universe

> This entry is about the notion in [[mathematics]]/[[logic]]/[[type theory]]. For the notion of the same name in [[physics]] see at _[[observable universe]]_.

***

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### Context
#### Universes
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#### Foundations
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[[!include foundations - contents]]
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# Contents
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## Idea

A universe is a realm within which (some conceived part, naively and virtually all, of) [[mathematics]] may be thought of as taking place.  Universes can be purely metamathematical, but we can also reflect upon them and bring them into mathematics.  There are several different kinds of 'universes'. For a physical notion of universe see [[observable universe]].

## Categories as universes

Much of [[ordinary mathematics]] can be thought of as taking place [[internalization|inside]] "the archetypical [[category]] [[Set|SET]] of sets".  Typically, the properties of $SET$ are formulated in [[first-order logic]] using a [[set theory]] such as [[ZFC]] or (more directly) [[ETCS]].

We can generalise this from $SET$ to any other [[category]] $C$.  Without further assumptions on the category, there is in general very little mathematics that can be formulated inside it, but a few extra properties and structures are usually sufficient to provide something interesting.  This is the general topic of _[[internalisation]]_.  The attitude to take is that any *specific* category is merely one [[model]], while a general *class* of categories is a [[theory]] (really a [[doctrine]], or [[2-theory]]).

We can also use [[higher category theory|higher categories]] instead of mere categories here.  Even for ordinary mathematics, this means starting with $\infty$-[[infinity-Grpd|GRPD]] instead of $SET$.

## Universes as classes

The idea of the [[large category]] $SET$ as the universe of mathematics has an analogue in [[class theory]]. In a [[material class theory]], the __von Neumann universe__ $V$ is the [[class]] of all [[axiom of foundation|well-founded]] [[pure sets]].

More explicitly: for every [[ordinal number]] $\alpha$, we have a [[set]] $V_\alpha$ (the von Neumann universe of __rank__ $\alpha$), defined [[recursion|recursively]] using the operations of [[power set]] and (material) [[union]] as
$$ V_\alpha \coloneqq \bigcup_{\beta \lt \alpha} \mathcal{P}V_\beta .$$
Then $V$ itself is the union of all of the $V_\alpha$.

A similar but more complicated definition allows us to define the universe $L$ of [[constructible set]]s, called the [[constructible universe]].

See also a [Wikipedia article](https://secure.wikimedia.org/wikipedia/en/wiki/Universe_%28mathematics%29) written largely by [[Toby Bartels]] in another lifetime.

In a [[structural class theory]], such as the [[category with class structure]] found in [[algebraic set theory]], a **universe** is a class $U$ with a [[monic]] class [[map]] $\mathcal{P}(U) \hookrightarrow U$ from the [[power object|powerclass]] of $U$ to $U$. In particular, every [[universal class]] is a universe.

## Grothendieck universes

A [[Grothendieck universe]] is a set $U$ such that the following properties hold for $U$:

1. if $x \in a \in U$, then $x \in U$;
1. if $a, b \in U$, then $\{a, b\}$ and $a\times b$ are elements of $U$;
1. if $a \in U$, then $\cup a$ and $P(a)$ are elements of $U$;
1. the set $\omega$ of all natural numbers is an element of $U$;
1. if $f:a \to b$ is surjective with $a \in U$ and $b \subseteq U$, then $b \in U$.

## Universes inside $SET$

If set theory is the [[foundation of mathematics]], then the universes above are part of metamathematics rather than mathematics itself.  However, we can also look for [[small categories]] or sets that exist within set theory and have the properties of a universe.

There is already a hint of this in the hierarchy $V_\alpha$ out of which the von Neumann universe is built.  For some values of $\alpha$, $V_\alpha$ might be a sufficiently large and complete collection of [[sets]] in which to do ordinary mathematics.  From the [[nPOV]], we can instead look at the category $Set_\alpha$ of these sets and the [[functions]] between them, although it\'s more common to think about $Set_\kappa$ for some [[cardinal number]] $\kappa$.

An [[infinite set]] is a simple example, as [[finite mathematics]] can be done inside it.  Going beyond this, a [[Grothendieck universe]] is a set within which other infinite sets exist but which is complete under the operations of [[material set theory]] that are needed for ordinary mathematics.  The [[structural set theory|structural]] analogue is a [[universe in the topos]] $SET$.

In general, we need some axioms to state the existence of such universes; we can think of these as [[large cardinal]] axioms.  (The existence of an infinite set is the [[axiom of infinity]]; the existence of a Grothendieck universe is the existence of an [[inaccessible cardinal]].)

The use of such universes is convenient when we want to work with [[large categories]] "as if they were small."  That is, if we redefine "small" to mean "element of some fixed universe," then the categories of all small structures of some sort will still be sets (rather than [[proper classes]]) in the bigger ambient universe, and thus we can for instance form [[functor categories]] between them freely.  We do sometimes then need a way to "move a category" from one universe to another; see [[universe enlargement]].

## Reflection

In [[logic]], we use [[reflection principle]]s (see [Wikipedia](https://secure.wikimedia.org/wikipedia/en/wiki/Reflection_principle)) to systematically identify features of our meta-universe and see what is needed to prove the existence of an internal universe with these features.

This follows the following outline:

1.  We assume that we understand _a priori_ what it means to use [[first-order logic]] (or some other finitary base logic).

2.  Using this base logic, we can formulate a foundational [[set theory]] that describes a meta-universe such as $SET$.

3.  Of course, $SET$ cannot be described from inside itself.  But there may be objects in $SET$ ([[internal categories]]) that _look like_ $SET$ itself.

4.  We add an additional axiom to our set theory stating that such objects, the _internal universes_, exist.  These are then [[models]] of our original set theory.

5.  Now we are using a new, stronger set theory; repeat.

## Universes in type theory

Set theory is not the only [[foundation of mathematics]].  For example, there are also various foundational [[type theories]], closely related to [[structural set theory]].  Then we have a meta-universe of all types, and we can also add axioms for the existence of internal [[type universes]].

## Related concepts

* [[constructible universe]]

* [[universe enlargement]]

* [[universe polymorphism]]

* [[type of types]]

  * [[subobject classifier]], [[object classifier]], [[partial map classifier]]

* [[reflective subuniverse]]

* [[G-universe]]

## References

### General

For universes in [[class theory]] and [[algebraic set theory]], see

* [[Steve Awodey]]. *Notes on algebraic set theory*, Notes for lectures given at the Summer School on Topos Theory, Haute-Bodeux, Belgium. May 29 to June 5, 2005. Carnegie Mellon University Technical Report No. CMU-PHIL-170. June 2005. ([pdf](https://www.phil.cmu.edu/projects/ast/Papers/bnotes.pdf))

### In type theory

* [[Martin Hofmann]], section 2.1.6 of _Syntax and semantics of dependent types_ ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.8985))

[[!redirects universe]]
[[!redirects universes]]
[[!redirects universe > history]]

[[!redirects von Neumann universe]]
[[!redirects von Neumann universes]]

[[!redirects set of all sets]]


universe - contents

**[[universe]]**

(in [[category theory]]/[[type theory]]/[[computer science]])

**of all [[homotopy types]]**

* [[type of types]], [[object classifier]],

  * [[univalence]]

* [[codomain fibration]]

**of [[homotopy n-types]]**

* [[type of n-types]]

**of [[0-truncated object|0-truncated types]]/[[h-sets]]**

* [[type of h-sets]], [[universe in a topos]]

**of [[(-1)-truncated object|(-1)-truncated types]]/[[h-propositions]]

* [[type of propositions]], [[subobject classifier]]

### resizing

* [[universe enlargement]]


universe enlargement

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## Idea

Let $\mathbf{U}$ and $\mathbf{V}$ be set-theoretic [[universes]] (such as [[Grothendieck universes]] or [[universe in a topos|universes in an ambient topos]]) with $\mathbf{V}$ "bigger than" $\mathbf{U}$, and let $C$ be a [[category]] "in the world of $\mathbf{U}$," i.e. whose objects are $\mathbf{U}$-[[small category|small]] and which itself is $\mathbf{U}$-[[large category|large]].  Its **universe enlargement** is supposed to be a category "in the world of $\mathbf{V}$" which is "the same" or at least similar to $C$, and perhaps better behaved.

## Notation

An enlargement of a specific category, such as [[Set]], [[Cat]], [[Grp]], or [[Top]], is often denoted by writing its name in all capitals: SET, CAT, GRP, TOP.

As a notation for the operation on a general category $C$, the notation $\Uparrow C$ has been proposed.

## Definitions

Since the enlargement is not well-specified by the above intuitive description, there are several different definitions, which are often equivalent but not always.

Henceforth we fix two universes $\mathbf{U}$ and $\mathbf{V}$ with $\mathbf{U}\in \mathbf{V}$.  We will refer to sets in $\mathbf{U}$ as *small*, sets in $\mathbf{V}$ as *large*, and sets not necessarily in $\mathbf{V}$ (or in the next larger universe $\mathbf{W}$) as *very large*.

### Models of a theory

If $C$ is the category of [[small set|small]] [[models]] of some [[theory]], then we can take $\Uparrow C$ to be the category of large [[models]] of the same theory.  For instance:

* the [[large category]] [[Set]] of [[small sets]] is the category of small models of the theory with one sort and no operations or relations.  The category of large models of this theory is just the very large category $SET$ of [[large set|large sets]].

* Similarly, from categories such as [[Grp]], [[Cat]], and [[Top]] of small [[groups]], [[categories]], and [[topological spaces]], we obtain the categories $GRP$, $CAT$, and $TOP$ of large groups, categories, and topological spaces.

Note that the theory could be [[algebraic theory|algebraic]], as for $Grp$, or [[essentially algebraic theory|essentially algebraic]], as for $Cat$, or even [[higher-order logic|higher-order]], as for $Top$.  We will refer to this as the **logical enlargement**.  In practice, this enlargement usually suffices, but for theoretical reasons it would be nice to have a construction which works on *any* large category.

### Locally presentable enlargement
 {#LocallyPresentableEnlargement}

For any large category $C$, the very large [[presheaf category]] $[C^{op},SET]$ contains $C$ as a [[full subcategory]].  Furthermore, $[C^{op},SET]$ is $SET$-bicomplete, i.e. it has all large [[limits]] and [[colimits]], whether or not $C$ was $Set$-complete or cocomplete, and the [[Yoneda embedding]] $C \hookrightarrow [C^{op},SET]$ [[preserved limit|preserves]] all limits that exist in $C$.

However, it hardly preserves any *colimits* (since it is a [[free cocompletion]], after all).  If we want an enlargement of $C$ which preserves some class $\Phi$ of colimits in $C$, then we can restrict to the full subcategory of $[C^{op},SET]$ consisting of the presheaves $C^{op}\to SET$ which preserved limit the colimits $\Phi$ (i.e. take them to limits in $SET$).  Let's denote this category by $\Uparrow_\Phi C$.  Since [[representable functors]] preserve all limits, the Yoneda embedding of $C$ factors through $\Uparrow_\Phi C$, and all the colimits in $\Phi$ are preserved by this restricted embedding.

Moreover, $\Uparrow_\Phi C$ is closed under limits in $[C^{op},SET]$, so it is itself $SET$-complete.  And as long as the (diagrams sizes of the) colimits in $\Phi$ are at most *large*, then (by theorems of Day etc.) $\Uparrow_\Phi C$ is reflective in $[C^{op},SET]$, and hence also $SET$-cocomplete.  Since usually, the only colimits we might care about in a large category $C$ are *small*, a natural choice for $\Phi$ is the class of all small colimits existing in $C$.

We will refer to this $\Uparrow_\Phi C$ as the **locally presentable enlargement**, since it is a [[locally presentable category]] relative to $\mathbf{V}$.  (In fact, it is locally $\kappa$-presentable, where $\kappa$ is the [[cardinality]] of the first universe $\mathbf{U}$.)  Moreover, we have the following (where $\lambda Cts(A,B)$ denotes the category of $\lambda$-small-limit preserving functors).

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###### Theorem
Suppose $C$ is locally presentable relative to $\mathbf{U}$, so that there is a small cardinal $\lambda$ and a small category $A$ with $\lambda$-small colimits such that $C\simeq \lambda Cts(A^{op},Set)$.  If $\Phi$ denotes the class of all small colimits in $C$, then $\Uparrow_\Phi C \simeq \lambda Cts(A^{op},SET)$.
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Note that in this case, $\lambda Cts(A^{op},SET)$ is the category of large models of the theory (namely $A^{op}$) of which $C$ is the category of small ones.  Thus, the theorem says that when $C$ is locally presentable, the locally presentable enlargement is the same as the logical one.

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###### Proof
Write $C' \coloneqq \lambda Cts(A^{op},SET)$.  Then we have a full embedding $C\hookrightarrow C'$, which preserves limits and colimits since they are calculated in the same way in both categories.

Now every object of $C$ is a small colimit of objects of $A$, and the objects of $A$ are all $\lambda$-presentable in $C$ and in $C'$.  Thus, every object of $C$ is $\kappa$-presentable in $C'$, where $\kappa$ is the size of the universe $\mathbf{U}$.  Conversely, since $C'$ is locally $\lambda$-presentable (relative to $\mathbf{V}$), any $\kappa$-presentable object in it is a $\kappa$-small colimit of $\lambda$-presentable objects---so since $C$ is closed under small colimits in $C'$, it must consist exactly of the $\kappa$-presentable objects.  However, since $C'$ is locally $\lambda$-presentable, it is also locally $\kappa$-presentable, so this implies that $C' \simeq \kappa Cts(C^{op},SET)$.  But $\kappa Cts(C^{op},SET)$ is precisely $\Uparrow_\Phi C$.
=--

However, if $C$ is not locally presentable, then its logical and locally-presentable enlargements can disagree.  In fact, they will *usually* disagree in this case, since if $C$ is not locally presentable relative to $\mathbf{U}$, its logical enlargement will usually not be locally presentable relative to $\mathbf{V}$ either, whereas its locally-presentable enlargement is, by definition, locally presentable relative to $\mathbf{V}$.  For example, the logical and locally presentable enlargements of $Top$ are quite different.

### Accessible enlargement

There is a simple modification of the locally-presentable enlargement which makes it agree with the logical enlargement for all [[accessible categories]], not just locally presentable ones.  Namely, instead of $\kappa Cts(C^{op},SET)$, we consider $\kappa Flat(C^{op},SET)$, the category of $\kappa$-[[flat functors]] (functors that "would preserve all $\kappa$-small limits if they existed").  We call this the **accessible enlargement**.

Essentially the same proof as for the theorem above shows that if $C$ is accessible relative to $\mathbf{U}$, hence $C\simeq \lambda Flat(A^{op},Set)$, then the accessible enlargement is equivalent to $\lambda Flat(A^{op},SET)$, which is of course the logical enlargement.

Note that if $C$ has all small limits, then its accessible and locally-presentable enlargements are the same, since a functor defined on a category with $\kappa$-small limits is $\kappa$-continuous iff it is $\kappa$-flat.  However, if $C$ does not have small limits, then its locally-presentable enlargement *does* have all large limits, whereas its accessible enlargement does not.

Thus, when $C$ is "poorly behaved," its accessible enlargement is "closer to it" (in particular, identical to its logical enlargement if $C$ is accessible) and thus equally poorly behaved, whereas its locally-presentable enlargement is further from it, but better behaved (complete and cocomplete).  For some purposes one may want one of these behaviors; for other purposes one may want the other.

### Monoidal enlargements

One may further ask for an enlargement which inherits whatever additional structure $C$ has, such as [[monoidal category|monoidal structure]].  The logical enlargement will usually inherit such structure trivially, while the [[Day convolution]] theorem implies that when $C$ is monoidal, so is its locally-presentable enlargement---and moreover, the latter is [[closed monoidal category|closed]] even if $C$ is not!

## Examples

* [[inter-universal Teichmüller theory]]

## Related concepts

* [[universe polymorphism]]

## References

* Sections 2.6 and 3.11--3.12 of [Basic Concepts of Enriched Category](http://tac.mta.ca/tac/reprints/articles/10/tr10abs.html).

* [Blog post](http://golem.ph.utexas.edu/category/2010/11/universe_enlargement.html)

* [[Zhen Lin Low]], [MO comment on universe enlargement](http://mathoverflow.net/a/130639/381)

[[!redirects universe enlargements]]
[[!redirects change of universe]]
[[!redirects changes of universe]]

[[!redirects universe extension]]


universe in a topos

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# Contents
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## Idea

A **universe** in a [[topos]] is a [[topos theory|topos-theoretic]] version of the notion of [[Grothendieck universe]]; see that page for general motivation and applications.

To free the notion from membership-based set theory, we must replace _sets of sets_ by _families of sets_, just as in passing from [[power set]]s to [[power object]]s we must replace sets of subsets by families of subsets.

## Definition

A **universe** in a topos $\mathcal{E}$ is a morphism $el\colon E \to U$ satisfying the axioms to follow.  We think of $el\colon E \to U$ as an $U$-indexed family of objects/sets (fibers of $el$ being those objects), and we define a morphism $a\colon A \to I$ (regarded as an $I$-indexed family of objects) to be **$U$-small** if there exists a morphism $f\colon I \to U$ and a [[pullback]] square
$$\array{A & \to & E\\
  ^a\downarrow && \downarrow^{el}\\
  I& \underset{f}{\to} & U.}
$$

The arrow $f$ is sometimes called the **name** of $a\colon A \to I$, since in the case when $I=*$, the arrow $f:* \to U$ _points_ at the term in the universe $U$ representing the object $A$. (See also this [discussion](http://nforum.ncatlab.org/discussion/6564/elementary-formulation-of-groupautomorphism-group/?Focus=53083#Comment_53083) and references there.)

Note that $f$ is not, in general, unique: a universe can contain many isomorphic sets.  With this definition, the pullback of a $U$-small morphism is automatically again $U$-small.  We say that an object $X$ is $U$-small if $X\to 1$ is $U$-small.

The axioms which must be satisfied are:

1. Every [[monomorphism]] is $U$-small.

2. The composite of $U$-small morphisms is $U$-small.

3. If $f\colon A \to I$ and $g\colon B \to A$ are $U$-small, then so is the [[dependent product]] $\Pi_f g$ (where $\Pi_f$ is the right adjoint to $f^*\colon \mathcal{E}/I \to \mathcal{E}/A$).

4. The [[subobject classifier]] $\Omega$ is $U$-small.

Note that since $0\to \Omega$ is a monomorphism, (1) and (4) imply that the [[initial object]] $0$ is $U$-small.  A _[[predicative mathematics|predicative]] universe_ is a morphism $el\colon E \to U$ where instead of (4) we assume merely that $0$ is $U$-small; this makes sense in any [[locally cartesian closed category]].  In a topos, the generic subobject $1\to \Omega$ is a predicative universe, and of course a morphism is $\Omega$-small if and only if it is a monomorphism.

If we assume only (1)--(3), then the identity morphism $1_0\colon 0 \to 0$ of the initial object would be a universe, for which it itself is the only $U$-small morphism.  On the other hand, if $\mathcal{E}$ has a [[natural numbers object]] $N$, we may additionally assume that $N$ is $U$-small, to ensure that all universes contain "infinite" sets.

Note that any object isomorphic to a $U$-small object is $U$-small; thus in the language of [[Grothendieck universe]]s this notion of smallness corresponds to _essential_ smallness.  Roughly, we may say that (1) corresponds to transitivity of a Grothendieck universe, (3) and (4) correspond to closure under power sets, and (2) corresponds to closure under indexed unions.

## Example: universes in $SET$

We spell out in detail some implications of these axioms for the case
that the [[topos]] in question is the Categeory of Sets according to
[[ETCS]], to be denoted $SET$.

Write $*$ for [[generalized the|the]]
[[terminal object]] in $SET$, the singleton set. Notice that for each
ordinary element $u \in U$, i.e. $* \stackrel{u}{\to} U$, there is the
set $E_u $ over $u$, defined as the [[pullback]]
$$\array{
  E_u &\to& E
  \\
  \downarrow && \downarrow
  \\
  * &\stackrel{u}{\to}& U
}$$

We think of $E$ as being the disjoint union over $U$ of the $E_u$.  In the language of indexed categories, this is precisely the case: the object $E\in SET$ is the indexed coproduct of the $U$-indexed family $(E\to U) \in SET/U$.

* By the definition of $U$-smallness and the notation just introduced,
an object $S$ in $SET$, regarded as a $*$-indexed family
$S \to *$, is $U$-small precisely if it is isomorphic to
one of the $E_u$.

* If $S$ is a $U$-small set by the above and if
$S_0 \hookrightarrow S$ is a [[monomorphism]] so that $S_0$
is a subset of $S$, it follows from 1) and 2) that the comoposite
$(S_0 \hookrightarrow S \to *) = (S_0 \to *)$ is $U$-small, hence that
$S_0$ is $U$-small. So: a subset of a $U$-small set is $U$-small.

  * In particular, let $\emptyset$ be the [[initial object]], which is
  a subset $\emptyset \hookrightarrow \Omega$ of $\Omega = \mathbf{2}$. So: the
  empty set is $U$-small.

* Let $S$, $T$ and $K$ be objects of $SET$, regarded as $*$-indexed families
$f\colon S \to *$, $T \to *$ and $K \to *$. Notice that
$(SET\downarrow S)(f^* K, f^* T) \simeq (SET\downarrow S)(\array{K \times S \\ \downarrow^{p_2} \\ S}, \array{T \times S \\ \downarrow^{p_2} \\ S})$
is canonically isomorphic to $SET(K \times S, T)$. Since $\Pi_f$ is
defined to be the [[right adjoint]] to $f^*\colon SET \to SET \downarrow S$
it follows that $\Pi_f f^* T \simeq T^S$  is the function set of functions
from $S$ to $T$. By 3), if $S$, $T$ are $U$-small then so is
the function set $T^S$.

  * Since by 4) $\Omega = \mathbf{2}$ is $U$-small and for every $S$ the function
  set $\mathbf{2}^S \simeq P(S)$ is the power set of $S$, it follows that
  the power set of a $U$-small set is $U$-small.

* Let $I$ be a $U$-small set, in that $I \to *$ is $U$-small, and let
$S \to I$ be $U$-small, to be thought of as an $I$-indexed family of
$U$-small sets $S_i$, where $S_i$ is the [[pullback]] $\array{
  S_i &\to& S
  \\
  \downarrow && \downarrow
  \\
  * &\stackrel{i}{\to}& I
}$, so that $S$ is the disjoint union of the $S_i$: $S = \bigsqcup_{i \in I} S_i$.
By axiom 2) the composite morphism $(S \to I \to *) = (S \to *)$ is $U$-small,
hence $S$ is a $U$-small set, hence the $I$-indexed union of $U$-small sets
$\bigsqcup_{i \in I} S_i$ is $U$-small.

By standard constructions in [[set theory]] from these
properties the following further closure properties of the universe $U$
follow.

* For $I$ a $U$-small set and $S \to I$ an $I$-indexed family of
$U$-small sets $S_i$, the cartesian product $\prod_{i \in I} S_i$
is $U$-small, as it is a subset of $P(I \times S)$.

In **summary**

* the sets $\emptyset, *, \mathbf{2}$ are $U$-small;

* a subset of a $U$-small set is $U$-small;

* the power set $P(S)$ of any $U$-small set is $U$-small;

* the function set $T^S$ for any two $U$-small sets is $U$-small;

* the [[coproduct|union]] of a $U$-small family of $U$-small sets is $U$-small.

* the [[product|product]] of a $U$-small family of $U$-small sets is $U$-small.

## Axioms of universes

Just as [[ZFC]] and other material [[set theory|set theories]] may be augmented with axioms guaranteeing the existence of [[Grothendieck universe]]s, so may [[ETCS]] and other structural set theories be augmented with axioms guaranteeing the existence of universes in the above sense.  For example, the counterpart of Grothendieck's axiom

* For every set $s$ there exists a universe $U$ containing $s$, i.e. $s\in U$

would be

* For every morphism $a\colon A \to I$ in $\mathcal{E}$, there exists a universe $el\colon E \to U$ such that $a$ is $U$-small.

## Consequences

One can show, from the above axioms, that the $U$-small morphisms are closed under finite [[coproduct]]s and under [[quotient object]]s.  See the reference below.

## In terms of indexed categories

Recall that an $\mathcal{E}$-[[indexed category]] is a [[pseudofunctor]] $\mathcal{E}^{op}\to \Cat$.  The fundamental $\mathcal{E}$-indexed category is the _self-indexing_ $\mathbb{E}$ of $\mathcal{E}$, which takes $I\in \mathcal{E}$ to the [[over category|slice category]] $\mathbb{E}^I = \mathcal{E}/I$ and $x\colon I \to J$ to the [[base change]] functor $x^*$.

An **internal full subcategory** of $\mathcal{E}$ is a full sub-indexed category $\mathbb{F}$ of $\mathbb{E}$ (that is, a collection of full subcategories $\mathbb{F}^I\subset \mathbb{E}^I$ closed under reindexing) such that there exists a generic $\mathbb{F}$-morphism, i.e. a morphism $el\colon E \to U$ in $\mathbb{F}^U$ such that for any $a\colon A \to I$ in $\mathbb{F}^I$, we have $a \cong f^*(el)$ for some $f\colon I \to U$.  In this case (since $\mathcal{E}$ is [[locally cartesian closed category|locally cartesian closed]]) there exists an [[internal category]] $U_1 \;\rightrightarrows\; U$ in $\mathcal{E}$ such that $\mathbb{F}$ is equivalent, as an indexed category, to the indexed category represented by $U_1 \;\rightrightarrows\; U$.

An internal full subcategory is an **internal full subtopos** if each $\mathbb{F}$ is a [[logical functor|logical]] subtopos of $\mathbb{E}$ (closed under finite limits, exponentials, and containing the subobject classifier).  A universe in $\mathcal{E}$, as defined above, can then be identified with an internal full subtopos satisfying the additional axiom that $U$-small morphisms are closed under composition.

## In the internal logic

In a topos with a universe, we can talk about small objects in the [[internal logic]] by instead talking about elements of $U$.  We can then rephrase the axioms of a universe in the internal logic to look more like the usual axioms for a Grothendieck universe, with the morphism $el\colon E \to U$ interpreted as a "family of objects"  $(S_u)_{u\colon U}$:

1. for all $u$ in $U$, if $X$ is a [[subset]] of $S_u$ (in the sense that there exists an [[injection]] $X \embedsin S_u$), then there is a $v$ in $U$ such that $X \cong S_v$;
2. for all $u$ in $U$, there is a $v$ in $U$ such that the [[power set]] $P(S_u) \cong S_v$;
3. there is a $v$ in $U$ such that $S_v$ is [[empty set|empty]];
4. for all $u$ in $U$ and all functions $f\colon S_u \to U$, there is a $v$ in $U$ such that the [[disjoint union]] $\biguplus_{i\colon S_u} S_{f(i)} \cong S_v$.

In a [[well-pointed topos]], such as a model of [[ETCS]], these "internal" axioms are equivalent to their "external" versions that refer to [[global element]]s of $U$.

+-- {: .query}
[[Mike Shulman|Mike]]: I haven't actually checked anything in this section, but it seems probable.
=--

## Related concepts

* [[type universe]]

* [[object classifier in an (infinity,1)-topos]]

* [[Grothendieck universe]]

## References

The construction of universes in presheaf toposes was first discussed in the following short note which might serve as a concise introduction to the more general reference that follows:

* [[Martin Hofmann]], [[Thomas Streicher]], _Lifting Grothendieck Universes_ , ms. University of Darmstadt (unpublished). ([pdf](http://www.mathematik.tu-darmstadt.de/~streicher/NOTES/lift.pdf))

The general case is considered here:

* [[Thomas Streicher]], _Universes in Toposes_ , pp.78-90 in L. Crosilla, P. Schuster (eds.), _From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics_ , Oxford UP 2005. ([ps](http://www.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.ps.gz),[pdf](http://www.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.pdf))

See also

* Christian Maurer, _Universes in Topoi_ , pp.285-296 in F. W. Lawvere, C. Maurer, G. C. Wraith (eds.), _Model Theory and Topoi_ , LNM **445** Springer Heidelberg 1975.

[[!redirects universe in a topos]]
[[!redirects universes in a topos]]
[[!redirects universes in toposes]]
[[!redirects universe in the topos]]
[[!redirects universes in the topos]]
[[!redirects universes in the toposes]]
[[!redirects internal full subcategory]]
[[!redirects internal full subcategories]]
[[!redirects internal full subtopos]]
[[!redirects internal full subtoposes]]
[[!redirects internal full subtopoi]]


universe polymorphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Universes
+-- {: .hide}
[[!include universe - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

> This page is about the concept in [[type theory]]. For the analogous concept in [[computer science]] see at *[[polymorphism]]*. For the concept introduced by [[Shinichi Mochizuki]] in [[inter-universal Teichmüller theory]], see at *[[poly-morphism]]*.

#Contents#
* table of contents
{:toc}

## Idea

In ([[homotopy type theory|homotopy]]) [[type theory]], by
*universe polymorphism* one refers to the ability for [[definitions]] to be implicitly duplicated at different [[type universe]]-levels, with the [[types]] they contain. Both Russell and Tarski style universes can be polymorphic or not.

## Examples and applications

* [[Coq]] uses Russell style universes and used to have a sort of "fake" universe polymorphism, but recent versions of Coq have real universe polymorphism (thanks to Mathieu Sozeau).

* [[Agda]] uses non-cumulative Russell style universes with universe polymorphism.

* The [[HoTT Book]] (first edition) uses Russell style universes with universe polymorphism.

## Related concepts

* [[universe enlargement]]

* [[typical ambiguity]]

## References

* [[Robert Harper]], Rober Pollack, _Type Checking with Universes_ (1991) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.38.8166))

* [[Mike Shulman]], [Universe Polymorphism and Typical Ambiguity](https://golem.ph.utexas.edu/category/2012/12/universe_polymorphism_and_typi.html), blog post 2012

* Matthieu Sozeau and Nicolas Tabareau, *Universe polymorphism in Coq*, In: Klein G., Gamboa R. (eds) Interactive Theorem Proving. ITP 2014. Lecture Notes in Computer Science, vol 8558. Springer, Cham. [doi](https://doi.org/10.1007/978-3-319-08970-6_32)

* [[András Kovács]], *Generalized Universe Hierarchies and First-Class Universe Levels*, 2021, [arxiv:2103.00223](https://arxiv.org/abs/2103.00223).

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

[[!redirects universe polymorphisms]]


universes in SEAR

# Universes in SEAR
* table of contents
{: toc}

This page develops a theory of universes in the [[structural set theory]] called [[SEAR]], parallel to the theory of [[Grothendieck universes]] for [[ZF]] and the theory of a [[universe in a topos]].  See also [[categories in SEAR]].

+-- {: .query}
[[David Roberts]]: Can we talk about Grothendieck universes or analogous size-related mechanisms? (this is a naive question, since there may be (no need for/obvious way to do) this in a structural set theory that I am unaware of.

[[Mike Shulman]]: Universes can be defined in any structural set theory in a fairly straightforward way, modulo the usual translation from "set of sets" to "family of sets."  There is some stuff from a more ETCS-like perspective at [[universe in a topos]]; I'm sure that this can be rephrased in SEAR (if you want to take a shot, feel free!).

[[David Roberts]]: I'd certainly like to, subject to real-world time constraints. It seems to me that something could be done to the axiom of collection if we had a axiom of universes (for every set there is a universe acting as a universal family for it, or similar).

[[Mike Shulman]]: I'm not sure that Collection can really be eliminated using universes, but maybe it can.
=--

## Using families

+-- {: .query}
[[David Roberts]]: Some sketchy thoughts - a universe will be a relation $U \looparrowright E$ satisfying axioms analogous to those at [[universe in a topos]]. There are few points that need translating into SEAR-language first so we can define smallness of a family of sets $I \looparrowright A$

* Pullback - unless I am mistaken, the appropriate version is this: Given a function $I \to U$, the family $I \looparrowright E$ should be regarded as the \'pullback\' family. This family should be isomorphic to the family $I \looparrowright A$. This leads us to wonder how to define two families to be isomorphic.

* Dependent product - this is tricker and I'm not going to do that right now.
=--

Fix a [[family of sets]] $U \looparrowright E$.

+-- {: .num_defn}
###### Definition
A family $a:I \looparrowright A$ is called _$U$-small_ if there is a function $f:I \to U$ such that for the composite relation $I \looparrowright E$ ...
=--

A set $A$ is called $U$-small if the relation $\mathbf{1}\looparrowright A$, which holds of $*$ and $a$ for all elements $a$ of $A$, is a $U$-small family.

For ease of language, we define a composite of families to be the composite relation of the two relations defining the families.

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

A family of sets $U \looparrowright E$ is called a _universe_ if the following axioms are satisfied:

* The composite of two $U$-small families is $U$-small
* The relation $f^o:B\looparrowright A$ opposite to an injective function $f:A \to B$ is a $U$-small family
* $P\mathbf{1}$ is a $U$-small set
* (Something about dependent products, once these are defined)
=--

There is an optional axiom, given the axiom of infinity from before:

* The [[natural numbers object]] is a $U$-small set

Without this axiom we do not have that universes contain infinite sets.

Given a universe $U \looparrowright E$, we want to define a (meta-)category $Set_U$...

## Using mere sets

+-- {: .query}
[[Mike Shulman]]: The main idea of SEAR is that in a well-pointed topos, global elements suffice for everything.  In particular, we can usually dispense with families of things and consider only single ones.  So here's a formulation that I think is more in the spirit of SEAR.
=--

Given a family $E:U\looparrowright X$, recall that for $u\in U$ we let $E_u$ be a tabulation of $\{x\in X | E(u,x) \}\subseteq X$.  We say that a set $A$ is **$U$-small** (although really, it would be more appropriate to say "$E$-small") if there exists a $u\in U$ with $A\cong E_u$.  Now we say that $U$ (equipped with $E$) is a **universe** if it satisfies:

* There exists an inhabited $U$-small set.
* The tabulation of any relation between two $U$-small sets is $U$-small.
* If $A$ is $U$-small, then so is $P A$.
* (Replacement) If $f:A\to B$ is a function such that $B$ is $U$-small and for each $b\in B$, the set $f^{-1}(b)$ is $U$-small, then $A$ is $U$-small.
* (Collection) If $p:B\twoheadrightarrow A$ is a surjective function where $A$ is $U$-small, then there exists a $U$-small set $C$, a surjection $q:C\twoheadrightarrow A$ and a function $f:C\to B$ such that $q = p\circ f$.

The optional axiom is the same:

* There exists an infinite $U$-small set (in the sense of Axiom 4).

The first two properties of a universe clearly imply that the $U$-small sets satisfy Axioms 0--2 of SEAR all by themselves.  In particular, $\emptyset$ and $1$ are $U$-small, any subset of a $U$-small set is $U$-small, and if $A$ and $B$ are small then so is $A\times B$.  Thus from the additional assumption of the smallness of powersets, we can conclude that if $A$ and $B$ are small, so is the set $B^A$ of functions from $A$ to $B$.  It also follows that coproducts and quotient sets preserve smallness.  (For a notion of "predicative universe" we would replace the smallness of powersets with the smallness of function sets, and we might have to separately assume smallness of coproducts and quotients, I'm not sure.)

In order to show that the $U$-small sets model SEAR all by themselves, it remains to show:

+-- {: .num_theorem #univ-coll}
###### Theorem
The $U$-small sets for a universe $U$ satisfy the Collection axiom of SEAR.
=--
+-- {: .proof}
###### Proof
Let $A$ be a $U$-small set and $P$ a property as in the Collection axiom.  Let $P'$ be a property such that $P'(a,X)$ holds iff $P(a,X)$ holds *and* $X$ is $U$-small.  By the Collection axiom for all sets applied to $P'$, there exists a set $B$, a function $p:B\to A$, and a $B$-indexed family of sets $M:B\looparrowright Y$ satisfying the desired properties: (1) for any $b\in B$, we have $P(p(b),M_b)$ *and* $M_b$ is $U$-small, and (2) for any $a\in A$, if there exists a $U$-small set $X$ with $P(a,X)$, then $a\in im(p)$.  Our goal is to replace $B$ and $Y$ by $U$-small sets $B'$ and $Y'$ with a relation $M':B'\looparrowright Y'$ maintaining the truth of these statements.

Now the induced function $\overline{p}:B\to im(p)$ is surjective, and since $im(p)$ is a subset of $A$ it is $U$-small.  Thus, by the "Collection" property of a universe, there exists a $U$-small set $C$, a surjection $q:C\twoheadrightarrow im(p)$ and a function $f:C\to B$ such that $q = \overline{p} \circ f$.  We take $B'=C$ and the composite $C \to im(p)\to A$ as the function $p'$; we claim that the composite relation $M\circ f$ still satisfies (1) and (2) above.  (We have not replaced $Y$ yet, so this is still an intermediate step.)  Property (1) follows since $p'(c) = p(f(c))$ and $(M\circ f)_c \cong M_{f(c)}$, while property (2) follows since $im(p) = im(p')$ by construction.

Now consider a tabulation $B' \overset{r}{\leftarrow} Z \overset{s}{\to} Y$ of $M\circ f$.  Note that for any $b\in B'$ we have $r^{-1}(b) \cong (M\circ f)_b$, and therefore in particular $r^{-1}(b)$ is $U$-small by (1).  Since $B'$ is also $U$-small, the "Replacement" property of a universe implies that $Z$ is also $U$-small.  But the family $r^o:B'\looparrowright Z$ is equivalent to $M\circ f$, in the sense that $(r^o)_b \cong (M\circ f)_b$ for all $b\in B'$, so it also satisfies (1) and (2); thus we can take $Y'=Z$ and $M'=r^o$.
=--

Note that both the "Replacement" and "Collection" properties of a universe are required to show the SEAR Collection axiom for $U$-small sets.  The naming of these two properties appears to be traditional, however.  See in particular the book _Algebraic Set Theory_ by Joyal and Moerdijk.

What about families and dependent products?  We define a function, considered as a family of sets, to be **$U$-small** if each of its fibers is.  Note that then we can state the Replacement property as "the composite of $U$-small functions is $U$-small."  Now for functions $g:C\to B$ and $f:B\to A$, the fiber of the dependent product $\Pi_f g$ over $a\in A$ is the subset of the function set $\Big((f g)^{-1}(a)\Big)^{f^{-1}(a)}$ consisting of functions $s: f^{-1}(a) \to (f g)^{-1}(a)$ such that $g\circ s = \id$.  Thus, if $f$ and $g$ are $U$-small, so is $f g$ (by the "Replacement" property), and thus so is this fiber.  Hence $U$-small functions are closed under dependent products.

Also of interest is the following, which should be compared to the "descent" axiom in Joyal-Moerdijk.

+-- {: .num_theorem #univ-descent}
###### Theorem
The following are equivalent for a function $f:B\to A$.
1. $f$ is $U$-small.
2. There exist a surjection $p:C\to A$ and a function $g:C\to U$ and two pullback squares
$$\array{B &\overset{q}{\leftarrow} & P & \overset{k}{\to} & {|E|}\\
   ^f\downarrow && \downarrow^j && \downarrow\\
   A &\underset{p}{\twoheadleftarrow} & C & \underset{g}{\to} & U}$$
3. (If the axiom of choice holds) There exists a function $g:A\to U$ and a pullback square
$$\array{B & \overset{}{\to} & {|E|}\\
   ^f\downarrow && \downarrow\\
   A & \underset{g}{\to} & U}$$
=--
+--{: .proof}
###### Proof
Clearly the third condition implies the second.  The second implies the first, since pullbacks have isomorphic fibers.  And in the presence of AC, the second implies the third, since the surjection $p$ has a section.  Thus the difficulty is in showing that the first implies the second.

Suppose that $f:B\to A$ is $U$-small.  This means that for any $a\in A$, there exists a $u\in U$ and a bijection $f^{-1}(a)\cong E_u$.  Let $C$ be the subset of $A\times U\times P(B\times |E|)$ consisting of those triples $(a,u,h)$ such that $h$ is a bijection $f^{-1}(a) \overset{\cong}{\to} E_{u}$, and let $p:C\to A$ and $g:C\to U$ be the projections.  Since $f$ is $U$-small, $p$ is surjective.  Define $P$ to be a pullback of $f$ along $p$, with projections $q:P\to B$ and $j:P\to C$, and define $k:P\to {|E|}$ by (informally speaking) $k(x) = h(q(x))$ where $j(x) = (a,u,h)$.  Since each $h$ is a bijection, $k$ makes $P$ a pullback of ${|E|}$.
=--

[[!redirects universe in SEAR]]
[[!redirects universes in SEAR]]
[[!redirects Grothendieck universe in SEAR]]
[[!redirects Grothendieck universes in SEAR]]


unknot

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

#Contents#
* table of contents
{:toc}

## Idea

The trivial [[knot]].

## Definition

An *unknot* is any [[knot]] [[isotopy|isotopic]] to the flat [[circle]] embedded in the $(x,y)$-plane of $\mathbb{R}^3$.

## References

See also

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

Computation of the perturbative [[Wilson loop observable]] ([[universal Vassiliev invariant]]) of the [[unknot]] ("[[Wheels theorem]]"):

* [[Dror Bar-Natan]], Thang T Q Le, [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))

following

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of  the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))

On [[machine learning]] the detection of unknots:

* [[Sergei Gukov]], [[James Halverson]], [[Fabian Ruehle]], [[Piotr Sułkowski]], *Learning to Unknot*, Mach. Learn.: Sci. Technol. **2** (2022) 025035 [[arXiv:2010.16263](https://arxiv.org/abs/2010.16263), [doi:10.1088/2632-2153/abe91f](https://iopscience.iop.org/article/10.1088/2632-2153/abe91f)]

Exposition in:

* [[Sergei Gukov]], *Machine learning and hard problems in topology*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#GukovMar2023)]

[[!redirects unknots]]


unknotting number

## Idea##
If you had a piece of string possibly tangled up, and could, at a crossing, pull one part of the string through the other, then, intuitively, repeating this enough times, the string would become unknotted. At the mathematical level, there is a corresponding notion of a _crossing change_ on a diagram

+--{: .un_defn}
######Definition######
A crossing change in a diagram exchanges an overpass and underpass at a crossing, as below:
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 (The central arrow should be a left-right arrow, but the arrowheads do not come out!)

Crossing changes will usually alter the isotopy type of the diagram.

+--{: .un_lemma}
######Lemma
Let $D$ be a diagram with $c$ crossings, then changing at most $c/2$ crossings of $D$ produces a diagram of the unknot.
=--

+--{: .un_defn}
######Definition######
The unknotting number, $u(K)$, is the smallest number of crossing changes required to obtain the unknot from some diagram of the knot.
=--
Of course, we know that $u(K)\leq c(K)/2$, but the natural difficulty of calculating $u(K)$ is made worse by the following result of Beiler (1984).

_The unknotting number of a knot does not necessarily occur in a minimal diagram._

Beiler gave an example of a minimal diagram for a 10 crossing knot, which cannot be unknotted with fewer than 3 crossing changes, yet for which there is a 14 crossing diagram, which is  isotopic to it, yet can be unknotted with just 2 crossing changes.


unnatural isomorphism

# Unnatural isomorphisms #
* table of contents
{:toc}

## Idea

A [[natural isomorphism]] is the "correct" notion of [[isomorphism]] between [[functors]].  Informally, one sometimes speaks of two [[objects]] being "naturally isomorphic" if they are the images of two functors that are naturally isomorphic.  However, sometimes one encounters functors that are "pointwise isomorphic" but *not* naturally.  There is not much to say *mathematically* about such "unnatural isomorphisms", but some discussion of them can help the intuitive understanding of natural isomorphisms and their importance.

On this page we will adhere to the convention that an "unnatural isomorphism" means an isomorphism that is not *necessarily* natural (but might be), similarly to how a "noncommutative [[ring]]" might happen to be commutative.  The word "unnatural" is not to be taken pejoratively or as suggesting that unnatural isomorphisms are rare; in [[cardinality]] terms there are many more unnatural isomorphisms than natural ones, although in mathematical practice surprisingly many isomorphisms are indeed natural.  Other possible terms with the same meaning are "pointwise isomorphic" or "objectwise isomorphic".

The term "natural" is used for traditional reasons, cf. [EilenbergMac Lane45](#EilenbergMacLane45), and is an example of the common phenomenon that concepts branded _natural_, _normal_ or _regular_ tend to be rather non-generic or non-random.  In fact, [Eilenberg and MacLane](#EilenbergMacLane45) comment (cf. p. 233, first paragraph in loc. cit) on the term, associating it with an intuition of _simultaneity_, rather than genericity.  This intuition is in tune with conceiving of naturality as generalized _commutatitivity_.

## Definition

### Between functors

Let $F,G:C\to D$ be [[functors]].  An *unnatural isomorphism* between $F$ and $G$ consists of, for each [[object]] $c\in C$, an isomorphism $F(c) \cong G(c)$, satisfying no further conditions.

In other words, it is an [[unnatural transformation]] whose components are isomorphisms, or equivalently that is an isomorphism in the category of unnatural transformations.

Note that if each component of a [[natural transformation]] is an isomorphism, then it is necessarily a natural isomorphism.  (Depending on the definition of "natural isomorphism", this could be a tautology; but the non-tautologous statement is that if each component of a natural transformation is an isomorphism then it is an isomorphism in the functor category.  This statement, though non-tautologous, is easy to prove for 1-categories, but more difficult for higher categories.)

More generally, an unnatural isomorphism (or transformation) might be natural with respect to only *some* morphisms in the domain, such as for instance the isomorphisms.  Some people have proposed calling transformations that are natural only with respect to isomorphisms "canonical".

### Between objects

If one says that two *objects* $A$ and $B$ are "unnaturally isomorphic", it might mean that they are the images of two functors that are unnaturally isomorphic, but it might also mean only that there is no obvious way even to make them the images of two functors. For instance, this seems to be the meaning in Section 1.1 of [Strickland](#Strickland1998).

Somewhat relatedly, the [[complex numbers]] $\mathbb{C}$ and the [[p-adic complex numbers]] $\mathbf{C}_p$ are "unnaturally isomorphic". More precisely, there is no way of making constructions such as "algebraic closure of a field" or "injective hull of an $R$-module" functorial, if the desired embeddings into these constructions are to be natural, and this precludes speaking of any two such constructions as "naturally isomorphic". We touch on this again below, but see [here](#AHRT).

## Reasons for unnaturality

Intuitively, there are several reasons that an isomorphism might fail to be natural.

* Two functors of opposite variances (one covariant and one contravariant) *cannot* be naturally isomorphic in the usual sense.  However, they can be [[dinatural transformation|dinaturally isomorphic]], so this is not really a "reason for a failure of naturality".

* In defining an isomorphism, it may be necessary to make certain arbitrary choices.  If making these choices differently would yield a different isomorphism, then usually the isomorphism will not be natural.

## Making the unnatural natural

Unnatural isomorphisms are not very well-behaved mathematically, so if we have an unnatural isomorphism we may want to "make it natural".  There are several ways to do this:

1. Use a different definition that includes more structure, such as [[derivators]] instead of [[triangulated categories]].  See for instance this [Wikipedia discussion](https://en.wikipedia.org/wiki/Triangulated_category#Are_there_better_axioms.3F) on axioms for triangulated categories.

2. Relatedly, pass to a [[higher categories|higher-categorical]] context in which a non-commuting square can commute up to a higher transformation; see [[pseudonatural transformation]] and [[lax natural transformation]].

3. Cut down to a smaller class of morphisms in the domain category for which the naturality square does commute.

## Natural, canonical, and functorial

The use of adjectives like _canonical_, _invariant_, _functorial_, _natural _ is sometimes satirized. For example, as quoted in [MO2010](#MathOverflow2010), André Weil writes:

> I can assure you, at any rate, that (...) my results are invariant, probably canonical, perhaps even functorial. (*Oeuvres*, vol. 2, page 558)

which may be subtly mocking.

These three terms can be distinguished in the following way:

* The definition of a single function or operation is "canonical" if there are "no arbitrary choices" involved in it, or more precisely if the end result is independent of any arbitrary choices that we had to make in defining it.  In category theory, we generally consider something "canonical" if this independence is "up to unique isomorphism" in an appropriate sense, such as the definition of [[generalized the|the]] limit functor on a category with limits.
* The definition of a function taking the [[objects]] of a category into the objects of another is _functorial_ if we have also supplied a function on morphisms making it into a [[functor]], or more loosely if such a function exists (although in practice one always has a particular such function in mind).
* The definition of a function taking the objects of one category into morphisms of another is _natural_ if it forms a natural isomorphism between two functors (generally assumed to be given).

In particular, the adjectives "functorial" and "natural" in general apply to different  classes of things.

See [[non-canonical isomorphism]] for examples of *natural* isomorphisms that are not *canonical*.

## Examples

1. Let $U: Vect\to Aff$ be the functor taking the underlying [[affine space]] of a [[vector space]], and $D:Aff\to Vect$ the functor constructing the vector space of displacements of an affine space.  Then there is a *natural* isomorphism $D \circ U \cong Id_{Vect}$, but only an *unnatural* isomorphism "$U \circ D \cong Id_{Aff}$".  In particular, these functors do not form an [[equivalence of categories]]. A similar phenomenon occurs with [[groups]] and [[heaps]], and with [[torsors]] of groups.

1. As a particular example of the last phenomenon: there is a bijective correspondence between [[linear order|linear orderings]] of a set $S$ and [[permutations]] of $S$. Indeed, the set $Lin(S)$ is a [[torsor]] over $Perm(S)$: there is a group action $\alpha: Perm(S) \times Lin(S) \to Lin(S)$ such that for any *choice* of linear ordering $L$, the map $\alpha(-, L): Perm(S) \to Lin(S)$ is a bijection. But, there is no canonical choice of ordering $L$.

1. [[species|Species]] are functors for which unnatural isomorphisms are sometimes discussed. For example, the species of linear orders is unnaturally isomorphic to the species of permutations; for a related example, see the commentary under this MathOverflow answer [MO2014](#GesselTrimble2014). [Yorgey2014](#Yorgey2014) explicitly uses the term: his definition 3.3.4 introduces an "equipotence between species (...) as an "unnatural" isomorphism between [the two species in question]".

1. The term _unnatural isomorphism_ is often used in [[homological algebra]], in particular in discussions of splittings of exact sequences. An example is [Hilton-Stammbach, Theorem 4.3](#HiltonStammbach71), which explicitly uses the term to describe an isomorphism between Hom- and Ext-functors.  More generally, _unnatural transformations_ also often occur in homological algebra.

1. It is often said that while there is an isomorphism between the [[field]] of [[complex numbers]] $\mathbb{C}$ and the completion $\mathbf{C}_p$ (with respect to the canonical absolute value) of the algebraic closure of the [[p-adic numbers]] $\mathbb{Q}_p$, this isomorphism is not natural. This statement, while somewhat loose since there are no functors specified, can be made formally precise along the lines of [Adámek, Herrlich, Rosický, and Tholen](#AHR). The abstract reads: "In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are injective. In particular, assigning to a field its algebraic closure, to a poset or Boolean algebra its MacNeille completion, and to an $R$-module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation."

1. If $C$ and $D$ have only one object (or slightly more generally, one [[isomorphism class]] of objects), then any two functors $F,G:C\to D$ are unnaturally isomorphic.  For example, for any [[ring]] $R$, any two endofunctors of the category of rank-one free $R$-[[modules]] are unnaturally isomorphic.

1. See [MathOverflow2013](#MathOverflow2013).

## Related pages

* [[unnatural transformation]]
* [[funny tensor product]]

## References

* {#AHRT} [[Adamek|Ji?í Adámek]], [[Horst Herrlich]], [[Rosicky|Ji?í Rosický]], and [[Walter Tholen]], *Injective Hulls are not Natural*, algebra universalis
Volume 48, Issue 4 (December 2002), 379–388. ([Citeseer link](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.8.194))

* {#EilenbergMacLane45} [[Samuel Eilenberg|S. Eilenberg]], [[Saunders Mac Lane|S. Mac Lane]], _General Theory of Natural Equivalences_,  Transactions of the American Mathematical Society Vol. 58, No. 2 (Sep., 1945), pp. 231-294 ([JSTOR](http://www.jstor.org/stable/1990284))

* {#HiltonStammbach71} [[Peter Hilton|P. Hilton]],  U. Stammbach, _A course in homological algebra_, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4.

* {#MathOverflow2010} Konrad Waldorf (https://mathoverflow.net/users/3473/konrad-waldorf), What is the definition of "canonical"?, URL (version: 2013-11-13): [https://mathoverflow.net/q/19644](https://mathoverflow.net/q/19644)

* {#MathOverflow2013} Bugs Bunny (https://mathoverflow.net/users/5301/bugs-bunny), Example of an unnatural isomorphism, URL (version: 2013-08-14): [https://mathoverflow.net/q/139388](https://mathoverflow.net/q/139388)

* {#MathOverflow201308} Almeo Maus (https://mathoverflow.net/users/29853/almeo-maus), Unicity of Yoneda isomorphism, URL (version: 2013-08-24): [https://mathoverflow.net/q/140303](https://mathoverflow.net/q/140303)

* {#GesselTrimble2014} [[Todd Trimble]] (https://mathoverflow.net/users/2926/todd-trimble), What are some examples of interesting uses of the theory of combinatorial species?, URL (version: 2011-03-23): [https://mathoverflow.net/q/59259](https://mathoverflow.net/q/59259)

* {#Rezk16} [[Charles Rezk]], _Stuff about quasicategories_, Lecture Notes for course at University of Illinois at Urbana-Champaign, 2016, version May 2017,([pdf](http://math.uiuc.edu/~rezk/595-fal16/quasicats.pdf))

* {#Strickland1998} [[Neil Strickland|N. P. Strickland]], Morava E-Theory, Topology, Vol. 37, No. 5 (1998), 757--779

* {#Yorgey2014} [[Brent Yorgey|B. A. Yorgey]], Combinatorial Species and Labelled Structures,  Dissertation, University of Pennsylvania, 2014

[[!redirects unnatural isomorphism]]
[[!redirects unnatural isomorphisms]]
[[!redirects unnaturally isomorphic]]
[[!redirects unnaturally isomorphic functors]]
[[!redirects objectwise isomorphism]]
[[!redirects objectwise isomorphisms]]
[[!redirects pointwise isomorphism]]
[[!redirects pointwise isomorphisms]]


unnatural transformation

# Unnatural transformations

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{: toc}

## Definition

Given [[categories]] $C,D$ and [[functors]] $F,G\colon C \to D$, an __unnatural transformation__ from $F$ to $G$ consists of, for each object $x$ of $C$, a morphism from $F(x)$ to $G(x)$ in $D$.

Thus a [[natural transformation]] is an unnatural transformation that satisfies an [[extra property]], namely naturality.

## Related pages

* [[unnatural isomorphism]]
* [[funny tensor product]]

[[!redirects unnatural transformations]]


unnaturally isomorphic > history

< [[unnaturally isomorphic]]


unordered pair

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# Unordered pairs
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## Idea

The _unordered pair_ of $x$ and $y$, denoted $\{x,y\}$, has the property that $\{x_1,y_1\} = \{x_2,y_2\}$ if and only if $x_1 = x_2$ and $y_1 = y_2$ or $x_1 = y_2$ and $y_1 = x_2$.  In other words, the unordered pair $\{x,y\}$ is the same as the _[[ordered pair]]_ $(x,y)$, except that presentation order does not matter.

## Definition

In [[structural set theory]] and [[material set theory]] with [[ordered pairing structure]], given a [[set]] $A$, the set of unordered pairs of elements of $A$ is the [[quotient set]] $A \times A / \sim$, where $\sim$ is a [[binary relation]] on $A \times A$ where $(x, y) \sim (y, x)$ for all elements $x \in A$ and $y \in A$. An **unordered pair** is just an element of said quotient set.

In [[material set theory]] without [[ordered pairing structure]], an unordered pair is usually just defined as a [[pair set]].

### In type theory

In [[dependent type theory]], let $A$ be an [[h-set]]. Then the type of unordered pairs of elements in $A$ is a [[higher inductive type]] $\mathrm{UnorderedPairs}(A)$ generated by the constructors

* $\{-,-\}:A \times A \to \mathrm{UnorderedPairs}(A)$
* $a:A, b:A \vdash \mathrm{eq}_{\{-,-\}}(a, b):\{a,b\} =_{\mathrm{UnorderedPairs}(A)} \{b,a\}$
* $a:A, c(a):\{a,a\} =_{\mathrm{UnorderedPairs}(A)} \{a,a\} \vdash K(a):\mathrm{eq}_{\{-,-\}}(a, a) =_{\{a,a\} =_{\mathrm{UnorderedPairs}(A)} \{a,a\}} c(a)$

## Relation to ordered pairs

The unordered pair $\{x,y\}$ should not be confused with the [[ordered pair]] $(x,y)$. In particular, $\{x,y\} = \{y,x\}$, while $(x,y) \neq (y,x)$ (if $x \neq y$).

In [[material set theory]], $\{x,y\}$ has a direct definition, but $(x,y)$ must be coded in a complicated way (traditionally as $\big\{\{x\}, \{x,y\}\big\}$). In [[structural set theory]], $(x, y)$ has a direct definition, and $\{x,y\}$ must be coded in a complicated way (traditionally as an element of the quotient set $A \times A / (x, y) \sim (y, x)$).

However, the two are somewhat related:

*  As just seen, the usual encoding of an ordered pair of [[pure sets]] as a pure set (due to Kuratowski) involves unordered pairs.  Conversely, the set $\big(\!\big({A \atop 2}\big)\!\big)$ of unordered pairs of elements of $A$ may be constructed as a [[quotient set]] of the set $A^2$ of ordered pairs of elements of $A$ (by the [[equivalence relation]] generated by declaring that $(x,y) \sim (y,x)$).
*  Just as unordered pairs (through the [[axiom of pairing]]) are needed to get anywhere in material set theory, so some axiom related to ordered pairs is needed to get anywhere in [[structural set theory]].  (In [[ETCS]], this is the axiom of the ([[Cartesian product|Cartesian]]) [[product]]; in [[SEAR]], this is the axiom of tabulations.)

Despite the name of the [[axiom of pairing]], the term '[[pairing]]' in the $n$Lab usually refers to *ordered* pairs.

## See also

* [[pair set]]
* [[axiom of pairing]]

[[!redirects unordered pair]]
[[!redirects unordered pairs]]
[[!redirects unordered pairing]]
[[!redirects unordered pairings]]


unramifiable polynomial

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\tableofcontents

## Definition

Given a notion of [[real number]] $\mathbb{R}$, let $\sigma_j:\mathbb{R}^n \to \mathbb{R}$ denote the $j$-th [[elementary symmetric polynomial|elementary symmetric]] [[polynomial function]] in $\mathbb{R}$,
and the polynomials functions
$e_j:\mathbb{R}^n \to \mathbb{R}$ as
$$e_j(x_1, x_2 \ldots x_n) \coloneqq \prod_{k \neq j} (x_k - x_j)$$
Let us define the polynomial function $f:\mathbb{R}^{n+1} \to \mathbb{R}$ as
$$f(y, x_1, x_2 \ldots x_n) \coloneqq \prod_{j = 0}^n (y + x_j) = \sum_{j = 0}^n \sigma_{n - j}(x_1, x_2, \ldots x_n) y^j$$
Now, $\sigma_j(e_1(x_1, x_2 \ldots x_n), e_2(x_1, x_2 \ldots x_n), \ldots e_n(x_1, x_2 \ldots x_n))$ is by definition also a symmetric polynomial function. Thus, there exists a polynomial function $d_j:\mathbb{R}^n \to \mathbb{R}$ such that
$$d_j(\sigma_1(x_1, x_2, \ldots x_n), \sigma_2(x_1, x_2, \ldots x_n), \ldots \sigma_n(x_1, x_2, \ldots x_n)) = \sigma_j(e_1(x_1, x_2 \ldots x_n), e_2(x_1, x_2 \ldots x_n), \ldots e_n(x_1, x_2 \ldots x_n))$$

The function $f$ is **unramifiable** if there exists a $j$ such that $d_j(\sigma_1(x_1, x_2, \ldots x_n), \sigma_2(x_1, x_2, \ldots x_n), \ldots \sigma_n(x_1, x_2, \ldots x_n))$ is invertible in $\mathbb{R}$.

## Related concepts

* [[simple root]]

* [[elementary symmetric polynomial]]

* [[fundamental theorem of algebra]]

## References

* {#Ruitenberg91} [[Wim Ruitenberg]], *Constructing Roots of Polynomials over the Complex Numbers*, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract, **84** Centre for Mathematics and Computer Science, Amsterdam (1991) 107–128 [[pdf](https://www.mscsnet.mu.edu/~wim/publica/roots_new.pdf), [[Ruitenberg-Roots.pdf:file]]]

[[!redirects unramifiable polynomial]]
[[!redirects unramifiable polynomials]]


unramified morphism

#Contents#
* table of contents
{:toc}

## Idea

### General

The notion of _unramified morphism_ of algebraic [[schemes]] is a geometric generalization of the notion of an unramified [[field extension]]. The notion of ramification there is in turn motivated by branching phenomena in [[number fields]], which involve branchings similar to those occurring in [[Riemann surfaces]] over the [[complex numbers]].

So "unramified" means "not branching". In the context of [[differential geometry]] unramified maps correspond to [[immersions]].

A weaker (infinitesimal) version is the notion of _[[formally unramified morphism]]_.

### Historical remarks

The basic picture is one from Riemann surfaces: the power $z\mapsto z^n$ has a branching point around $z=0$. Dedekind and Weber in 19th century considered more generally algebraic curves over more general fields, and proposed a generalization of a Riemann surface picture by considering valuations and in this analysis the phenomenon of branching occurred again.

## Definition
 {#Definition}

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###### Definition

A [[morphism]] $f \colon X\longrightarrow Y$ of [[schemes]] is __unramified__ if it is [[locally of finite presentation]], and if for every point $x\in X$ we have $\mathfrak{m}_{f(x)}\mathcal{O}_{X,x}=\mathfrak{m}_x$ and the induced morphism of [[residue fields]]
$$
\mathcal{O}_{Y,f(x)}/\mathfrak{m}_{f(x)}\to\mathcal{O}_{X,x}/\mathfrak{m}_{x}
$$
is a [[finite number|finite]] and [[separable field extension|separable]] [[extension of fields]].

=--

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

A morphism $f:X\to Y$ of schemes is __[[formally unramified]]__ if for every [[infinitesimal object|infinitesimal thickening]]
$T\to T'$ of schemes over $Y$, the canonical morphism of sheaves of sets over $T$
$$
U\mapsto (Sch_{/Y})(U,X)
$$
to
$$
U\mapsto (Sch_{/Y})(U',X)
$$
is injective, where $U' = U$ as the open set, but as a scheme it is the open subscheme of the thickening $T'$.

=--

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

In this condition, it is sufficient to consider the
thickenings of  affine $Y$-schemes. Thus $f$ is formally unramified if for each morphism of $Y$-schemes $T\to X$ which has an extension to a morphism $T'\to X$ the extension is unique.

=--

## Properties and usage

The notion of unramified morphism is stable under base change and composition.

An [[etale morphism]] of schemes is, by one of the equivalent definitions, a morphism that is [[flat morphism|flat]] and unramified.

Every [[open immersion of schemes]] is formally etale hence a fortiori formally unramified. A morphism which is locally of finite type is unramified iff the diagonal morphism $X\to X\times_Y X$ is an open immersion.

Characterization: a morphism is formally unramified iff the module $\Omega_{Y/X}$ of relative [[Kahler differential]]s is zero.

## Related concepts

* [[decidable object]]

* [[ramification of ideals]]

* [[formally unramified morphism]]

## Literature

* Ogus, _Unramified morphisms_, [pdf](http://math.berkeley.edu/~ogus/Math%20_256A--08/unramif.pdf), notes from a course on algebraic geometry at Berkeley

* [[EGA]] IV

* [[James Milne]], p. 20 of _[[Lectures on Étale Cohomology]]_

[[!redirects unramified morphisms]]
[[!redirects unramified]]

[[!redirects unramified morphism of schemes]]


unsorted set theory

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# Unsorted set theory
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## Overview
 {#Overview}

There are a number of ways to present a [[set theory]]; one of the most basic decisions when it comes to presenting a set theory is whether the probable [[sets]] and probable [[elements]] should be regarded as the same kind of objects, as in [[unsorted set theory]], or as fundamentally different objects, as in [[two-sorted set theory]] or [[three-sorted set theory]]. In the former, there is a global membership relation $\in$ which is defined on the entire [[domain of discourse]]. In the latter, there are two domains of discourse, one representing the probable sets and the other representing the probable elements or atoms, and for the membership relation $a \in b$, $a$ is required to be a probable element and $b$ is required to be a probable set.

Furthermore, in an unsorted set theory, there is also a global equality predicate, whereby it is meaningful to ask, given any two objects, whether those two objects are equal. Additionally, a set's identity here is determined by its elements, in other words the [[axiom of weak extensionality]] holds for the global membership relation with respect to the equality relation.

Unsorted set theories come in both [[material set theory]] and [[structural set theory]] flavors. For example, [[ZFC]], [[ZFA]], and [[New Foundations]] are unsorted [[material set theories]], while [[fully formal ETCS]] is an unsorted [[structural set theory]].

## Notions of sets and elements

First of all, in order to give a definition pertaining to all unsorted set theories, we need to know: what is a set theory? Probably no theory can *intrinsically* be called a set theory; it is only given that distinction by our intent to regard some of its objects of study as “sets” and others as their “elements.” Thus we make the following definition.

\begin{definition}
A notion of set and element in a unsorted first-order theory consists of:

* A binary predicate $(-)=(-)$ called *equality*

* A unary predicate $\mathrm{set}(-)$ called *being a set*

* A unary predicate $\mathrm{element}(-)$ called *being an element*

* A binary predicate $(-)\in(-)$ called *membership*

such that

* if $a \in b$, then $a$ is an element and $b$ is a set

$$\forall a.\forall b.(a \in b) \to \mathrm{element}(a) \wedge \mathrm{set}(b)$$

* weak extensionality for sets and elements: for all $a$ and for all $b$, if a and b are sets, then $a = b$ if and only if, for all $c$, if $c$ is an element, then $c \in a$ if and only if $c \in b$

$$\forall a.\forall b.(\mathrm{set}(a) \wedge \mathrm{set}(b)) \implies ((a = b) \iff \forall c.(\mathrm{element}(c) \implies ((c \in a) \iff (c \in b))))$$
\end{definition}

## Examples

### Pure set theories

In [[pure set]] theories, such as [[ZFC]] and [[New Foundations]], $\mathrm{set}(a)$ and $\mathrm{element}(a)$ is $\top$ for all objects $a$, $\in$ is the primitive membership relation and $=$ is defined through the axiom of extensionality.

### Set theories with urelements

In set theories with [[urelements]], such as [[ZFA]], only $\mathrm{element}(a)$ is $\top$ for all objects $a$, $\in$ is the primitive membership relation, $\mathrm{set}$ is the primitive sethood predicate, and $=$ is defined through the axiom of extensionality.

### Categorical set theories

In an unsorted categorical set theory, such as [[Todd Trimble]]'s [[fully formal ETCS]], $=$ is a primitive, and there is a specified symbol $1$ representing the [[identity morphism]] of the [[terminal object]] and a specified symbol $0$ representing the [[identity morphism]] of the [[initial object]], as well as $s$ and $t$ representing the [[identity morphisms]] of the [[source]] and [[target]]. The elements are functions with domain $1$, $\mathrm{element}(a) \coloneqq (\mathrm{dom}(a) = 1)$ There are many possible notions of sets, three of which include:

* sets as [[identity functions]] $\mathrm{set}(a) \coloneqq \mathrm{dom}(a) = a$ or $\mathrm{set}(a) \coloneqq \mathrm{codom}(a) = a$

* sets as [[functions]] into the [[singleton]] $\mathrm{set}(a) \coloneqq (\mathrm{codom}(a) = 1)$

* sets as functions out of the [[empty set]] ($\mathrm{set}(a) \coloneqq (\mathrm{dom}(a) = 0)$).

Thus, there are many different definitions of the membership relation $\in$, which for the above notions of set are:

* for sets as identity functions, the membership relation is defined as
$$a \in b \coloneqq \mathrm{element}(a) \wedge \mathrm{set}(b) \wedge \mathrm{codom}(a) = b$$

* for sets as functions into the singleton, the membership relation is defined as
$$a \in b \coloneqq \mathrm{element}(a) \wedge \mathrm{set}(b) \wedge \mathrm{codom}(a) = \mathrm{dom}(b)$$

* for sets as functions out of the empty set, the membership relation is defined as
$$a \in b \coloneqq \mathrm{element}(a) \wedge \mathrm{set}(b) \wedge \mathrm{codom}(a) = \mathrm{codom}(b)$$

For the first two membership relations $\in$, the singleton $1$ is a [[Quine atom]], and any set in bijection with $1$ is a [[Quine atom]]. For the third membership relation, there are no [[Quine atoms]]. While all the membership relations defined above are [[weakly extensional relations]], all three membership relations defined above could be proven not to be [[strongly extensional relations]], which means that the [[axiom of foundation]] fails in any of the three unsorted presentations of categorical set theory.

### Allegorical set theories

## See also

* [[unsorted set theory]]
* [[two-sorted set theory]]
* [[dependently sorted set theory]]

* [[structural set theory]]
* [[material set theory]]
* [[structurally presented set theory]]

[[!redirects unsorted set theory]]
[[!redirects unsorted set theories]]

[[!redirects one-sorted set theory]]
[[!redirects one-sorted set theories]]

[[!redirects single-sorted set theory]]
[[!redirects single-sorted set theories]]


unstable K-theory

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#Contents#
* table of contents
{:toc}

## Idea

Given that the (degree-shifted, [[reduced cohomology|reduced]]) [[topological K-theory]] group of a [[topological space]] $X$ can be computed as a [[colimit]] over sets of [[homotopy classes]] of maps (cf. at *[[stable unitary group]]*)

$$
  \tilde{K}^1(X)
  \;=\;
  [X,U]
  \;=\;
  \underset{\underset{n}{\longrightarrow}}{lim}
  \,
  \big[X,U(n)\big]
  \,
$$

one may think of this as a [[stabilization]] of generalized [[nonabelian cohomology]] theories (in the sense of [FSS23, §2](nonabelian+cohomology#FSS20)) $U_n(X)$ whose classifying spaces are the [[unitary groups]] $U(n)$. For this reason, we may call the groups $[X,U(n)] \,\coloneqq\, \pi_0 Map\big(X,U(n)\big)$ the **unstable** $\tilde{K}^1$-theory groups of $X$, at stage $n$.

## Definition

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

Let $X$ be a [[topological space]] ([[CW-complex]]), and $U(n)$ the [[unitary group]] in dimension $n$. The $n$-*unstable*  $\tilde{K}^1$*-theory group* $U_n(X)$ of $X$ is defined as the [[homotopy classes]] of maps, hence the [[connected components]] of the [[mapping space|space of maps]] from $X$ to (the [[underlying]] topological space of) $U(n)$:

$$
  U_n(X)
  \;\coloneqq\;
  [X,U(n)]
  \,\coloneqq\,
  \pi_0 Map\big(X,\, U(n)\big)
  \,.
$$

=--

From this definition one can see that, if $X$ is a finite-[[dimension|dimensional]] [[CW complex]] then $U_n(X) = \tilde{K}^1 (X)$ for sufficiently large $n$.

## Properties

The following properties are proven in [Hamanaka and Kono 2003](#HK03):

+-- {: .num_defn #LimitToKtheory}
###### Proposition
(Theorem 1.1 of [op.cit.](#HK03))
\linebreak
Let $\text{dim}(X)\leq 2n$. Then there exists an [[exact sequence]] of the form:

$$
  \tilde{K}^0(X)
    \xrightarrow{\Theta}
  H^{2n}(X;\mathbb{Z})
   \to
  U_n(X)
   \to
  \tilde{K}^1 (X)
    \to
  1
  \,,
$$

or, put differently, defining $N_n (X) \coloneqq coker(\Theta)$ (see Section 3 of [op.cit.](#HK03) for the definition of $\Theta$):

$$
  1 \to N_n (X) \to U_n(X) \to \tilde{K}^1 (X) \to 1
  \,.
$$

=--

In particular, Prop. \ref{LimitToKtheory} shows that in general $U_n(X)$ is neither abelian nor does it [[injection|inject]] into $\tilde{K}^1 (X)$.

+-- {: .num_defn #LimitToKtheory}
###### Proposition
(Theorem 1.2 of [op.cit.](#HK03))

For $dim(X)\leq 2n$, the [[cokernel]] group $N_n(X)$ is a [[finite group|finite]] [[abelian group]] where the order of any element divides $n!$.

=--

## Related concepts

* [[topological K-theory]]

* [[unstable Cohomotopy]]

## References

* {#HK03} [[Hiroaki Hamanaka]], [[Akira Kono]]: *On $[X, U(n)]$ when $\text{dim}(X)$ is $2n$*, Journal of Mathematics of Kyoto University **43** 2 (2003) 333-348 [[doi:10.1215/kjm/1250283730](https://doi.org/10.1215/kjm/1250283730)[

* [[Hiroaki Hamanaka]], [[Akira Kono]]: *An application of unstable K-theory*, Journal of Mathematics of Kyoto University **44** 2 (2004) 451-456 [[doi:10.1215/kjm/1250283560](https://doi.org/10.1215/kjm/1250283560)]

* [[Hiroaki Hamanaka]]: *Adams $ e $-invariant, Toda bracket and $[X, U (n)]$*, Journal of Mathematics of Kyoto University **43** 4 (2003) 815-827 [[doi:10.1215/kjm/1250281737](https://doi.org/10.1215/kjm/1250281737)]

[[!redirects unstable topological K-theory]]


up quark

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

#Contents#
* table of contents
{:toc}

## Idea

In the [[standard model of particle physics]] the _up quark_ is the up-type [[quark]] in the first [[generation of fermions]].

## Related concepts

[[!include flavors of fundamental particles -- table]]

## References

See also

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

[[!redirects up quarks]]

[[!redirects up-quark]]
[[!redirects up-quarks]]


up set

In a [[partial order|poset]] or even [[preorder|proset]], the __up set__ of an element $x$ is the set
$$ x{\uparrow} = \{ y \;|\; x \leq y \} .$$

In a [[quasiorder]], the __strict up set__ of $x$ is the set
$$ x\dot{\uparrow} = \{ y \;|\; x \lt y \} .$$

If you think of a poset $P$ as a [[category]], then the up set of $x$ is the [[coslice category]] $x / P$.

An up set in the [[opposite category|opposite]] $P^{op}$ of $P$ is a [[down set]] in $P$.

Note: The term 'up set' is also often used for an [[upper set]], a more general concept.  In the terminology above, the up set of $x$ is the upper set [[Moore closure|generated by]] $x$.

[[!redirects upset]]
[[!redirects up-set]]
[[!redirects up set]]
[[!redirects upsets]]
[[!redirects up-sets]]
[[!redirects up sets]]


Uploaded Files

A list of all files uploaded to the nLab can be found [here](http://ncatlab.org/nlab/file_list).

category: meta


upper bound

+-- {: .rightHandSide}
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### Context
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
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# Contents
* table of contents
{: toc}

## Definitions

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

For $(S, \leq)$ a [[preorder]] and $A \hookrightarrow S$ a [[subset]],
an **upper bound** of $A$ in $S$ is an [[element]] $x$ of $S$ such that $y \leq x$ whenever $y \in A$.
=--

Dually (in the [[opposite poset]]):

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

For $(S, \leq)$ a [[preorder]] and $A \hookrightarrow S$ a [[subset]],
a **lower bound** of $A$ in $S$ is an [[element]] $x$ of $S$ such that $x \leq y$ whenever $y \in A$.
=--

## Related concepts

* [[Zorn's lemma]]

[[!redirects upper bound]]
[[!redirects upper bounds]]

[[!redirects lower bound]]
[[!redirects lower bounds]]


upper central series

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

#Contents#
* table of contents
{:toc}

## Idea

The **upper central series** of a [[group]], $G$, is an ascending [[sequence]] of [[subgroups]]

$$
  Z^0(G) \leq Z^1(G) \leq Z^2(G) \leq \ldots,
$$

where the zeroth member is the [[trivial group]], the first member is the [[center]], the second member is the second center, and so on.

$$
  Z^0(G)
  \,=\, 1,
  \;\;
  Z^1(G) = Z(G),
  \;\;
  Z^2(G)/Z^1(G) = Z(G/Z^1(G)),
  \;\;
  Z^3(G)/Z^2(G) = Z(G/Z^2(G)),
  \;\;
  \ldots
$$

The indexing may be continued to the [[ordinals]], where the member indexed by a limiting ordinal is the union of all previous members.

For a [[nilpotent group]], the upper central series reaches the whole group in finitely many steps, and is the fastest ascending [[central series]]. It has the same length then as the [[lower central series]], although they need not coincide.

## Related concepts

* [[lower central series]]

## References

See also:

* Wikipedia, *[Central series](https://en.wikipedia.org/wiki/Central_series)*

* [Groupprops](http://groupprops.subwiki.org/wiki/Upper_central_series)


upper half-plane

#Contents#
* table of contents
{:toc}

## Idea

The part of the [[complex plane]] of [[positive number|positive]] [[imaginary part]]  is often called the _[[upper half plane]]_, sometimes denoted
$\mathfrak{H}$.

Accodingly the _lower half plane_ is the subset of complex numbers with negative imaginary part.

## Related concepts

* [[contour integral]]

* [[Möbius group]]

* [[modular form]], [[automorphic form]]

* [[Siegel upper half-space]]

* [[Drinfeld upper half-plane]]

* [[moduli space of elliptic curves]], [[moduli space of Riemann surfaces]]

## References

* Wikipedia, _[Upper half plane](http://en.wikipedia.org/wiki/Upper_half-plane)_

[[!redirects upper half-planes]]

[[!redirects upper half plane]]
[[!redirects upper half planes]]

[[!redirects lower half plane]]
[[!redirects lower half planes]]

[[!redirects lower half-plane]]
[[!redirects lower half-planes]]


upper interval topology

The __upper interval topology__ on a [[proset]] $X$ is given by taking the [[down sets]] in $X$ as generating the [[closed subsets]].


upper set

+-- {: .rightHandSide}
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#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

In a [[poset]] or even [[proset]], an __upper set__ $U$ is a [[subset]] that is 'upwards closed'; that is,

*  whenever $x \leq y$ and $x \in U$, then $y \in U$.

Upper sets form a [[Moore collection]] and so one can speak of the upper set _generated_ by an arbitrary subset $A$:
$$ A{\uparrow} = \{ y \;|\; \exists x,\; x \in A \;\wedge\; x \leq y \} .$$

Sometimes an upper set is called a 'filter', but that term can also mean [[filter|something more specific]] (and always does in a [[lattice]]).

An upper set is also sometimes called an 'up set', but that term can also mean something more specific: the [[up set]] of $x$ is the upper set generated by $x$.

An [[inhabited subset|inhabited]], [[open subset|open]] upper set of [[rational numbers]] (or equivalently of [[real numbers]]) determines precisely an [[upper real number]].

The upper sets form a [[topological structure]] on (the [[underlying set]] of) the proset, called the [[Alexandroff topology]].

## See also

* [[posite]]
* [[lower set]]
* [[ideal]]
* [[filter]]
* [[copresheaf]]
* [[presheaf]]

[[!redirects upper set]]
[[!redirects upper sets]]
[[!redirects upper subset]]
[[!redirects upper subsets]]

[[!redirects upward closed subset]]
[[!redirects upward closed subsets]]

[[!redirects (0,1)-copresheaf]]
[[!redirects (0,1)-copresheaves]]


upsilon-meson

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

#Contents#
* table of contents
{:toc}

## Idea

The _upsilon-meson_ $\Upsilon$ is a [[heavy meson|heavy]] [[meson]] which
the the [[bottom quark|bottom]]-form of _[[quarkonium]]_: a [[bound state]] of a [[bottom quark]] with its [[antiparticle]], .

## References

See also:

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

On [[Dalitz decays]] of upsilon-mesons:

* Li-Min Gu, Hai-Bo Li, Xin-Xin Ma, Mao-Zhi Yang, _Study of the electromagnetic Dalitz decays $\psi(\Upsilon) \to \eta_c(\eta_b) l^+ l^-$_, Phys. Rev. D 100, 016018 (2019) ([arXiv:1904.06085](https://arxiv.org/abs/1904.06085))

[[!redirects upsilon-mesons]]

[[!redirects ϒ-meson]]
[[!redirects ϒ-mesons]]

[[!redirects ϒ meson]]
[[!redirects ϒ mesons]]

[[!redirects upsilon meson]]
[[!redirects upsilon mesons]]


urelement

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

* table of contents
{:toc}

## Idea

An **urelement**, also called an **atom**, in a [[material set theory]] is a thing which is not a [[set]].  Many common material set theories, such as [[ZFC]], do not by default allow urelements; thus everything is a set and all sets are [[pure set|pure]].  However, they can usually be easily modified to allow urelements --- one such version is [[ZFA]].  Quine\'s [[New Foundations]] is consistent with the [[axiom of choice]] only if it allows urelements.

[[!redirects urelement]]
[[!redirects urelements]]
[[!redirects urelemente]]
[[!redirects ur-element]]
[[!redirects ur-elements]]
[[!redirects Urelement]]
[[!redirects Urelemente]]


Urs Frauenfelder

* [Institute page](https://www.uni-augsburg.de/en/fakultaet/mntf/math/prof/geom/frauenfelder/)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=120240)

## Selected writings

On [[cyclic loop spaces]]:

* [[Urs Frauenfelder]], *Dihedral homology and the moon*, J. Fixed Point Theory Appl. **14** (2013) 55–69 ([arXiv:1204.4549](https://arxiv.org/abs/1204.4549), [doi:10.1007/s11784-013-0146-z](https://doi.org/10.1007/s11784-013-0146-z))

category: people


Urs Hartl

* [webpage](http://www.math.uni-muenster.de/u/urs.hartl/)

category: people