Cantor-Schroeder-Bernstein theorem

# Contents
* automatic table of contents goes here
{: toc}

## Statement

The _Cantor--Schroeder--Bernstein_ theorem says that the usual [[order relation]] on [[cardinalities]] of [[set]]s is [[antisymmetric relation|antisymmetric]]. In other words, define an order on sets by $X \leq Y$ if there exists a [[monomorphism]] $f\colon X \to Y$. Then, if both $X \leq Y$ and $Y \leq X$, there exists an [[isomorphism]] of sets $X \cong Y$.

The result is really only interesting in the absence of the [[axiom of choice]] ($AC$).  With $AC$, it is a trivial corollary of the [[well-ordering theorem]].  However, the theorem actually requires only [[excluded middle]], although it does not hold in [[constructive mathematics]] --- indeed, it is actually *equivalent* to excluded middle (at least assuming the [[axiom of infinity]]).

## Proof

We prove that the Cantor--Schroeder--Bernstein theorem holds in a [[Boolean topos]]. The theorem is not however [[intuitionistic logic|intuitionistically]] valid, in that it fails in some [[topos]]es, such as the topos $Set^{\bullet \to \bullet}$ (the [[arrow category]] of $Set$); see Example \ref{counterexample} below.

Throughout we use ordinary [[set theory|set-theoretic]] reasoning which can be translated into the formal theory of toposes. (This can be formalized via the [[Mitchell–Benabou language]], for instance.)

First, let's try a little pedagogy. Somehow functions $h: X \to Y, h^{-1}: Y \to X$ are to be cooked up from injections $f: X \to Y$ and $g: Y \to X$, so we might guess $h$ is to be defined as $f$ at least part of the time, and $h^{-1}$ as $g$ another part of the time. An ideal situation would be to have a set-up

$$f: A \stackrel{\sim}{\longrightarrow} B$$

$$\,$$

$$C \stackrel{\sim}{\longleftarrow} D: g$$

where $A, C$ are complementary subsets in $X$ and $B, D$ are complementary subsets in $Y$; then $h$ could be defined as $f$ on $A$ and as $g^{-1}$ on $C$, and everything works out fine. How can we achieve this?

If we have this situation, then apparently $B = f(A)$ (the direct image of $A$), and $D = \neg B = \neg f(A)$ (the complement of $f(A)$), and then $C = g(D) = g(\neg f(A))$, and finally $A, C$ are required to be complementary, so we would need

$$A = \neg g(\neg f(A)).$$

In other words, $A$ would be a [[fixed point]] of a suitable operation built from direct image and complementation operators. In fact, if we find such a fixed point $A$, then the plan above would work without a hitch.

Perhaps the simplest fixed-point theorem for this purpose on the market is

+-- {: .num_lemma}
###### Lemma (Knaster--Tarski fixed-point theorem)
Let $\phi\colon P X \to P X$ be an order-preserving map. Then there exists $A$ in $P X$ for which $\phi(A) = A$.
=--

+-- {: .proof}
###### Proof
Let $A$ be the (internal) [[intersection]] of $U = \{T \in P X : \phi(T) \leq T\}$. Since $A \leq T$ for every $T$ in $U$, we have $\phi(A) \leq \phi(T) \leq T$ for every $T$ in $U$. Hence $\phi(A) \leq A$ by definition of $A$. Applying $\phi$ again, we get $\phi \phi(A) \leq \phi(A)$. Hence $\phi(A)$ belongs to $U$. But then $A \leq \phi(A)$ by definition of $A$.
=--

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

The preceding proof is valid in _any_ topos (and so holds for $Set$ even intuitionistically). It can be seen as a specialization to posets of a result of Lambek on the [[initial algebra|initial]] [[algebra of an endofunctor]], saying that the structure maps of such initial algebras are necessarily isomorphisms. Here the initial algebra $A$ is (by construction) an initial fixed point.

=--

+-- {: .proof}
###### Proof of Cantor--Schroeder--Bernstein

Suppose given two monos $f: X \to Y$, $g: Y \to X$. Let $\exists_f: P X \to P Y$ denote direct image or [[existential quantification]] along $f$, and let $\neg_X: P X \to P X$ denote [[negation]]. Then the composite

$$\phi = \neg_X \exists_g \neg_Y \exists_f: P X \to P X$$

is order-preserving, and so has a fixed point $A$ by the Knaster-Tarski lemma. Now define $h: X \to Y$ by the rule

$$\array{
h(x) & = & f(x) & \text{ if }\; x \in A \\
h(x) & = & g^{-1}(x) & \text{ if }\; x \notin A
}$$

(the multi-line definition is where we use the Boolean condition). The second line makes sense because $\neg A$ is in the image of $g$. The inverse of $h$ is

$$\array{
j(y) & = & f^{-1}(y) & \; \text{if} \; y \in \exists_f(A) \\
j(y) & = & g(y) & \; \text{if} \; y \notin \exists_f(A)
}$$

That $j$ is inverse to $h$ uses the fact that $\neg A = \exists_g \neg \exists_f(A)$. The rest is obvious.
=--

This classic proof is substantially the proof given in Johnstone's [[Elephant]], D4.1.11. The Boolean condition is not *strictly speaking* necessary, i.e., the [[principle of excluded middle]] ($EM$) does not logically follow from the Cantor--Schroeder--Bernstein statement since, for example, using notation as above, monic endomorphisms like $f \circ g: Y \to Y$ and $g \circ f: X \to X$ are isomorphisms in

$$FinSet^C,$$

but this is a non-Boolean topos provided that $C$ is any [[finite category]] that is not a [[groupoid]], e.g., $C = \mathbf{2}$. But that being said, $EM$ is certainly the most natural supposition to make. In fact EM does constructively follow from the Cantor-Schroeder-Bernstein statement provided that a [[natural numbers object]] exists; see [Pradic and Brown, 2019](#PB19).
A proof that in a topos CSB+NNO implies Booleanness is outlined in [Freyd  1994](#Freyd94).

### Alternative construction of a fixed point {#alt}

In some schools of thought, the proof using the Knaster-Tarski lemma would be criticized because that lemma makes use of an [[impredicative]] construction. However, the application made of it in the proof of the CSB theorem is only to ensure that the operator $\neg_X \exists_g \neg_Y \exists_f: P X \to P X$ has a fixed point. This objection can be countered by shopping around for a different fixed-point theorem, one which is predicatively and constructively valid.

A time-honored way of constructing a fixed point of an operator $\phi$ is by taking a limit of a sequence of iterates of $\phi$ that converges, provided that $\phi$ preserves the limit. To this end, we find that specializing Adámek's theorem (see [[initial algebra of an endofunctor]]) suits our purposes perfectly.

+-- {: .num_lemma #chain}
###### Lemma
If $g: Y \to X$ is monic, then the operator $\exists_g: P Y \to P X$ preserves limits of inverse chains $\omega^{op} \to P Y$ (i.e. intersections of decreasing sequences).
=--

+-- {: .proof}
###### Proof
More generally, $\exists_g$ preserves [[connected limits]], because it lifts through the inclusion $i: P X \downarrow \exists_g(1) \hookrightarrow P X$ to an isomorphism $P Y \stackrel{\sim}{\to} P X \downarrow \exists_g(1)$ (here $1$ denotes the top element of $P Y$, aka $Y$), and $i$ preserves connected limits.

In more detail: by [[Frobenius reciprocity]], we have $\exists_g T \wedge S = \exists_g(T \wedge g^\ast S)$ for elements $S$ of $P X$ and $T$ of $P Y$. Putting $T = 1$, we get $\exists_g 1 \wedge S = \exists_g g^\ast S$, and so the composite

$$P X \downarrow \exists_g 1 \stackrel{g^\ast}{\to} P Y \stackrel{\exists_g}{\to} P X \downarrow \exists_g 1$$

is the identity. But since $g$ is monic, $g^\ast \exists_g: P Y \to P Y$ is also the identity, which completes the proof.
=--

+-- {: .num_cor}
###### Corollary
$\forall_g = \neg_X \exists_g \neg_Y: P Y \to P X$ preserves colimits of $\omega$-chains.
=--

Naturally the left adjoint $\exists_f$ also preserves such colimits. So by the corollary, the composite $\neg_X \exists_g \neg_Y \exists_f: P X \to P X$ preserves colimits of $\omega$-chains.

Putting now $A_0 = 0$ (the bottom element of $P X$), $A_1 = \neg_X \exists_g \neg_Y \exists_f(0)$, and generally

$$\label{An} A_n = (\neg_X \exists_g \neg_Y \exists_f)^n(0),$$

we have $A_n \subseteq A_{n+1}$ (apply the monotone operator $(\neg_X \exists_g \neg_Y \exists_f)^n$ to the inclusion $A_0 = 0 \subseteq A_1$), and so $\neg_X \exists_g \neg_Y \exists_f$ preserves the union of the chain $A_0 \subseteq A_1 \subseteq \ldots$,

$$\label{A} A = \bigcup_{n \geq 0} (\neg_X \exists_g \neg_Y \exists_f)^n(0),$$

which implies that $A$ is a fixed point of $\neg_X \exists_g \neg_Y \exists_f$, as desired. In fact this $A$ is the minimal fixed point, just as in the conclusion of the Knaster-Tarski lemma. (Cf. [[initial algebra of an endofunctor]], especially Adámek's theorem.)

### Beta reduced proof

The preceding proofs are sometimes considered too abstract to easily visualize, but this is slightly misleading: the second proof, involving the construction of a minimal fixed point as a countable limit, can be "[[beta-reduced]]" to produce one of the standard "concrete" proofs.

In a nutshell, the minimal fixed point of the operator $\neg \exists_g \neg \exists_f: P X \to P X$ can be expressed as an alternating series of iterated direct images:

$$\label{series} X - g Y + g f X - g f g Y + \ldots$$

where $-$ stands for set-theoretic difference $\setminus$ and $+$ stands for the union $\cup$. The meaning of the infinite series is that we have a increasing sequence of those finite alternating sums with an even number of terms, starting with the empty sum (which is $0$, the empty set):

$$0$$

$$\,$$

$$X - g Y,$$

$$\,$$

$$X - g Y + g f X - g f g Y,$$

etc., and the infinite series is interpreted as the countable union of this increasing sequence. Note that we have to be careful about the order of the appearance of $+$ and $-$, but alternatively, letting $\oplus$ be the addition in the Boolean *ring* $P X$ (symmetric difference), we could write also the series as $X \oplus g Y \oplus g f X \oplus \ldots$ in the ring, where we do not need to be fussy about order.

Most of this is a routine calculation, which for the most part boils down to the following observation:

+-- {: .num_lemma}
###### Lemma
If $B, D$ are elements of $P Y$, with $D \leq B$, then $\exists_g(B - D) = \exists_g B - \exists_g D$. (With a similar statement for $\exists_f$.)
=--

The proof is left to the reader, but in brief, the injectivity of $g$ implies that $\exists_g$ preserves binary intersections and relative complements.

From here, if we write

$$\neg \exists_g \neg \exists_f(B) = X - g(Y - f B) = X - g Y + g f B = X \oplus g Y \oplus g f B,$$

then it is easily verified by induction that, referring to equation (eq:An),

$$A_n = (\neg \exists_g \neg \exists_f)^n 0 = \bigoplus_{j=0}^n (g f)^j X \oplus \bigoplus_{j=0}^n g(f g)^j Y.$$

Thus, according to equation (eq:A), the minimal fixed point $A$ is the union of the $A_n$ which is how we are interpreting the series (eq:series).

Now we set up a comparison with one of the standard proofs involving a [[back-and-forth argument]], say the one given in [Wikipedia](https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem#Proof) that is attributed to Julius König. The minimal fixed point is a union of finitary approximations

$$A_1 = X - g Y,$$

$$\,$$

$$A_2 = X - g Y + g f X - g f g Y,$$

etc. Elements in $A_1$ are those which have no inverse images under $g$. Elements in $A_2$ are elements in $X$ to which $(g f)^{-1}$ can be applied at most once before we hit an element of $X$ with no inverse image under $g^{-1}$. Elements in the union $A_1 \cup A_2 \cup \ldots$ are those which survive at most finitely many applications of $(g f)^{-1}$ before hitting an element of $X$ with no inverse image under $g$. In the terminology of the Wikipedia article, such elements $x$ in $X$ are called "$X$-stoppers", and these are exactly the elements for which $h(x)$ (where recall $h$ is the bijection under construction) is defined to be $f(x)$ in the Wikipedia article. For elements $x$ not in this fixed point $A$ (the non-$X$-stoppers), our proof of CSB (via a minimal fixed point) defined $h(x)$ to be $g^{-1}(x)$, the same prescription that is used in the Wikipedia article.

Other prescriptions are possible. For example, one could dually construct a *maximal* fixed point of the operator $\neg \exists_g \neg \exists_f: P X \to P X$, using Lemma \ref{chain} to note that $\exists_f$ and the right adjoint $\forall_g = \neg \exists_g \neg$ preserve limits of inverse $\omega$-chains, so that the maximal fixed point or terminal algebra of the endofunctor $\neg \exists_g \neg \exists_f$ could be constructed as an intersection $\bigcap_{n \geq 1} (\neg \exists_g \neg \exists_f)^n(1)$. This could also be written as a series

$$X - g Y + g f X - g f g Y + g f g f X - \ldots$$

except this time the series is interpreted as an *intersection* of a *decreasing* sequence whose partial sums have an *odd* number of terms:

$$1 = X,$$

$$\,$$

$$\neg \exists_g \neg \exists_f(1) = X - g Y + g f X,$$

$$\,$$

$$(\neg \exists_g \neg \exists_f)^2(1) = X - g Y + g f X - g f g Y + g f g f X,$$

etc. The difference between this and the minimal fixed point is the set $\bigcap_{n \geq 1} (g f)^n(X)$, consisting of elements $x$ of $X$ that belong to a doubly infinite sequence or a cyclic sequence (in the terminology of the Wikipedia article). As remarked in that article, for such $x$ we have an option to define $h(x)$ as $f(x)$ or $g^{-1}(x)$; in the present article we defined $h(x) = f(x)$ for all $x$ belonging to whichever fixed point $A$ is used, which includes points in doubly infinite sequences or cyclic sequences if $A$ is the maximal fixed point. In that case the remaining $x$ (belonging to the complement of the maximal fixed point) are mapped to $g^{-1}(x)$.

## Failure in toposes and constructive mathematics

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

Counterexample \ref{counterexample2} below shows that the CSB theorem fails in Brouwer\'s [[intuitionistic mathematics]] even for $Set$ (since every function between the sets $[0, 1]$ and $\mathbb{R}$ must be continuous by Brouwer\'s continuity principle!). See also the discussion in [Mac Lane-Moerdijk](#MM), VI.9, on toposes that realize Brouwer's theorem.
=--

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

As mentioned above, the Cantor-Schroeder-Bernstein theorem fails in the arrow category $Set^\to$, whose objects are functions $X_0 \to X_1$ between sets and whose morphisms are commutative squares. For example, let $X$ be the object $f: \mathbb{N} \to \mathbb{N}$ that takes $n \in \mathbb{N}$ to $\mathrm{int}(n/2)$, where $\mathrm{int}(x)$ is the greatest integer less than or equal to $x$; let $Y$ be the object $g: \mathbb{N} \to \mathbb{N}$ that takes $n$ to $\mathrm{Int}((n+1)/2)$, where $\mathrm{Int}(x)$ is the least integer greater than or equal to $x$. Pretty clearly $X$ and $Y$ are non-isomorphic, because $g^{-1}(0)$ has cardinality $1$ whereas all fibers of $f$ have cardinality $2$. But, just by drawing pictures of these objects, it is easy to construct monomophisms $i: X \to Y$ and $j: Y \to X$ (e.g., define $i_0(n) = n+1$ and $i_1(n) = n+1$ for all $n$, and define $j_0(n) = n+1$ for $n \gt 0$, $j_0(0) = 0$, and $j_1(n) = n$ for all $n$).
=--

Nor can one have internal existence of an isomorphism between $X$ and $Y$ in this last example, since internal existence implies external existence as soon as the terminal object is (externally) projective.

In fact, the CSB theorem is equivalent in [[constructive mathematics]] (with the [[axiom of infinity]]) to the [[law of excluded middle]].  This was shown in [Pradic and Brown, 2019](#PB19) using the [[principle of omniscience]] for the [[extended natural numbers]].

## In other categories

The CSB property holds in some other [[categories]] of interest (but arguably fails in many more). Some examples follow:

+-- {: .num_example #model}
###### Example
The CSB property holds in the category of [[vector spaces]] and in the category of [[algebraically closed field]]s. See also this [MO post](http://mathoverflow.net/questions/1058/when-does-cantor-bernstein-hold) by John Goodrick, where model-theoretic criteria come into play, sometimes under strengthenings of the notion of monomorphism (e.g., [[elementary embedding]], [[split monomorphism]]) Some slides by Goodrick [here](http://settheory.mathtalks.org/wp-content/uploads/2012/06/jonh_goodrick.pdf) go into more detail, giving connections between CSB and [[stability theory|stable theories]] in the sense of [[Shelah]].
=--

+-- {: .num_example #counterexample2}
###### Counterexample
On the other hand, the CSB property obviously fails in [[Top]], since we have [[embeddings]] $\mathbb{R} \cong (0,1) \to [0,1] \to \mathbb{R}$, yet $[0,1] \ncong \mathbb{R}$. It fails in [[Grp]] (e.g., the free group on countably many generators embeds in the free group on two generators).
=--

More examples and discussion can be found at this Secret Blogging Seminar [post](http://sbseminar.wordpress.com/2007/10/30/theme-and-variations-schroeder-bernstein/).

In a [celebrated work](#Gowers), [[Timothy Gowers]] gave a negative solution in the case of [[Banach spaces]].

Martin Escardó has announced a proof of CSB in the base theory of [[Martin-Löf dependent type theory]] with the added assumption of [[function extensionality]] and [[excluded middle]], and supplied an [[Agda]] proof ([Escardó 2020](#Escardo20)). There is a slight subtlety in that "injection" is replaced by "embedding", where the latter is defined to be a map whose fibres are [[subsingletons]]. This implies that CSB holds for [[homotopy types]]/$\infty$-groupoids.

## Name

The CSB theorem was first stated by [[Georg Cantor]], but his proof relied on the [[well-ordering theorem]].  The modern (choice-free) theorem was proved (independently) by [[Felix Bernstein]] and [[Ernst Schröder]].  It has been variously named after two or three of these in almost every possible combination, although Cantor (when mentioned at all) seems always to be mentioned first.

[Wikipedia](http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem#History) reports that [[Richard Dedekind]] had an (unpublished) proof in 1887, well before any announced proofs by Cantor, Schroeder, or Bernstein in 1895, 1896, 1897 respectively.

## References

* [[Peter Johnstone]], _[[Sketches of an Elephant]]: A Topos Theory Conpendium_, Vol. I, Clarendon Press, Oxford (2002)

* {#Gowers} [[Timothy Gowers]], _A Solution to the Schroeder-Bernstein Problem for Banach Spaces_, Bulletin of the London Mathematical Society, Volume 28, Issue 3 (1996), 297-304 [(abstract)](http://blms.oxfordjournals.org/content/28/3/297.abstract)

* {#MM} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_, Springer-Verlag 1992.

* {#Freyd94}[[Peter Freyd]], _Cantor-Bernstein - message to the category theory mailing list_, February 1994. ([link](https://www.mta.ca/~cat-dist/archive/1994/94-2))

* {#PB19} Pierre Pradic, Chad E. Brown, *Cantor-Bernstein implies Excluded Middle*, 2019 ([arxiv:1904.09193](https://arxiv.org/abs/1904.09193))

* {#Escardo20} Martín Hötzel Escardó, _The Cantor-Schröder-Bernstein Theorem for ∞-groupoids_, 2020 ([blog post](https://homotopytypetheory.org/2020/01/26/the-cantor-schroder-bernstein-theorem-for-%e2%88%9e-groupoids/), [Agda proof](https://www.cs.bham.ac.uk/~mhe/agda-new/CantorSchroederBernstein.html), [arXiv:2002.07079](https://arxiv.org/abs/2002.07079))

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cap product

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

#Contents#
* table of contents
{:toc}

## Idea

In [[ordinary homology]]/[[ordinary cohomology]] represented as [[singular homology]]/[[singular cohomology]], then the _cap product_ operation over a [[topological space]] $X$ is the pairing

$$
  H^p(X)\otimes H_{p+q}(X) \longrightarrow H_q(X)
$$

given by combining the [[Kronecker pairing]] of the cohomology class with the image of the homology class under [[diagonal]] and using the [Eilenberg-Zilber theorem](Eilenberg-Zilber+theorem#ProdTopSp).

More generally, in [[relative cohomology|relative]] [[generalized (Eilenberg-Steenrod) cohomology]]/[[generalized homology]] represented by a [[ring spectrum]] $E$, then the cap product

$$
  E^p(X,A) \otimes E_{(p+q)}(X, A) \longrightarrow E_q(X)
$$

is given on representatives $(\alpha,\beta)$

$$
  [X/A \overset{\alpha}{\longrightarrow} \Sigma^{p} E]
  \,,
  \;\;\;
  [S^{p+q} \overset{\beta}{\to} E \wedge X/A]
$$

(where $X/A$ denotes the [[homotopy cofiber]]/[[mapping cone]] by a map $A \to X$) by the element represented by the composite

$$
  S^{p+q}
   \overset{\beta}{\longrightarrow}
  E \wedge X/A
   \overset{id \wedge \Delta}{\longrightarrow}
  E \wedge X/A \wedge X
   \overset{id \wedge \alpha \wedge id}{\longrightarrow}
  E \wedge \Sigma^p E \wedge X
    \overset{\mu \wedge id}{\longrightarrow}
  \Sigma^p E \wedge X
$$

([e.g. Kochman 96, p. 136](#Kochman96))

This pairing induces a corresponding pairing on [[Atiyah-Hirzebruch spectral sequences]] (in the manner discussed at [[multiplicative cohomology theory]]/[[multiplicative spectral sequence]]), such that on the second page it restricts to the cap prodct in ordinary cohomology.

([e.g. Kochman 96, p. 138](#Kochman96))

## Related concepts

* [[cup product]]

* [[intersection pairing]]

* [[Poincaré duality]]

* [[fundamental class]]

## References

* {#Kochman96} [[Stanley Kochman]], (4.2.1) and prop. 4.2.10 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

[[!redirects cap products]]


carbon

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[chemical element]] with [[atomic number]] 6.

## Properties

### Skyrmion model

At least some aspects of the carbon [[atomic nucleus]] may be modeled by [[Skyrmions]]

([Lau-Manton 14](#LauManton14)):

\begin{center}
<img src="https://ncatlab.org/nlab/files/SkyrmionB12.jpg" width="300">
\end{center}

> graphics grabbed form [Lau-Manton 14](#LauManton14)

## Related entries

* [[diamond]]

* [[nitrogen-vacancy center in diamond]]

## References

See also:

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

Discussion of the carbon [[atomic nucleus]] via the [[Skyrmion model]]:

* {#LauManton14} P.H.C. Lau, [[Nicholas Manton]], _States of Carbon-12 in the Skyrme Model_,  Phys. Rev. Lett. 113, 232503 (2014) ([arXiv:1408.6680](https://arxiv.org/abs/1408.6680))

* Christoph Adam, [[Carlos Naya]], Andrzej Wereszczyński, *Carbon-12 in the generalized Skyrme model* [[arXiv:2401.08778](https://arxiv.org/abs/2401.08778)]


cardinal arithmetic

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#### Arithmetic
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[[!include arithmetic geometry - contents]]
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=--
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#Contents#
* table of contents
{:toc}

## Idea

Cardinal arithmetic is an [[arithmetic]] with [[cardinals]], which generalizes the ordinary [[arithmetic]] of [[natural numbers]] to non-[[finite set|finite numbers]].

The idea is that on [[finite sets]], whose [[cardinalities]] are [[natural numbers]], the usual operations of [[arithmetic]] -- [[addition]], [[multiplication]] and [[exponentiation]] -- are represented on [[finite sets]] by basic operations of [[set theory]], namely forming [[disjoint union]] of sets, forming [[Cartesian product]] of sets and forming [[function sets]] of sets.

| $\phantom{A}$[[natural numbers]]$\phantom{A}$ | $\phantom{A}$[[finite sets]]$\phantom{A}$ |
|---------------------|-----------------|
| $\phantom{A}$[[addition]]$\phantom{A}$ | $\phantom{A}$[[disjoint union]]$\phantom{A}$ |
| $\phantom{A}$[[multiplication]]$\phantom{A}$ | $\phantom{A}$[[Cartesian product]]$\phantom{A}$ |
| $\phantom{A}$[[exponentiation]]$\phantom{A}$ | $\phantom{A}$[[function sets]]$\phantom{A}$ |

(Here passing from the left to the right column is an example of _[[vertical categorification|categorification]]_, while passing from right to left is an example _[[decategorification]]_.)

But the operations on sets on the right directly generalize from [[finite sets]] to general sets, hence from sets whose [[cardinality]] is a [[natural number]] to [[infinite cardinals]]. Therefore cardinal arithmetic is also called a _[[transfinite arithmetic]]_.

## Definitions

For $S$ a [[set]], write ${|S|}$ for its [[cardinality]]. Then the standard operations in the [[category]] [[Set]] induce [[arithmetic]] operations on [[cardinal numbers]]:

For $S_1$ and $S_2$ two sets, the **sum** of their cardinalities is the cardinality of their [[disjoint union]], the [[coproduct]] in $Set$:
$$
  {|S_1|} + {|S_2|} \coloneqq {|S_1 \amalg S_2|}
  \,.
$$

More generally, given any family $(S_i)_{i: I}$ of sets indexed by a set $I$, the **sum** of their cardinalities is the cardinality of their disjoint union:
$$
  \sum_{i: I} {|S_i|} \coloneqq {|\coprod_{i: I} S_i|}
  \,.
$$

Likewise, the **product** of their cardinalities is the cardinality of their [[cartesian product]], the [[product]] in $Set$:
$$
  {|S_1|} \, {|S_2|} \coloneqq {|S_1 \times S_2|}
  \,.
$$

More generally again, given any family $(S_i)_{i: I}$ of sets indexed by a set $I$, the **product** of their cardinalities is the cardinality of their cartesian product:
$$
  \prod_{i: I} {|S_i|} \coloneqq {|\prod_{i: I} S_i|}
  \,.
$$

Also, the **exponential** of one cardinality raised to the power of the other is the cardinality of their [[function set]], the [[exponential object]] in $Set$:
$$
  {|S_1|}^{|S_2|} \coloneqq {|Set(S_2,S_1)|}
  \,.
$$

In particular, we have $2^{|S|}$, which (assuming the law of [[excluded middle]]) is the cardinality of the [[power set]] $P(S)$.  In [[constructive mathematics|constructive]] (but not [[predicative mathematics|predicative]]) mathematics, the cardinality of the power set is $\Omega^{|S|}$, where $\Omega$ is the cardinality of the set of [[truth values]].

The usual way to define an ordering on cardinal numbers is that ${|S_1|} \leq {|S_2|}$ if there exists an [[injection]] from $S_1$ to $S_2$:
$$
  ({|S_1|} \leq {|S_2|})
  \;:\Leftrightarrow\;
  (\exists (S_1 \hookrightarrow S_2))
  \,.
$$
Classically, this is almost equivalent to the existence of a [[surjection]] $S_2 \to S_1$, except when $S_1$ is [[empty set|empty]].  Even restricting to [[inhabited sets]], these are not equivalent conditions in [[constructive mathematics]], so one may instead define that ${|S_1|} \leq {|S_2|}$ if there exists a [[subset]] $X$ of $S_2$ and a surjection $X \to S_1$.  Another alternative is to require that $S_1$ (or $X$) be a [[decidable subset]] of $S_2$.  All of these definitions are equivalent using [[excluded middle]].

This order relation is [[antisymmetric relation|antisymmetric]] (and therefore a [[partial order]]) by the [[Cantor–Schroeder–Bernstein theorem]] (proved by Cantor using the [[well-ordering theorem]], then proved by Schroeder and Bernstein without it).  That is, if $S_1 \hookrightarrow S_2$ and $S_2 \hookrightarrow S_1$ exist, then a bijection $S_1 \cong S_2$ exists.  This theorem is not constructively valid, however.

The well-ordered cardinals are [[well-order|well-ordered]] by the ordering $\lt$ on [[ordinal numbers]].  Assuming the [[axiom of choice]], this agrees with the previous order in the sense that $\kappa \leq \lambda$ iff $\kappa \lt \lambda$ or $\kappa = \lambda$.  Another definition is to define that $\kappa \lt \lambda$ if $\kappa^+ \leq \lambda$, using the successor operation below.

The __[[successor]]__ of a well-ordered cardinal $\kappa$ is the smallest well-ordered cardinal larger than $\kappa$.  Note that (except for finite cardinals), this is different from $\kappa$\'s successor as an [[ordinal number]].  We can also take successors of arbitrary cardinals using the operation of [[Hartog's number]], although this won\'t quite have the properties that we want of a successor without the axiom of choice.

### Remarks

* It is traditional to write [[ℵ]]${}_0$ for the first [[infinite cardinal]] (the cardinality of the [[natural numbers]]), $\aleph_1$ for the next (the first uncountable cardinality), and so on.  In this way every cardinal (assuming choice) is labeled $\aleph_\mu$ for a unique [[ordinal number]] $\mu$, with $(\aleph_\mu))^+ = \aleph_{\mu^+}$.

* For every cardinal $\pi$, we have $2^\pi \gt \pi$ (this is sometimes called [[Cantor's theorem]]).  The question of whether $2^{\aleph_0} = \aleph_{1}$ (or more generally whether $2^{\aleph_\mu} = \aleph_{\mu^+}$) is called Cantor's continuum problem; the assertion that this is the case is called the (generalized) [[continuum hypothesis]]. It is known that the continuum hypothesis is undecidable in [[ZFC]].

## Properties

+-- {: .num_theorem}
###### Theorem
For every [[infinite cardinal]] $\pi$ we have (using the axiom of choice) $\pi + \pi = \pi$ and $\pi \cdot \pi = \pi$, so addition and multiplication are [[idempotent]].
=--

+-- {: .proof}
###### Proof
Since $\pi \leq \pi + \pi = 2 \cdot \pi \leq \pi \cdot \pi$, it suffices to prove $\pi \cdot \pi \leq \pi$. Establishing this is closely analogous to establishing one of the standard bijections $\mathbb{N} \cong \mathbb{N} \times \mathbb{N}$ -- not the one that enumerates along successive diagonals  $x + y = k$ which are spheres in an $L^1$ [[p-norm|metric]] (and which involves additive structure), but one which enumerates along successive spheres in an $L^\infty$ metric (which involves only order structure).

With this hint in mind, regard $\pi$ as the least [[ordinal]] in its cardinality class (using the [[well-ordering theorem]]), and suppose $\pi$ is a minimal counterexample to the statement. Thus ${|\alpha|}^2 = {|\alpha|}$ for all ordinals $\alpha \lt \pi$. Consider $\pi^3 = \pi \times \pi \times \pi$ in [[lexicographic order]]. This is a well-ordered set, as is the subset

$$S = \{(x, y, z) \in \pi^3: x = \max(y, z)\}$$

under the order inherited from $\pi^3$. Clearly $S \cong \pi^2$ as sets. Under the supposition ${|\pi|} \lt {|\pi|}^2$, the ordinal $\pi$ is isomorphic to one of the [[lower set|initial segments]]

$$S(a, b, c) \coloneqq \{(x, y, z) \in S: (x, y, z) \lt (a, b, c)\}$$

of $S$, say $S(\alpha, \beta, \gamma)$. But then (regarding $\alpha$ as an ordinal less than $\pi$)

$${|\pi|} = {|S(\alpha, \beta, \gamma)|} \leq {|S(\alpha, \alpha, \alpha)|} = {|\alpha|}^2 = {|\alpha|} \lt {|\pi|}$$

which gives a contradiction.
=--

+-- {: .num_remark}
###### Remark
Conversely, if (relative to ZF or any other standard set theory) we assume that any infinite set can be put in bijection with its cartesian square, then every set can be well-ordered, a result originally due to Tarski. Details may be found at [[Hartogs number]].
=--

+-- {: .num_cor}
###### Corollary
If one of two non-zero cardinals $\kappa, \lambda$ is infinite, then

$$\kappa + \lambda = \kappa \cdot \lambda = \max \{\kappa, \lambda\}.$$
=--

+-- {: .proof}
###### Proof
Clearly, $\kappa \leq \kappa + \lambda$ and $\lambda \leq \kappa + \lambda$, so $\max \{\kappa, \lambda\} \leq \kappa + \lambda$, and similarly for multiplication instead of addition, assuming the cardinals are non-zero.
Letting $\mu = \max \{\kappa, \lambda\}$, we also have

$$\kappa \cdot \lambda \leq \mu \cdot \mu = \mu$$

and similarly for addition instead of multiplication.
=--

## References

Traditional lecture notes include

* Alexandru Baltag, _Axiomatix set  theory lecture 8: Cardinal arithmetic_ ([pdf](http://www.vub.ac.be/CLWF/SS/Sets8.pdf))

* _Cardinal arithmetic_ ([pdf](https://www.math.ksu.edu/~nagy/real-an/ap-b-card.pdf))

See also

* Planetmath, _[cardinal arithmetic](http://planetmath.org/cardinalarithmetic)_

The point of view of [[categorification]] is amplified in

* [[Emily Riehl]], _Categorifying cardinal arithmetic_ ([pdf](http://www.math.jhu.edu/~eriehl/arithmetic.pdf))


cardinal number

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _cardinal numbers_ (or just _cardinals_) constitute a generalisation of [[natural number|natural numbers]] to [[numbers]] of possibly infinite magnitudes.  Specifically, cardinal numbers generalise the concept of 'the number of ...'.  In particular, the number of natural numbers is the first infinite cardinal number.

## Definition

Naïvely, a **cardinal number** should be an [[isomorphism]] class of [[sets]], and the **cardinality** of a set $S$ would be its isomorphism class.  That is:

1. every set has a unique cardinal number as its cardinality;
2. every cardinal number is the cardinality of some set;
3. two sets have the same cardinality if and only if they are isomorphic as sets.

Then a **finite cardinal** is the cardinality of a [[finite set]], while an **infinite cardinal** or **transfinite cardinal** is the cardinality of an [[infinite set]].  (If you interpret both terms in the strictest sense, then there may be cardinals that are neither finite nor infinite, without some form of the [[axiom of choice]]).

Taking this definition literally in material [[set theory]], each cardinal is then a [[proper class]] (so one could not make further sets using them as elements).  For this reason, in axiomatic set theory one usually defines a cardinal number as a particular representative of this [[equivalence class]].  There are several ways to do this:

* The __cardinality__ of a set $S$ is the smallest possible ordinal rank of any [[well-order]] on $S$.  In other words, it is the smallest [[ordinal number]] (usually defined following von Neumann) which can be put in bijection with $S$.  A __cardinal number__ is then any cardinality, i.e. any ordinal which is not in bijection with any smaller ordinal.

  * On well-orderable sets, this cardinality function satisfies (1--3), but one needs the [[axiom of choice]] (precisely, the [[well-ordering theorem]]) to prove that every set is well-orderable.  This approach is probably the most common one in the presence of the axiom of choice.

  * In the absence of [[excluded middle]], when the "correct" definition of well-order is different from the usual one (and so "the least ordinal such that ..." may not exist), a better definition of the cardinality of $S$ is as the set of all [[ordinal numbers]] less than the ordinal rank of every [[well-order]] on $S$.

* Alternatively, we can define the __cardinality__ of a set $X$ to be the set of all well-founded [[pure sets]] that are isomorphic as sets to $X$ and such that no pure set of smaller hereditary rank (that is, which occurs earlier in the [[von Neumann hierarchy]]) is isomorphic to $X$.

  * On those sets that are isomorphic to some well-founded set, this cardinality function satisfies (1--3), but one needs some assumption to prove that every set is isomorphic to a well-founded set.  (This will follow directly from the [[axiom of foundation]]; it will also follow from the [[axiom of choice]], since then every set is isomorphic to a von Neumann ordinal.)

In the absence of the appropriate axioms, the definitions above can still be used to define __well-ordered cardinals__ and __well-founded cardinals__, respectively.

From the perspective of structural [[set theory]], it is [[evil]] to care about distinctions between isomorphic objects, and unnecessary to insist on a canonical choice of representatives for isomorphism classes.  Therefore, from this point of view it is natural to simply say:

* A __cardinal__ is a [[set]] (that is, an object of [[Set]]).

However, one still may need sets of cardinals, that is sets that serve as the target of a cardinality function satisfying (1--3) on any (small) collection of sets.  One can construct this as a [[quotient set]] of that collection.

Lowercase Greek letters starting from $\kappa$ are often used for cardinal numbers.

### In homotopy type theory

A **cardinal** in [[homotopy type theory]] is an element of the type of cardinals $\kappa \colon Card_\mathcal{U}$ relative to a universe $\mathcal{U}$. The type of cardinals is defined as the set-truncation of the type of sets $Set_\mathcal{U}$ relative to $\mathcal{U}$, $Card_\mathcal{U} \coloneqq \left| Set_\mathcal{U} \right|_0$ with $Set_\mathcal{U}$ defined as

$$Set_\mathcal{U} \coloneqq \sum_{A:\mathcal{U}} \prod_{a:A} \prod_{b:A} \prod_{p:a =_A b} \prod_{q:a =_A b} p =_{a =_A b} q$$

## Cardinal arithmetic

### Definition

For $S$ a [[set]], write ${|S|}$ for its cardinality. Then the standard operations in the [[category]] [[Set]] induce [[arithmetic]] operations on cardinal numbers ("[[cardinal arithmetic]]"):

For $S_1$ and $S_2$ two sets, the **sum** of their cardinalities is the cardinality of their [[disjoint union]], the [[coproduct]] in $Set$:
$$
  {|S_1|} + {|S_2|} \coloneqq {|S_1 \amalg S_2|}
  \,.
$$

More generally, given any family $(S_i)_{i: I}$ of sets indexed by a set $I$, the **sum** of their cardinalities is the cardinality of their disjoint union:
$$
  \sum_{i: I} {|S_i|} \coloneqq {|\coprod_{i: I} S_i|}
  \,.
$$

Likewise, the **product** of their cardinalities is the cardinality of their [[cartesian product]], the [[product]] in $Set$:
$$
  {|S_1|} \, {|S_2|} \coloneqq {|S_1 \times S_2|}
  \,.
$$

More generally again, given any family $(S_i)_{i: I}$ of sets indexed by a set $I$, the **product** of their cardinalities is the cardinality of their cartesian product:
$$
  \prod_{i: I} {|S_i|} \coloneqq {|\prod_{i: I} S_i|}
  \,.
$$

Also, the **exponential** of one cardinality raised to the power of the other is the cardinality of their [[function set]], the [[exponential object]] in $Set$:
$$
  {|S_1|}^{|S_2|} \coloneqq {|Set(S_2,S_1)|}
  \,.
$$

In particular, we have $2^{|S|}$, which (assuming the law of [[excluded middle]]) is the cardinality of the [[power set]] $P(S)$.  In [[constructive mathematics|constructive]] (but not [[predicative mathematics|predicative]]) mathematics, the cardinality of the power set is $\Omega^{|S|}$, where $\Omega$ is the cardinality of the set of [[truth values]].

The usual way to define an ordering on cardinal numbers is that ${|S_1|} \leq {|S_2|}$ if there exists an [[injection]] from $S_1$ to $S_2$:
$$
  ({|S_1|} \leq {|S_2|})
  \;:\Leftrightarrow\;
  (\exists (S_1 \hookrightarrow S_2))
  \,.
$$
Classically, this is almost equivalent to the existence of a [[surjection]] $S_2 \to S_1$, except when $S_1$ is [[empty set|empty]].  Even restricting to [[inhabited sets]], these are not equivalent conditions in [[constructive mathematics]], so one may instead define that ${|S_1|} \leq {|S_2|}$ if there exists a [[subset]] $X$ of $S_2$ and a surjection $X \to S_1$.  Another alternative is to require that $S_1$ (or $X$) be a [[decidable subset]] of $S_2$.  All of these definitions are equivalent using [[excluded middle]].

This order relation is [[antisymmetric relation|antisymmetric]] (and therefore a [[partial order]]) by the [[Cantor–Schroeder–Bernstein theorem]] (proved by Cantor using the [[well-ordering theorem]], then proved by Schroeder and Bernstein without it).  That is, if $S_1 \hookrightarrow S_2$ and $S_2 \hookrightarrow S_1$ exist, then a bijection $S_1 \cong S_2$ exists.  This theorem is not constructively valid, however.

The well-ordered cardinals are [[well-order|well-ordered]] by the ordering $\lt$ on [[ordinal numbers]].  Assuming the [[axiom of choice]], this agrees with the previous order in the sense that $\kappa \leq \lambda$ iff $\kappa \lt \lambda$ or $\kappa = \lambda$.  Another definition is to define that $\kappa \lt \lambda$ if $\kappa^+ \leq \lambda$, using the successor operation below.

The __[[successor]]__ of a well-ordered cardinal $\kappa$ is the smallest well-ordered cardinal larger than $\kappa$.  Note that (except for finite cardinals), this is different from $\kappa$\'s successor as an [[ordinal number]].  We can also take successors of arbitrary cardinals using the operation of [[Hartog's number]], although this won\'t quite have the properties that we want of a successor without the axiom of choice.

### Properties

* It is traditional to write [[ℵ]]${}_0$ for the first infinite cardinal (the cardinality of the [[natural numbers]]), $\aleph_1$ for the next (the first uncountable cardinality), and so on.  In this way every cardinal (assuming choice) is labeled $\aleph_\mu$ for a unique [[ordinal number]] $\mu$, with $(\aleph_\mu))^+ = \aleph_{\mu^+}$.

* For every cardinal $\pi$, we have $2^\pi \gt \pi$ (this is sometimes called [[Cantor's theorem]]).  The question of whether $2^{\aleph_0} = \aleph_{1}$ (or more generally whether $2^{\aleph_\mu} = \aleph_{\mu^+}$) is called Cantor's continuum problem; the assertion that this is the case is called the (generalized) [[continuum hypothesis]]. It is known that the continuum hypothesis is undecidable in [[ZFC]].

* For every transfinite cardinal $\pi$ we have (using the axiom of choice) $\pi + \pi = \pi$ and $\pi \cdot \pi = \pi$, so addition and multiplication are [[idempotent]].

## Properties of cardinals

* A transfinite cardinal $\pi$ is a **[[regular cardinal]]** if no set of cardinality $\pi$ is the union of fewer than $\pi$ sets of cardinality less than $\pi$.  Equivalently, $\pi$ is regular if given a function $P \to X$ (regarded as a family $\{P_x\}_{x\in X}$) such that ${|X|} \lt \pi$ and ${|P_x|} \lt \pi$ for all $x \in X$, then ${|P|} \lt \pi$.  Again equivalently, $\pi$ is regular if the category $\Set_{\lt\pi}$ of sets of cardinality $\lt\pi$ has all [[colimits]] of size $\lt\pi$.  The [[successor]] of any infinite cardinal, such as $\aleph_1$, is a regular cardinal.

* A cardinal is called **singular** if it is not regular.  For instance, $\aleph_\omega = \bigcup_{n\in \mathbb{N}} \aleph_n$ is singular, more or less by definition, since $\aleph_n \lt \aleph_\omega$ and ${|\mathbb{N}|} = \aleph_0 \lt \aleph_\omega$.

* A **limit cardinal** is one which is not a successor of any other cardinal.  Note that _every_ infinite cardinal is a limit [[ordinal number|ordinal]] (in the picture where cardinals are identified with certain ordinals).

* A **strong limit cardinal** is a cardinal $\pi$ such that if $\lambda \lt \pi$, then $2^\lambda \lt \pi$, for any cardinal $\lambda$. Since $\lambda^+ \le 2^\lambda$, any strong limit is a limit. Conversely, assuming the [[continuum hypothesis]], every limit is a strong limit.  Since $2^\lambda$ is the cardinality of the [[power set]] $P(\lambda)$, a cardinal $\pi$ is a strong limit iff the category $\Set_{\lt\pi}$ is an elementary [[topos]].

* An **[[inaccessible cardinal]]** is any (usually uncountable) regular strong limit cardinal.  A **weakly inaccessible cardinal** is a regular limit cardinal.

* A cardinal $\kappa$ is a **[[measurable cardinal]]** if some (hence any) set of cardinality $\kappa$ admits a two-valued [[measure]] which is $\kappa$-additive, or equivalently an [[ultrafilter]] which is $\kappa$-complete.

## Related concepts

* [[groupoid cardinality]]

* [[counting measure]]

* [[ordinal number]]

## References

The original article is

* {#Cantor1895} [[Georg Cantor]], _[[Beiträge zur Begründung der transfiniten Mengenlehre]]_, Math. Ann. 46 (1895) pp.481-512, reprinted from p. 282 on in [[Ernst Zermelo]] (ed.), _Georg Cantor -- Gesammelte Abhandlungen mathematischen und philosophischen Inhalts_, Springer Berlin 1932 ([online English translation](https://archive.org/details/contributionstof00cant))

The book

* Peter J Cameron, _Sets, Logic and Categories_  (ISBN: 1-85233-056-2 )

contains a very readable account of ZFC and the definitions of both ordinal and cardinal numbers.

Any serious reference on set theory should cover cardinal numbers.  The long-established respected tome is

* Thomas Jech _[Set Theory](http://books.google.com/books?id=pLxq0myANiEC)_;

there are also some references listed at

* [MathWorld](http://mathworld.wolfram.com/CardinalNumber.html) and

* [the English Wikipedia](http://en.wikipedia.org/wiki/Cardinal_number).

A good introduction to infinite cardinals is:

* Frank R. Drake, _Set Theory: An Introduction to Large Cardinals_, Studies in Logic and the Foundations of Mathematics, vol. 76, Elsevier, 1974.

For a much deeper treatment, which assumes most of the material in Drake as a prerequisite, see:

* Akihiro Kanamori, _The Higher Infinite: Large Cardinals in Set Theory from Their Beginning, Springer, 2003.

This is a very readable and freely available historical introduction:

* Akihiro Kanamori and M. Magidor, The evolution of large cardinal axioms in set theory, in _Higher Set Theory_, Springer Lecture Notes in Mathematics 669, 1978.  ([pdf](http://math.bu.edu/people/aki/e.pdf))

*Standard* approaches start with a [[material set theory]], such as ZFC, whereas the approach here uses [[structural set theory]] as emphasized above.  Since cardinality is isomorphism-invariant, it is easy to interpret the standard material structurally, although the basic definitions will be different.  There does not seem to be a standard account of cardinality from within structural set theory.

For cardinal numbers in [[homotopy type theory]], see chapter 10 of

* {#UFP13} Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

For a critical discussion of the history of the meaning of Cantor's "Kardinalen", see

* [[William Lawvere]], _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_,

which argues that Cantor's original meaning of _set_ was more like what today is _[[cohesive]] set_ and that his _Kardinalen_ refer to the _underlying set_ (see at _[[flat modality]]_).

[[!redirects cardinal number]]
[[!redirects cardinal numbers]]
[[!redirects cardinal]]
[[!redirects cardinals]]
[[!redirects cardinality]]
[[!redirects cardinalities]]

[[!redirects limit cardinal]]
[[!redirects limit cardinals]]

[[!redirects strong limit cardinal]]
[[!redirects strong limit cardinals]]


Cardy condition

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Cardy condition_ is part of the [[sewing constraint]] consistency condition on data that gives an [[open string|open]]-[[closed string|closed]] [[2d CFT]] or [[2d TQFT]].

It is the algebraic reflection of the fact that the following two [[cobordisms]] are equivalent (as topological as well as as conformal cobordisms):

1. an [[open string]] coming in, closing to a [[closed string]] and then opening up again to an open string (the "zip-unzip cobordism");

1. an [[open string]] coming in, splitting into two open strings, these crossing each other (with the endpoints of one of them at the same time making a full rotation), then both merging again to one open string.

(e.g. [Lauda-Pfeiffer 05 (3.44)](#LaudaPfeiffer05), [Kong 06, figure 3](#Kong06))

[[cardy_condition.jpg:pic]]

For instance in the classification of open-closed [[2d TQFT]] with [[coefficients]] in [[Vect]] via [[Frobenius algebras]], this means that for $O$ the Frobenius algebra of open string states and for $C$ the commutative Frobenius algebra of closed string states, then the canonical [[linear function]]

$$
  O \longrightarrow C \longrightarrow O
$$

is equal to the canonical map

$$
  O \stackrel{\Delta}{\longrightarrow} O \otimes O
  \stackrel{\tau}{\longrightarrow}
  O \otimes O
  \stackrel{\mu}{\longrightarrow}
  O
  \,,
$$

where $\mu$ is the [[coalgebra|coproduct]], $\mu$ the product and $\tau$ the [[braided monoidal category|braiding]] (e.g. [Lauda-Pfeiffer 05 (2.14)](#LaudaPfeiffer05)).

## Related concepts

* [[modular invariance]]

## References

Discussion in [[2d TQFT]] includes

* {#LaudaPfeiffer05} [[Aaron Lauda]], [[Hendryk Pfeiffer]] (2005), Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, _Topology Appl._ **155**, 623-666. ([arXiv:0510664](http://arxiv.org/abs/math/0510664))

* {#MooreSegal06} [[Gregory W. Moore]], [[Graeme Segal]] (2006), D-branes and K-theory in 2D topological field theory. ([arXiv:hep-th/0609042](https://arxiv.org/abs/hep-th/0609042))

* {#MooreSegal02} [[Gregory W. Moore]], [[Graeme Segal]] (2002), Lectures on Branes, K-theory and RR Charges. Lecture notes from the Clay Institute School on Geometry and String Theory held at the Isaac Newton Institute, Cambridge, UK. Available [here](http://www.physics.rutgers.edu/%7egmoore/clay1/clay1.html); related notes from the ITP Miniprogram ‘The Duality Workshop’ at Santa Barbara available [here](http://online.itp.ucsb.edu/online/mp01/moore1/).

Discussion in [[2d CFT]] includes

* {#Kong06} [[Liang Kong]], _Cardy condition for open-closed field algebras_ ([arXiv:math/0612255](http://arxiv.org/abs/math/0612255))

[[!redirects Cardy conditions]]


Carl A. Gunter

## Selected writings

On [[syntax]] and [[semantics]] of [[programming languages]]:

* [[Carl A. Gunter]], §2-3 in: *Semantics of Programming Languages -- Structures and Techniques*, MIT Press (1992) [[ISBN:9780262570954](https://mitpress.mit.edu/9780262570954/semantics-of-programming-languages/)]

## Related entries

* [[denotational semantics]], [[domain theory]]

* [[simple type theory]]

category: people

[[!redirects Carl Gunter]]


Carl Brans

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

* [website](http://www.loyno.edu/~brans/)

category: people


Carl Eckart

* [Wikipedia entry](https://en.wikipedia.org/wiki/Carl_Eckart#References)

* Walter Munk, Rudolph Preisendorfer: *Carl Henry Eckart 1902--1973*, Biographical Memoir (1976) $[$[pdf](http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/eckart-carl.pdf)$]$

## Selected writings

Early article on the equivalence of the [[Heisenberg picture]] and the [[Schrödinger picture]] of [[quantum mechanics]]:

* [[Carl Eckart]], *Operator Calculus and the Solution of the Equations of Quantum Dynamics*, Phys. Rev. **28** 4  (1926) 711-726 $[$[doi:10.1103/PhysRev.28.711](https://doi.org/10.1103/PhysRev.28.711)$]$

category: people

[[!redirects Carl Henry Eckart]]
[[!redirects Carl H. Eckart]]


Carl Faith

* [Wikipedia entry (deutsch)](https://de.wikipedia.org/wiki/Carl_Faith)

## Selected writings

On basic [[algebra]] and [[category theory]] for [[homological algebra]]:

* [[Carl Faith]], _Algebra: Rings, Modules, Categories_, vol. 1,  Grundlehren der mathematischen Wissenschaften **190**, Springer (1973) [[doi:10.1007/978-3-642-80634-6](https://doi.org/10.1007/978-3-642-80634-6)]

category: people


Carl Gauss

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

* [Wikipedia entry (de)](https://de.wikipedia.org/wiki/Carl_Friedrich_Gau%C3%9F)

* [Encyclopedia.com entry](https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/carl-friedrich-gauss)

## Selected writings

On [[number theory]] and [[quadratic forms]]:

* [[Carl Gauss]],  _[[Disquisitiones Arithmeticae]]_,  1801 ([pdf scan](http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?action=pdf&metsFile=PPN235993352&divID=LOG_0001&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN235993352%7C&targetFileName=PPN235993352_LOG_0001.pdf&))

The article after which the *[[Gauss law]]* in [[electromagnetism]] is named:

* {#Gauss1877} [[Carl F. Gauß]], *Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum, methodo nova tractata*, Commentationes Societatis Regiae Scientiarum Göttingensis Recentiores. Comm. Class. Math. **2** (1813) 1–24 [[PPN35283028X_0002_2NS](http://resolver.sub.uni-goettingen.de/purl?PPN35283028X_0002_2NS)]

  reprinted in: *Carl Friedrich Gauss -- Werke* **5**, Springer (1877) 2-22 [[doi:10.1007/978-3-642-49319-5_1](https://doi.org/10.1007/978-3-642-49319-5_1)]

On the [[fundamental theorem of algebra]]:

* {#Gauss1799} [[Carl Gauss]], _Demonstratio nova theorematis functionem algebraicam rationalem integramunius variabilis in factores reales primi vel secundi gradus resolvi posse_, Dissertation, Helmstedt (1799); Werke 3, 1–30 (1866) ([English transl. pdf](http://www.jon-arny.com/httpdocs/Gauss/Gauss%20Fundamental%20Theorem.pdf)))

* {#Gauss1866} [[Carl Gauss]], _Demonstratio nova altera theorematis omnem functionem algebraicamrationalem integram unius variabilis in factores reales primi vel secundi gradus resolviposse_, Comm. Soc. Reg. Sci. Göttingen 3, 107–142 (1816); Werke 3, 33–56 (1866)

  _Another new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree_ translated by [[Paul Taylor]] and B. Leak (1983) ([web](http://www.paultaylor.eu/misc/gauss-web))

## Related $n$Lab entries

* [[differential geometry of curves and surfaces]]

* [[Gauss decomposition]], [[quantum Gauss decomposition]]

* [[fundamental theorem of algebra]]

* _[[Disquisitiones Arithmeticae]]_

* [[Gauss diagram]]

* [[Gauss-Bonnet theorem]]

category: people

[[!redirects Gauss]]
[[!redirects Gauß]]
[[!redirects Carl Gauss]]
[[!redirects Carl Gauß]]
[[!redirects Karl Gauss]]
[[!redirects Karl Gauß]]
[[!redirects Carl Friedrich Gauss]]
[[!redirects Carl Friedrich Gauß]]
[[!redirects Karl Friedrich Gauss]]
[[!redirects Karl Friedrich Gauß]]

[[!redirects Carl F. Gauß]]

[[!redirects Johann Carl Friedrich Gauss]]
[[!redirects Johann Carl Friedrich Gauß]]
[[!redirects Johann Karl Friedrich Gauss]]
[[!redirects Johann Karl Friedrich Gauß]]


Carl Gustav Jacob Jacobi

[[!redirects Carl Jacobi]]

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

## Eelated $n$Lab entries

* [[Jacobi identity]]

* [[Jacobi theta function]], [[Jacobi form]]

* [[Jacobi triple product]]

* [[Schur function]]

(...)

category: people


Carl McTague

* [webpage](http://www.mctague.org/carl/)

## related $n$Lab entries

* [[elliptic genus]], [[Witten genus]]

* [[string bordism]]

category: people


Carl Sagan

## Selected writings

On Science, civilization

* {#Sagan97} [[Carl Sagan]]  (with Ann Druyan), *The Demon-Haunted World: Science as a Candle in the Dark*, Ballantine Books (1997) [[ISBN:9780345409461](https://www.bookfinder.com/book/The_Demon-Haunted_World-Science_as_a_Candle_in_the_Dark/0345409469/)]

and the [[observable universe]]:

* [[Carl Sagan]], *Cosmos*, Ballantine Books (1985) [Wikipedia entry, [GoogleBooks entry](https://books.google.de/books/about/Cosmos.html?id=_-XhL6_xsVkC&redir_esc=y)]

## Interviews

On science and society:

* {#AWayOfThinking} [[Carl Sagan]], *A way of thinking*, interview by Charlie Rose (May 27, 1996) [[video recording](https://www.youtube.com/watch?v=U8HEwO-2L4w&t=17s)]

## Quotes

### On scepticism
 {#OnScepticism}

From the [last interview](#AWayOfThinking):

> [3:25](https://youtu.be/U8HEwO-2L4w?t=205) There's two kinds of dangers. One is what I just talked about. That we've arranged a society based on science and technology in which nobody understands anything about science and technology, and this combustible mixture of ignorance and power, sooner or later, is going to blow up in our faces. I mean, who is running the science and technology in a democracy if the people don't know anything about it?

> [3:51](https://youtu.be/U8HEwO-2L4w?t=231) And the second reason that I'm worried about this is that science is more than a body of knowledge. It's a way of thinking. A way of skeptically interrogating the universe with a fine understanding of human fallibility.

> [4:07](https://youtu.be/U8HEwO-2L4w?t=247) If we are not able to ask skeptical questions, to interrogate those who tell us that something is true, to be skeptical of those in authority, then we're up for grabs for the next charlatan political or religious who comes ambling along.

> [4:27](https://youtu.be/U8HEwO-2L4w?t=267) It's a thing that Jefferson laid great stress on. It wasn't enough, he said, to enshrine some rights in a Constitution or a Bill of Rights. The people had to be educated, and they had to practice their skepticism and their education. Oherwise we don't run the government—the government runs us.

cf. [[Richard Feynman]] [on doubt](Richard+Feynman#OnDoubt).

From [Sagan 1997, p. 33](#SaganDruyan97):

> science is more than a body of knowledge; it
is a way of thinking. I have a foreboding of an America in my children's or grandchildren's time -- when the United States is a service and information economy; when nearly all the manufacturing industries have slipped away to other countries; when awesome technological powers are in the hands of a very few, and no one representing the public interest can even grasp the issues; when the people have lost the ability to set their own agendas or knowledgeably question those in authority; when, clutching our crystals and nervously consulting our horoscopes, our critical faculties in decline, unable to distinguish between what feels good and what's true, we slide, almost without noticing, back into superstition and darkness...

category: people


Carl-Friedrich Bödigheimer

* [institute page](https://www.math.uni-bonn.de/people/cfb/)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=25707)

## Selected writings

On [[stable splitting of mapping spaces]] via [[configuration spaces of points]]:

* {#Boedigheimer87} [[Carl-Friedrich Bödigheimer]], *Stable splittings of mapping spaces*, in: *Algebraic topology*, Lecture Notes in Mathematics **1286**, Springer (1987) 174-187 [[pdf](http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf), [[BoedigheimerStableSplittings87.pdf:file]], [doi:10.1007/BFb0078741](https://doi.org/10.1007/BFb0078741)]

On the [[ordinary cohomology]] of [[configuration spaces of points]] (such as in [[surfaces]] and [[knot complements]]):

* [[Carl-Friedrich Bödigheimer]], [[Fred Cohen]], L. Taylor, _On the homology of configuration spaces_, Topology Vol. 28 No. 1, p. 111-123 1989 ([pdf](https://core.ac.uk/download/pdf/82124359.pdf))

## Related entries

* [[stable splitting of mapping spaces]]

* [[configuration space (mathematics)]]

* [[Goodwillie calculus]]

category: people

[[!redirects Carl-Friedrich Boedigheimer]]
[[!redirects Carl-Friedrich Bodigheimer]]


Carles Casacuberta

* [website](http://atlas.mat.ub.es/personals/casac/)

## Selected writings

On [[localization of a space|localization of spaces]] with respect to sel-maps of the [[circle]]:

* [[Carles Casacuberta]], [[Georg Peschke]], *Localizing with respect to self maps of the circle*, Trans. Amer. Math. Soc. **339** (1993) 117-140  [[doi:10.1090/S0002-9947-1993-1123451-X](https://doi.org/10.1090/S0002-9947-1993-1123451-X), [pdf](http://www.ub.edu/topologia/casacuberta/articles/cpes.pdf)]

On relative [[completions of groups]]:

* [[Carles Casacuberta]], [[An Descheemaeker]], *Relative group completions*, Journal of Algebra, **285** 2 (2005) 451-469 [[doi:10.1016/j.jalgebra.2004.10.023](https://doi.org/10.1016/j.jalgebra.2004.10.023)]

## Related entries

* [[cohomology localization]]

* [[Vopěnka's principle]]

* [[idempotent monad]]

category: people


Carlo A. Rossi

* [InSpire page](https://inspirehep.net/authors/1064914)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=209798)

## Selected writings

On the [[Duflo isomorphism]]:

* [[Damien Calaque]], [[Carlo A. Rossi]], _Lectures on Duflo isomorphisms in Lie algebra and complex geometry_, European Math. Soc. (2011) [[doi:10.4171/096](https://doi.org/10.4171/096), [pdf](https://imag.umontpellier.fr/~calaque/lecturenotes-LecturesDufloETH)]

[[!redirects Carlo Rossi]]


Carlo Angiuli

* [personal page](https://www.carloangiuli.com/).

## Publications

* _Computational Higher Type Theory I: Abstract Cubical Realizability_, [[Carlo Angiuli]], [[Robert Harper]], [[Todd Wilson]], [arxiv](https://arxiv.org/abs/1604.08873), 2016
* _Computational Higher Type Theory II: Dependent Cubical Realizability_, [[Carlo Angiuli]], [[Robert Harper]], [arxiv](https://arxiv.org/abs/1606.09638), 2016
* _Computational Higher Type Theory III: Univalent Universes and Exact Equality_, [[Carlo Angiuli]], [[Kuen-Bang Hou]], [[Robert Harper]], [arxiv](https://arxiv.org/abs/1712.01800), 2017

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.FCSD.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)).

On [[universe polymorphism]] in [[dependent type theory]] via [[monads]]:

* [[Favonia]], [[Carlo Angiuli]], [[Reed Mullanix]], *An Order-Theoretic Analysis of Universe Polymorphism*, Proceedings of the ACM on Programming Languages **7** 57 (2023) 1659–1685 [[doi:10.1145/3571250](https://doi.org/10.1145/3571250)]

category: people


Carlo Becchi

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

## related $n$Lab entries

* [[BRST]], [[BRST complex]], [[BV-BRST formalism]]

category: people


Carlo Collari

* [webpage](https://poisson.phc.dm.unipi.it/~collari/)

## Selected writings

On [[Khovanov homology]] and [[knot theory]]:

* [[Carlo Collari]],  _The Functoriality of Khovanov Homology and the Monodromy of Knots_, 2013 ([pdf](https://core.ac.uk/download/pdf/19204099.pdf), [[CollariKhovanovHomology.pdf:file]])

Identifying [[schreiber:Fundamental weight systems are quantum states|more]] [[weight systems]] on [[horizontal chord diagrams]] which are [[state on a star-algebra|quantum states]]:

* {#Collari2023} [[nLab:Carlo Collari]], *A note on weight systems which are quantum states*, Can. Math. Bull. (2023, in print) [[arXiv:2210.05399](https://arxiv.org/abs/2210.05399), [doi:10.4153/S0008439523000206](https://doi.org/10.4153/S0008439523000206)]

Review:

* [[Carlo Collari]], *Weight systems which are quantum states*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#CollariMar2023), [[Collari-WeightSystems.pdf:file]]]

category: people


Carlo Mazza

* [website](http://www.math.ias.edu/~carlo/) (?)

category: people


Carlo Rovelli

* [website](http://www.cpt.univ-mrs.fr/~rovelli/)

## related $n$Lab entries

* [[multisymplectic geometry]]

category: people


Carlos Batista

* [webpage](https://carlosbatistas.wixsite.com/fisica)

## Selected writings

On [[SL(2,H)]] and [[spin representations]] of [[Spin(5,1)]] (see also _[[supersymmetry and division algebras]]_):

* [[Joás Venâncio]], [[Carlos Batista]], _Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes_, Adv. Appl. Clifford Algebras **31** 71 (2021) ([arXiv:2007.04296](https://arxiv.org/abs/2007.04296), [doi:10.1007/s00006-021-01172-1](https://doi.org/10.1007/s00006-021-01172-1))

category: people


Carlos Contou-Carrere

C. Contou Carrère was a student of [[Alexander Grothendieck]]. His thesis is

* _Géométrie des groupes réductifs, lieu singulier et résolutions équivariantes de leurs variétés de Schubert_

Later he will develop the theory of [[Jacobian variety|relative Jacobians]], this work is summarized in

* _[Jacobiennes généralisées globales relatives](https://link.springer.com/chapter/10.1007/978-0-8176-4575-5_2)_, _[Jacobienne locale d'une courbe formelle relative](http://www.numdam.org/item/RSMUP_2013__130__1_0)_

a work for which Grothendieck had a deep admiration.

* [Webpage](https://www.coulomb.univ-montp2.fr/spip.php?page=pageperso&nom=CONTOU%20CARRERE&prenom=Carlos&lang=fr)

* [Mathematics genealogy](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=31175)


Carlos Mafra

* [webpage](https://www.southampton.ac.uk/maths/about/staff/crm1n16.page)

## Selected writings

On [[string scattering amplitudes]] for the [[Berkovits superstring|Berkovits-formulation]] of the [[superstring]]:

* {#MafraSchlotterer18a} [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude I: Pure spinors and superfield kinematics_ ([arXiv:1812.10969](https://arxiv.org/abs/1812.10969))

* {#MafraSchlotterer18b} [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude II: Worldsheet functions and their duality to kinematics_ ([arXiv:1812.10970](https://arxiv.org/abs/1812.10970))

* {#MafraSchlotterer18c} [[Carlos Mafra]], [[Oliver Schlotterer]], _Towards the $n$-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure_ ([arXiv:1812.10971](https://arxiv.org/abs/1812.10971))

* [[Carlos R. Mafra]], [[Oliver Schlotterer]], *Tree-level amplitudes from the pure spinor superstring* [[arXiv:2210.14241](https://arxiv.org/abs/2210.14241)]

Review of the [[pure spinor]] [[Berkovits superstring]]:

* [[Nathan Berkovits]], [[Carlos R. Mafra]], *Pure spinor formulation of the superstring and its applications*, in *[[Handbook of Quantum Gravity]]* (2023) [[arXiv:2210.10510](https://arxiv.org/abs/2210.10510)]

On [[2-loop]] [[string scattering amplitudes]] of the [[superstring]]:

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors_ ([arXiv:2006.05270](https://arxiv.org/abs/2006.05270))

* [[Eric D'Hoker]], [[Carlos Mafra]], [[Boris Pioline]], [[Oliver Schlotterer]], _Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality_ ([doi:2008.08687](https://arxiv.org/abs/2008.08687))

## Related $n$Lab entries

* [[superstring scattering amplitudes]]

* [[KLT relations]]

* [[pure spinor]]

category: people

[[!redirects Carlos R. Mafra]]


Carlos Mochon

* [spire page](https://inspirehep.net/authors/1033283)

## Selected writings

On [[topological quantum computation]] with [[anyons]] descrived by [[Dijkgraaf-Witten theory]]:

* [[Carlos Mochon]], *Anyons from non-solvable finite groups are sufficient for universal quantum computation*, Phys. Rev. A **67** 022315 (2003) [[arXiv:quant-ph/0206128](https://arxiv.org/abs/quant-ph/0206128), [doi:10.1103/PhysRevA.67.022315](https://doi.org/10.1103/PhysRevA.67.022315)]

* [[Carlos Mochon]], *Anyon computers with smaller groups*, Phys. Rev. A **69** 032306 (2004) [[arXiv:quant-ph/0306063](https://arxiv.org/abs/quant-ph/0306063), [doi:10.1103/PhysRevA.69.032306](https://doi.org/10.1103/PhysRevA.69.032306)]

category: people


Carlos Naya

## Selected writings

On [[skyrmions]] as [[model (in theoretical physics)|models]] of [[atomic nuclei]] including beyond the [[pion]] also the [[rho-meson]]:

* {#NayaSutcliffe18a} [[Carlos Naya]], [[Paul Sutcliffe]], _Skyrmions and clustering in light nuclei_, Phys. Rev. Lett. 121, 232002 (2018) ([arXiv:1811.02064](https://arxiv.org/abs/1811.02064))

* {#NayaSutcliffe18b} [[Carlos Naya]], [[Paul Sutcliffe]], _Skyrmions in models with pions and rho_, JHEP 05 (2018) 174 ([arXiv:1803.06098](https://arxiv.org/abs/1803.06098))

  APS Synopsis: _[Revamping the Skyrmion Model](https://physics.aps.org/synopsis-for/10.1103/PhysRevLett.121.232002)_, 2018

Discussion of [[model (in theoretical physics)|models]] of [[neutron stars]] by [[Skyrmions]]:

* C. Adam, [[Carlos Naya]], J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, _BPS Skyrmions as neutron stars_, Physics Letters B
Volume 742, 6 March 2015, Pages 136-142 ([arXiv:1407.3799](https://arxiv.org/abs/1407.3799))

* C. Adam, [[Carlos Naya]], J. Sanchez-Guillen, R. Vazquez, A. Wereszczynski, _Neutron stars in the BPS Skyrme model: mean-field limit vs. full field theory_, Phys. Rev. C 92, 025802 (2015) ([arXiv:1503.03095](https://arxiv.org/abs/1503.03095))

* [[Carlos Naya]], _Neutron stars within the Skyrme model_, 	Int. J. Mod. Phys. E 28, 1930006 (2019) ([arXiv:1910.01145](https://arxiv.org/abs/1910.01145))

More on [[Skyrmions]]:

* [[Carlos Naya]], K. Oles, _Background fields and self-dual Skyrmions_ ([arXiv:2004.07069](https://arxiv.org/abs/2004.07069))

On [[Skyrmion]] models for the [[carbon]] 12 [[atomic nucleus]]:

* Christoph Adam, [[Carlos Naya]], Andrzej Wereszczyński, *Carbon-12 in the generalized Skyrme model* [[arXiv:2401.08778](https://arxiv.org/abs/2401.08778)]

category: people


Carlos Núñez

* [webpage](http://www.swan.ac.uk/staff/academic/physicalsciences/nunezc/)

## Selected writings

On [[flavor branes]]:

* [[Carlos Nunez]], A. Paredes, A.V. Ramallo, _Flavoring the gravity dual of $\mathcal{N}=1$ Yang-Mills with probes_, JHEP 0312:024, 2003 ([arXiv:hep-th/0311201](https://arxiv.org/abs/hep-th/0311201))

On [[Seiberg duality]]:

* Eduardo Conde, [[Jérôme Gaillard]], [[Carlos Núñez]], Maurizio Piai, Alfonso V. Ramallo, _Towards the string dual of tumbling and cascading gauge theories_ ([arXiv:1112.3346](http://arxiv.org/abs/1112.3346))

On the [[BMN matrix model]] and [[nonabelian T-duality]]:

* [[Yolanda Lozano]], [[Carlos Nunez]], Salomon Zacarias, _BMN Vacua, Superstars and Non-Abelian T-duality_, JHEP 09 (2017) 008 ([arXiv:1703.00417](https://arxiv.org/abs/1703.00417))

On [[black brane|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] as [[defect field theory]] "inside" [[AdS7-CFT6]]:

* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], $1/4$ BPS $AdS_3/CFT_2$ ([arxiv:1909.09636](https://arxiv.org/abs/1909.09636))

* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _Two dimensional $N=(0,4)$ quivers dual to $AdS_3$ solutions in massive IIA_ ([arxiv:1909.10510](https://arxiv.org/abs/1909.10510))

* [[Yolanda Lozano]], [[Niall Macpherson]], [[Carlos Nunez]], [[Anayeli Ramirez]], _$AdS_3$ solutions in massive IIA, defect CFTs and T-duality_ ([arxiv:1909.11669](https://arxiv.org/abs/1909.11669))

On [[AdS2/CFT1]]:

* [[Yolanda Lozano]], [[Carlos Nunez]], [[Anayeli Ramirez]], Stefano Speziali, _New $AdS_2$ backgrounds and $\mathcal{N}=4$ Conformal Quantum Mechanics_ ([arXiv:2011.00005](https://arxiv.org/abs/2011.00005))

category: people

[[!redirects Carlos Nunez]]


Carlos Perez-Pardavila

* [InSpire page](https://inspirehep.net/authors/1915265)

* [GoogleScholar page](https://scholar.google.com/citations?user=0QFFSpgAAAAJ)

## Selected writings

Argument that "islands" can resolve the [[black hole information paradox]] only for massive gravity:

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Inconsistency of islands in theories with long-range gravity*, J. High Energ. Phys. **2022** 182 (2022) [doi:10.1007/JHEP01(2022)182]

and on what to do about it:

* [[Hao Geng]], [[Andreas Karch]], [[Carlos Perez-Pardavila]], [[Suvrat Raju]], [[Lisa Randall]], [[Marcos Riojas]], [[Sanjit Shashi]], *Information Transfer with a Gravitating Bath*, SciPost Phys. **10** 103 (2021)  \[arXiv:2012.04671, [doi:10.21468/SciPostPhys.10.5.103](https://scipost.org/10.21468/SciPostPhys.10.5.103)\]

category: people


Carlos Prieto

## Selected writings

On [[algebraic topology]] and [[homotopy theory]]:

* [[Marcelo Aguilar]], [[Samuel Gitler]], and [[Carlos Prieto]], *Algebraic topology from a homotopical viewpoint*, Springer Science & Business Media, 2008 ([doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586))

On [[equivariant homotopy theory]]:

* {#MarzantowiczPrieto04} Waclaw Marzantowicz, [[Carlos Prieto]], _The unstable equivariant fixed point index and the equivariant degree_, Jourmal of the London Mathematical Society **69**  1 (2004) 214-230 [[pdf](http://hopf.math.purdue.edu/Marzantowicz-Prieto/Marprieto.pdf), doi:10.1112/S0024610703004721]

## Related entries

* [[representation sphere]]

category: people


Carlos S. Shahbazi

* [insitute page](https://grk1670.math.uni-hamburg.de/researchers/shahbazi/)

## Selected writings

On [[generalized Spin(7)-manifold|generalized]] [[Spin(7)-manifolds]] and [[M-theory on G2-manifolds]]:

* [[Mariana Graña]], [[Carlos S. Shahbazi]], [[Marco Zambon]], _$Spin(7)$-manifolds in compactifications to four dimensions_, JHEP11(2014)046 [[arXiv:1405.3698](http://arxiv.org/abs/1405.3698)]

On [[G-structures]] in [[M-theory]]:

* [[Carlos S. Shahbazi]], _M-theory on non-Kähler manifolds_, JHEP 09 (2015) 178 [[arXiv:1503.00733](https://arxiv.org/abs/1503.00733)]

On [[contact geometry]] in [[D=6 supergravity]]:

* [[Ángel Murcia]], [[Carlos S. Shahbazi]], *Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity*, Journal of Geometry and Physics **158** (2020) [[arXiv:1912.08723](https://arxiv.org/abs/1912.08723), [doi:10.1016/j.geomphys.2020.103868](https://doi.org/10.1016/j.geomphys.2020.103868)]

Discussion of [[Killing spinors]] on [[globally hyperbolic Lorentzian manifolds]]:

* [[Ángel Murcia]], [[Carlos S. Shahbazi]], *Parallel spinors on globally hyperbolic Lorentzian four-manifolds*, Ann Glob Anal Geom **61** (2022) 253–292 [[arXiv:2011.02423](https://arxiv.org/abs/2011.02423), [doi:10.1007/s10455-021-09808-y](https://doi.org/10.1007/s10455-021-09808-y)]

On [[duality-symmetric abelian Yang-Mills theory]] ("[[premetric electromagnetism]]") in the generality allowing "[[U-duality]]-twists" among several abelian gauge fields, motivated by application to [[D=4 supergravity]]:

* [[Calin Lazaroiu]], [[Carlos S. Shahbazi]], *The duality covariant geometry and DSZ quantization of abelian gauge theory*, Advances in Theoretical and Mathematical Physics **26** (2022) 2213–2312 [[arXiv:2101.07236](https://arxiv.org/abs/2101.07236), [doi:10.4310/ATMP.2022.v26.n7.a5](https://dx.doi.org/10.4310/ATMP.2022.v26.n7.a5)]

* [[Calin Lazaroiu]], [[Carlos S. Shahbazi]], *The geometry and DSZ quantization of four-dimensional supergravity*, Letters in Mathematical Physics  **113** 4 (2023) [[arXiv:2101.07778](https://arxiv.org/abs/2101.07778), [doi:10.1007/s11005-022-01626-y](https://doi.org/10.1007/s11005-022-01626-y)]

On variants of [[differential concretification|differentially concretified]] [[infinity-stack|higher]] [[moduli stacks]] of [[ordinary differential cohomology]] (higher [[bundle gerbes with connection]]) with application to [[higher gauge theory]]:

* [[Severin Bunk]], [[Carlos S. Shahbazi]], *Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology* [[arXiv:2304.06633](https://arxiv.org/abs/2304.06633)]

[[!redirects Carlos Shahbazi]]
[[!redirects C. S. Shahbazi]]


Carlos Simpson

* [website](http://math.unice.fr/~carlos/)

#Contents#
* table of contents
{:toc}

## Selected writings

* _Systems of Hodge bundles and uniformization_, Ph.D. Thesis, Harvard Univ., Cambridge, MA, 1987; J. Amer. Math. Soc. 1 (1988), no. 4, 867--918; [MR90e:58026](http://www.ams.org/mathscinet-getitem?mr=90e:58026), [doi](https://doi.org/10.2307/1990994)
* _Some families of local systems over smooth projective varieties_, Ann. Math., Sec. Ser. __138__:2 (1993) 337--425
[doi](https://doi.org/10.2307/2946614)

On a [[de Rham theorem]] for [[algebraic stack|algbraic]] [[infinity-stacks|$\infty$-stacks]] via an [[infinitesimal shape modality]] realizing the [[de Rham stack]]:

* {#SimpsonTeleman} [[Carlos Simpson]], [[Constantin Teleman]], _deRham theorem for $\infty$-stacks_ (1997) [[pdf](http://math.berkeley.edu/~teleman/math/simpson.pdf), [[SimpsonTeleman-deRham.pdf:file]]]

* _Homotopy over the complex numbers and generalized de Rham cohomology_, in: Moduli of Vector Bundles, edited by Masaki Maruyama, CRC Press 1996

* _Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology_, arXiv:[alg-geom/9712020](https://arxiv.org/abs/alg-geom/9712020)

On Deligne's [[lambda-connection]]s and Hodge moduli space

* C. T. Simpson, _The Hodge filtration on nonabelian cohomology_, in Algebraic geometry-Santa
Cruz 1995, 217--281
* Carlos T. Simpson, _A weight two phenomenon for the moduli space of rank one local systems on
open varieties_ [arXiv:0710.2800](https://arxiv.org/abs/0710.2800)
* C. T. Simpson, _Iterated destability modifications for vector bundles with connection, Contemp. Math. __522__ (2010) 183--206
* [[Carlos Simpson]], _On the notion of lambda-connection_, [slides](https://math.univ-cotedazur.fr/~carlos/slides/deligneconf/slideA.html)
* C. T. Simpson, _Geometricity of the Hodge filtration on the ∞-stack of perfect complexes over $X_{DR}$_, Mosc. Math. J. 9 (2009) no. 3, 665--721 arXiv:[math.AG/0510269](https://arxiv.org/abs/math/0510269) [journal](http://www.mathjournals.org/mmj/vol9-3-2009/cont9-3-2009.html)

More on [[nonabelian Hodge theory]]:

* _[[Higgs bundles]] and [[local systems]]_, Publ. Mathématiques de l'IHÉS __75__ (1992), p. 5-95 ([numdam:PMIHES_1992__75__5_0](http://www.numdam.org/item?id=PMIHES_1992__75__5_0), [MR94d:32027](http://www.ams.org/mathscinet-getitem?mr=94d:32027))
* _Algebraic aspects of higher nonabelian Hodge theory_, arXiv:[math.AG/9902067](https://arxiv.org/abs/math/9902067)

An early conception of [[presentable (infinity,1)-categories|presentable $\infty$-categorues]] and  [[(infinity,1)-toposes|$\infty$-toposes]] (the term seems to originate here) via (what later were called) [[model toposes]]:

* {#Simpson99} [[Carlos Simpson]],  *A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi* ([arXiv:math/9903167](http://arxiv.org/abs/math/9903167))

On [[secondary characteristic classes]]:

* Jaya N. Iyer, [[Carlos Simpson]], _Regulators of canonical extensions are torsion: the smooth divisor case_ ([arXiv:0707.0372](https://arxiv.org/abs/0707.0372))

On [[homotopy theory]] and [[higher category theory]]:

* [[Carlos Simpson]], *[[Homotopy Theory of Higher Categories]]*, Cambridge University Press 2011 ([ISBN:9780521516952](https://www.cambridge.org/in/academic/subjects  /mathematics/geometry-and-topology/homotopy-theory-higher-categories-segal-categories-ini-categories-and-beyond?format=HB)), [hal:00449826](https://hal.archives-ouvertes.fr/hal-00449826/document), [http://arxiv.org/abs/1001.4071](http://arxiv.org/abs/1001.4071) )

Lectures on [[stack]]s:

* Nicole Mestrano, Carlos Simpson, _Stacks_, 32 pp. [pdf](https://hal.science/hal-03347136/document)

On [[descent]]:

In January 2009 Carlos Simpson wrote the following short note

* C. Simpson, _Descent_ ([pdf](http://math.unice.fr/~carlos/slides/ihesAGjan09.pdf))

The following reproduces the text unabridged, but equipped with hyperlinks on keywords to $n$Lab entries as far as existent:

"
The notion of [[descent]], piecing together a global picture out of local pieces and glueing data, permeates Grothendieck's work. The history of this idea dates to the middle ages with mapmakers drawing an ever more precise picture of the world, as modern terminology of "[[atlas]]es" and "[[chart]]s" reminds us. It is crucial to the notion of [[cohomology]], where we first meet higher glueing data.

[[descent|Descent]] comes into Grothendieck's philosophy and work in a myriad of forms, starting with his papers _Technique de descente et théorèmes
d'existence en géométrie algébrique_ . The technical requirements of this theory incited him to introduce the notion of [[fibered category]]. He extended
the domain of application of this point of view in a revolutionary way by introducing the notion of "[[Grothendieck topology]]", integrating all of
[[Galois theory]] and giving us [[etale cohomology]]. A further transformation occured with the notion of [[topos]]. Another incarnation of the idea was cohomological
[[descent]] used in Deligne's papers on Hodge theory, where [[simplicial object]]s enter in a way which differs significantly from their original occurences in
[[algebraic topology]].

Between the 1960's and the 1980's, the notion of descent for objects of a [[category]], slowly gave way to a notion of "higher descent" for objects in generalized categorical situations. Examples include Breen's calculation
of etale Ext groups using the cohomology of simplicial [[Eilenberg-MacLane space|Eilenberg-MacLane]]
presheaves, and the theory of twisted complexes of Toledo and Tong. [[Jim Stasheff|Stasheff]] and Wirth [[gerbe|investigated  higher cocycles]] in the 1960's although Wirth's thesis was only recently [made public](http://golem.ph.utexas.edu/category/2006/09/wirth_and_stasheff_on_homotopy.html). In algebraic topology, Quillen's theory
of [[model category|model categories]] was gradually applied to simplicial diagrams. Illusie introduced the notion of [[model structure on simplicial presheaves|weak equivalence of]] [[simplicial presheaf|simplicial presheaves]] on a [[site|Grothendieck site]].

Grothendieck came out of isolation with the manuscript [[Pursuing Stacks|La poursuite des champs]], which at its start refers to a letter from Joyal to Grothendieck
developing the [[model structure on simplicial sheaves|model category structure on simplicial sheaves]]. This led to Jardine's [[model structure on simplicial presheaves|model category structure for simplicial presheaves]] enhancing
Illusie's weak equivalences. We enter into the modern period in which Jardine's model structure and its variants have been used and developped with
applications in a wide range of mathematics including Thomason's work in K-theory and then Voevodsky's theory of $A^1$-homotopy and [[motive]]s. [[algebraic stack|Algebraic
stack]]s, the first step in the "higher descent!" direction, are now used without restraint in all of [[algebraic geometry]].

In Grothendieck's vision as set out in "[[Pursuing Stacks|La poursuite des champs]]", higher descent is just the same as usual descent, but for [[infinity-stack|n-stacks]] of [[n-category|n-categories]]
over a [[site]]. The theory of [[2-category|2-categories]] was developped early on by Benabou,
having occured also in the book of Gabriel and Zisman. The theory of [[strict omega-category|strict n-categories]] was thoroughly investigated by Street and the Australian
school, and Brown and Loday introduced other related algebraic objects which could model [[homotopy type]]s. Grothendieck set out the goal of finding
an adequate theory of [[omega-category|weak n-categories]] where composition would be associative only up to a coherent system of higher equivalences. Similar ideas
were being developped by Dwyer and Kan in algebraic topology, and Cordier and [[Tim Porter|Porter]] in [[category theory]]. Several definitions of [[weak omega-category|weak n-categories]] have been proposed, by Baez-Dolan, Tamsamani, Batanin and others. It is now well understood that the [[homotopy coherent category theory|homotopy coherence]] problems inherent in [[higher category theory|higher categories]], are basically the same as those which were studied by [[topology|topologists]] for [[delooping hypothesis|delooping machines]]. Segal's simplicial approach and May's operadic approaches
play important roles in all of the current definitions. Maltsiniotis
points out that Batanin's definition is the closest to Grothendieck's original idea. The topologists, notably Rezk and Bergner, have developped model
structures on simplicial categories and simplicial spaces, and Joyal gives a model structure on the restricted Kan complexes originally defined by
Boardman and Vogt in the 1960's. Cisinski and Maltsiniotis have built on
a somewhat different direction of "[[Pursuing Stacks|La poursuite des champs]]" which aims to
characterize the algebraic models for [[homotopy theory]].

My own work in this area is inspired by the phrase in "[[Pursuing Stacks|La poursuite des champs]]" where Grothendieck forsees [[infinity-stack|n-stacks]] as the natural coefficients for higher [[nonabelian cohomology]]. With Hirschowitz, we have developped the
notion of [[infinity-stack|n-stack]] based on Tamsamani's definition of [[n-category]], and proven
that the association $U \mapsto \{n-stacks on U\}$ is an $(n+1)$-stack.

The theory of "[[derived algebraic geometry]]" originated by Kontsevich, Kapranov and Ciocan-Fontanine is now cast by Toen, Vezzosi and [[Jacob Lurie|Lurie]]
in a foundational framework which relies on [[higher category theory|higher categories]] and [[infinity-stack|higher
stacks]] for glueing. In the future derived geometry should be a key ingredient in Hodge theory for
higher [[nonabelian cohomology]], to be compared with
[[Katzarkov]], Pantev and [[Bertrand Toen|Toen]]'s [[Hodge theory]] on the [[schematic homotopy type]]. The latter is a higher categorical version of Grothendieck's reinterpretation
of [[Galois theory]], forseen in "[[Pursuing Stacks|La poursuite des champs]]", or really
its Tannakian counterpart. Grothendieck also mentionned, somewhat cryptically, a potential application to stratified spaces. The respective theses of
[[Treumann]] and Dupont go in this direction by using exit-path n-categories to classify constructible complexes of sheaves.

More recent developments include [Hopkins and Lurie's proof](http://golem.ph.utexas.edu/category/2006/09/wirth_and_stasheff_on_homotopy.html) of a part of the [[generalized tangle hypothesis|Baez-Dolan system of conjectures ]] relating higher categories to
[[TQFT|topological]] [[quantum field theory]]. And [[derived algebraic geometry]] permits
us to imagine a local notion of descent as was explained to me by [[David Ben-Zvi]]: using the derived non-transverse intersections, Schlessinger-[[Jim Stasheff|Stasheff]]-
Deligne-Goldman-Millson theory should be viewed as descent for the inclusion of a [[point]] into a local formal space, with the neighborhood intersection being the [[geometric infinity-function theory|derived loop space of Ben-Zvi and Nadler]].
"

## Related $n$Lab entries

* [[Simpson's conjecture]]

* [[n-category]]

* [[Pursuing Stacks]]

* [[descent]]

* [[nonabelian cohomology]]

* [[infinity-stack]]

* [[Higgs bundle]]

category: people


Carlton Caves

* [personal page](http://info.phys.unm.edu/~caves/)

## Selected writings

On [[quantum Bayesianism]]/[[QBism]]:

* {#CFS2001} [[Carlton Caves]], [[Christopher Fuchs]], [[Ruediger Schack]], *Unknown Quantum States: The Quantum de Finetti Representation* (2001) [[arXiv:quant-ph/0104088](http://arxiv.org/abs/quant-ph/0104088)]

category: people


Carmen Quinteriro

## Selected writings

On [[accessible categories]] and [[sketches]] in the generality of [[enriched category theory]]:

* [[Francis Borceux]], [[Carmen Quinteriro]], *Enriched accessible categories*, Bull. Austral. Math. Soc. __54__ (1996) 489-501 [[doi:10.1017/S0004972700021900](https://doi.org/10.1017/S0004972700021900)]

* [[Francis Borceux]], [[Carmen Quinteriro]], [[Jiří Rosický]], *A theory of enriched sketches*, Theory and Applications of Categories, **4** 3 (1998) 47-72 [[tac:4-03](http://www.tac.mta.ca/tac/volumes/1998/n3/4-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/1998/n3/n3.pdf)]

category: people


Carmen Rovi

* [personal page](https://www.mathi.uni-heidelberg.de/~crovi/)

## Selected writings

Relating ([[invertible TQFT|invertible]]) [[topological quantum field theories]] to [[cutting and pasting of manifolds|cut-and-paste-ivariants]]:

* [[Carmen Rovi]], Matthew Schoenbauer, *Relating Cut and Paste Invariants and TQFTs*, The Quarterly Journal of Mathematics **73** 2 (2022) 579–607 [[arXiv:1803.02939](https://arxiv.org/abs/1803.02939), [doi:10.1093/qmath/haab044](https://doi.org/10.1093/qmath/haab044)]

Survey:

* [[Carmen Rovi]], *Relating cut and paste invariants and TQFTS*, [talk](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#RoviArp2023) at [[CQTS]] (Apr. 2023) [video: [YT](https://www.youtube.com/watch?v=lD8wQhwyrdg)]

category: people


carrying

# Contents
* tabe of contents
{: toc}

## Idea

In the context of [[arithmetic]], _carrying_ is part of the operation of representing [[addition]] of [[natural numbers]] by digits with respect to a base.

## In terms of commutative algebra

Given the [[rig]] of natural numbers $\mathbb{N}$, there exists a [[free object|free]] commutative $\mathbb{N}$-algebra $\mathbb{N}[b]$ on one generator $b$ called the base. Since multiplication in a [[commutative algebra]] is [[power-associative]], there exists a right $\mathbb{N}$-[[action]] on $\mathbb{N}[b]$ $(-)^{(-)}:\mathbb{N}[b]\times\mathbb{N}\to\mathbb{N}[b]$ called the power, and every element in $\mathbb{N}[b]$ could be written as a [[polynomial]]

$$p = \sum_{n=0}^{k} a(n) b^n$$

When the algebra is quotiented out by the relation $b \sim 10$, the resulting [[quotient object|quotient algebra]] is [[isomorphic]] to the original rig of natural numbers $\mathbb{N}$. This means that every natural number could be expressed a polynomial with base ten,

$$p = \sum_{n=0}^{k} a(n) 10^n$$

There is a [[canonical]] such polynomial, one where all natural numbers in the [[sequence]] $a(n) \lt 10$ in the polynomial. Carrying arises from adding two canonical polynomials, when the sum $a_1(n) + a_2(n) \geq 10$ and the polynomial is no longer canonical; in order to make the polynomial canonical again, one would have to take the sum $a_1(n) + a_2(n)$ [[modulo]] 10 and add 1 to the sum $a_1(n+1) + a_2(n+1)$ in the next power of ten. This means there ought to be another representation of the digits in terms of integers modulo 10.

## In terms of cohomology

Write $\mathbb{Z}/10$ for the [[abelian group]] of [[addition]] of [[integers]] modulo 10. In the following we identify the elements as

$$
  \mathbb{Z}/{10} = \{0,1,2, \cdots, 9\}
  \,,
$$

as usual.

Being an abelian group, every [[delooping]] [[n-groupoid]] $\mathbf{B}^n (\mathbb{Z}/{10})$ exists.

Carrying is a 2-[[cocycle]] in the [[group cohomology]], hence a morphism of [[infinity-groupoids]]

$$
  c : \mathbf{B} (\mathbb{Z}/{10}) \to \mathbf{B}^2 (\mathbb{Z}/{10})
  \,.
$$

It sends

$$
  \array{
    && \bullet
    \\
    & {}^{\mathllap{a}}\nearrow
     &\Downarrow^=&
    \searrow^{\mathrlap{b}}
    \\
    \bullet &&\stackrel{a+b mod 10}{\to}&&
  }
  \;\;\;
  \mapsto
  \;\;\;
  \array{
    && \bullet
    \\
    & {}^{\mathllap{id}}\nearrow
     &\Downarrow^{c(a,b)}&
    \searrow^{\mathrlap{id}}
    \\
    \bullet &&\stackrel{id}{\to}&& \bullet
  }
  \,,
$$

where

$$
  c(a,b) =
  \left\{
    \array{
       1 & a + b \geq 10
       \\
       0 & a + b \lt 10
       \,.
    }
  \right.
$$

The [[central extension]] classified by this 2-cocycle, hence the [[homotopy fiber]] of this morphism is $\mathbb{Z}/{100}$

$$
  \array{
    \mathbf{B} (\mathbb{Z}/{100}) &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B} (\mathbb{Z}/{10}) &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^2 (\mathbb{Z}/{10})
  }
  \,.
$$

That now carries a 2-cocycle

$$
  \mathbf{B} (\mathbb{Z}/{100}) \to \mathbf{B}^2 (\mathbb{Z}/{10})
  \,,
$$

and so on.

$$
  \array{
    \vdots
    \\
    \downarrow
    \\
    \mathbf{B} (\mathbb{Z}/{1000})
    &\stackrel{c}{\to}&
    \mathbf{B}^2 (\mathbb{Z}/{10})
    \\
    \downarrow
    \\
    \mathbf{B} (\mathbb{Z}/{100})
    &\stackrel{c}{\to}&
    \mathbf{B}^2 (\mathbb{Z}/{10})
    \\
    \downarrow
    \\
    \mathbf{B} (\mathbb{Z}/{10})
    &\stackrel{c}{\to}&
    \mathbf{B}^2 (\mathbb{Z}/{10})
  }
$$

This tower can be viewed as a sort of "[[Postnikov tower]]" of $\mathbb{Z}$ (although it is of course not a Postnikov tower in the usual sense).  Note that it is not "convergent": the limit of the tower is the ring of $10$-[[p-adic numbers|adic integers]] $\mathbb{Z}_{10}$.  This makes perfect sense in terms of carrying: the $10$-adic integers can be identified with "decimal numbers" that can be "infinite to the left", with addition and multiplication defined using the usual carrying rules "on off to infinity".

## References

* [[Dan Isaksen]], _A cohomological viewpoint on elementary school arithmetic_, The American Mathematical Monthly, Vol. 109, No. 9. (Nov., 2002), pp. 796-805. ([jstor](http://links.jstor.org/sici?sici=0002-9890%28200211%29109%3A9%3C796%3AACVOES%3E2.0.CO%3B2-2))


Carsten Andrew Lütken

* [at Technology Innovation Institute (TII), Abu Dhabi](https://www.tii.ae/team/prof-dr-andrew-lutken-about)

* [at Dept. of Physics, Univerity of Oslo](https://www.mn.uio.no/fysikk/english/people/aca/lutken/)

## Selected writings

On [[D-branes]] at an [[ADE-singularity]] and their [[open string]] [[worldsheet]] [[2d CFT]]s (see also [here](ADE+singularity#ReferencesTypeIIStringsOnADE)):

* [[Wolfgang Lerche]], [[Carsten Andrew Lütken]], [[Christoph Schweigert]], *D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories*, Nucl.Phys. B **622** (2002) 269-278  ([doi:10.1016/S0550-3213%2801%2900613-7](https://doi.org/10.1016/S0550-3213%2801%2900613-7), [arXiv:hep-th/0006247](https://arxiv.org/abs/hep-th/0006247))

category: people

[[!redirects Carsten Andrew Lutken]]

[[!redirects Carsten Lütken]]
[[!redirects Carsten Lutken]]

[[!redirects Carsten A. Lütken]]
[[!redirects Carsten A. Lutken]]

[[!redirects Andrew Lütken]]
[[!redirects Andrew Lutken]]

[[!redirects C. A. Lütken]]
[[!redirects C. A Lutken]]

[[!redirects C. Andrew Lütken]]
[[!redirects C. Andrew Lutken]]


Carsten Butz

* [Mathematics genealogy page](http://www.genealogy.ams.org/id.php?id=69259)

## Selected writings

On [[first-order set theory]] and [[categorical logic]]:

* [[Steve Awodey]], [[Carsten Butz]], [[Alex Simpson]], [[Thomas Streicher]], *Relating first-order set theories, toposes and categories of classes*, Annals of Pure and Applied Logic **165** 2 (2014) 428-502 [[doi:10.1016/j.apal.2013.06.004](https://doi.org/10.1016/j.apal.2013.06.004)]

## Related entries

* [[classifying topos of a topological groupoid]]

category: people


Carsten Timm

* [home page](https://tu-dresden.de/mn/physik/itp/cmt/die-professur/inhaber)

* [GoogleScholar page](https://scholar.google.de/citations?user=FEYtmW4AAAAJ)

## Selected writings

* {#Timm20} [[Carsten Timm]], *Theory of Superconductivity*, lecture notes (2020-2023) [[pdf](https://tu-dresden.de/mn/physik/itp/cmt/ressourcen/dateien/skripte/Skript_Supra.pdf?lang=en), [[Timm-Superconductivity.pdf:file]]]

category: people


Cartan calculus

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _Cartan calculus_ are the structures and relations present in an [[inner derivation Lie 2-algebra]].

The classical examples considers for $X$ a [[smooth manifold]] the [[de Rham complex]] $(\Omega^\bullet(X), d_{dR})$ of [[differential forms]] on $X$, a [[cochain complex]] with the structure of a [[dg-algebra]].

(There are of course other differential geometric structures named after Cartan, see also at [[equivariant de Rham cohomology]] the section [The Cartan model](equivariant+de+Rham+cohomology#TheCartanModel).)

Every [[vector field]] $v \in \Gamma(T X)$ of $X$ induces a [[derivation]] on this [[dg-algebra]] of degree $-1$
$$
  \iota_v
    \,\colon\,
  \Omega^\bullet(X) \to \Omega^{\bullet-1}(X)
$$
given by [[tensor contraction|contraction]] of forms with $v$.

As every degree -1-map, this induces a [[chain homotopy]]
$$
  0
    \stackrel{\iota_v}{\to}
  [d_{dR},\iota_v]
  \,\colon\,
  \Omega^\bullet(X) \to \Omega^\bullet(X)
  \,.
$$

One finds that:

* $[d_{dR},\iota_v] = \mathcal{L}_v$ is the [[Lie derivative]] on forms along $v$;

* $[d_{dR},\mathcal{L}_v]=0$;

* $[\mathcal{L}_v, \mathcal{L}_w] = \mathcal{L}_{[v,w]}$;

* $[\mathcal{L}_v, \iota_w] = \iota_{[v,w]}$;

* $[\iota_v, \iota_w] = 0$;

* $[d_{dR},\iota_v]=0$.

These relations are sometimes called **Cartan calculus**. The first one is sometimes called **Cartan's magic formula** or _[[Cartan's homotopy formula]]_.

## In $\infty$-Lie theory

The relations of Cartan calculus are precisely those in an [[inner derivation Lie 2-algebra]].

This allows to generalize Cartan calculus to $\infty$-Lie algebroids, see the section As inner derivations at [[Weil algebra]].

There is also the full [[automorphism ∞-Lie algebra]] of any [[dg-algebra]], which subsumes the inner derivation algebras. This is the context in wich the calculus of [[derived brackets]] lives.

## Related concepts

* [[horizontal differential form]]

* [[basic differential form]]

* [[equivariant de Rham cohomology]]

* [[noncommutative differential calculus]] (in the sense of Tsygan, Tamarkin and Nest)

## References

Named after [[Élie Cartan]].

Cartan calculus on [[diffeological spaces]] requires a nontrivial condition, which is explored and developed in

* [[Christian Blohmann]], _Elastic diffeological spaces_, [arXiv:2301.02583](https://arxiv.org/abs/2301.02583).

For the closely related Cartan model of [[equivariant de Rham cohomology]] see the references [there](equivariant+de+Rham+cohomology#References).

See also

* [[Planet Math]], _[Cartan calculus](https://planetmath.org/cartancalculus)_

The expression _Cartan calculus_ is also used for [[noncommutative geometry]]-analogues such as for [[quantum groups]], see

* P. Schupp, _Cartan calculus: differential geometry for quantum groups_,  Quantum groups and their applications in physics (Varenna, 1994),  507--524, Proc. Internat. School Phys. Enrico Fermi, 127, IOS, Amsterdam, 1996.