well-founded coalgebra

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The notion of _well-founded coalgebra_ is due to [[Paul Taylor]] (with antecedents in the work of Gerhard Osius). Our account is largely based on ([Taylor, section 6.3)](#Taylor99)), and on his paper [The General Recursion Theorem](#tTaylor96), although in some cases we work with slightly different hypotheses.

## Definition

Let $C$ be a [[finitely complete category]], and let $T$ be an [[endofunctor]] on $C$. We will suppose that $T$ is [[taut functor|taut]], i.e., preserves [[pullbacks]] of [[monomorphisms]] (preserves [[limits]] of [[cospans]] in which one of the cospan arrows is [[monomorphism|monic]]).  In particular, this implies that $T$ preserves monos.  A helpful example to keep in mind is the covariant power-set functor $P: Set \to Set$.

+-- {: .un_def}
######Definition
Let $\theta: X \to T X$ be a $T$-[[coalgebra for an endofunctor|coalgebra]] structure on $X$. A [[subobject]] $i: U \hookrightarrow X$ in $C$ is $\theta$-**inductive** if in the pullback

$$\array{
H & \stackrel{j}{\to} & X \\
\downarrow & & \downarrow^\mathrlap{\theta} \\
T U & \underset{T i}{\to} & T X
}$$

the map $j$ factors through $i$ (note that $j$ is monic since $T$ preserves monos).

A coalgebra $(X, \theta)$ is **well-founded** if the only inductive subobject of $X$ is $X$ itself.
=--

**Example:** Suppose $X$ is an initial algebra for the endofunctor $T$, with algebra structure map $\alpha: T X \to X$. By Lambek's theorem, $\alpha$ is invertible, so that $X$ carries a coalgebra structure $\theta = \alpha^{-1}: X \to T X$. It is easy to check that an inductive subobject gives a subalgebra $i: U \to X$, but a subalgebra of an initial algebra must be the initial algebra itself. Hence $(X, \theta)$ is well-founded.

Our goal in this article is to show that one can perform inductive arguments and recursive constructions in the abstract context of well-founded coalgebras. An interesting challenge is to make precise what is meant by a recursive construction of a map $\phi: X \to A$, where the problem is to show how to build $\phi$ from the ground up, as it were. Stages in the recursive construction of a morphism will be _partial_ maps, defined as usual as spans

$$\array{
 & & D & & \\
 &  ^\mathllap{i} \swarrow & & \searrow^\mathrlap{f} \\
A & & & & B
}$$

for which the arrow $i$ is _monic_. Composition of partial maps is effected by span composition, for which we need only finite cocompleteness (we do not need $C$ to be a [[regular category]], as we would if we were composing general relations between $A$ and $B$). Notice that since $T$ preserves the pulling back of monos, it carries partial maps to partial maps and also preserves partial map composition.

We denote a partial map by a dashed arrow

$$f: A \dashrightarrow B$$

(generally without explicitly mentioning the domain $D$ of $f$), or sometimes by a harpoon notation.

### Connection with initial algebras

One way of constructing the [[initial algebra of an endofunctor]], $(X, \alpha: T X \to X)$, is by constructing first some [[fixed point]] of $T$, that is, an object $Y$ together with an isomorphism $\xi: Y \cong T Y$. (For example, it might be the terminal coalgebra, whose existence is sometimes easy to establish.) Then, inside $Y$ consider the system of well-founded subcoalgebras of $\xi$. The colimit of this system, assuming it exists, will be the initial algebra. (More theory to develop here.)

The connection with initial algebras goes a little further. Firstly, an initial $T$-algebra is a *Peano $T$-algebra* in the sense of the following definition:

+-- {: .num_defn}
###### Definition
A $T$-algebra $(X, \alpha: T X \to X)$ is *semi-Peano* if every $T$-subalgebra [[monomorphism|inclusion]] $i: Y \hookrightarrow X$ is an isomorphism, and *Peano* if in addition $\alpha$ is an isomorphism.
=--

If $X$ is initial and $i: Y \to X$ is a subalgebra, then there is a unique algebra map $r: X \to Y$, and $i r = 1_X: X \to X$ by initiality, whence $i r i = i$ and then $r i = 1_Y: Y \to Y$ as $i$ is monic. Hence initial algebras are semi-Peano, and Peano by Lambek's theorem.

Secondly, a functor $T: E \to E$ induces, for any object $X$ of $E$, a functor between slices $T_\ast: E/X \to E/ T X$, and so if $(X, \theta: X \to T X)$ is a coalgebra, we may form an endofunctor on $E/X$:

$$E/X \stackrel{T_\ast}{\longrightarrow} E/ T X \stackrel{\theta^\ast}{\longrightarrow} E/X$$

and of course the terminal object $1_X: X \to X$ is automatically and uniquely a $\theta^\ast T_\ast$-algebra. The next two propositions then follow immediately from the definitions of inductive subobject and well-founded coalgebra:

+-- {: .num_prop}
###### Proposition
A subobject $i: U \to X$ of a $T$-coalgebra $(X, \theta)$ is inductive precisely when $i$ is a $\theta^\ast T_\ast$-subalgebra of the terminal object $1_X$ of $E/X$.
=--

+-- {: .num_prop}
###### Proposition
A $T$-coalgebra $X$ is well-founded precisely when the terminal $\theta^\ast T_\ast$-algebra $1_X$ is semi-Peano.
=--

## Examples

### Well-founded relations

The prototype of the notion of well-founded coalgebra is a well-founded [[relation]], which is essentially the same thing as a well-founded coalgebra over the covariant power-set functor $P: Set \to Set$. In brief, a binary [[relation]] $\prec$ on a set $X$ corresponds to a coalgebra $\theta: X \to P X$, by saying $y \prec x$ if and only if $y \in \theta(x)$. See [[well-founded relation]] for more information.

Most of the constructions of this article are well-illustrated by checking them against the background of this prototypical case.

## Induction and recursion

"One _proves_ things by [[induction]]; one _defines_ things by [[recursion]]." This slogan is not mere pedantry; it is meant to underline a difference between these processes.

A [[proof]] by induction, say a proof of a [[property]] or [[predicate]] $R(x)$ where $x$ varies over a domain $X$, proceeds by showing that the subobject $i: U \hookrightarrow X$ defined by $R$ is an inductive subset with respect to a relation on $X$. It follows that $R$ is universally true on $X$, if the relation is well-founded. The same idiom applies more generally to well-founded coalgebras.

A recursive construction on the other hand involves a _dependency_ on prior stages of the construction. A typical application is to define a map $f: X \to A$ by recursion with respect to a well-founded relation, where $f(x)$ is specified in three stages:

* Consider the collection $\theta(x) = \{y: y \prec x\}$ of all elements preceding $x$;

* Pass to the values $f(y)$ defined earlier in the construction, giving a subset $P(f)(\theta(x)) = \{f(y): y \prec x\}$ of $A$;

* Apply a given operation $\phi: P(A) \to A$ to this subset $(P(f) \circ \theta)(x)$, to obtain $f(x)$.

In the last step, the operation may be only partially defined on $P(A)$. In fact, the map $f$ itself may be only partially defined; $f(x)$ is defined only if $\phi((P(f) \circ \theta) (x))$ is defined "when we call for it".

An inductive argument is used to show that $f$, so far as it is defined, is uniquely determined. The recursive equation that uniquely determines $f$ (to the extent that it exists, of course) is

$$f = \phi \circ P(f) \circ \theta.$$

Letting $\downarrow x$ denote the down-closure of an element $x \in X$ with respect to $\prec$ (and containing $x$), we may then form the set

$$\{x \in X: \exists_{g: \downarrow x \to A} (g = \phi \circ P(g) \circ \theta)\}$$

whence the existence of a map $f: X \to A$ satisfying the recursive equation might indeed be proven by appeal to an inductive argument.

However, notice that this set is defined by a construction in [[dependent type theory]]. For other categories $C$, we might not have the luxury of interpreting dependent types, so there it wouldn't be right to conflate recursion with induction.

It is nevertheless true that one can prove, in rather considerable generality, both an induction principle and recursion theorem for well-founded coalgebras; this will occupy us in the following sections.

## Related concepts

* [[well-founded relation]]

## References

* [[Paul Taylor]], _Practical Foundations of Mathematics_ , Cambridge University Press (1999).
 {#Taylor99}

* [[Paul Taylor]], _Towards a unified treatment of induction_ , I: the general recursion theorem (1996). ([pdf](http://www.paultaylor.eu/ordinals/towuti.pdf))
 {#Taylor96}

[[!redirects well-founded coalgebras]]


well-founded relation

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#### Relations
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## Idea {#idea}

A (binary) [[relation]] $\prec$ on a [[set]] $S$ is called _well-founded_ if it is valid to do [[induction]] on $\prec$ over $S$.

## Definitions {#defn}

Let $S$ be a [[set]], and let $\prec$ be a [[binary relation]] on $S$.  A [[subset]] $A$ of $S$ is __$\prec$-inductive__ if
$$ \forall (x\colon S),\; (\forall (t\colon S),\; t \prec x \;\Rightarrow\; t \in A) \;\Rightarrow\; x \in A. $$
The relation $\prec$ is __well-founded__ if the only $\prec$-inductive subset of $S$ is $S$ itself.

Note that this is precisely what is necessary to validate induction over $\prec$: if we can show that a statement is true of an element $x$ of $S$ whenever it is true of everything that precedes ($\prec$) $x$, then it must be true of everything in $S$.  In the presence of [[excluded middle]] it is equivalent to other commonly stated definitions; see _Formulations in classical logic_ below.

### Formulations in classical logic

While the definition above follows how a well-founded relation is generally *used* (namely, to prove properties of elements of $S$ by induction), it is complicated.  Two alternative formulations are given by the following:

1. The relation $\prec$ has __no infinite descent__ (usually attributed to [[Pierre de Fermat]]) if there exists no [[sequence]] $\cdots \prec x_2 \prec x_1 \prec x_0$ in $S$.  (Such a sequence is called an _infinite descending sequence_.)

2. The relation $\prec$ is __classically well-founded__ if every [[inhabited set|inhabited]] subset $A$ of $S$ has a member $x \in A$ such that no $t \in A$ satisfies $t \prec x$.  (Such an $x$ is called a _minimal_ element of $A$.)

In [[classical mathematics]], both of these conditions are equivalent to being well-founded.  In constructive mathematics, we may prove that a well-founded relation has no infinite descent (see Proposition \ref{empty}), but not the converse, and that a classically well-founded relation is well-founded (see Proposition \ref{classical}), but not the converse.

The classical notion of well-foundedness enforces classical logic onto us, in the following sense.

+-- {: .num_prop #LEM}
###### Proposition
If $(X, \prec)$ is an [[inhabited subset|inhabited]] well-founded relation in a classical sense, then the unrestricted [[excluded middle|law of excluded middle]] holds.
=--

For a [[topos]]-theoretic proof see [here](http://ncatlab.org/toddtrimble/published/classical+well-foundedness).

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

Suppose there are $x$ and $y$ such that $y\prec x$, and let $Q$ be an arbitrary proposition. Consider then a set $P \subset X$ defined as $P = \{ x \} \cup \{ a \mid a \prec x \;\wedge\; Q\}$. Clearly, the set $P$ is inhabited, thus by classical well-foundedness it has a minimal element $x_0$. By intuitionistic reasoning, either $x_0$ is in $\{ x \}$, i.e. $x_0 = x$, or $x_0 \in \{ a \mid a \prec x \;\wedge\; Q\}$, i.e. $x_0 \prec x \;\wedge\; Q$. In the latter case we immediately see that $Q$ holds. So, suppose that $x_0 = x$ is the minimal element of $P$; we will show that $\neg Q$ holds. For suppose that $Q$ holds; then $y \in P$ and $y \prec x = x_0$, violating the condition that $x_0$ is a minimal element of $P$.

Since $Q$ was an arbitrary proposition, we can deduce $\forall Q. (Q \vee \neg Q)$.
=--

+-- {: .num_remark #LEM_rem}
###### Remark
We note that classical well-foundedness is really too strong for constructive (i.e., intuitionistic) mathematics, due to Proposition \ref{LEM}. On the other hand, the infinite descent condition is too weak to be of much use in constructive mathematics. It is the inductive notion of well-foundedness that is just right.
=--

Note however that in [[predicative mathematics]], the definition of well-founded may be impossible to even state, and so either of these alternative definitions would be preferable, provided classical logic is used.

Even in constructive predicative mathematics, (1) is strong enough to establish the [[Burali-Forti paradox]] (when applied to [[linear orders]]).  In [[material set theory]], (2) is traditionally used to state the [[axiom of foundation]], although the impredicative definition could also be used as an axiom scheme (and must be in constructive versions).  In any case, either (1) or (2) is usually preferred by classical mathematicians as simpler.

To round out the discussion we prove the following two results, both valid in intuitionistic mathematics:

+-- {: .num_prop #empty}
###### Proposition
If $(X, \prec)$ is a well-founded relation and $A \subseteq X$ has no minimal element, then $A$ is empty.
=--

This result makes it trivial to infer (under classical logic) that classical well-foundedness is a consequence of well-foundedness. It also shows that well-foundedness rules out infinite descent (intuitionistically), since an infinite descent sequence has no minimal element.

+-- {: .proof}
###### Proof
Let $U = \{u \in X: u \notin A\; \wedge \; (\forall x: X),\; x \prec u \Rightarrow x \notin A\}$. Clearly $U \cap A = \emptyset$. We show $U$ is inductive, so that under well-foundedness $U = X$ and $A = X \cap A = U \cap A = \emptyset$, as desired.

So, suppose $z$ is an element such that $y \in U$ whenever $y \prec z$. We must show $z \in U$. Claim: $z \notin A$. For if $z \in A$, then $z$ would be a minimal element of $A$ (as $y \prec z \Rightarrow y \in U \Rightarrow y \notin A$). But this negates the assumption that $A$ has no minimal element.

Thus $z \notin A$, and $y \prec z \Rightarrow y \in U \Rightarrow y \notin A$, so that $z \in U$. This completes the proof.
=--

+-- {: .num_prop #classical}
###### Proposition
In intuitionistic set theory, classical well-foundedness implies (inductive) well-foundedness.
=--

+-- {: .proof}
###### Proof
Let $\prec$ be a classically well-founded relation on $X$, and let $U$ be an inductive subset. We must show that every element $x \in X$ belongs to $U$. Since $U$ is inductive, it suffices to show that every $u \prec x$ belongs to $U$, i.e. we may assume given a $u$ such that $u\prec x$ and show $u\in U$. But under this assumption we have that $\prec$ is inhabited, so according to Remark \ref{LEM}, the law of [[excluded middle]] follows and we might as well then argue classically. The argument is well-known but we include it for completeness: under classical well-foundedness, if an inductive subset $U$ is not the entirety of $X$, then the complement $\neg U$ has a minimal element $y$. In that case, $v \prec y$ implies $v \in \neg\neg U = U$, but then $y \in U$ since $U$ is inductive, contradiction. Hence $U = X$ and in particular $u \in U$, which is what we wanted.
=--

To bring us full circle: in classical set theory we may prove that if $(X, \prec)$ has no infinite descent, then $\prec$ is classically well-founded. For suppose an inhabited subset $P \subseteq X$ (say with an element $x \in P$) failed to have a least element. Then we can find an infinite descent sequence $x_n \in P$ with $x_0 = x$, by choosing at each stage $x_{n+1} \in P$ such that $x_{n+1} \prec x_n$. Technically this requires the use of [[dependent choice]], but generally this is felt to be a mild choice principle (that is true even for intuitionistic mathematics).

### Coalgebraic formulation

Many inductive or recursive notions may also be packaged in [[coalgebra|coalgebraic]] terms. For the concept of well-founded relation, first observe that a binary relation $\prec$ on a set $X$ is the same as a coalgebra structure $\theta\colon X \to P(X)$ for the covariant [[power-set]] endofunctor on $Set$, where $y \prec x$ if and only if $y \in \theta(x)$.

In this language, a [[subset]] $i\colon U \hookrightarrow X$ is $\prec$-inductive, or $\theta$-inductive, if in the [[pullback]]

$$\array{
H & \stackrel{j}{\to} & X \\
\downarrow & & \downarrow^\mathrlap{\theta} \\
P U & \underset{P i}{\to} & P X
}$$

the map $j$ factors through $i$. (Note that $j$ is necessarily [[monomorphism|monic]], since $P$ preserves monos.) Unpacking this a bit: for any $x \in X$, if $\theta(x) = V$ belongs to $P U$, that is if $\theta(x) \subseteq U$, then $x \in U$. This says the same thing as $\forall_{x\colon X} (\forall_{y\colon X} y \prec x \Rightarrow y \in U) \Rightarrow x \in U$.

Then, as usual, the $P$-coalgebra $(X, \theta)$ is well-founded if every $\theta$-inductive subset $U \hookrightarrow X$ is all of $X$.

Other relevant notions may also be packaged; for example, the $P$-coalgebra $X$ is [[extensional relation|extensional]] if $\theta\colon X \to P X$ is monic. See also [[well-founded coalgebra]].

### Simulations

Given two sets $S$ and $T$, each equipped with a well-founded relation $\prec$, a [[function]] $f\colon S \to T$ is a __[[simulation]]__ of $S$ in $T$ if

1.  $f(x) \prec f(y)$ whenever $x \prec y$ and

1.  given $t \prec f(x)$, there exists $y \prec x$ with $t = f(y)$.

Then sets so equipped form a [[category]] with simulations as [[morphisms]].  See [[extensional relation]] for more uses of simulations.

For example, a simulation $X \to Y$ between two [[well-ordered sets]] is an isomorphism of $X$ onto an [[lower set|initial segment]] of $Y$.

In coalgebraic language, a simulation $S \to T$ is simply a $P$-coalgebra homomorphism $f\colon S \to T$. Condition (1), that $f$ is merely $\prec$-preserving, translates to the condition that $f$ is a [[colax morphism]] of coalgebras, in the sense that there is an inclusion

$$\array{
X & \stackrel{\theta_X}{\to} & P X \\
 ^\mathllap{f} \downarrow & \swArrow & \downarrow^\mathrlap{P f} \\
Y & \underset{\theta_Y}{\to} & P Y.
}$$

+-- {: .num_prop}
###### Proposition
If $(X, \prec)$ is well-founded and $(Y, \prec)$ is (weakly) extensional, then there is at most one simulation $f: X \to Y$.
=--

+-- {: .proof}
###### Proof
It is enough to show that if $f, g: X \to Y$ are $P$-coalgebra morphisms, then $U = \{u \in X: f(u) = g(u)\}$ is an inductive subset. Suppose $y \in X$ is an element such that $f(u) = g(u)$ whenever $u \prec y$. Then $f(\{u: u \prec y\}) = g(\{u: u \prec y\})$; since $f, g$ are $P$-coalgebra maps, this is the same as the equality $\{x: x \prec f(y)\} = \{x: x \prec g(y)\}$. By extensionality, we infer $f(y) = g(y)$, so that $y \in U$. Thus $U$ is inductive.
=--

## Properties

Every well-founded relation is [[irreflexive relation|irreflexive]]; that is, $x \nprec x$.  Sometimes one wants a reflexive version $\preceq$ of a well-founded relation; let $x \preceq y$ if and only $x \prec y$ or $x = y$.  Then the requirement that $x$ be a minimal element of a subset $A$ states that $t \preceq x$ only if $t = x$.  But infinite descent or direct proof by induction still require $\prec$ rather than $\preceq$.

A [[well-order|well order]] may be defined as a well-founded [[linear order]], or alternatively as a [[transitive relation|transitive]], [[extensional relation|extensional]], well-founded relation.

A [[well-quasi-order]] is a well-founded [[preorder]] (referring to the reflexive version of well-foundedness above) that in addition has no infinite [[antichain|antichains]].

The [[axiom of foundation]] in material [[set theory]] states precisely that the membership relation $\in$ on the proper class of all [[pure sets]] is well-founded.  In structural set theory, accordingly, one uses well-founded relations in building structural models of well-founded pure sets.

## Examples

Let $S$ be a [[finite set]].  Then any relation on $S$ whose [[transitive closure]] is irreflexive is well-founded.

Let $S$ be the set of [[natural numbers]], and let $x \prec y$ if $y$ is the [[successor]] of $x$: $y = x + 1$.  That this relation is well-founded is the usual principle of _mathematical induction_.

Again let $S$ be the set of natural numbers, but now let $x \prec y$ if $x \lt y$ in the usual order.  That this relation is well-founded is the principle of _strong induction_.

More generally, let $S$ be a set of [[ordinal numbers]] (or even the proper class of all ordinal numbers), and let $x \prec y$ if $x \lt y$ in the usual order.  That this relation is well-founded is the principle of _transfinite induction_.

Similarly, let $S$ be a set of [[pure sets]] (or even the proper class of all pure sets), and let $x \prec y$ if $x \in y$.  That this relation is well-founded is the _[[axiom of foundation]]_.

Let $S$ be the set of [[integers]], and let $x \prec y$ mean that $x$ properly divides $y$: $y/x$ is an integer other than $\pm{1}$.  This relation is also well-founded, so one can prove properties of integers by induction on their proper divisors.

[[!redirects well-founded relation]]
[[!redirects well-founded relations]]
[[!redirects well founded relation]]
[[!redirects well founded relations]]

[[!redirects classically well-founded relation]]
[[!redirects classically well-founded relations]]
[[!redirects classically well founded relation]]
[[!redirects classically well founded relations]]

[[!redirects relation without infinite descent]]
[[!redirects relations without infinite descent]]
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[[!redirects Fermat's method of infinite descent]]
[[!redirects Fermat\'s method of infinite descent]]
[[!redirects Fermat's method of infinite descent]]
[[!redirects Fermat method of infinite descent]]
[[!redirects method of infinite descent]]
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well-generated triangulated category

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##Idea

A well generated [[triangulated category]] is a generalization of the notion of [[compactly generated triangulated category]] which was introduced by [Neeman, 2001](#Neeman).  The following definition is from ([Krause](#Krause)) and is somewhat shorter and more natural than Neeman's original definition.

##Definition

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

Let $T$ be a [[triangulated category]] with arbitrary [[coproducts]]. Then $T$ is **well-generated** in the sense of Neeman if and only if there exists a [[set]] $S_0$ of [[objects]] satisfying:

1. an object $X$ of $T$ is [[zero object|zero]] if $[S,X]=0$ for all $S\in S_0$;

1. for every set of maps $X_i\to Y_i$ in $T$, the induced map $[S,\coprod_I X_i]\to[S,\coprod_I Y_i]$ is surjective for all $S\in S_0$ whenever $[S,X_i]\to[S,Y_i]$ is surjective for all $i$ and all $S\in S_0$.

1. the objects of $S_0$ are $\alpha$-[[compact object|small]] for some [[cardinal]] $\alpha$.

=--

We recall that an object $S$ in a triangulated category is $\alpha$-small if every map $S\to\coprod_J X_j$ factors through $\coprod_I X_j$ for some $I \subseteq J$ with $\vert I\vert \lt \alpha$.

##References

* [[Henning Krause]], _On Neeman's well generated triangulated categories_, Documenta Mathematica __6__ (2001) ([pdf](https://www.math.uni-bielefeld.de/documenta/vol-06/07.pdf)).
 {#Krause}

* Henning Krause, _Localization theory for triangulated categories_, [arXiv:0806.1324](https://arxiv.org/abs/0806.1324)

* [[Amnon Neeman]], _Triangulated Categories_, Annals of Mathematics Studies 148, Princeton University Press (2001).
 {#Neeman}

[[!redirects well generated triangular category]]
[[!redirects well-generated triangulated categories]]


well-order

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#### $(0,1)$-Category theory
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# Contents
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## Idea

A _well-order_ on a set $S$ is a [[relation]] $\prec$ that allows one to interpret $S$ as an [[ordinal number]] $\alpha$ and $\prec$ as the relation $\lt$ on the ordinal numbers less than $\alpha$.  In particular, one can do [[induction]] on $S$ over $\prec$ (although the more general [[well-founded relations]] also allow this).

The _[[well-ordering theorem]]_ states precisely that every set may be equipped with a well-order.  This theorem follows from the [[axiom of choice]], and is equivalent to it in the presence of [[excluded middle]].

## Definition

A [[binary relation]] $\prec$ on a [[set]] $S$ is a __well-order__ if it is [[transitive relation|transitive]], [[extensional relation|extensional]], and [[well-founded relation|well-founded]].  A set equipped with a well-order is called a __well-ordered set__, or (following '[[partial order|poset]]') a __woset__.  Actually, the term 'well-ordered' came first; 'well-order' is a back formation, which explains the strange grammar.

Other definitions of a well-order may be found in the literature; they are equivalent given [[excluded middle]], but the definition above seems to be the most powerful in [[constructive mathematics]].  Specifically:

*  a well-order is precisely a [[well-founded relation|well-founded]] [[pseudo-order]] $\prec$;
*  a well-order is precisely a [[total order]] $\preceq$ whose strict version $\prec$ (defined by $x \prec y$ iff $x \preceq y$ and $x \ne y$) is [[well-founded relation|well-founded]];
*  (assuming also [[dependent choice]]) a well-order is precisely a pseudo-order $\prec$ with no infinite descending sequence $\cdots \prec x_2 \prec x_1 \prec x_0$;
*  (assuming also [[dependent choice]]) a well-order is precisely a total order $\preceq$ such that every infinite descending sequence $\cdots \preceq x_2 \preceq x_1 \preceq x_0$ has $x_i = x_{i^+}$ for some $i$ (and hence for infinitely many $i$);
*  a well-order on $S$ is precisely a [[pseudo-order]] $\prec$ with the property that every [[inhabited subset]] $U$ of $S$ has a least element (an element $\bot_U$ such that no $x \in U$ satisfies $x \prec \bot_U$;
*  a well-order on $S$ is precisely a [[total order]] $\preceq$ with the property that every [[inhabited subset]] $U$ of $S$ has a least element (an element $\bot_U$ such that every $x \in U$ satisfies $\bot_U \preceq x$.

The really interesting thing here is that every well-order is a pseudo-order; it is a constructive theorem that every pseudo-order is weakly extensional (and so extensional if well-founded) and transitive.  (For a [[weak counterexample]], take the set of [[truth values]] with $x \prec y$ iff $y$ is true and $x$ is false; this is a well-order that\'s a pseudo-order iff [[excluded middle]] holds.)  For the other equivalences, we\'re simply using well-known classical equivalents for well-foundedness and the classical correspondence between a pseudo-order $\prec$ and its [[reflexive closure]] $\preceq$.

For reference, a __classical well-order__ is any order satisfying the last definition (a total order that is classically well-founded).  A classically well-ordered set is a [[choice set]], and so if any set with at least $2$ [[denial inequality|unequal]] elements has a classical well-order, then [[excluded middle]] follows.  (But there are still interesting examples of classically well-ordered sets in constructive mathematics, such as the [[Higgs object|Higgs set]].)

### Well-orders are pseudo-orders

As stated above, well-founded extensional transitive relations $\prec$ on a set $X$ are pseudo-orders, assuming [[classical logic]].

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

Order $X \times X$ [[lexicographic order|lexicographically]]: $(a, b) \prec (a', b')$ if either $a \prec a'$ in $X$ or $a = a'$ and $b \prec b$ in $X$. It is not hard to see that the lexicographic order is well-founded (and in fact a well-order, although we do not need this).  Now let $A \subset X \times X$ be the set of pairs $(x, y)$ of non-equal elements $x$ and $y$ that are incomparable in $X$, and suppose $A$ is inhabited. Then $A$ has a minimal element $(a, b)$ (using excluded middle). Then, for every $x \prec b$, either $a \preceq x$ or $x \preceq a$. If the former holds for some $x$, then $a \prec b$ follows by transitivity, contradiction. Hence $x \prec a$ for every $x \prec b$. Now let $a'$ be minimal such that $x \prec a' \preceq a$ for every $x \prec b$.

Claim:
$$ \{x: x \prec a'\} = \{x: x \prec b\}. $$
We know already the right side is contained in the left. In the other direction, suppose $x \prec a'$. Since $x \prec a$ and $(a, b)$ was chosen minimal in the lexicographic order, $x$ and $b$ are comparable. If $b \preceq x \prec a'$, this contradicts minimality of $a'$. Thus $x \prec b$, i.e., the left side is contained in the right.

But now, by extensionality, $a' = b$, whence $b \preceq a$, contradiction. Therefore $A$ was empty, so that $X$ is [[connected relation|connected]] and therefore (being already transitive and irreflexive and using excluded middle again) a pseudo-order.
=--

## Examples

* Any [[finite set|finite]] [[pseudo-ordered set]] $\{x_1 \lt \cdots \lt x_n\}$ is well-ordered.

* The set of [[natural numbers]] is well-ordered under the usual order $\lt$.

* More generally, any set of [[ordinal numbers]] (or even the [[proper class]] of all ordinal numbers) is well-ordered under the usual order $\lt$ (which, constructively, may not be a pseudo-order).

* The [[cardinal numbers]] of well-orderable sets (the well-orderable cardinals), forming a [[retract]] of the ordinals, are well-ordered.  So by the well-ordering theorem, the class of *all* cardinal numbers is well-ordered.

* A special case of the well-ordering theorem is the existence of a well-order on the set of [[real numbers]]; this is enough for many applications of the [[axiom of choice]] to [[analysis]].

## Interpretation as an ordinal number

Any well-ordered set $S$ defines an [[ordinal number]] $\alpha$ and an order isomorphism $r$ between $S$ and the set of ordinal numbers less than $\alpha$; as such, $S$ may be identified (up to isomorphism of wosets) with the von Neumann ordinal $\alpha$.  The idea is that the minimal element $\bot$ of $S$ itself (if any) is mapped to the ordinal number $0$, the minimal element of $S \setminus \{\bot\}$ (if any) is mapped to $1$, and so on; after which the next element of $S$ (if any) is mapped to $\omega$, and so on; and so on.

This may be defined immediately (and constructively) as a recursively defined function from $S$ to the class of all ordinal numbers:
$$ r(x) = \sup_{t \prec x} r(t)^+ ;$$
the validity of this sort of recursive definition is precisely what the well-foundedness of $\prec$ allows.  Here, $\beta^+$ is the [[successor]] of the ordinal number $\beta$, and $\sup$ is the [[supremum]] operation on ordinal numbers (which is the [[union]] of von Neumann ordinals).

Since $S$ is a set, the image of $r$ in the class of all ordinals is also a set (by the [[axiom of replacement]]), and one can now prove that $r$ is an order isomorphism between $S$ and the set of ordinals less than the next ordinal, $\alpha \coloneqq (\im r)^+$.

## Simulations

Given two well-ordered sets $S$ and $T$, a [[function]] $f\colon S \to T$ is a __[[simulation]]__ of $S$ in $T$ if

*  $f(x) \prec f(y)$ whenever $x \prec y$ and
*  given $t \prec f(x)$, there exists $y \prec x$ with $t = f(y)$.

Note that any simulation of $S$ in $T$ must be unique.  Thus, well-ordered sets and simulations form a category that is in fact a (large) [[preorder]], whose reflection in the category of [[posets]] is in fact the poset of [[ordinal numbers]].

## Successor

A well-ordered set $S$ comes equipped with a **[[successor]]**, which is a [[partial function|partial map]]
$
  succ\colon S \to S
  \,
$
that sends $a \in S$ to the lowest element of the subset $S_a := \{ s \in S, a \prec s\}$, whenever this set is inhabited.

+-- {: .num_defn}
###### Definition
A **limit** well-order is a well-order $S$ whose successor map is a [[total function]].
=--

Similarly, one may define a successor [[functor]] on the [[category]] of well-ordered sets, taking $S$ to the well-order obtained by freely adjoining a (new) top element to $S$. Since this category (which is [[thin category|thin]]) can be regarded as itself a well-ordered [[proper class]], this is a special case of the successor operation above.  (Hence the ordinal of all ordinals is a limit ordinal.)

## Related entries

* [[well-quasi-order]]
* [[marked extensional well-founded relation]]

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well-ordering theorem

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#### Foundations
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## Idea

The **well-ordering theorem** is a famous result in [[set theory]] stating that every [[set]] may be [[well-order|well-ordered]].

Fundamental for [[Georg Cantor| G. Cantor's]] approach to [[ordinal arithmetic]] it was an open problem until [[Ernst Zermelo|E. Zermelo]] gave a [[proof]] in 1904 using the [[axiom of choice]] (to which it is in fact equivalent).

Hence the well-ordering theorem is one of the many equivalent formulations of the [[axiom of choice]] like also e.g. [[Zorn's lemma]].

## Statement and proof

Given any [[set]] $S$, there exists a [[well-order]] $\prec$ on $S$.

The first proof was given in ([Zermelo 1904](#Zermelo04)). Within the nLab, the article [[Zorn's lemma]] gives a standard informal proof that can be formalized in [[ZFC]] under [[classical logic]], as well as the easy argument that conversely, the well-ordering principle (in its classical "least element" form; see below) implies the [[axiom of choice]] and Zorn's lemma.

\begin{remark}
That the well-ordering theorem is more of a *theorem* in need of a proof, while the axiom of choice is more of an *axiom* to be assumed without proof is, of course, a matter of opinion, but it\'s reflected in Jerry Bona\'s famous quotation:

> The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about [[Zorn's lemma]]?

\end{remark}

## Consequences

If every set can be well-ordered, then the natural map from [[ordinal number]]s to [[cardinal number]]s is a [[surjection]].  Since these form proper classes, the ordinary axiom of choice will not split this surjection automatically; however, we can easily split it by assigning each cardinal number to the *smallest* ordinal number with that cardinality.  (This does require [[excluded middle]], however.)  In this way, a cardinal number may be *defined* to be an ordinal number that is _initial_: such that no smaller ordinal number has the same cardinality.

As in the argument above, the [[axiom of choice]] follows; given any [[surjection]] $f: A \to B$, place a well-ordering on $A$ and then split $f$ by mapping an element $y$ of $B$ to the smallest element $x$ of $A$ such that $y = f(x)$.  Again, this uses [[excluded middle]] to show that such a smallest element exists, so the well-ordering principle does not (seem to) imply the axiom of choice constructively.

To get the large (or "global") axiom of choice (that any surjection between proper classes splits), we need a large well-ordering theorem: that every proper class can be well-ordered.  The large principles do not follow from the small ones.

## In constructive mathematics

In [[constructive mathematics]], the well-ordering principle is also equivalent to the axiom of choice. This was [proved by Andrew Swan](https://hott.zulipchat.com/#narrow/stream/228519-general/topic/inductive.20well-ordering.20gives.20excluded.20middle.3F/near/246863644) (2021) and formalized in [Agda by Tom de Jong](https://www.cs.bham.ac.uk/~mhe/TypeTopology/WellOrderingTaboo.html) (2021) and in [Coq by Dominik Kirst](https://github.com/dominik-kirst/sierpinski-hott/blob/master/coq/orders_lem.v) (2021).

[[Zorn's lemma]] is, on the other hand, constructively weaker than the axiom of choice, as it doesn't even imply excluded middle. But together with excluded middle it implies choice. However, Zorn's lemma is not particularly useful without excluded middle.

## Related entries

* [[Zorn's lemma]]

* [[axiom of choice]]

* [[Hartogs number]]

## History

**Georg Cantor** first developed [[set theory]] in the context of studying well-ordered sets of [[real number]]s whence the validity of the well-ordering principle became important for his theory of [[ordinal numbers]]. In his 1883 paper he calls it a _'fundamental and weighty law of thought that is remarkable for his generality'_ and promised to come back to it later ([Cantor 1932](#Cantor32), p.169). In the following, he announced proofs but they failed to materialize so that he was forced to take it as an assumption.[^choice]

[^choice]:This contrasts with Cantor's attitude towards the axiom of choice which he used implicitly but never thematised explicitly. In fact, the full explicit awareness of the use of the axiom choice in mathematics had to await the controversy over Zermelo's well-ordering theorem in 1904 (with some anticipation by G. Peano and B. Levi earlier).

Consequently, the well-ordering principle ended as 'a very strange claim' second on [[Hilbert's problems|Hilbert's millennium list]] of open problems in mathematics in 1900. Then in 1904, the Hungarian mathematician J. König announced a proof that the [[continuum]] could not be well-ordered but had to retract the proof.

Soon afterwards in 1904, [[Ernst Zermelo]] finally gave a proof using the [[axiom of choice]] following a suggestion by E. Schmidt. The proof, albeit correct, was met with heavy criticism by prominent mathematicians so that Zermelo published a new proof and a defense of the contested axiom of choice in 1908.

The attempt to make explicit the set-theoretic assumptions in the proof led him to publish his axioms for set theory in the same year which became later a part of [[Zermelo-Fraenkel set theory]]. Hence the 1904ff controversy proved to become a _decisive watershed for the development of modern mathematics_: triggering the advent of set-theoretic [[foundation of mathematics]] and putting the problem of non-constructive methods of proof on the agenda.

## References

The original proof is in

* [[Ernst Zermelo]], _Beweis, daß jede Menge wohlgeordnet werden kann_, Mathematische Annalen **59** (1904) pp.514-516. ([gdz](http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=28526))
 {#Zermelo04}

The second proof together with an eloquent defense of the axiom of choice can be found in

* [[Ernst Zermelo]], _Neuer Beweis für die Möglichkeit einer Wohlordnung_ , Mathematische Annalen **65** (1908) pp.107-128. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002261952)) {#Zermel08}

Cantor's texts are collected together with comments by Zermelo in

* [[Ernst Zermelo]] (ed.), _Georg Cantor - Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts_ , Springer Berlin 1932. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN237853094)) {#Cantor32}

English versions of Zermelo's papers are in

* J. van Heijenoort (ed.), _From Frege to Gödel - A Source Book in Mathematical Logic 1879-1931_ , Harvard UP 1967.

On the relation between AC and the well-ordering principle in general toposes see

* [[Peter Freyd]], _Choice and Well-ordering_ , APAL **35** (1987) pp.149-166.

* M.-M. Mawanda, _Well-ordering and Choice in Toposes_ , JPAA **50** (1988) pp.171-184.

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vol. II_, Oxford UP 2002. (D4.5, pp.987-998)

* J. Todd Wilson, "An Intuitionistic version of Zermelo's proof that every choice set can be well-ordered", J. Symbolic Logic, 66:3 (2001), 1121--1126; ([JSTOR](http://www.jstor.org/stable/2695096): paywalled), [formalization in the Lean Theorem Prover] (https://github.com/lambdacalculator/lean-choice).

The proof that Zorn's Lemma doesn't imply excluded middle (and hence doesn't imply choice without assuming excluded middle):

* {#Bell} [[John Lane Bell]], *Zorn’s lemma and complete Boolean algebras in intuitionistic type theories*, The Journal of Symbolic Logic, 62(4):1265–1279, 1997 ([doi:10.2307/2275642](https://doi.org/10.2307/2275642))

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well-pointed endofunctor

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\tableofcontents

## Definition

\begin{definition}\label{PointedEndofunctor}
An [[endofunctor]] $S \colon \mathcal{A}\to \mathcal{A}$ is called [[pointed endofunctor|pointed]] if it is equipped with a [[natural transformation]] $\sigma \colon Id_\mathcal{A} \to S$ from the [[identity functor]].
\end{definition}

(Beware that is *not* quite the notion of a [[pointed object]] in the endo-[[functor category]], since the [[identity functor]] is not in general a [[terminal object]] there. See the remark [there](pointed+endofunctor#MeaningOfPointedness).)

\begin{definition}
A pointed endofunctor $(S, \sigma)$ (Def. \ref{PointedEndofunctor}) is called **well-pointed** if $S\sigma = \sigma S$ as natural transformations $S \longrightarrow S \circ S$.
\end{definition}

The dual notion is known as a **well-copointed** endofunctor.

## Properties

Well-pointed endofunctors have a particularly well-behaved free algebra sequence; see *[[transfinite construction of free algebras]]*.

## References

* {#Kelly} [[Max Kelly]], _A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on._  Bull. Austral. Math. Soc. 22 (1980), 1--83. doi:[10.1017/S0004972700006353](http://dx.doi.org/10.1017/S0004972700006353)

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well-pointed topical dagger 2-poset

[[!redirects well-pointed dagger 2-posets]]

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## Contents ##
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## Definition ##

A **well-pointed topical dagger 2-poset** is an [[elementarily topical dagger 2-poset]] $C$ such that for every object $A \in Ob(C)$ and $B \in Ob(C)$ and [[map in a dagger 2-poset|maps]] $f \in Map_C(A, B)$, $g \in Map_C(A, B)$ and $x \in Map_C(\mathbb{1}, A)$, $f \circ x = g \circ x$ implies $f = g$.

## Examples ##

The dagger 2-poset [[Rel]] of [[sets]] and [[relations]] is a well-pointed topical dagger 2-poset.

## See also ##

* [[elementarily topical dagger 2-poset]]


well-pointed topological space

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## Definition
 {#Definition}

A [[pointed topological space]] $(X,x)$ is called *well-pointed* if the base-point inclusion $\{x\} \xhookrightarrow{\;} X$ is a closed [[Hurewicz cofibration]] (e.g. [tom Dieck 2008, p. 102](#tomDieck2008)). If the topological space is [[Hausdorff topological space|Hausdorff]], then closedness is implied (by [this Prop.](Hurewicz+cofibration#HurewiczCofibrationsInCGWHSpacesAreClosed)) and one may require just a Hurewicz cofibration (eg. [Bredon 1993, VII, Def. 1.8](#Bredon93)).

A [[topological group]] is called well-pointed if it is so at its [[neutral element]], hence if $\{\mathrm{e}\} \xhookrightarrow{\;} G$ is a closed [[Hurewicz cofibration]].

A [[simplicial topological group]] is well-pointed if all its component groups are.

A key property of well-pointed topological groups (in the [[convenient category of topological spaces|convenient]] context of [[compactly generated weak Hausdorff spaces]]) is that the [[nerves]] of their [[delooping groupoids]] are [[good simplicial topological spaces]] (by [this Ex.](Hurewicz+cofibration#KifiedProductsOfCofibrationsWithCompactlyGeneratedSpaces)). Similarly, the underlying simplicial topological spaces of well-pointed simplicial topological groups are [[good simplicial topological space|good]] ([this Prop.](simplicial+topological+group#RealizationOfWellPointedIsWellPointed)). These facts explain the key role of well-pointedness in [[classifying space]]-theory, where it ensures that plain [[topological realization of simplicial topological spaces]] coincides, up to [[weak homotopy equivalence]], with [[fat geometric realization]] and hence with the [[homotopy colimits]].

## Examples

\begin{example}\label{ParacompactBanachManifoldsAreWellPointed}
  Every [[locally Euclidean topological space|locally Euclidean]] [[Hausdorff space]] is well-pointed, in particular every [[topological manifold]] is well pointed. In fact, every [[paracompact topological space|paracompact]] [[Banach manifold]] is well-pointed.
\end{example}
(Immediate by the discussion of [examples of Hurewicz cofibrations](Hurewicz+cofibration#Examples)).

\begin{example}\label{PointInclusionIntoPUH}
**(the [[projective unitary group]] [[PU(ℋ)]])**
\linebreak
  The [[projective unitary group]] [[PU(ℋ)]] on an infinite-dimensional [[separable Hilbert space|separable]] [[Hilbert space]] is:

* a [[Banach Lie group]] in its [[norm topology]], and as such [[well-pointed topological group|well-pointed]] by Ex.\ref{ParacompactBanachManifoldsAreWellPointed};

* no longer a Banach space in its weak/strong [[operator topology]], but nevertheless still [[well-pointed topological group|well-pointed]] in this case, by [this Prop.](projective+unitary+group+on+a+Hilbert+space#WellPointedInOperatorTopology).

\end{example}

## References

Textbook accounts:

* {#Bredon93} [[Glen Bredon]], Section VII.1 of: _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0),  [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

* {#tomDieck2008} [[Tammo tom Dieck]],  _Algebraic topology_. European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86))

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well-pointed topos

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## Definition

An [[elementary topos]] $E$ is **well-pointed** if

1. the [[terminal object]] 1 is a [[separator|generator]];

   * equivalently: the [[global section]] functor $\Gamma : E \to Set = E(1, -)$
   is a [[faithful functor]];

   * equivalently: if $f, g: a \rightarrow b$ are morphisms such that $f x = g x$ for all [[global element]]s $x: 1 \to a$, then $f = g$;

   * equivalently: the family of all global elements $x: 1 \to a$ is a [[jointly epimorphic family]].

2. and $E$ is nondegenerate, i.e., 1 is not an [[initial object]].

## Examples

* The category [[Set]].  Here the [[global section]] functor is even (isomorphic to) the identity functor.

* Any model of [[ETCS]].

* Every [[coherent category]] satisfying the [[axiom of finiteness]] is a well-pointed topos equivalent to [[FinSet]].

## Properties

### Strong generation

In a well-pointed topos, the terminal object is even an [[extremal generator]],

  * equivalently: the global section functor $E(1,-)$ is a [[conservative functor]].
  * equivalently: if $f:A\to B$ induces a bijection $E(1,A) \cong E(1,B)$, then $f$ is an isomorphism.
  * equivalently: if $m:A\rightarrowtail B$ is a [[monomorphism]] such that every global element $1\to B$ factors through $A$, then $m$ is an isomorphism.

To prove the last version, let $\chi_A : B\to \Omega$ be the classifying map of $A$, and let $\top : B\to \Omega$ be the classifying map of the maximal subobject.  If every global element $b:1\to B$ factors through $A$, then $\chi_A b = \top b$ for all such $b$.  Hence $\chi_A = \top$ by well-pointedness, so $A$ is isomorphic to the maximal subobject.

Note that in any category with [[equalizers]], an extremal generator is automatically a generator, since the equalizer of two parallel morphisms is the maximal subobject just when the two morphisms are equal.  So the above statements are also equivalent to well-pointedness.

### Boolean properties

Assuming that one accepts [[excluded middle]] in one\'s metalogic, a well-pointed topos is also [[Boolean topos|Boolean]].  To see this, let $U\rightarrowtail A$ be a subobject and $\neg U$ its [[Heyting algebra|Heyting complement]].  Then by definition, the global elements of $\neg U$ are precisely the global elements of $A$ that do not factor through $U$.  But then every global element of $A$ factors through $U \cup \neg U$, hence by strong generation $U\cup \neg U = A$.

Similarly, a well-pointed topos is [[two-valued topos|two-valued]].  In other words, that is, the only [[global element|global elements]] of the [[subobject classifier]] are $\top$ and $\bot$ (and these are distinct, by nondegeneracy) -- or equivalently, the only [[subterminal objects]] are $0$ and $1$.  To see this, note by strong generation a subobject of any object is uniquely determined by the global elements that factor through it.  But $1$ has only one global element, so it only has two possible global elements.

Finally, a well-pointed topos has [[split supports]].  For the support of any object $A$ must be either $0$ or $1$ by two-valuedness.  If it is $0$ then $A\cong 0$ also and its support is split.  Otherwise, $A\ncong 0$; now consider the two coproduct inclusions $inl, inr : A \rightrightarrows A+A$.  Their pullback is $0$ by disjointness.  Since $A\ncong 0$, we have $inl\neq inr$, hence by well-pointedness there is a global element $a:1\to A$ such that $inl a \neq inr a$.  Thus $a$ splits the support of $A$.

Conversely, we have the following:

\begin{theorem}\label{BooleanTwovaluedSupportssplit}
If $E$ is nondegenerate (i.e. $1\ncong 0$), Boolean, two-valued, and split supports, then it is well-pointed.
\end{theorem}
\begin{proof}
This is Proposition 9.33 on p. 314 of [Johnstone, Topos Theory](#Johnstone77), attributed there to [Freyd](#Freyd70).  Let $A\rightarrowtail B$ be a monomorphism such that every global element of $B$ factors through it.  Then $\forall_B A$ is a subterminal object, hence either $1$ or $0$.  If it is $0$, then its complement $\neg\forall_B A = \exists_B \neg A$ is $1$, which is to say that $\neg A$ is well-supported.  Hence, since supports split, $\neg A$ has a global element, which is impossible since every global element of $B$ factors through $A$.  Hence it must be that $\forall_B A = 1$, which implies $A=B$.
\end{proof}

Thus, in external classical logic, a topos is well-pointed if and only if it is nondegenerate, Boolean, two-valued, and has split supports.  In particular, a nondegenerate two-valued topos satisfying the external [[axiom of choice]] is well-pointed.

### Logical properties

The main point of a well-pointed topos in logic is the equivalence of *external* and *internal* properties. In particular, a statement in the [[internal logic]] of the topos will be satisfied if and only if it holds for all global terms. (For the 'only if' part, it is necessary that the topos be nondegenerate.)

### Well-pointedness and concrete categories

The category [[Set]] of sets and functions is both well-pointed and concrete. However, not every [[concrete category]] is a well-pointed category: the category $Field$ of [[fields]] and field [[homomorphisms]] is concrete, but is not well-pointed because it doesn't have a [[terminal object]]. Moreover, not every well-pointed category is a concrete category, as well-pointed categories are not required to be concrete categories: most models of [[ETCS]] aren't defined to be concrete.

The distinction between well-pointed categories and concrete categories is the distinction between [[elements]] and [[global elements]] in a concrete category, as it is not true that elements and global elements (if they exist) coincide in general.

## Generalizations
{#general}

### In constructive mathematics

To maintain this logical result in [[constructive mathematics]] (that is, without excluded middle in the metalogic), one must add  the following requirements:

3. the terminal object is [[indecomposable object|indecomposable]], and
4. the terminal object is [[projective object|projective]].

These are analogues, for [[disjunction]] and [[existential quantification]], of the nondegeneracy requirement (which is about [[falsehood]]).  Classically, they follow from the two conditions given above.

Incidentally, a well-pointed topos in a constructive metalogic is still "two-valued" in the sense that a global element of the subobject classifier is false if and only if it is not true.  However, it is not two-valued in the (classically equivalent) sense that every global element of the subobject classifier is either true or false.

### In a pretopos or coherent category

If $E$ is only a [[pretopos]] or a [[coherent category]], we have to strengthen the condition that 1 is a generator to the condition that 1 is an [[extremal generator]], i.e. for any monomorphism $m:A\to B$, if every global element $1\to B$ factors through $m$, then $m$ is an isomorphism.  In a category with a subobject classifier (such as a topos), any generator is a extremal generator.

This strengthening is important in [[predicative mathematics]], where the category of sets (and in general, a [[Grothendieck topos|category of sheaves]]) is a pretopos but need not be a topos. But of course, the same applies whenever one is studying an arbitrary pretopos, not just a predicative version of $Set$.

### In a lextensive category

If $E$ is only a [[lextensive category]], such as in a [[category of sets]] without [[quotient sets]] (as commonly found in the [[syntactic category]] of an [[extensional type theory]]), then $E$ is well-pointed if $1$ is only a [[initial object|noninitial]] [[indecomposable object|indecomposable]] [[extremal generator]].

### In more general categories

Do we know what these should be in any more general situations?

[[Mike Shulman]]: Well, the pretopos version makes sense in any [[coherent category]], and I would bet that it's the right notion in that generality.  In a [[regular category]] one might just want to assert that $1$ is a (regular-)projective extremal generator, which would probably be enough for regular logic.  And in a category with mere finite limits, being a extremal generator is all one could ask for, and that'd probably be enough for finite-limit logic.

### Well-pointed $(\infty,1)$-toposes {#Infty1Version}

> [[Urs Schreiber|Urs]]: an attempt

One might like to say that "[[∞Grpd]] is essentially the unique [[(∞,1)-topos]] with all small limits and colimits that is well-pointed ."

Possibly one should say: an $(\infty,1)$-topos $\mathbf{H}$ is _well-pointed_ if the terminal object is not the initial one and the  [[global section]] [[(∞,1)-functor]] $\Gamma : \mathbf{H} \to \infty Grpd$ is faithful...

...which should mean that on hom-$\infty$-groupoids it is a [[monomorphism in an (∞,1)-category]]...

...which should mean that for all $X,Y \in \mathbf{H}$ the image of the morphism $\Gamma_{X,Y} : \mathbf{H}(X,Y) \to Func(\Gamma(X),\Gamma(Y))$ in the [[homotopy category]] identifies $\mathbf{H}(X,Y)$ as a [[direct sum]]mand of $Func(\Gamma(X),\Gamma(Y))$.

## As an axiom schema of separation

Well-pointedness in a [[Boolean category]] (if using [[classical logic]]) or a [[Heyting category]] (if using [[intuitionistic logic]]) could also be represented, in addition to the [[terminal object]] being a [[strong generator]], by a version of the [[axiom schema of bounded separation]].

Let us define the following

* A $\Delta_0$-variable is a [[global element]] variable.
* A $\Delta_0$-context is a context only containing $\Delta_0$-variables
* An $\Delta_0$-atomic formula is an equality of global elements
* A $\Delta_0$-quantifier is a quantifier over a $\Delta_0$-variable.
* A formula whose only atomic subformulas are $\Delta_0$-atomic and whose only quantifiers are $\Delta_0$-quantifiers is a $\Delta_0$-formula.

A [[Boolean category]] or [[Heyting category]] $\mathcal{E}$ is well-pointed if

* the [[terminal object]] $1 \in \mathrm{Ob}(\mathcal{E})$ is a strong generator: given objects $A \in \mathrm{Ob}(\mathcal{E})$ and $B \in \mathrm{Ob}(\mathcal{E})$ and [[monomorphism]] $m:A \hookrightarrow B$, if for every [[global element]] $x:1 \to B$ there exists a global element $y:1 \to A$ such that $m \circ y = x$, then $m$ is an [[isomorphism]].

* the [[axiom schemata of bounded separation]] holds for [[global elements]]: for any $\Delta_0$-formula $\phi(x)$ with global element free variable $x:1 \to B$, there exists an object $A \in \mathrm{Ob}(\mathcal{E})$ and a monomorphism $m:A \hookrightarrow B$ such that for any global element $x:1 \to B$, $\phi(x)$ holds if and only if there exists a global element $y:1 \to A$ such that $m \circ y = x$.

In fact, given that $\mathcal{E}$ is a [[finitely complete category]], we could use this as the definition of a **well-pointed Boolean category** or **well-pointed Heyting category**, and thus in a foundational categorical [[set theory]]:

* Improper subsets: given a subset $A \subseteq B$ with chosen [[injection]] $m:A \hookrightarrow B$, if for every element $x \in B$ there exists an element $y \in A$ such that $m(y) = x$, then $A$ is [[generalized the|the]] [[improper subset]] of $B$ and $m$ is a [[bijection]].

* Bounded separation: for any $\Delta_0$-formula $\phi(x)$ with free variable $x \in B$, there exists a subset $A \subseteq B$ with chosen injection $m:A \hookrightarrow B$ such that for every element $x \in B$, $\phi(x)$ holds if and only if there exists an element $y \in A$ such that $m(y) = x$.

## Related entries

* [[ETCS]]
* [[concrete category]]

## References

* {#MM} [[Sheaves in Geometry and Logic]], Sections VI.1 and 10. Thm. \ref{BooleanTwovaluedSupportssplit} in this article is a strengthening of SGL's Prop. VI.1.7. SGL's Section VI.10 is a comparison of well-pointed toposes to RZC (Restricted Zermelo with Choice).

* {#Freyd70} [[Peter Freyd]], _Aspects of Topoi_, Bull. Austral. Soc. Math. no.7 pp.1-76,467-480 (1970).

* {#Johnstone77} [[Peter Johnstone]], _Topos Theory_, Academic Press New York 1977. (also available as Dover reprint, Minneola 2014).  See Section 9.3.

For constructive well-pointedness of [[Heyting categories]] as a structural [[axiom schemata of separation]] in addition to the [[terminal object]] being a [[strong generator]], see:

* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).

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well-powered abelian category

[[!redirects locally small abelian category]]

A __locally small abelian category__ is an abelian category which is [[well-powered category|well-powered]].
In other words, (equivalent classes of) [[subobject]]s of any set form a set.

Remark. In older literature and occasionally in some contemporary literature in the subject of [[abelian categories]] locally small signifies what is now standardly called [[well-powered category]]. As every abelian category is a [[locally small category]] in the usual sense (all Homs are sets), the terminology locally small abelian category (if the modifier is understood as nontrivial) does not lead to ambiguities (hence it denotes a well-powered abelian category).

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well-powered category

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## Definition

A [[category]] $C$ is **well-powered** if every object has a [[small category|small]] [[partial order|poset]] of [[subobject]]s.

Assuming that by "subobject" we mean (an [[equivalence class]] of) [[monomorphism|monomorphisms]], this means that for every object $X$, the (generally [[large category|large]]) [[preorder|preordered]] set of monomorphisms with codomain $X$ is equivalent to a small poset, or equivalently that this preordered set is essentially small.  Variations exist that use notions of subobject other than monomorphisms.

If $C^{op}$ is well-powered, we say that $C$ is **well-copowered** (although "cowell-powered" is also common).

## Properties

### Relation to local smallness

A well-powered category with binary [[product|products]] is always [[locally small category|locally small]], since morphisms $f: A \to B$ can be identified with particular subobjects of $A \times B$ (their [[graph of a function|graphs]]).

Conversely, any locally small category with a [[subobject classifier]] must obviously be well-powered.  In particular, a [[topos]] is locally small if and only if it is well-powered.

There are interesting conditions and applications of the preorder on the sets of subobjects in well-powered categories, cf. e.g. [[property sup]].

## Examples

* Every [[Grothendieck topos]] (indeed, every [[locally small category|locally small]] [[elementary topos]]) is well-powered (by the existence of a [[subobject classifier]] and the smallness of [[hom sets]]).

* More generally, every [[locally presentable category]] is well-powered, since it is a full reflective subcategory of a presheaf topos, so its subobject lattices are subsets of those of the latter.

* Indeed, every category $C$ with a small [[dense subcategory]] $A$ is well-powered, since the [[restricted Yoneda embedding]] $C \to [A^{op},Set]$ is then fully faithful and preserves monomorphisms, so it embeds the subobject posets of $C$ as sub-posets of those of $[A^{op},Set]$.  This includes in particular all [[accessible categories]].

* On the other hand, a locally small category with a [[strong generator]] can fail to be well-powered; a counterexample is [here](https://math.stackexchange.com/a/254794).

* Every locally presentable category, indeed every [[accessible category]] with [[pushouts]], is well-*copowered*.  This is shown in [Adamek-Rosicky, Proposition 1.58 and Theorem 2.49](#AR).  Whether this is true for all accessible categories depends on what [[large cardinal]] properties hold: by Corollary 6.8 of Adamek-Rosicky, if [[Vopenka's principle]] holds then all accessible categories are well-copowered, while by Example A.19 of Adamek-Rosicky, if all accessible categories are well-copowered then there exist arbitrarily large [[measurable cardinals]].

## References

* {#AR} [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)

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well-quasi-order

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## Definition

In [[classical mathematics]], a __well-quasi-order__ is a [[preorder]] $(P, \leq)$ such that for any infinite sequence $x_i$ in $P$, there exist $i, j$ with $i \lt j$ and $x_i \leq x_j$. Another way of expressing this condition is:

* There are no infinite [[antichain|antichains]] in $P$: if $x_i$ is a sequence in $P$, then there exist some pair of elements $x_i$, $x_j$ that are comparable, i.e., either $x_i \leq x_j$ or $x_j \leq x_i$, and

* There is no strictly decreasing sequence in $P$: if $x_0 \geq x_1 \geq \ldots$ in $P$, then there exists $n$ such that $x_i \leq x_{i+1}$ for all $i \geq n$.

One motivation for this notion is given by the following theorem:

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###### Theorem
Let $(X, \leq)$ be a quasi-order (i.e., a [[preorder]]). Define a [[partial order]] on the [[power set]] $P(X)$ by $A \leq^+ B$ if $\forall_{a\colon A} \exists_{b\colon B} (a \leq b)$. Then $\leq^+$ is a [[well-founded relation]] if and only if $\leq$ is a well-quasi-ordering.
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## Example

The collection of [[graph|finite simple graphs]] is well-quasi-ordered by the [[graph minor]] relation. This is the celebrated _Robertson-Seymour Graph Minor Theorem_.

## Related entries

* [[well-order]]

## References

* [Wikipedia article](http://en.wikipedia.org/wiki/Well-quasi-ordering)

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Wen-Tsun Wu

* [biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/Wu_Wen-Tsun.html)

category: people


Wendong Liang

* [personal page](https://wendongl.github.io/)

## Selected writings

On [[gs-monoidal categories]] in [[category theory|category theoretic]] [[probability theory]]:

* [[Tobias Fritz]], [[Wendong Liang]], *Free gs-monoidal categories and free Markov categories*, Appl. Categ. Structures **31** 21 (2023)  [[arXiv:2204.02284](https://arxiv.org/abs/2204.02284), [doi:10.1007/s10485-023-09717-0](https://doi.org/10.1007/s10485-023-09717-0)]

category: people

[[!redirects LiangvWendong]]


Wendy Lowen

__Wendy Lowen__ is a mathematician at Universiteit Antwerpen, working mainly in algebra, categorical algebra and algebraic geometry.

Webpage: [publications](https://win.uantwerpen.be/~wlowen/publications.html)

* Wendy Lowen, [[Michel Van den Bergh]], _On compact generation of deformed schemes_, Adv. Math. __244__ (2013) 441--464 [doi](https://doi.org/10.1016/j.aim.2013.04.024)
* Wendy Lowen, _A generalization of the Gabriel-Popescu theorem_, J. Pure and Appl. Alg. __190__ (1) (2004): 197--211, [doi](https://doi.org/10.1016/j.jpaa.2003.11.016) [MR2043328](https://www.ams.org/mathscinet-getitem?mr=2043328)
* [[Dmitry Kaledin]], Wendy Lowen, _Cohomology of exact categories and (non-)additive sheaves_, Adv. Math. __272__ (2015) 652--698 [arXiv:1102.5756](https://arxiv.org/abs/1102.5756)

category: people


Werner Burau

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=21539)

## Selected writings

Introducing the [[braid representation]] now known as the [[Burau representation]]:

* [[Werner Burau]], *Über Zopfgruppen und gleichsinnig verdrillte Verkettungen*, Abh. Math. Semin. Univ. Hambg. **11** (1935) 179–186 $[$[doi:10.1007/BF02940722](https://doi.org/10.1007/BF02940722)$]$

category: people


Werner Greub

* [website]() ?

category: people


Werner Heisenberg

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

## Selected writings

Introducing [[quantum mechanics]]:

* [[Werner Heisenberg]], *Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen*, Zeitschrift für Physik **33** (1925) 879–893 $[$[doi:10.1007/BF01328377]( https://doi.org/10.1007/BF01328377), [Engl. pdf](http://users.mat.unimi.it/users/galgani/arch/heis25ajp.pdf)$]$

Introducing the [[Schwinger effect]] of [[vacuum polarization]]:

* {#HeisenbergEuler36} [[Werner Heisenberg]], [[Hans Euler]],  _Folgerungen aus der Diracschen Theorie des Positrons_, Z. Physik 98, 714–732 (1936) ([doi:10.1007/BF01343663](https://doi.org/10.1007/BF01343663))

On [[quantum physics]] and the *[[potentiality]]* of [[quantum states]]:

* {#Heisenberg1958} [[Werner Heisenberg]], *Physics and Philosophy -- The Revolution in Modern Science*, Harper & Brothers (1958) [[archive](https://archive.org/details/physics-and-philosophy-the-revolution-in-modern-scirnce-werner-heisenberg-f.-s.-c.-northrop)]

category: people

[[!redirects Heisenberg]]


Werner Nahm

* [webpage](http://www.th.physik.uni-bonn.de/People/werner/)

* [wikipedia entry](http://de.wikipedia.org/wiki/Werner_Nahm)

## Selected writings

On the [[Nahm transform]]:

* [[Werner Nahm]], _The construction of all self-dual multi-monopoles by the ADHM method_, In: Craigie et al. (eds.), _Monopoles in quantum theory_, Singapore, World Scientific 1982 ([spire:178340](http://inspirehep.net/record/178340)

* [[Werner Nahm]], _Self-dual monopoles and calorons_, In: G. Denardo, G. Ghirardi and T. Weber (eds.), _Group theoretical methods in physics_, Lecture Notes in Physics 201. Berlin, Springer-Verlag 1984 ([doi:10.1007/BFb0016145](https://doi.org/10.1007/BFb0016145))

On the classification of [[supersymmetry]] [[super Lie algebras]]:

* [[Werner Nahm]], _[[Supersymmetries and their Representations]]_, Nucl.Phys. B135 (1978) 149  ([spire:120988](https://inspirehep.net/record/120988), [pdf](http://cds.cern.ch/record/132743/files/197709213.pdf))

## Related $n$Lab entries

* [[Yang-Mills instanton]]

* [[ADHM construction]]

* [[magnetic monopole]]

* [[Nahm transform]]

* [[caloron]]

* [[supersymmetry]]

category: people


Werner Porod

* [Institute page](https://www.physik.uni-wuerzburg.de/tp2/personen/prof-dr-werner-porod/)

* [Spire page](https://inspirehep.net/authors/992946?ui-citation-summary=true)

## Selected writins

On the [[Higgs field]] in [[holographic QCD]]:

* [[Johanna Erdmenger]], [[Nick Evans]], [[Werner Porod]], [[Konstantinos Rigatos]], *Gauge/gravity dual dynamics for the strongly coupled sector of composite Higgs models*, JHEP **58** (2021) [[arXiv:2010.10279](https://arxiv.org/abs/2010.10279), doi:10.1007/JHEP02(2021)058]

category: people