countable ordinal

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## Idea

In [[classical mathematics]], a _countable_ [[ordinal]] is the [[order type]] of a [[well ordering|well-ordered]] [[set]] whose [[cardinality]] is countable (i.e., either [[finite set|finite]] or [[natural numbers|denumerable]]).

Countable ordinals play an important role in [[recursion]] theory, [[computability theory]], and in [[proof theory]], where they are used for example to measure the [[consistency]] strength of formal [[theories]] via [[ordinal analysis]]. They also play a role in [[philosophy of mathematics]], for example to discuss what it means for a notion to be "[[predicativity|predicative]]".

## Definitions

As indicated above, countable ordinals can be defined structurally as countable well-ordered sets (or "[[wosets]]"). As is the case with any ordered set, a woset $W$ can be regarded as a [[coalgebra over an endofunctor|coalgebra]] over the covariant [[power-set]] functor,

$$\theta: W \to P(W)$$

$$\,$$

$$x \mapsto \{y: y \lt x\}.$$

A coalgebra map $f: W \to W'$ between wosets, aka a [[simulation]], amounts to an inclusion of the domain as an initial segment of the codomain.

The [[category]] of countable wosets and simulations, thus ordered by inclusion, is a [[preorder]] and in fact equivalent to a woset. In [[material set theory]], this woset is often identified with the [[first uncountable ordinal]] (well-ordered set under the membership relation), often denoted $\omega_1$. [[structural set theory|Structurally]], we can define $\omega_1$ up to [[isomorphism]] as a [[skeleton]] of the category of countable ordinals, and so the study of countable ordinals becomes a study of the order structure of $\omega_1$.

This $\omega_1$ has no top element (of course not: otherwise $\omega_1$ would be a countable ordinal), and much of the interest and fun of the subject lies in playing the childhood game of seeing who can name the bigger number, or in this case the bigger countable ordinal. A key fact is that $\omega_1$ is countably cocomplete, and therefore behaves like a self-contained [[universe]] with respect to countably cocontinuous operations. There is quite a lot of scope in what one can build up to using such operations. After playing the larger ordinal game for a while, and becoming impressed by the sizes of ordinals one can define -- while recognizing at the same time that one is "never getting anywhere" relative to $\omega_1$ itself -- it is hard not to become awestruck by the staggering immensity of $\omega_1$.

Anyone who is unimpressed by the size of large countable ordinals has almost surely never seriously played with them. For that matter, people who are unimpressed by large _finite_ numbers have probably never played around with them either, or at least not all that imaginatively! (And interestingly, one can use large countable ordinals reflectively to construct enormous integers, or one can use large [[uncountable ordinals]] reflectively to construct enormous countable ordinals.)

## Fixed points and the Veblen hierarchy

Here we discuss the use of [[fixed point]] theory in describing the lower stages of the Veblen hierarchies of countable ordinals, which will lead up to the Feferman-Schütte ordinal. We freely allow ourselves to reason classically and use the [[principle of excluded middle]].

The ordinal $\omega_1$ is countably [[cocomplete category|cocomplete]], and we are primarily interested in functors or order-preserving maps $f: \omega_1 \to \omega_1$ that preserve [[colimits]] of countable (finite or denumerable) diagrams -- or countable chains, without loss of generality. We'll call such maps _continuous_, for short. If $x \leq f(x)$ for all $x \in \omega_1$, then we say that $f$ is _inflationary_.

Preservation of colimits of finite nonempty chains comes for free, so we can ignore that. The colimit of the empty chain is the bottom element $0$, and we will make a point of including preservation of that.

The first result is a baby form of a general theorem due to [[Adamek|Adámek]] on [[initial algebras of endofunctors]].

+-- {: .num_prop #fix}
###### Proposition
Suppose $f: \omega_1 \to \omega_1$ is inflationary and continuous. Then every $x \in \omega_1$ has an upper bound that is a fixed point under $f$.
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###### Proof
For any $x$, the colimit $x'$ of $x \leq f(x) \leq f^2(x) \leq \ldots$ exists, and since $f$ preserves countable colimits, $x'$ is a fixed point of $f$.
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+-- {: .num_lemma}
###### Lemma
If $f: \omega_1 \to \omega_1$ is order-preserving and satisfies $f(\beta) = \sup \{f(\alpha): \alpha \lt \beta\}$ for all limit (i.e., non-successor) ordinals $\beta$, then $f$ is continuous.
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The converse of this result is trivially true.

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###### Proof
For the case $\beta = 0$, the set $\{f(\alpha): \alpha \lt \beta\}$ is empty and thus has $0$ as its supremum, so $f(0) = 0$.

Suppose that $\beta$ is the colimit of a chain $x_1 \leq x_2 \leq \ldots$ with countably infinite image. Then this chain is [[cofinal functor|cofinal]] in the set $\{\alpha: \alpha \lt \beta\}$, and similarly $f(x_1) \leq f(x_2) \leq \ldots$ is cofinal in $\{f(\alpha): \alpha \lt \beta\}$. Therefore the colimit of this chain is the same as the [[supremum]] of $\{f(\alpha): \alpha \lt \beta\}$, which is $f(\beta)$.
=--

+-- {: .num_prop}
###### Proposition
Suppose $f: \omega_1 \to \omega_1$ is inflationary and continuous. Define $f': \omega_1 \to \omega_1$ to be the function that takes each $x \in \omega_1$ to the least fixed point of $f$ that is greater than $f'(y)$ for all $y \lt x$. Then $f'$ is also inflationary and continuous.
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###### Proof
By definition, $f'(y) \lt f'(x)$ whenever $y \lt x$, so $f'$ is order-preserving. That $f'$ is inflationary is proved by [[induction]]: given that $\alpha \leq f'(\alpha)$ for all $\alpha \lt \beta$, we have $\alpha \leq f'(\alpha) \lt f'(\beta)$ i.e. $\alpha \lt f'(\beta)$ for all $\alpha \lt \beta$, whence $\beta \leq f'(\beta)$. To prove $f'$ is continuous, let $\beta$ be a non-successor; by continuity of $f$ we have

$$f(\sup \{f'(\alpha): \alpha \lt \beta\}) = \sup \{f(f'(\alpha)): \alpha \lt \beta\} = \sup \{f'(\alpha): \alpha \lt \beta\}$$

so that this supremum is in fact a fixed point of $f$, and thus the least fixed point of $f$ greater than every $f'(\alpha)$ for $\alpha \lt \beta$. It therefore follows by definition of $f'$ that

$$f'(\beta) = \sup \{f'(\alpha): \alpha \lt \beta\}$$

and so, by the preceding lemma, $f'$ is continuous.
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###### Definition
The _Veblen hierarchy_ is a family of functions $f_\alpha: \omega_1 \to \omega_1$, one for each $\alpha \in \omega_1$, defined as follows.

* Define $f_0: \omega_1 \to \omega_1$ by $f_0(x) = \sup \{y+2: y \lt x\}$.

This is inflationary, and continuous by the lemma.

* Then define $f_\alpha: \omega_1 \to \omega_1$ by $f_{\alpha + 1} = f_\alpha^'$ and $f_\beta = \sup_{\alpha \lt \beta} f_\alpha$ at limit ordinals $\beta \gt 0$.

By induction and the preceding proposition, $f_\alpha$ is inflationary and continuous at successor cardinals $\alpha$, and also at limit cardinals $\alpha \gt 0$ since countable colimits of inflationary and continuous maps are also inflationary and continuous.
=--

It is worth contemplating the growth of the first few $f_\alpha$. For $f_0$, we have $f_0(\alpha) = \alpha + 1$ if $\alpha$ is a successor, and $f_0(\alpha) = \alpha$ if $\alpha$ is a limit ordinal.

This means that next we have $f_1(0) = 0$, $f_1(1) = \omega$, $f_1(2) = 2 \omega$, ..., $f_1(\omega) = \omega^2$, and so on. The first fixed point past $0$ is at $x = \omega^\omega$.

Next, we have $f_2(0) = 0$, $f_2(1) = \omega^\omega$, $f_2(2) = \omega^{2\omega}$, ..., $f_2(\omega) = \omega^{\omega^2}$, and so on. The first fixed point past $0$ is the same as the first fixed point of $x \mapsto \omega^x$ (a countably iterated exponential stack), typically denoted as $\epsilon_0$.

We pause to note that $\epsilon_0$ is sometimes considered the first reasonable thing to call a "large countable ordinal". An [[ordinal analysis]] first carried by by Gerhard Gentzen asserts, roughly speaking, that the consistency of [[Peano arithmetic]] is provable from [[primitive recursive arithmetic]] (which is weaker than PA) augmented by the principle of induction up to $\epsilon_0$.

Pressing on, we have $f_3(0)= 0$, $f_3(1) = \epsilon_0$, $f_3(2) = \epsilon_1$ (the first fixed point of $x \mapsto \omega^x$ past $\epsilon_0$), and so on. The first fixed point past $0$ of $f_3$ is the limit of $\epsilon_0, \epsilon_{\epsilon_0}, \epsilon_{\epsilon_{\epsilon_0}}, \ldots$, sometimes indicated by an infinite descending stack of subscripts $\epsilon$. This ordinal is of size that far, far surpasses any human powers of visualization.

And clearly we are only just getting started. Each successive $f_n(1)$ dwarfs its predecessor $f_{n-1}(1)$ to an indescribable degree. To speak nothing of $f_\omega(1)$, and so on.

## Ordinal notations and the Feferman-Schütte ordinal

Despite such colossal sizes, it is possible to effectively notate such ordinals using nothing more than finite binary trees, all the way up to the Feferman-Schütte ordinal, which seems to be the first of the "large countable ordinals" that the experts consider big enough to honor with human names. (There are others much further down the pike, such as the Bachmann-Howard ordinal or the considerably larger, almost godlike Church-Kleene ordinal.)

Before introducing this, we prove the following lemmas.

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###### Lemma
If $\alpha \lt \beta$, then every value $f_\beta(x)$ is a fixed point of $f_\alpha$.
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###### Proof
This is clear if $\alpha + 1 = \beta$. Suppose $f_\gamma(x)$ is a fixed point of $f_\alpha$ whenever $\alpha \lt \gamma \lt \beta$. If $\beta$ is a successor $\gamma + 1$, then we have

$$f_\beta(x) = f_\gamma(f_\beta(x)) = f_\alpha(f_\gamma(f_\beta(x))) = f_\alpha(f_\beta(x)),$$

and if $\beta$ is a limit, then

$$f_\alpha(f_\beta(x)) = f_\alpha(\sup_{\alpha \lt \gamma \lt \beta} f_\gamma(x)) = \sup_{\alpha \lt \gamma \lt \beta} f_\alpha(f_\gamma(x)) = \sup_{\alpha \lt \gamma \lt \beta} f_\gamma(x) = f_\beta(x)$$

which completes the proof.
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###### Lemma
For all $y \in \omega_1$, we have $y \leq f_y(1)$.
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###### Proof
By induction. Suppose $\alpha \leq f_\alpha(1)$ for all $\alpha \lt \beta$. Since $1 \leq f_\beta(1)$, we have

$$\alpha \leq f_\alpha(1) \leq f_\alpha(f_\beta(1)) = f_\beta(1)$$

since $f_\beta(1)$ is by the preceding lemma a fixed point of $f_\alpha$. Since $f_\beta(1)$ is both a limit ordinal and an upper bound for every $\alpha \lt \beta$, it follows that $\beta \leq f_\beta(1)$.
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Notice that by definition, $f_\beta(\gamma)$ is continuous in the argument $\beta$, since we defined $f_\beta = \sup_{\alpha \lt \beta} f_\alpha$ at limit ordinals $\beta$. Therefore the function $x \mapsto f_x(1)$ is inflationary (by lemma \ref{inflat}) and continuous. Applying proposition \ref{fix}, it has a least fixed point. This fixed point is called the **Feferman-Schütte ordinal**, denoted by $\Gamma_0$.

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###### Theorem
If $\alpha \gt 0$ is less than $\Gamma_0$, then there exist $\beta, \gamma$, both less than $\alpha$, such that $\alpha = f_\beta(\gamma)$.
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###### Proof
If $\alpha$ is not a limit ordinal, then $\alpha = \gamma + 1$ and hence $\alpha = f_0(\gamma)$. Obviously $\alpha$ cannot be a value of $f_\beta$ for $\beta \gt 0$, since all such values are limit ordinals.

If $\alpha$ is a limit ordinal and $\alpha \lt \Gamma_0$, then there is some $\beta \lt \alpha$ for which $\alpha$ is _not_ a fixed point of $f_\beta$. Otherwise we would have $f_\beta(\alpha) = \alpha$ for all $\beta \lt \alpha$, and this would force $f_\alpha(\alpha) = \alpha$ (by definition of $f_\alpha$). And in that case

$$\alpha \leq f_\alpha(1) \leq f_\alpha(\alpha) = \alpha$$

so that $\alpha$ is a fixed point of $x \mapsto f_x(1)$, contradicting $\alpha \lt \Gamma_0$.

Thus, for $\alpha$ a limit ordinal, take $\beta$ to be the least ordinal such that $\alpha \lt f_\beta(\alpha)$. Note that $\beta$ is a successor, $\beta = \beta' + 1$, because we have $\alpha = f_\gamma(\alpha)$ for any $\gamma \lt \beta$, and this would imply $\alpha = f_\beta(\alpha)$ if $\beta$ were a limit.

We have $\alpha \leq f_\beta(\gamma)$ for some $\gamma \lt \alpha$ (for if $f_\beta(\gamma) \lt \alpha$ for all $\gamma \lt \alpha$, we could infer $\alpha$ is a fixed point of $f_\beta$). Let $\gamma$ be the least ordinal (less than $\alpha$) such that $\alpha \leq f_\beta(\gamma)$. Then in fact this last inequality is an equality. For suppose $\gamma = \delta + 1$; then by definition of $\gamma$, we have

$$f_\beta(\delta) = f_{\beta' + 1}(\delta) \lt \alpha$$

and we also have that $\alpha = f_{\beta'}(\alpha)$ by definition of $\beta$. Since $f_\beta(\gamma) = f_{\beta' + 1}(\gamma) = f_{\beta' + 1}(\delta + 1)$ is by definition the least $f_{\beta'}$-fixpoint greater than $f_{\beta' + 1}(\delta)$, we immediately infer $f_\beta(\gamma) \leq \alpha$, as required. If on the other hand $\gamma$ is a limit, we have

$$f_\beta(\delta) \lt \alpha$$

for all $\delta \lt \gamma$, and we again infer $f_{\beta}(\gamma) \leq \alpha$, as required.
=--

This theorem gives a canonical finite binary tree representation of any $\alpha \lt \Gamma_0$, by recursion. The binary tree $\tau(0)$ of $0$ consists of just $0$ as root, with no child-pair. Given that every $\beta$ less than $\alpha$ has been assigned a binary tree $\tau(\beta)$, we define $\tau(\alpha)$ to have the child-pair $(\tau(\beta), \tau(\gamma))$ where

* $\beta$ is the least ordinal less than $\alpha$ such that $\alpha \lt f_\beta(\alpha)$, and

* $\gamma$ is the least ordinal less than $\alpha$ such that $\alpha \leq f_\beta(\gamma)$.

The resulting binary tree $\tau$ is finite for every $\alpha \lt \Gamma_0$, since it has binary branching and there are no infinite branches or paths up the tree (since that would yield an infinite descent $\alpha \gt \alpha_1 \gt \alpha_2 \gt \ldots$ with no least element, contradicting well-orderedness).

This finite binary tree notation is one of many possible ordinal notations for naming countable ordinals. It is more expressive the Cantor normal form that provides a notation for ordinals $\alpha$ less than $\epsilon_0$, where one writes $\alpha$ uniquely in the form

$$\alpha = \omega^\beta + \gamma$$

with $\beta, \gamma \lt \alpha$ and $\gamma \lt \gamma + \omega^\beta$, and continues recursively.

+-- {: .num_remark}
###### Remark
However, Cantor normal form has the virtue of uniqueness. For the binary operation $(\beta, \gamma) \mapsto f_\beta(\gamma)$, it is possible to have two distinct pairs $(\beta, \gamma)$, $(\beta', \gamma')$ which are equivalent in the sense that

$$f_\beta(\gamma) = f_{\beta'}(\gamma');$$

we counteracted this by using the pair $(\beta, \gamma)$ which was least in its equivalence class with respect to [[lexicographic order]].
=--

## Fundamental sequences and the Bachmann-Howard ordinal

### Fast-growing functions on $\mathbb{N}$

## Recursive ordinals

Of course, having reached $\Gamma_0$ as the least fixed point of $x \mapsto f_x(1)$, there is nothing stopping one from starting all over again with a new Veblen hierarchy, defining $g_0(x) = f_x(1)$ and defining $g_\alpha$ analogously to how we defined $f_\alpha$.

In one or another way we can continue extending ordinal notation schemes, but all such schemes are recursive by definition, and at some point we cannot go any further:

+-- {: .num_defn}
###### Definition
A countable ordinal $\alpha$ is **recursive** if there is a computable relation on $\mathbb{N}$ (or a [[recursive subset]] of $\mathbb{N} \times \mathbb{N}$) that well-orders $\mathbb{N}$, so that the resulting woset is isomorphic to $\alpha$.
=--

There are countably many computable relations on $\mathbb{N}$, and so the supremum of the recursive ordinals in $\omega_1$ exists. It is called the **Church-Kleene ordinal**. It is the first non-recursive ordinal.

## Related concepts

* [[uncountable ordinal]]
* [[countable set]]
* [[ordinal analysis]]

## References

For a light-hearted introduction with many useful references, see

* [[John Baez]], Week 236, This Week's Finds in Mathematical Physics, July 26, 2006. ([web](http://math.ucr.edu/home/baez/week236.html))

A very readable account of countable ordinals up to the Feferman-Schütte ordinal is

* Jean Gallier, *What's so Special about Kruskal's Theorem and the Ordinal $\Gamma_0$?, A Survey of Some Results in Proof Theory*, Annals of Pure and Applied Logic, Vol. 53, 199-260 (1991). ([Part II](ftp://ftp.cis.upenn.edu/pub/papers/gallier/kruskal2.pdf))

An application of coalgebra theory to ordinal notation systems is described here:

* Peter Hancock, Ordinal notation systems. ([web](http://homepages.inf.ed.ac.uk/v1phanc1/ordinal-notations.html))

[[ordinal analysis|Ordinal analysis]] as measuring consistency strengths of theories is well-described in this paper:

* [[Michael Rathjen]], The art of ordinal analysis, International Congress of Mathematicians, II, Eur. Math. Soc., Zurich, pp. 45–69. ([pdf](http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_03.pdf))

[[!redirects countable ordinals]]


countable random graph

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## Idea

The _countable random graph_, also known as the _Rado graph_, is the "generic finite [[graph]]": it contains every finite graph as an induced subgraph, models the almost-sure first-order theory of the class of finite graphs, and is axiomatized by the _extension property_: given any two finite subsets of vertices, there exists a vertex not in either subset which is connected to all the points in one subset and disconnected from the other.

## Definition

The usual probabilistic construction is given by taking a countably infinite set of vertices and percolating the edges: for any pair of vertices, connect them with probability $\frac{1}{2}$; almost surely the end result will be _the_ countable random graph, which is unique up to isomorphism.

## Remarks

- The extension property implies that every formula in the random graph has infinite [[Vapnik-Chervonenkis dimension|Vapnik–Chervonenkis dimension]].

- Here is an example of a way that the automorphisms of a first-order structure are not generally captured by its theory: a finite graph will almost surely have no nontrival automorphisms while the countable random graph is $\omega$-categorical and thus has infinitely many nontrivial automorphisms.

## Related concepts

* [[omega-categorical structure]]
* [[Fraisse limit]]

## References

- P.J. Cameron, [The Random Graph](http://math.ipm.ac.ir/combin/useful_material/Cameron-Rgraphs.pdf)

[[!redirects Rado graph]]
[[!redirects Rado graphs]]
[[!redirects countable random graph]]
[[!redirects countable random graphs]]


countable set

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# Countable sets
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## Idea

Let $S$ be any [[set]], and let $\mathbf{N}$ be the [[set]] of [[natural number]]s.  Let $|S|$ be the [[cardinality]] of $S$, and let $\aleph_0$ be the cardinality of $\mathbf{N}$.

Then $S$ is:

*  __denumerable__ if ${|S|} = \aleph_0$,
*  __countable__ if ${|S|} \leq \aleph_0$, and
*  __uncountable__ if ${|S|} \gt \aleph_0$.

Note that the first two terms are not entirely standardised; some authors use them interchangeably for one or the other concept.

## Definitions

### Countable sets

If you accept the [[axiom of choice]], then there is really nothing more to say than what was above.  In weaker [[foundations]], more care may be needed.  The following definitions should be more than enough for practical use in [[constructive mathematics]]:

* A [[set]] $S$ is __denumerable__ if there exists a [[bijection]] between $S$ and $\mathbf{N}$; or in other words if $S$ is the same as $\mathbf{N}$ [[up to isomorphism]] in [[Set]].
* $S$ is __denumerably indexed__ if there exists a [[surjection]] to $S$ from $\mathbf{N}$ (or equivalently any denumerable set); or in other words if $S$ a [[quotient set]] of a $\mathbf{N}$, up to isomorphism.
* $S$ is __split denumerably indexed__ if there exists a [[split surjection]] to $S$ from $\mathbf{N}$.
* $S$ is __countable__ if there exists a bijection between $S$ and a [[detachable subset|detachable]] [[lower subset]] of $\mathbf{N}$.
* $S$ is __countably indexed__ if there exists a [[surjection]] to $S$ from a countable set.
* $S$ is __split countably indexed__ if there exists a [[split surjection]] to $S$ from a countable set.
* $S$ is __subcountable__ (or _subdenumerable_) if there exists an [[injection]] from $S$ to a countable set, or equivalently to a denumerable set; or in other words if $S$ is a [[subset]] of $\mathbf{N}$, up to isomorphism.
* $S$ is __subcountably indexed__ (or _subdenumerably indexed_) if $S$ is a [[subquotient]] of $\mathbf{N}$, up to isomorphism.
* $S$ is __split subcountably indexed__ (or _split subdenumerably indexed_) if there exists a [[split surjection]] to $S$ from a subcountable set, or equivalently if there exists an [[injection]] from $S$ to a split countably indexed set or equivalently to a split denumerably indexed set.

Of course, the terms are even less standardised here.  Certainly it is common for 'countable' to mean countably indexed, and yet it may also (as in classical mathematics) mean denumerable.  In at least some contexts, 'enumerable' (without the initial 'd') means denumerably indexed.

Not only does the difference between 'denumerable' and 'countable' disappear when we add the prefix 'sub' (for obvious reasons), but it almost disappears when we add the suffix 'ly indexed'!  Specifically, a set is denumerably indexed if and only if it is both [[inhabited set|inhabited]] and countably indexed; and a set is countably indexed if and only if it is either [[empty set|empty]] or denumerably indexed.  That countability involves *lower* subsets of $\mathbb{N}$ is also almost irrelevant here; any *inhabited* detachable subset of $\mathbb{N}$ is denumerably indexed regardless of its closure under order.

On the other hand, it is key that countability involves *detachable* subsets of $\mathbb{N}$, so that not every subcountable set is necessarily countable (or even countably indexed).  Indeed, the set of [[computable numbers]] is subcountably indexed, while the set of [[real numbers]] is uncountable (and thus not countably indexed); yet in [[Russian constructivism]], these are the same set!  One reason to focus on countably indexed sets is that this is about the most general concept that definitely does *not* hold of the set of real numbers.

In addition, in the presence of [[countable choice]], the difference between *denumerably indexed* and *split denumerably indexed*, *countably indexed* and *split countably indexed*, and *subcountably indexed* and *split subcountably indexed* disappear, since every surjection out of $\mathbb{N}$ and every subset of $\mathbb{N}$ has a [[section]] by [[countable choice]].

Under the [[BHK interpretation]] of constructive mathematics, one only has the split versions of denumerably indexed, countably indexed, subcountably indexed.

### Uncountable sets

There are a number of different definitions of an uncountable set, which all differ in the absence of [[countable choice]]:

* $S$ is __indenumerable__ if every [[function]] $f$ from $\mathbf{N}$ to $S$ is not a [[bijection]].
* $S$ is __split indenumerably indexed__ if every [[function]] $f$ from $\mathbf{N}$ to $S$ is not a [[split surjection]].
* $S$ is __indenumerably indexed__ if every [[function]] $f$ from $\mathbf{N}$ to $S$ is not a [[surjection]].
* $S$ is __strongly indenumerably indexed__ if every [[function]] $f$ from $\mathbf{N}$ to $S$ is strongly non-surjective in the sense that there exists an element of $S$ that is not in the [[image]] of $f$.
* Suppose $S$ is an [[inequality space]]. Then $S$ is __$\#$-indenumerably indexed__, or the pair $(S,\#)$ is __strongly indenumerably indexed__ (or just _indenumerable_ if context is clear) if every [[function]] $f$ from $\mathbf{N}$ to $S$ (or equivalently any variation as in the previous paragraph) is $\#$-strongly non-surjective in the sense that there exists an element of $S$ that is $\#$-distinct from every element in the [[image]] of $f$.
* $S$ is __uncountable__ if every [[function]] $f$ from a detachable subset of $\mathbf{N}$ to $S$ is not a [[bijection]].
* $S$ is __split uncountably indexed__ if every [[function]] $f$ from a detachable subset of $\mathbf{N}$ to $S$ is not a [[split surjection]].
* $S$ is __uncountably indexed__ if every [[function]] $f$ from a detachable subset of $\mathbf{N}$ to $S$ is not a [[surjection]].
* $S$ is __strongly uncountably indexed__ if every [[function]] $f$ from a detachable subset of $\mathbf{N}$ to $S$ is strongly non-surjective in the sense that there exists an element of $S$ that is not in the [[image]] of $f$.
* Suppose $S$ is an [[inequality space]]. Then $S$ is __$\#$-uncountably indexed__, or the pair $(S,\#)$ is __strongly uncountably indexed__ (or just _uncountable_ if context is clear) if every [[function]] $f$ from a detachable subset of $\mathbf{N}$ to $S$ (or equivalently any variation as in the previous paragraph) is $\#$-strongly non-surjective in the sense that there exists an element of $S$ that is $\#$-distinct from every element in the [[image]] of $f$.

Similarly as the case for above, the terms are even less standardised here.  Certainly it is common for 'uncountable' to mean uncountably indexed, and yet it may also (as in classical mathematics) mean indenumerable.

We make the distinction between uncountable and uncountably indexed and between indenumerable and indenumerably indexed because in [[neutral constructive mathematics]] it is consistent that there exist a set which is both indenumerable and indenumerably indexed. However, similarly to countably indexed sets, in the presence of [[countable choice]], the difference between *uncountably indexed* and *split uncountably indexed* and *indenumerably indexed* and *split indenumerably indexed* disappear.

There is a lot of freedom in defining uncountably indexed sets. You can start with countable and require that any function from them to $S$ be non-surjective (in the strong sense used in the definition above). Or you can start with denumerable or denumerably indexed sets and require that any function from them to $S$ be non-surjective, then add the additional clause that $S$ must be [[inhabited set|inhabited]]. Or you can compromise, and start with a detachable subset of $\mathbb{N}$ without requiring it to be a lower subset, since this is enough to prove $S$ inhabited and make the order-closure irrelevant. All of these definitions are equivalent!

## Examples and Properties

In [[classical mathematics]], the set of [[real numbers]] is uncountable. Even in [[constructive mathematics]], the [[MacNeille real numbers]], the [[Dedekind real numbers]], and the Cauchy real numbers are all uncountable, in the sense of having a bijection with a countable set.

More generally, it is impossible for the real numbers to be both subcountable and countably indexed at the same time; subcountability implies that the real numbers have [[decidable equality]] because the natural numbers have decidable equality and injections preserve and reflect decidable equality, from which one can derive [[WLPO]] allowing one to prove that the real numbers are uncountably indexed by any of the usual arguments, resulting in a contradiction.

In fact, [[WLPO]] itself is sufficient to prove that the real numbers are uncountably indexed. However, it is consistent that, absent [[WLPO]], the Dedekind real numbers are denumerably indexed and thus *not* uncountably indexed (cf [Bauer 2022](#Bauer22)).

Every denumerable set is [[infinite set|infinite]], and hence so is every set with a denumerable subset.  Classically (usng [[dependent choice]] $DC$), we have the converse: a set is infinite if and only if it has a denumerable subset.  If a set has a denumerable subset, then its [[power set]] is uncountable.  Even without $DC$, using [[excluded middle]] instead, the power set of an infinite set is uncountable.

In contrast, the set of [[algebraic numbers]] is denumerable.  Also, the set of [[computable numbers]] is subcountably indexed (and infinite, so denumerable in classical mathematics).  Combined with the uncountability of the real numbers, this gives the existence of transcendental numbers (constructively) and uncomputable numbers (classically).

Classically, every set is countably indexed [[xor]] uncountably indexed.  Even constructively, a countably indexed set cannot be uncountably indexed. In [[Russian constructivism]], however, there are subcountably indexed sets that are also uncountably indexed. In particular, the set of computable numbers is subcountably indexed, while the set of real numbers is uncountably indexed, yet in Russian constructivism, these are the same set! In some formalisms of Russian constructivism, *every* set is subcountably indexed.

The [[empty set]] is countable, although not denumerably indexed.  Conversely, any uncountable set must be [[inhabited set|inhabited]], as must be any denumerably indexed set.  Even constructively, a countable set is empty xor inhabited.

Every [[finite set]] is countable, every finitely indexed set is countably indexed, every subfinite set is subcountable, and every subfinitely indexed set is subcountably indexed.  Conversely, every uncountable set is [[infinite set|infinite]].

Classically, a denumerable set is precisely an [[infinite set|infinite]] countable set; sometimes this is written as a _countably infinite set_.  Thus classically, a countable set is finite xor denumerable.  Even constructively, a countable set is denumerable if and only if it is not finite.

Classically (using [[countable choice]]), [[countable unions of countable sets are countable]].

## History

Arguably, [[set theory]] as such begins with the 1873 correspondence between [[Georg Cantor]] and [[Richard Dedekind]] on countability.  Although they tacitly used [[excluded middle]] throughout, generally speaking, their results were constructive if taken to be about countably indexed sets.  They worked out the countability of the algebraic numbers and the uncountability of the real numbers, including both constructive and nonconstructive proofs of the existence of transcendental numbers.  (This was the same year that [[Charles Hermite]] published a proof that $\mathrm{e}$ is transcendental, although [[Joseph Liouville]] had shown how to artificially construct transcendental numbers as early as 1851.)

In subsequent correspondence with Cantor, [[Karl Weierstrass]] stressed the usefulness of the countability of the algebraic numbers but downplayed the notion of uncountability.  Still, he urged Cantor to publish.  Since Weierstrass and the [[finitism|finitist]] [[Leopold Kronecker]] were senior editors at Krelle\'s Journal, Cantor wrote his 1874 article carefully.  He showed the denumerability of the algebraic numbers explicitly and titled the article as if that were the only subject.  He then added a footnote in the proofreading stages, using an [[interval]]-based argument, showing how a transcendental number could be constructed from this denumeration.

Cantor\'s argument was essentially a constructive proof that the set of real numbers is uncountable, as it could apply to any denumerably indexed subset. He tacitly used [[LLPO]], but it\'s straightforward to avoid this using binary [[meets]] and [[joins]] of real numbers, which are constructive. He also used a [[Dedekind cut]] to construct the final transcendental number, and it is this that Kronecker would have most likely objected to, since it is an irreducibly infinite procedure to construct a single number.  However, the argument is acceptable to any non-finitist constructivist.

In the correspondence with Dedekind, it was Cantor who first considered and proved the uncountability of the real numbers but Dedekind who first considered and proved the countability of the algebraic numbers, which was the ostensible subject of Cantor\'s article.  Furthermore, it was Dedekind who first wrote the interval-based proof that Cantor used in his footnote that was the real point of the article.  However, Cantor gave Dedekind no credit.  This strained their relationship for a few years afterward.

The [[diagonal argument]] also shows the uncountability of the real numbers (nonconstructively through [[Cantor's Theorem]] on the cardinality of power sets, constructively through a slight generalization), but Cantor did not publish this until 1891.  Interpreted finitistically as an algorithm to construct, from a sequence of real numbers, an approximation of a hypothetical number not in the sequence, the diagonal argument is more efficient; pedagogically, it is easier to explain.  However, it was not the first.

## Related concepts

* [[countable ordinal]]

* [[countable cover]]

## References

* Wikipedia, _[Cantor's first set theory article](https://en.wikipedia.org/wiki/Cantor%27s_first_set_theory_article)_

which discusses both the origin of the 1874 article and the questions of constructivity that arose in its aftermath.

That in the absence of [[countable choice]] it is consistent that the [[Dedekind real numbers]] are denumerably indexed:

* {#Bauer22} [[Andrej Bauer]], *The countable reals* ([video](https://www.youtube.com/watch?v=4CBFUojXoq4)).

[[!redirects countable set]]
[[!redirects countable sets]]

[[!redirects countably infinite set]]
[[!redirects countably infinite sets]]

[[!redirects denumerable set]]
[[!redirects denumerable sets]]

[[!redirects subcountable]]
[[!redirects subcountable set]]
[[!redirects subcountable sets]]

[[!redirects subdenumerable]]
[[!redirects subdenumerable set]]
[[!redirects subdenumerable sets]]

[[!redirects countably indexed]]
[[!redirects countably indexed set]]
[[!redirects countably indexed sets]]

[[!redirects denumerably indexed]]
[[!redirects denumerably indexed set]]
[[!redirects denumerably indexed sets]]

[[!redirects uncountable]]
[[!redirects uncountable set]]
[[!redirects uncountable sets]]

[[!redirects uncountably indexed]]
[[!redirects uncountably indexed set]]
[[!redirects uncountably indexed sets]]

[[!redirects indenumerable]]
[[!redirects indenumerable set]]
[[!redirects indenumerable sets]]

[[!redirects indenumerably indexed]]
[[!redirects indenumerably indexed set]]
[[!redirects indenumerably indexed sets]]

[[!redirects countable family]]
[[!redirects countable families]]

[[!redirects numerable]]
[[!redirects numerable set]]
[[!redirects numerable sets]]


countable unions of countable sets are countable

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## Statement

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###### Proposition
**(a countable union of countable sets is countable, aka the countable union theorem)**

Assuming the [[axiom of countable choice]] then:

Let $I$ be a [[countable set]] and let
$\{S_i\}_{i \in I}$ be an $I$-[[dependent type|dependent set]] of [[countable sets]] $S_i$. Then the [[disjoint union]]

$$
  \underset{i \in I}{\cup} S_i
$$

is itself a countable set.

=--

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

[[classical mathematics|Classical]] proof: we may assume all the $S_i$ are [[nonempty set|nonempty]]. For each $i \in I$, choose a [[surjection]] $f_i: \mathbb{N} \to S_i$ (this requires the [[axiom of countable choice]]) and also a surjection $f: \mathbb{N} \to I$. Then we have a composite surjection

$$\mathbb{N} \stackrel{pair}{\to} \mathbb{N} \times \mathbb{N} \stackrel{f \times 1}{\to} I \times \mathbb{N} \cong \underset{i \in I}{\cup} \mathbb{N} \stackrel{\underset{i \in I}{\cup} f_i}{\to} \underset{i \in I}{\cup} S_i$$

where for $pair: \mathbb{N} \to \mathbb{N} \times \mathbb{N}$ we may take for example the function that is inverse to $(x, y) \mapsto \binom{x+y+1}{2} + y$.

For [[constructive mathematics|constructive mathematicians]] who accept the [[axiom of countable choice]], the proof is only slightly more elaborate. Here we define a set to be *countable* if it is a [[quotient]] of (is enumerated by) a [[decidable subset]], i.e., a [[complemented subobject]] of $\mathbb{N}$. Thus, supposing we have chosen surjections out of decidable subsets $(f_i: J_i \to S_i)_{i \in I}$, and a surjection $J \to I$ out of a decidable subset $J$, we have a diagram (switching to $\sum$ to denote disjoint sums)

$$\array{
 & & & & J \cdot \mathbb{N} & \hookrightarrow & \mathbb{N} \cdot \mathbb{N} \cong \mathbb{N} \times \mathbb{N} & \stackrel{pair^{-1}}{\to} & \mathbb{N} \\
 & & & & \downarrow \mathrlap{f \cdot \mathbb{N}} & & & & \\
\sum_{i \in I} J_i & \hookrightarrow & \sum_{i \in I} \mathbb{N} & \cong & I \cdot \mathbb{N} & & & & \\
\mathllap{\sum_{i \in I} f_i} \downarrow & & & & & & & & \\
\sum_{i \in I} S_i & & & & & & & &
}$$

whose [[limit]] might be denoted $\sum_{j \in J} K_j$ where $K_j \coloneqq J_{f(j)}$. This is certainly a complemented subobject of $\mathbb{N}$, the complement being formed as $\left(\sum_{j \in J} \neg K_j\right) + (\neg J) \cdot \mathbb{N}$, and the limit obviously maps surjectively onto $\sum_{i \in I} S_i$, as desired.
=--

The implication countable choice $\Rightarrow$ countable union theorem cannot be reversed, as there are models of [[ZF]] where the latter holds, but countable choice fails. Further, the countable union theorem implies countable choice for countable sets, but this implication also cannot be reversed.

## Related statements

* [[images of unions are unions of images]]

* [[pre-images preserve unions and intersections]]

## References

* ProofWiki, _[Countable Union of Countable Sets is Countable](https://proofwiki.org/wiki/Countable_Union_of_Countable_Sets_is_Countable)

* _Countable Unions And The Axiom Of Countable Choice_, [MathOverflow discussion](https://mathoverflow.net/questions/74743/countable-unions-and-the-axiom-of-countable-choice)

[[!redirects countable union of countable sets is countable]]
[[!redirects countable union theorem]]
[[!redirects countable union axiom]]


countably compact metric spaces are equivalently compact metric spaces

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## Statement

Let $(X,d)$ be a [[metric space]], regarded as a [[topological space]] via its [[metric topology]]. Then the following are equivalent:

1. $(X,d)$ is a [[countably compact topological space]].

1. $(X,d)$ is a [[compact topological space]].

## Related statements

* [[sequentially compact metric spaces are equivalently compact metric spaces]]

* [[sequentially compact metric spaces are totally bounded]]

* [[Lebesgue number lemma]]

## References

* ProofWiki, _[Countably Compact Metric Space is Compact](https://proofwiki.org/wiki/Countably_Compact_Metric_Space_is_Compact)_


countably compact topological space

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## Definition

A [[topological space]] is called _countably compact_ if every [[open cover]] consisting of a [[countable set]] of [[open subsets]] (every [[countable cover]]) admits a [[finite cover|finite subcover]], hence if there is a [[finite set|finite]] [[subset]] of the open in the original cover which still cover the space.

## Examples

* A [[compact space]] is _a fortiori_ countably compact.

* A [[sequentially compact space]] is countably compact.

* The [[long line]] is an example of a countably compact space that is not compact.

## Properties

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###### Proposition
A countably compact space is a [[limit point compact space]].
=--

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###### Proof
Recall that a space is limit point compact if every [[closed subspace|closed]] [[discrete space|discrete]] [[subspace]] is [[finite set|finite]]; equivalently, if every [[countable set|countable]] closed discrete subspace is finite.

Suppose $X$ is countably compact and $A$ is a countable closed discrete subspace. For each $a \in A$, choose an open neighborhood $U_a$ such that $U_a \cap A = \{a\}$, and let $V_a$ be the open subset $U_a \cup \neg A$. Clearly we still have $V_a \cap A = \{a\}$. Also, $\{V_a: a \in A\}$ is a countable cover of $X$, hence admits a finite subcover $V_{a_1}, \ldots, V_{a_n}$. But then

$$A = \left(\bigcup_{i=1}^n V_{a_i} \right) \cap A = \bigcup_{i=1}^n (V_{a_i} \cap A) = \{a_1, \ldots, a_n\}$$

as was to be shown.
=--

+-- {: .num_prop}
###### Proposition
A space that is $T_1$ and limit point compact is countably compact.
=--

+-- {: .proof}
###### Proof
Consider any countable open cover $U_1, U_2, \ldots$ of $X$. Put $V_n = U_1 \cup \ldots \cup U_n$ so that $V_1 \subseteq V_2 \subseteq \ldots$, and discard repetitions, i.e., consider the maximal chain of *strict* inclusions

$$\emptyset = V_0 \subset V_{n_1} \subset V_{n_2} \subset \ldots$$

It suffices to show this maximal chain is finite. Rename it as $\emptyset = W_0 \subset W_1 \subset W_2 \subset \ldots$.

For each $W_n$ occurring in this chain ($n \geq 1$), pick a point $x_n \in W_n \setminus W_{n-1}$. Observe that if $m \lt n$, then $x_n \notin W_m$.

Since $X$ is $T_1$ (points are closed), the set $W_m \cap \neg \{x_1, \ldots, x_{m-1}\}$ is an open neighborhood of $x_m$ that does not contain $x_n$ whenever $n \gt m$, and does not contain $x_n$ whenever $n \lt m$. Thus every point $x_m$ is open relative to $A = \{x_1, x_2, \ldots\}$, i.e., $A$ is a discrete subspace.

Finally, any point $x \notin A$ belongs to some $W_n$, and then $W_n \cap \neg \{x_1, \ldots x_n\}$ is an open neighborhood of $x$ that doesn't intersect $A$. Thus $A$ is a closed discrete subspace, and is finite by limit point compactness. Therefore the maximal chain consisting of the sets $W_n$ is finite, as was to be shown.
=--

\begin{proposition}
If $X$ is countably compact and $f \colon X \to Y$ is a map in [[Top]], then the direct image $f(X)$ is a countably compact subset of $Y$.
\end{proposition}

\begin{proof}
The proof is just like the proof of the corresponding statement where the word "countably" is dropped. Suppose $\{U_n: n \in \mathbb{N}\}$ is a countable cover of $f(X)$, so that $f(X) \subseteq \bigcup_{n \in \mathbb{N}} U_n$. This is equivalent to

$$X \subseteq f^{-1}\left(\bigcup_{n \in \mathbb{N}} U_n\right) = \bigcup_{n \in \mathbb{N}} f^{-1}(U_n).$$
Thus $\{f^{-1}(U_n): n \in \mathbb{N}\}$ is a countable open cover of $X$, so by countable compactness there is a finite subcover $f^{-1}(U_{n_1}), \ldots, f^{-1}(U_{n_k})$ of $X$:

$$X \subseteq \bigcup_{i=1}^k f^{-1}(U_{n_i}) = f^{-1}\left(\bigcup_{i=1}^k U_{n_i}\right).$$
This is equivalent to $f(X) \subseteq \bigcup_{i=1}^k U_{n_i}$, as required.
\end{proof}

* In the case of metric spaces, [[countably compact metric spaces are equivalently compact metric spaces]].

## Related concepts

* [[countably paracompact topological space]]

* [[compact topological space]]

* [[countably compact topological space]]

* [[sequentially compact topological space]]

* [[paracompact topological space]]

* [[strongly compact topological space]]

[[!redirects countably compact]]

[[!redirects countably compact space]]
[[!redirects countably compact spaces]]


countably locally discrete set of subsets

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## Definition

Given a [[topological space]] $X$, then a set of [[subsets]] $\{S_i \subset X\}_{i \in I}$ is _countably locally discrete_ or _$\sigma$-locally discrete_ if it is a countable union of [[locally discrete set of subsets|locally discrete sets of subsets]].

Note that [[open covers of metric spaces have open countably locally discrete refinements]].

## Example

* Every [[metrisable topological space]] admits a [[topological base|base]] which is $\sigma$-locally discrete, see [[open covers of metric spaces have open countably locally discrete refinements]].

## Related concepts

* [[locally finite cover]]
* [[locally finite set of subsets]]
* [[locally discrete set of subsets]]
* [[countably locally finite set of subsets]]

## References

* Ryszard Engelking, _General Topology_, Heldermann Verlag Berlin, 1989.

* K. Kuratowski, _Topology_, 2014, vol. 1.

[[!redirects countably locally discrete sets of subsets]]
[[!redirects sigma-locally discrete sets of subsets]]
[[!redirects countably locally disjoint sets of subsets]]
[[!redirects sigma-locally disjoint sets of subsets]]


countably locally finite set of subsets

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## Definition

Given a [[topological space]] $X$, then a set of [[subsets]] $\{S_i \subset X\}_{i \in I}$ is _countably locally finite_ or _$\sigma$-locally finite_ if it is a countable union of [[locally finite set of subsets|locally finite sets of subsets]].

This property is used in [[Nagata-Smirnov metrization theorem]]:

[[!include Nagata-Smirnov metrization theorem - main theorem]]

## Related concepts

* [[locally finite cover]]
* [[locally finite set of subsets]]
* [[locally discrete set of subsets]]
* [[countably locally discrete set of subsets]]

## References

* [σ-Locally Finite Base](https://topology.jdabbs.com/properties/P000054), $\pi$-Base.

[[!redirects countably locally finite sets of subsets]]
[[!redirects sigma-locally finite sets of subsets]]


countably paracompact topological space

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## Definition

A [[topological space]] is _countably paracompact_ if for every [[countable cover]] has a [[refinement]] by a [[locally finite cover]].

## Related concepts

* [[paracompact topological space]]

* [[countably compact topological space]]

* [[Dowker space]]

[[!redirects countably paracompact topological spaces]]

[[!redirects countably paracompact space]]
[[!redirects countably paracompact spaces]]


countably tight space

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## Idea

The topology of a countably tight space is determined by countable subsets similar to how in a [[sequential space]] the topology is generated by sequences.

## Definition

A [[topological space]] $X$ is called _countably tight_ or _countably generated_ if a set $A \subset X $ is closed if (and hence only if) for each countable subset $D \subset X$ the intersection $A \cap D$ is closed in the relative topology on $D$.

## Categorical properties

The category of countably tight spaces is a [[coreflective subcategory]] of [[Top]]. More precisely, it is the coreflective hull of the subcategory of countable spaces.

## Related countability properties

[[!include topology - countability axioms]]

## References

* [Countably generated space](https://en.wikipedia.org/wiki/Countably_generated_space), Wikipedia


counterexample

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## Idea

In [[mathematics]], a *counterexample* is an [[example]] for the [[negation]] of a statement.

A counterexample to a [[conjecture]] is a way to satisfy the hypotheses of the conjecture such that the conclusion is a contradiction.

In [[constructive mathematics]] one considers also "[[weak counterexamples]]". See there for more.

## Examples

* [[counterexamples in algebra]]

* [[counterexamples in category theory]]

* [[Cauchy's mistake]]

## Related concepts

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

## References

Examples of counterexamples in [[topology]]:

* [[Lynn Arthur Steen]], [[J. Arthur Seebach]], *[[Counterexamples in Topology]]*, Springer 1970/1978 $[$[doi:10.1007/978-1-4612-6290-9](https://link.springer.com/book/10.1007/978-1-4612-6290-9)$]$

See also:

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

[[!redirects counterexamples]]

[[!redirects counter-example]]
[[!redirects counter-examples]]


counterexamples in algebra

This page lists [[counterexamples]] in [[algebra]].

#Contents#
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## Group Theory (including quasigroups, semigroups, etc)

1. A non-[[abelian group|abelian]] [[group]], all of whose [[subgroup]]s are [[normal subgroup|normal]]:

   $$
   Q \coloneqq \langle a, b | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle
   $$

1. A [[finitely presented group|finitely presented]], infinite, [[simple group]]

   [[Thomson's group]] T.

1. A [[group]] that is not the [[fundamental group]] of any [[3-manifold]].

   $$
   \mathbb{Z}^4
   $$

1. Two [[finite group|finite]] non-[[isomorphism|isomorphic]] groups with the same [[order profile]].

   $$
   C_4 \times C_4, \qquad C_2 \times \langle a, b, | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle
   $$

1. A [[quasigroup]] that is not isomorphic to any [[loop (algebra)|loop]].

   $\{a, b, c\}$ with multiplication table:

   $$
   \begin{matrix}
   * & a & b & c \\
   a & a & c & b \\
   b & c & b & a \\
   c & b & a & c
   \end{matrix}
   $$

1. A counterexample to the converse of [[Lagrange's theorem]].

   The [[alternating group]] $A_4$ has order $12$ but no [[subgroup]] of order $6$.

1. A [[finite group]] in which the product of two [[commutator]]s is not a commutator.

   $$
   G = \langle (a c)(b d), (e g)(f h), (i k)(j l), (m o)(n p), (a c)(e g)(i k), (a b)(c d)(m o), (e f)(g h)(m n)(o p), (i j)(k l)\rangle \subseteq S_{16}
   $$

1. A [[finitely generated group]] with a non-finitely generated subgroup.

   The free group on two generators $x$ and $y$ has commutator subgroup freely generated by $[x^n,y^m]$.

1. An Artinian but not Noetherian $\mathbb{Z}$-module.

   A [[Prüfer group]]. (The correct theorem is that an [[Artinian ring|Artinian]] _[[ring]]_ is [[Noetherian ring|Noetherian]].)

## Ring Theory

1. A [[ring]] that is right [[Noetherian ring|Noetherian]] but not left Noetherian:

   Matrices of the form $\begin{bmatrix} a & b \\ 0 & c \end{bmatrix}$ where $a \in \mathbb{Z}$ and $b,c \in \mathbb{Q}$.

1. A ring that is local commutative Noetherian but is not a [[Cohen-Macaulay ring]].

   $$
   k[x,y]/(x^2, x y)
   $$

1. A [[number ring]] that is a [[principal ideal domain]] that is not Euclidean.

   $$
   \mathbb{Q}(\sqrt{-19})
   $$

1. An [[epimorphism]] of [[rings]] that is not [[surjective]].

   $$
   \mathbb{Z} \to \mathbb{Q}
   $$

1. A [[ring]] whose [[spectrum|spec]] has non-[[open subset|open]] [[connected]] components.

   $$
   \prod_{n=1}^\infty \mathbb{F}_2
   $$

1. A non-[[Noetherian ring|Noetherian]] [[ring]] $A$ such that all local rings on $Spec(A)$ are Noetherian.

   $$
   \prod_{n=1}^\infty \mathbb{F}_2
   $$

1. A [[number field]] whose [[ring of integers]] is Euclidean but not norm-Euclidean.

   $$
   \mathbb{Q}(\sqrt{69})
   $$

## Hopf Algebras

1. A non-commutative and non-cocommutative [[Hopf algebra]]

   $$
   \begin{aligned}
   H &\coloneqq &\langle x, g | g^2 = 1, x^2 = 0, g x g = -x\rangle \\
   \Delta(g) &= &g \otimes g, \\
   \Delta(x) &= &x \otimes 1 + g \otimes x, \\
   \epsilon(g) &=& 1, \\
   \epsilon(x) &=& 0, \\
   S(g) &= &g, \\
   S(x) &= &- g x
   \end{aligned}
   $$

## Homological Algebra

1. An [[exact sequence]] that does not [[split sequence|split]]:

   $$
   0 \to \mathbb{Z} \stackrel{\times 2}{\to} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0
   $$

## Galois Theory

1. A [[polynomial]], solvable in [[radical]]s, whose [[splitting field]] is not a [[radical extension]] of $\mathbb{Q}$.

   Take any cyclic cubic; that is, any cubic with rational coefficients, irreducible over the [[rational number|rational]]s, with [[Galois group]] cyclic of order $3$.

2. A composition of two normal extensions need not be normal:

$$
   \mathbb{Q} \subset \mathbb{Q}(2^{1/2}) \subset \mathbb{Q}(2^{1/4})
   $$

## References

The initial import of counterexamples in this entry was taken from [this MO question](http://mathoverflow.net/questions/29006/counterexamples-in-algebra). See also [[counterexamples in category theory]].


counterexamples in category theory

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## Idea

In [[category theory]], there's often the sense that 'things just work', and that details can be skipped as everything is well behaved. Unfortunately, this is not quite true.

The [[counterexamples]] here should serve as specimens of possible pittfalls in category theory, and give an idea of what could go wrong.

## List

1. Sending an [[object]] $x \in \mathcal{C}$ of a [[category]] $\mathcal{C}$ to its [[automorphism group]] $Aut_{\mathcal{C}}(x)$  (or its [[endomorphism monoid]] $End_{\mathcal{C}}(x)$ ) does not in general extend to a [[functor]] from $\mathcal{C}$ to [[Groups]].

   It does however extend to a [[functor]] on the [[core]] of $\mathcal{C}$ (the maximal [[groupoid]] inside it, keeping only the [[isomorphisms]] of $\mathcal{C}$), where it sends [[morphisms]] (now constrained to be [[isomorphisms]]) to their [[conjugation action]]:

   $$
     \array{
       Core(\mathcal{C})
       &
       \overset{
         \;\;\;
         Aut_{\mathcal{C}}
         \;\;\;
       }{
         \longrightarrow
       }
       &
       Groups
       \\
       x
       &
       &
       Aut_{\mathcal{C}}(x)
       \\
       {}^{{}_{\mathllap{
         \simeq
       }}}
       \big\downarrow
       {}^{{}_{\mathrlap{
         \gamma
       }}}
       &
       \mapsto
       &
       \big\downarrow
       {}^{{}_{\mathrlap{
         ad_\gamma
         \colon
         g \mapsto \gamma \circ g \circ \gamma^{-1}
       }}}
       &
       \\
       y
       &
       &
       Aut_{\mathcal{C}}(y)
       \,.
     }
   $$

   For example, if $\mathcal{C} = \Pi_1(X)$ is the [[fundamental groupoid]] of a [[topological space]] (which thus coincides with its [[core]] already), then the [[automorphism groups]] of its objects $x \in X$ are the [[fundamental groups]] $\pi_1(X,x)$ at these basepoints, which famously *are* functorial under conjugation by paths in $X$.

1. Forming the [[center]] of a [[group]] does not extend to a [[functor]] from [[Groups]] to [[AbelianGroups]].

   It does extend to a functor on the [[core]], though:

   $$
     \array{
       Core(Groups)
       &
       \overset{
         Center
       }{
         \longrightarrow
       }
       &
       AbelianGroups
       \,.
     }
   $$

   Notice that this example is really a special case of the previous one (forming [[automorphism groups]]), or rather of a [[2-category theory|2-category theoretic]] version of it: The [[center]] of a [[group]] is the [[automorphism group]] in the [[endofunctor|endo]]-[[functor category]]  of the [[identity functor]] on the one-object [[delooping]]-[[groupoid]] $\mathbf{B}G$ of $G$:

   $$
     Center(G)
     \;\simeq\;
     Aut_{{}_{End(\mathbf{B}G)}}
     \big(
       id_{\mathbf{B}G}
     \big)
   $$

1. Composing a [[monadic functor]] with another monadic functor need not be monadic. For example, Torsion-free abelian groups are monadic over abelian groups, which are monadic over sets, but torsion-free abelian groups are not monadic over sets.

1. Taking the skeleton of a [[monoidal category]] does not in general result in a [[strict monoidal category]]. The argument in the case of Set is given in [this MO answer](https://mathoverflow.net/a/128629). In particular, one cannot in general replace a monoidal category with an equivalent category that is simultaneously  strict monoidal and skeletal.

1. The category of topological spaces and [[local homeomorphism | local homeomorphisms]] is [[locally cartesian closed category | locally cartesian closed]] but not [[cartesian closed category | cartesian closed]] since it does not have a terminal object.

1. There are functors $D:Aff\to Vect$ (taking the vector space of displacements) and $A:Vect \to Aff$ (taking the underlying affine space) between the categories of [[vector spaces]] and [[affine spaces]], and we have $D(A(V)) \cong V$ for any $V\in Vect$, and for any $U\in Aff$ there exists some isomorphism $A(D(U)) \cong U$ (after choosing a point in $U$ to serve as the origin), but the categories are not [[equivalence of categories|equivalent]] --- the second isomorphisms cannot be chosen naturally, not even after restricting to the [[core|cores]].

1. The opposite of the category of commutative [[von Neumann algebra | von Neumann algebras]] has a [[subobject classifier]] and it's [[finitely complete category | finitely complete]], but is not a [[topos]] since it is not [[cartesian closed]]. See [this MO question](https://mathoverflow.net/questions/384346/is-the-opposite-category-of-commutative-von-neumann-algebras-a-topos).

1. There is a 'wrong right adjoint' to the functor $F : Petri \to CMC$ (as described in [[Petri nets#semantics]]) which sends a [[commutative monoidal category]] $C$ to the [[Petri net]] whose source and target are $\eta \circ s : Mor(C) \to \mathbb{N}[Ob(C)]$ and $\eta \circ t : Mor(C) \to \mathbb{N}[Ob(C)]$ where $s$ and $t$ are the source and target of $C$ and $\eta$ is the unit for the monad $\mathbb{N}[-]$. In words, $C$ is sent to the [[Petri net]] whose source and target are the [[source]] and [[target]] of $C$ composed with $\eta$. The natural choice of isomorphism $Hom(FA, B) \cong Hom(A, GB)$, where $G$ is the functor just described, turns out not to yield well-defined morphisms of [[Petri nets]]. See Remark 4.4 in [Master 2020](#PN)

1. Let $C$ be a category without an [[initial object]] and let $R : C \to 1$ be the unique functor from $C$ into the [[terminal category]]. Then $R$ preserves all limits but does not have a left adjoint because this left adjoint would have to send the unique object of 1 to an initial object.

1.  Consider the functor $U\colon Group \to Set$ that sends a group $G$
to the set $\prod_{i\in I} hom(S_i, G)$,
where the groups $S_i$ are simple for all $i\in I$,
$I$ is the [[class]] of all [[ordinals]],
and the cardinality of $S_i$ eventually becomes bigger than any fixed cardinal as $i$ increases.
This is a limit-preserving functor between [[locally presentable categories]].
However, it does not have a [[left adjoint]].

1. [[dense functor | Dense functors]] are not closed under composition. For example, $\Delta_{\lt 2}$ is dense in the [[simplex category]] $\Delta$ and $\Delta$ is dense in $\mathbf{Cat}$, but $\Delta_{\lt 2}$ is not dense in $\mathbf{Cat}$.

## See also

* [[counterexamples in algebra]]

## References

The initial import of counterexamples in this entry was taken from [this Zulip discussion](https://mattecapu.github.io/ct-zulip-archive/stream/229111-general/topic/Counterexamples.20in.20Category.20Theory.html).

* {#PN} [[Jade Master]], 2020, _[Petri nets based on Lawvere theories](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/petri-nets-based-on-lawvere-theories/9E0F011860A1285DDC498B740774FBDC)_, Mathematical Structures in Computer Science, 30(7), 833-864


Counterexamples in Topology

This page is about the book:

* [[Lynn Arthur Steen]], [[J. Arthur Seebach]]:

  **Counterexamples in Topology**

  Springer 1970/1978

  [doi:10.1007/978-1-4612-6290-9](https://link.springer.com/book/10.1007/978-1-4612-6290-9)

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

on [[counterexamples]] in [[topology]].

category: reference


counterterm

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###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] one way to construct an [[S-matrix]] $\mathcal{S}$ via [[renormalization|("re"-)normalization]] of [[time-ordered products]]/[[Feynman amplitudes]] is to consider a sequence of [[UV cutoffs]] $\Lambda$ with corresponding [[effective S-matrices]] $\mathcal{S}_\Lambda$ and then form the [[limit of a sequence|limit]] as $\Lambda \to \infty$, which exists if in the course one applies suitable [[interaction vertex redefinitions]] $\mathcal{Z}_\Lambda$

$$
  \mathcal{S}(g S_{int} + j A) = \underset{\Lambda \to \infty}{\lim} (\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda)(g S_{int} + j A)
$$

(See at _[[effective QFT]]_ [this prop.](effective+quantum+field+theory#UVRegularization)).

This may be read as saying that as one "removes the UV-cutoff" the original [[interaction]] [[action functional]] $g S_{int} + j A$ is to be corrected by "counterterm-[[interactions]]"

$$
  S_{counter, \Lambda}
  \;\coloneqq\:
  \left(
    \mathcal{Z}_\Lambda - id
  \right)
  \left(
    g S_{int} + j A
  \right)
  \,.
$$

## Details

See at _[[effective action]]_ around [this remark](effective+quantum+field+theory#TermCounter).

## References

The rigorous formulation of [[renormalization|("re"-)normalization]] via [[UV cutoff]] and counterterms in [[causal perturbation theory]]/[[perturbative QFT]] is due to

* {#BrunettiDuetschFredenhagen09} [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], section 5.2 of _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))

* {#Duetsch10} [[Michael Dütsch]], _Connection between the renormalization groups of Stückelberg-Petermann and Wilson_, Confluentes Mathematici, Vol. 4, No. 1 (2012) 12400014 ([arXiv:1012.5604](https://arxiv.org/abs/1012.5604))

* {#DuetschFredenhagenKellerRejzner14} [[Michael Dütsch]], [[Klaus Fredenhagen]], [[Kai Keller]], [[Katarzyna Rejzner]], appendix A of _Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization_, J. Math. Phy. 55(12), 122303 (2014) ([arXiv:1311.5424](https://arxiv.org/abs/1311.5424))

reviewed in

* {#Duetsch18} [[Michael Dütsch]], section 3.8 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

See also

* [[Kevin Costello]], section 8 _Renormalisation and the Batalin-Vilkovisky formalism_ ([arXiv:0706.1533](http://arxiv.org/abs/0706.1533))

[[!redirects counterterms]]

[[!redirects counter-term]]
[[!redirects counter-terms]]


counting measure

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### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _counting measure_ on a [[measurable set]] (in particular on a [[countable set]]) is the [[measure]] which assigns to a [[subset]] the [[cardinality]] of this subset (hence which "counts" the number of elements of the subset).

## References

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


coupling constant

> This page is about the notion in [[physics]]/[[quantum field theory]]. For the concept of "coupling" in [[probability theory]], see *[[transport plan]]*.

***

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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the [[Lagrangian]] of a [[prequantum field theory]] the those terms that involve various [[field (physics)|fields]] describe the _coupling_ or _[[interaction]]_ of these fields with each other. Hence the prefactors of these terms encode the strength of this coupling, and hence are called _coupling constants_.

In [[perturbative quantum field theory]] the coupling constant (as well as [[Planck's constant]] which indicates the strength of [[quantum physics|quantum]] effects) is regarded as tiny, in fact as [[infinitesimal]], in that all [[observables]] are expressed as (generally non-converging, [[asymptotic series|asymptotic]]) [[formal power series]] in the coupling constant.

Beyond that approximation, in the realm of finite (large) coupling, one speaks of *[[non-perturbative field theory]]*.

## Example

* [[Yukawa interaction|Yukawa coupling constant]]

## Related concepts

* [[adiabatic switching]]

* [[renormalization]], [[running coupling constants]]

* [[complexified coupling constant]]

* [[gauge coupling unification]]

* [['t Hooft coupling]]

* [[string coupling constant]]

## References

See also

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

[[!redirects coupling constants]]

[[!redirects gauge coupling]]

[[!redirects coupling]]
[[!redirects couplings]]


coupling constants

[[!redirects coupling constants]]

#Contents#
* table of contents
{:toc}

## Idea

In the [[Lagrangian]] of a [[prequantum field theory]] the those terms that involve various [[field (physics)|fields]] describe the _coupling_ or _interaction_ of these fields with each other. Hence the prefactors of these terms encode the strength of this coupling, and hence are called _coupling constants_.

## Example

* [[renormalization]]

* [[Yukawa interaction|Yukawa coupling constant]]


Courant algebroid

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###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **Courant algebroid**  -- or better: **Courant Lie 2-algebroid** -- (named after [[Theodore Courant]]) is precisely a [[symplectic Lie n-algebroid|symplectic Lie 2-algebroid]] ([Roytenberg](#RoytenbergStructure)):

it is a [[L-infinity algebroid|Lie 2-algebroid]] $\mathfrak{P}$ whose [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{P})$ is equipped with the structure of a [[Poisson n-algebra|Poisson 3-algebra]] whose Poisson bracket

$$
  \{-,-\} : CE(\mathfrak{P})\otimes CE(\mathfrak{P})
  \to
  CE(\mathfrak{P})
$$

of degree -2 is non-degenerate.

Therefore the [[differential]] $d_{CE(\mathfrak{P})}$ on $CE(\mathfrak{P})$ has a [[Hamiltonian]] with respect to this bracket in that there is an element $\Theta \in CE(\mathfrak{P})$ such that

$$
  d_{CE(\mathfrak{P})} = \{\Theta, -\}
  \,.
$$

## History

The concept of Courant algebroids was originally introduced by [[Irene Dorfman]] and [[Ted Courant]] to study [[geometric quantization]] in the presence of constraints. Later it was considered by Liu, [[Alan Weinstein]] and [[Ping Xu]] in the study of [[double Lie algebroids]].

In these parts of the literature Courant algebroids are considered in the form of [[Lie algebroid]]s with relaxed axioms on the bracket. Even of this type there are two different definitions:

* in one there is a skew-symmetric bracket which fails to satisfy a Jacobi identity by a coherent term -- this is the Courant bracket definition proper;

* in the other there is a bracket which satisfies a Jacobi identity but is skew-symmetric only up to a correction term -- this is the Dorfman version.

So there are several different ways to present the structure encoded in a Courant algebroid. The picture that seems to be emerging is that the _true_ meaning of the notoin of Courant algebroids is given by the notion of [[n-symplectic manifold|2-symplectic manifold]]s.

Moreover, the way [[Lie algebroid]]s may be expressed in terms of [[Lie-Rinehart algebra]]s, Courant algebroids yield [[Courant-Dorfman algebra]]s.

***

(... need to say more about the way the Courant Lie algebroid is obtained from a Lie bialgebroid by derived brackets ...)

***

## Examples

### Lie algebras of compact type

A Courant Lie 2-algebroid over the point is precisely an ordinary [[Lie algebra]] $\mathfrak{g}$ that is equipped with a quadratic and non-degenerate [[invariant polynomial]].

### Standard Courant algebroid and $U(1)$-gerbes

The [[standard Courant algebroid]] of a [[manifold]] $X$ is the one which

* as a [[vector bundle]] with extra structure is $E = T X\oplus T^* X$, the fiberwise direct sum of the [[tangent bundle]] and the cotangent bundle; with

  * bilinear form

    $$
      \langle X + \xi , Y +\eta \rangle = \eta(X) + \xi(Y)
    $$

  for $X,Y \in \Gamma(T X)$ and $\xi, \eta \in \Gamma(T^* X)$

  * brackets

    $$
      [X + \xi, Y + \eta] = [X,Y] + \mathcal{L}_X \eta
      - \mathcal{L}_Y \xi + \frac{1}{2} d (\eta(X) - \xi(Y))
    $$

    where $\mathcal{L}_X \eta = \{d,\iota_X\} \eta$ denotes the [[Lie derivative]] of the 1-form $\eta$ by the vector field $X$.

* as a [[dg-manifold]] is $T^*[2] T[1] X$, the shifted cotangent bundle of the [[shifted tangent bundle]],

  where the [[differential]] is on each local [[coordinate patch]] $\mathbb{R}^n \simeq U \subset X$ with coordinates $\{x^i\}$ in degree 0, $\{d x^i\}$ and $\{\theta_i\}$ in degree 1 and $\{p_i\}$ in degree 2 given by

  $$
    \begin{aligned}
      d_C &= d_{dR} + p_i \frac{\partial}{\partial \theta_i}
      \\
      &= dx^i \frac{\partial}{\partial x^i }
       + p_i \frac{\partial}{\partial \theta_i}
    \end{aligned}
    \,.
  $$

Such a standard Courant algebroid may be understood as the higher analog of the  [[Atiyah Lie algebroid]] of a line bundle. See below in _[Relation to Atiyah groupoids](#RelationToAtiyahGroupoids)_.

## Properties

### Generalized complex geometry

The study of Courant algebroids is to a large extent known as [[generalized complex geometry]], where the Courant algebroid appears as the [[generalized tangent bundle]].

### Chern-Simons element and Courant $\sigma$-model

As every [[symplectic Lie n-algebroid]] the defining [[invariant polynomial]] on a Courant Lie 2-algebroid transgresses to a cocycle in [[∞-Lie algebroid cohomology]] and this transgression is witnessed by a [[Chern-Simons element]]. The [[schreiber:∞-Chern-Simons theory]] induced by this element is the [[Courant sigma-model]].

### Lagrangian submanifolds and Dirac structures

The [[Lagrangian dg-submanifolds]] of a Courant Lie 2-algebroid corespond to its [[Dirac structures]].

### Relation to Atiyah Lie 2-algebroid and quantomorphism 2-group
 {#RelationToAtiyahGroupoids}

We discuss how the following tower of notions works

| [[circle n-bundle]] with $(n-1)$-form connection | [[Lie n-algebra]] of [[group of bisections|of bisections]] of Atiyah Lie n-group |
|--|--|
|  [[circle bundle]] |  [[Lie algebra]] of [[sections]] of [[Atiyah Lie algebroid]] |
|  [[circle 2-bundle]] with 1-form connection | [[Lie 2-algebra]] of [[sections]] of [[Courant Lie 2-algebroid]]

For $n,k \in \mathbb{N}$ and $k \leq n$ write

$$
  \mathbf{B}^n U(1)_{conn^k}
  \coloneqq
  DK\left[
    U(1) \stackrel{d log}{\to}
    \Omega^1
    \stackrel{d}{\to}
    \cdots
    \stackrel{d}{\to}
     \Omega^{k-1}
    \stackrel{d}{\to} \Omega^k
     \to
     \underbrace{
     0
     \to 0
     \to \cdots
     \to 0
     }_{n-k}
  \right]
$$

for the [[smooth ∞-groupoid]] which is presented under the [[Dold-Kan correspondence]] by the [[sheaf]] of [[chain complexes]], as indicated (see also at _[differential cohomology diagram -- Examples -- Deligne coefficients](differential+cohomology+diagram#DeligneCoefficients)_). This is such that for $k = n$ we have the [[Deligne complex]], representing the [[moduli ∞-stack]] of [[circle n-bundles with connection]]

$$
  \mathbf{B}^n U(1)_{conn^n} \simeq \mathbf{B}^n U(1)_{conn}
$$

and for $k = 0$ we have the moduli $\infty$-stack for plain [[circle n-group]] [[principal ∞-bundles]]

$$
  \mathbf{B}^n U(1)_{conn^0} \simeq \mathbf{B}^n U(1)
  \,.
$$

For $k_2 \lt k_1$ there are evident truncation maps

$$
  \mathbf{B}^n U(1)_{conn^{k_1}}
  \to
  \mathbf{B}^n U(1)_{conn^{k_2}}
  \,.
$$

Now for $X \in $ [[SmthMfd]] $\hookrightarrow$ [[Smooth∞Grpd]] a [[smooth manifold]], a map

$$
  \nabla \;\colon\; X \to \mathbf{B}^n U(1)_{conn}
$$

modulates a [[circle n-bundle with connection]] ([[bundle gerbe|bundle (n-1)-gerbe]]), which we may think of as a _[[prequantum circle n-bundle]]_. Regarding this as an [[object]] in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}^n U(1)}$ this has an [[automorphism ∞-group]]. The [[concretification]] of this (...) is the [[quantomorphism n-group]] $QuantMorph(\nabla)$.

$$
  \mathbf{QuantMorph}(\nabla)
  \coloneqq
  conc\mathbf{Aut}(\nabla)
  =
  \left\{
    \array{
      X && \underoverset{\simeq}{\phi}{\to} && X
      \\
      & {}_{\mathllap{\nabla}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\nabla}}
      \\
      && \mathbf{B}^n U(1)_{conn}
    }
  \right\}
  \,.
$$

But we can also first forget the $n$-form pieces of the prequantum $n$-bundle away and consider

$$
  \nabla_{n-1}
    \;\colon\;
   X
    \stackrel{\nabla}{\to}
   \mathbf{B}^n U(1)_{conn}
   \simeq
   \mathbf{B}^n U(1)_{conn^n}
   \to
   \mathbf{B}^n U(1)_{conn^{n-1}}
  \,.
$$

For $n = 2$ this is sometimes known in the literature as a "bundle gerbe with connective structure but without curving".

The concretified [[automorphism ∞-group]] of that truncated connection is the [[n-group]] of [[group of bisections|of bisection]] of the [[Atiyah n-groupoid]]

$$
  conc \mathbf{Aut}(\nabla_{n-1}) \in Grp(Smooth \infty Grpd)
  \,.
$$

For $n = 1$ this is the [[group of bisections]] of the [[Atiyah Lie groupoid]] of the underlying [[circle principal bundle]] $\nabla_0 \colon X \to \mathbf{B} U(1)$. Hence its [[Lie differentiation]] is the [[Lie algebra]] of sections of the corresponding [[Atiyah Lie algebroid]].

For $n = 2$ the [[Lie differentiation]] of this [[Lie 2-group]] is the [[Lie 2-algebra]] of sections of the corresponding Courant Lie 2-algebroid. With a little bit of translation, this is what is shown in ([Collier](#Collier)).

Finally notice that the forgetful map $\mathbf{B}^n U(1)_{conn} \to \mathbf{B}^n U(1)_{conn^{n-1}}$ induces an [[homomorphism]] of [[∞-groups]]

$$
  conc \mathbf{Aut}(\nabla) \to conc \mathbf{Aut}(\nabla_{n-1})
$$

hence an embedding of the [[quantomorphism n-group]] into the $n$-[[group of bisections]] of the Atiyah n-groupoid. For $n = 2$ and after [[Lie differentiation]], this is an embedding of the [[Poisson Lie 2-algebra]] into the sections of the Courant Lie 2-algebroid. This embedding had been observed in ([Rogers](#Rogers)).

## Related concepts

[[!include higher Atiyah groupoid - table]]

* [[string algebroid]]

* [[symplectic Lie n-algebroid]]

  * [[symplectic manifold]]

  * [[Poisson Lie algebroid]]

  * **Courant algebroid**

* [[Dirac structure]]

* [[Courant sigma-model]]

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

## References

The original references in order of appearance are

* [[Pavol Ševera]], _Letters to A. Weinstein_ ([web](http://sophia.dtp.fmph.uniba.sk/~severa/letters/), [arXiv:1707.00265](https://arxiv.org/abs/1707.00265))

* [[Dmitry Roytenberg]], [[Alan Weinstein]], _Courant algebroids and strongly homotopy Lie algebras_, Lett. Math. Physics __46__(1):81-93, 1998.

* [[Dmitry Roytenberg]], _Courant algebroids, derived brackets and even symplectic supermanifolds_  PhD
thesis, University of California, Berkeley, 1999.
([math.DG/9910078](http://arxiv.org/abs/math.DG/9910078))

* [[Pavol Ševera]], _Some title containing the words "homotopy" and "symplectic", e.g. this one_, In Travaux
mathématiques. Fasc. XVI, chapter Trav. Math., XVI, pp. 121-137. Univ. Luxembourg, 2005.

* [[Dmitry Roytenberg]], _On the structure of graded symplectic supermanifolds and Courant algebroids_, pp. 169--185. in Contemporary Mathematics __315__, _Quantization, Poisson brackets and beyond_ (Manchester, 2001), Theodore Voronov, editor, Amer. Math. Soc. 2002.
([arXiv:math/0203110](http://arxiv.org/abs/math/0203110))
{#RoytenbergStructure}

* [[Dmitry Roytenberg]], _Quasi-Lie bialgebroids and twisted Poisson manifolds_ , Letters in Mathematical
Physics, 61(2):123-137 (2002) ([arXiv:math/0112152](http://arxiv.org/abs/math/0112152))

Another useful summary of the theory of Courant algebroids is in  of

* [[Marco Gualtieri]], [section 3](http://arxiv.org/PS_cache/math/pdf/0401/0401221v1.pdf#page=19) of: _Generalized complex geometry_ ([arXiv:math/0401221](http://arxiv.org/abs/math/0401221))

On the [[morphisms]] between Courant algebroids:

* [[Jan Vysoky]], _Hitchhiker's Guide to Courant Algebroid Relations_ ([arXiv:1910.05347](https://arxiv.org/abs/1910.05347))

A discussion of Courant algebroids with an eye towards the relation of the [[standard Courant algebroid]] to [[bundle gerbes]] is

* [[Paul Bressler]], Alexander Chervov, _Courant algebroids_ ([hep-th/0212195](http://arxiv.org/abs/hep-th/0212195))

The identification of the Lie 2-algebra of sctions of a Courant Lie 2-algebroid associated with a [[circle 2-bundle with connection]] as its Lie algebra of automorphisms after forgetting the "curving" is in

* [[Braxton Collier]], _Infinitesimal Symmetries of Dixmier-Douady Gerbes_ ([arXiv:1108.1525](http://arxiv.org/abs/1108.1525))
 {#Collier}

The embedding of the [[Poisson Lie 2-algebra]] of a given [[2-plectic geometry]] into the [[Lie 2-algebra]] of sections of the Courant Lie 2-algebroid of the corresponding [[prequantum 2-bundle]] is observed in

* [[Chris Rogers]], _Courant algebroids from categorified symplectic geometry_, ([arXiv:1001.0040](http://arxiv.org/abs/1001.0040))
 {#Rogers}

This is developed further in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_ ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))

See also

* Wikipedia, _[Courant algebroid](http://en.wikipedia.org/wiki/Courant_algebroid)_

The relation between the two different Lie-alebroid-like  definition of Courant algebroids, one with skew, the other with non-skew brackets inspired on the level of [[L-infinity algebra|Lie 2-algebras]] the treatment

* [[Dmitry Roytenberg]], _On weak Lie 2-algebras_ ([arXiv/0712.3461](http://arxiv.org/abs/0712.3461))

* [[Chris Rogers]], _2-plectic geometry, Courant algebroids, and categorified prequantization_, ([arxiv:1009.2975](http://arxiv.org/abs/1009.2975))

A proposal for a higher analog of the [[standard Courant algebroid]] with the [[generalized tangent bundle]] $T X \oplus T^* X$ replaced by $T X \oplus \wedge^n T^* X$ -- for a notion of standard [[higher Courant Lie algebroid]] -- is discussed in

* [[Marco Zambon]], _$L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids_, J. Symplectic Geometry, J. Sympl. Geom. __10__:4 (2012) 563--599 [arXiv:1003.1004](https://arxiv.org/abs/1003.1004) [Zbl:1260.53134](https://zbmath.org/?q=an:1260.53134) [doi](https://doi.org/10.4310/JSG.2012.v10.n4.a4)

The relation to [[schreiber:∞-Chern-Simons theory]] is discussed in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher Chern-Weil Derivation of AKSZ Sigma-Models]]_

Discussion of [[Riemannian geometry]] on Courant algebroids and relation to [[supergravity]] [[equations of motion]] is in

* [[Branislav Jurco]], [[Jan Vysoky]], _Courant Algebroid Connections and String Effective Actions_, Proceedings of Tohoku Forum for Creativity, Special volume: Noncommutative Geometry and Physics IV ([arXiv:1612.01540](https://arxiv.org/abs/1612.01540))

See also

* Xu Xiaomeng, _Twisted Courant algebroids and coisotropic Cartan geometries_ ([arXiv:1206.2282](http://arxiv.org/abs/1206.2282))

[[!redirects Courant Lie algebroid]]
[[!redirects Courant Lie algebroids]]

[[!redirects Courant algebroids]]

[[!redirects Courant Lie 2-algebroid]]
[[!redirects Courant Lie 2-algebroids]]

[[!redirects Courant bracket]]
[[!redirects Courant brackets]]


Courant L-infinity algebra

[[!redirects Courant L-∞ algebra]]

#Contents#
* table of contents
{:toc}

## Idea

The [[L-∞ algebras]] under [[Lie differentiation]] of the [[∞-groups of bisections]] of [[higher Courant groupoids]] (see there for the moment).

## Related concepts

[[!include slice automorphism groups in higher prequantum geometry - table]]


Courant Lie algebroid > history

< [[Courant Lie algebroid]]
[[!redirects Courant Lie algebroid -- history]]


Courant sigma-model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Every [[symplectic Lie n-algebroid]] $(\mathfrak{P},\omega)$ carries a specified [[invariant polynomial]] $\omega$. The [[action functional]] induced by the corresponding [[Chern-Simons element]] in [[schreiber:∞-Chern-Simons theory]] defined the corresponding [[AKSZ theory]] [[sigma model]].

For a [[Courant algebroid]] $(\mathfrak{P}, \omega)$ this is called the _Courant $\sigma$-model_ .

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[sigma-model]]

  * [[AKSZ sigma-model]]

    * [[Poisson sigma-model]]

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

    * **Courant sigma-model**

      * [[Chern-Simons theory]]

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

## References

The action functional was first deduced in

* [[Noriaki Ikeda]], _Chern-Simons Gauge Theory coupled with BF Theory_ Int. J. Mod. Phys. A18 (2003) 2689-2702 ([arXiv:hep-th/0203043](http://arxiv.org/abs/hep-th/0203043))

As an example of an [[AKSZ sigma-model]] it was later re-derived in

* [[Dmitry Roytenberg]], _AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories_ (2006) ([arXiv:hep-th/0608150](http://arxiv.org/abs/hep-th/0608150)).

Further discussion is in

* [[Alberto Cattaneo]], Jian Qiu, [[Maxim Zabzine]], _2D and 3D topological field theories for generalized complex geometry_ ([pdf](http://www.math.uzh.ch/fileadmin/math/preprints/GC_BV.pdf))

The interpretation in terms of [[infinity-Chern-Weil theory]] is discussed in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]_ (2011)

Relations to [[generalized complex geometry]] is discussed in

* [[Alberto Cattaneo]], Jian Qiub, [[Maxim Zabzine]], _2D and 3D topological field theories for generalized complex geometry_ ([pdf](http://www.math.uzh.ch/fileadmin/math/preprints/GC_BV.pdf))

[[!redirects Courant sigma model]]


Courant-Dorfman algebra

#Idea#

The notion of _Courant--Dorfman algeba_ is a generalization of the notion of [[Lie–Rinehart algebra]] from [[Lie algebroid]]s to [[Courant algebroid]]s.

#References #

* [[Dmitry Roytenberg]], _Courant-Dorfman algebras and their cohomology_ ([arXiv](http://arxiv.org/abs/0902.4862))

[[!redirects Courant–Dorfman algebra]]
[[!redirects Courant--Dorfman algebra]]


Cours d'Analyse

* [[Augustin Cauchy]]

  _Cours d'Analyse_

  1821

on [[infinitesimal analysis]]/[[epsilontic analysis]].

## Related entries

* [[Cauchy sum theorem]]

## References

* Wikipedia, _[Cours d'Analyse](https://en.wikipedia.org/wiki/Cours_d%27Analyse)_

category: reference


Course of Theoretical Physics

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

This page collects material related to the ten volume textbook series

* [[Lev Landau]], [[Evgeny Lifshitz]]:

  \linebreak

  **Course of Theoretical Physics**

  \linebreak

  $\sim$1950s

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

on [[theoretical physics]]

(often referred to simply as "Landau-Lifschitz").

The series was envisioned but not finished in writing by [[Lev Landau|Landau]] himself. It is used extensively by several generation of physicists, well known for clarity of the arguments. It reflects mainly the role of the main author, [[Lev Landau]]; his pupil Lifschitz helped with all volumes, while some other coauthors contributed to some volumes or revisions; volume 4 is written by Beresteckii, Lifschitz and Pitaevskii, without Landau.

Landau wrote most of volume 1, significant parts of volumes 2,3,5,6,9, and minor part in other volumes. Volume 6 on hydrodynamics reflects his own approach, where he derived most of hydrodynamics from scratch and on the way discovered some of the things classics overlooked. Notice that Lifschitz had a brother, also a theoretical physicist, who actually gave more significant contributions to condensed matter physics.

#Contents#
* table of contents
{:toc}

## I Mechanics

* [[mechanics]]

## II The classical theory of fields

(mainly electrodynamics and gravity; in Russian original the title is _Field theory_)

* [[classical field theory]]

## III Quantum mechanics

* [[quantum mechanics]]

## IV Quantum electrodynamics

* [[quantum electrodynamics]]

## V Statistical physics, Part I

* [[statistical physics]]

## VI Fluid mechanics (Russian original is _Hydrodynamics_)

* [[fluid mechanics]]

## VII Elasticity

* [[elasticity]]

  * [[stress tensor]]

  * [[strain tensor]]

## VIII Electrodynamics of continua
 {#VIII}

[[Lev Landau]], [[Evgeny Lifshitz]], *Electrodynamics of Continuous Media*, volume 8 of: *[[Course of Theoretical Physics]]*, Pergamon Press (1960, 1984) [[ISBN:9780750626347](https://shop.elsevier.com/books/electrodynamics-of-continuous-media/koch/978-0-08-057060-0), [archive](https://archive.org/details/ost-physics-landaulifshitz-electrodynamicsofcontinuousmedia/mode/2up),  [pdf](https://www.aldebaran.cz/studium/books/1984_Landau-Electrodynamics.pdf)]

* [[electrodynamics]] in [[dielectric media]]

* [[magnetohydrodynamics]]

* [[superconductivity]]

* [[electromagnetic waves]]

* ...

## IX Statistical physics, Part II

## X Physical kinetics

## References

* Wikipedia, _[Course of Theoretical Physics](http://en.wikipedia.org/wiki/Course_of_Theoretical_Physics)_

* Wikipedia, _[Lev Landau -- Course of Theoretical Physics](http://en.wikipedia.org/wiki/Lev_Landau#Landau_and_Lifshitz_Course_of_Theoretical_Physics)_

category: reference

[[!redirects Landau-Lifschitz]]


covariant derivative

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

#Contents#
* table of contents
{:toc}

## Idea

In an [[associated bundle]] with [[connection]] the _covariant derivative_ of a [[section]] is a measure for how that section fails to be constant with respect to the connection.

## Definition

### In the context of connections on $\infty$-groupoid principal bundles
 {#ForConnectionsOnInfinityBundles}

We give here a definition of covariant derivatives that is natural in the general context of [[∞-Chern-Weil theory]] in that it applies to [[connections on ∞-bundles]].

We start by describing this just for ordinary [[connections on a bundle]] and demonstrate how this general abstract definition reproduces the traditional definitions found in the literature.

The central statement is: a covariant derivative $\nabla \sigma$ of a section may be identified with the 1-form [[curvature]]-component of a [[Lie algebroid]]-valued connection, and the curvature equation

$$
  \nabla \nabla \sigma = F_\nabla \sigma
$$

is the [[Bianchi identity]] on its curvature 1-form.

#### Preliminaries on action Lie algebroid cohomology {#ActionLieAlgebroidCohomology}

Let $G$ be a [[Lie group]], $V$ a [[smooth manifold]] and $\rho : G \times V \to V$ a smooth [[action]]. Write $V//G$ for the corresponding [[action groupoid]], itself a Lie groupoid.  The [[Lie algebroid]] $Lie(V//G)$ corresponding to this is the [[action Lie algebroid]].

Below we shall define covariant derivatives as curvature components of [[∞-Lie algebroid valued forms]] with values in this action Lie algebroid. To prepare the ground for this, the following observation recalls some basic facts.

The [[Chevalley-Eilenberg algebra]] of the action Lie algebroid is

$$
  CE(Lie(V//G)) = (\wedge^\bullet_{C^\infty(V)} \mathfrak{g}^*, d_{\rho})
  \,,
$$

where the differential acts on functions $f \in C^\infty(V)$ by

$$
  d_\rho : f \mapsto \rho(-)(-)^* f \in C^\infty(V)\otimes \mathfrak{g}^*
  \,.
$$

Explicitly, for $t \in \mathfrak{g}$ this sends $f$ to the function
$(d_\rho f)(t)$ which is the derivative along $t \in T_e G$ of the function $G \times V \stackrel{\rho}{\to}V \stackrel{f}{\to} \mathbb{R}$.

Even more explicitly, if we choose local coordinates $\{v^k\} : \mathbb{R}^{dim V} \to V$ on a patch, and choose a basis $\{t^a\}$ of $\mathfrak{g}^*$ then we have that restricted to this patch the differential is on generators given by

$$
  d_\rho : f \mapsto \rho^\mu{}_a t^a \wedge \partial_k f
$$

$$
  d_\rho : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
  \,.
$$

Specifically for $V$ a finite dimensional [[vector space]], $\rho : G$ a _linear_ action, $\{v^k\}$ a choice of basis of that vector space and $f$ a _[[linear function]]_ $f= f_k v^k$ , we have that $(f_k := \partial_k f) \in \mathbb{R}^{dim V}$ are the components vector of the dual vector given by $V$ in this basis, and the above gives the [[matrix multiplication]] form of the action

$$
  d_\rho : v^k \mapsto t^a \rho_a{}^k{}_l  v^l
  \,.
$$

Notice for completeness that the equation $(d_\rho)^2 = 0$ is equivalent to the [[Jacobi identity]] of the Lie bracket and the action property of $\rho$:

$$
  d_\rho d_\rho v^k =
   (t^a \wedge t^b \rho_a{}^k{}_r \rho_b{}^r{}_l
    -
    \frac{1}{2}C^a{}_{b c}t^b \wedge t^c \rho_a{}^k{}_l )
     v^l
  \,.
$$

These local formulas shall be useful below for recognizing from our general abstract definition of covariant derivative the formulas traditionally given in the literature. For that notice that in the above local coordinates further restricting attention to linear actions, the [[Weil algebra]] of the action Lie algebroid is given by

$$
  W(Lie(V//G)) = (\wedge^\bullet_{C^\infty(\mathbb{R}^{dim V})} ( \Gamma(T^* \mathbb{R}^{dim V}) \oplus \mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W_\rho})
$$

where the differential is given on generators by

$$
  d_{W_\rho} : v^k \mapsto \rho_a{}^k{}_l t^a \wedge v^l + d_{dR} v^k
$$

$$
  d_{W_\rho} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
$$

and where the uniquely induced differential on the shifted generators -- the one encoding [[Bianchi identities]] -- is

$$
  d_{W_\rho}  : d_{dR} v^k
      \mapsto
        \rho_a{}^k{}_k r^a \wedge v^l
      - \rho_a{}^k{}_l t^a \wedge  d_{dR} v^l
$$

and

$$
  d_{W} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c
  \,.
$$

Notice that we may identify the [[delooping]] Lie groupoid $\mathbf{B}G$ of $G$ with the action groupoid of the trivial action on the point, $\mathbf{B}G \simeq *//G$.  On Lie algebroids this morphism is dually the inclusion

$$
  CE(Lie(V//G)) \leftarrow CE(\mathfrak{g})
$$

that is the identity on $\mathfrak{g}^*$.

#### Sections of bundles as groupoid principal bundles

For $X$ [[smooth manifold]], a $G$-[[principal bundle]] $P \to X$ is given by a  [[cocycle]] in [[?LieGrpd]] $g : X \to \mathbf{B}G$.

+-- {: .num_prop}
###### Proposition

The [[section | sections]] $\sigma$ of the corresponding $\rho$-[[associated bundle]] $P \times_\rho V$ are in natural [[bijection]] with the lifts to a $V//G$-cocycle

$$
  \array{
    && V//G
    \\
    & {}^{\mathllap{\sigma}}\nearrow & \downarrow
    \\
    X &\stackrel{g}{\to}& \mathbf{B}G
  }
  \,.
$$

=--

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

We may model the coycle $X \to \mathbf{B}G$ in the [[model structure on simplicial presheaves]] $[CartSp^{op}, sSet]_{proj,loc}$ by an [[anafunctor]] $X \stackrel{\simeq}{\leftarrow} C(U) \stackrel{g}{\to} \mathbf{B}G$ for $C(U)$ the [[Cech groupoid]] of a [[good open cover]] $\{U_i \to X\}$. This is a collection of [[smooth function | smooth functions]] $(g_{i j} : U_i \cap U_j \to G)$ such that

$$
  \left(
    \array{
      && (x,j)
      \\
      & \nearrow && \searrow
      \\
      (x,i) &&&& (x,k)
    }
  \right)
  \mapsto
  \left(
    \array{
      && \bullet
      \\
      & {}^{\mathllap{g_{i j}(x)}}\nearrow && \searrow^{\mathrlap{g_{j k}(x)}}
      \\
      \bullet &&\stackrel{g_{i k}(x)}{\to}&& \bullet
    }
  \right)
  \,.
$$

A lift of this cocycle through $V//G \to \mathbf{B}G$ is in addition a collection of smooth functions $\{sigma_i : U_i \to V\}$ such that on all $U_i \cap U_j$ the equation

$$
  \sigma_j = \rho(g_{i j})(\sigma_i)
$$

$$
  \left(
    \array{
      \sigma_i(x) &&\stackrel{\rho(g_{i k}(x))}{\to}&& \sigma_k(x)
    }
  \right)
$$

is satisfied. This identifies the $\sigma_i$ as precisely the components of a section $\sigma$ of $P \times_\rho G$ with respect to the local trivialization encoded by $g$.

=--

#### Definition of covariant derivative

+-- {: .num_lemma}
###### Observation

Given a [[connection on a bundle|connection]] $\nabla$ on the $G$-[[principal bundle]] with cocycle $g$, there is a unique connection $\nabla \sigma$ on the $V//G$-[[groupoid principal bundle]] that corresponds to a section $\sigma$ by the above proposition.

=--

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

The **covariant derivative** of a section $\sigma$ is the _1-form component_ $F_{\nabla \sigma}^1$ of the [[curvature]] of this groupoid-bundle connection.

=--

This 1-form curvature is literally the measure for the _non-flatness of the section_ . Whereas the 2-form curvature is a measure for the non-flatness of the connection.

#### Equivalence to more traditional definitions

We unwind this definition and find the traditional formulation of covariant derivatives as traditionally stated in the literature.

On a patch $U_i \hookrightarrow X$ the connection $\nabla$ is given by a morphism of [[dg-algebra | dg-algebras]]

$$
  \Omega^\bullet(U_i) \leftarrow W(\mathfrak{g}) : A_i
$$

for $W(\mathfrak{g})$ the [[Weil algebra]] of $\mathfrak{g}$.

The groupoid connection $\nabla \sigma$ on this patch is given by

$$
  \Omega^\bullet(U_i) \leftarrow W(Lie(V//G)) : \nabla \sigma_i
  \,.
$$

In degree 0 this is an algebra homomorphism

$$
  C^\infty(U_i) \leftarrow C^\infty(V) : \sigma_i
  \,.
$$

This is the dual of the local section $\sigma_i$ itself. In the case that $V$ is a vector space with chosen basis $\{v^k\}$, we have the corresponding components $(\sigma_i^k)$ of the local section.

Further, in degree 1 the connection is linear map

$$
  \Omega^1(U_i) \leftarrow \mathfrak{g}^* : A_i
  \,,
$$

which is the connection form itself, as well as a linear map

$$
  \Omega^1(U_i) \leftarrow \Omega^1(V) : F^1_{\nabla \sigma_i}
  \,,
$$

which is the curvature 1-form. The respect of these maps for the differential says that

$$
  F^1_{\nabla \sigma_i} = d_{dR} \sigma_i + \rho(A)(\sigma_i)
  \,.
$$

This is the familiar local formula for a covariant derivative as one finds it in the literature. We therefore write for short

$$
  \nabla_{(-)} \sigma_i := F^1_{\nabla sigma_i}
$$

If we keep $\nabla$ fixed and let $\sigma_i$ vary, then this may be thought of as a 1-form with values in [[endomorphism | endomorphisms]] of the space of sections

$$
  \nabla_{(-)} : \Gamma(P \times_\rho V) \to \Gamma(P \times_\rho V)
  \,.
$$

There is a [[Bianchi identity]] on every [[curvature]] component, induced from the respect for differentials of the dg-algebra morphism $\Omega^\bullet(U_i) \leftarrow W(Lie(V//G)) : F_\nabla$ on shifted generators.

From the discussion at [Action Lie algebroid cohomology](#ActionLieAlgebroidCohomology) above we read off the Bianchi identity for the 1-form curvature that we identified with the covariant derivative in the case of linear actions to be given in local coordinates (as above) by (suppressing the patch index ${}_i$)

$$
  d_{dR} \nabla_{(-)} \sigma^k = \rho_a{}^k{}_k F_A^a \wedge \sigma^k
  -
  \rho_a{}^k{}_l A^a d_{dR} \sigma^k
  ,.
$$

More invariantly we may write this as

$$
  \nabla_{(-)} \nabla_{(-)} \sigma = \rho(F_A)(\sigma)
$$

and this find the usual expression of the [[curvature]] of a connection as the square of the covariant derivative.

## Related concepts

* [[flat section]]

* [[exterior covariant derivative]]

[[!include gauge field - table]]

## References

* [[Mikio Nakahara]], Section 10.4 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

A discussion of covariant derivatives for a [[Levi-Civita connection]] in terms of [[synthetic differential geometry]] is in

* [[Gonzalo Reyes]], _Covariant derivation_ ([pdf](https://marieetgonzalo.files.wordpress.com/2006/04/covariant-derivation1.pdf))

[[!redirects covariant derivative]]
[[!redirects covariant derivatives]]

[[!redirects covariant differentiation]]