C*-algebra]].

## References

* Wikipedia, _[Cuntz algebra](http://en.wikipedia.org/wiki/Cuntz_algebra)_

[[!redirects Cuntz algebras]]