equality

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### Context
#### Equality and Equivalence
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[[!include equality and equivalence - contents]]
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#### Foundations
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[[!include foundations - contents]]
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# Contents
* table of contents
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## Idea

There is a lot of interesting stuff to be said about _equality_ in [[logic]], [[higher category theory]], and the [[foundations]] of mathematics, but it hasn\'t all been said here yet.

## Different kinds of equality
   {#DifferentKinds}

Here is a list of distinctions between different notions of _equality_, in different contexts, where possibly all the following pairs of notions are, in turn, "the same", just expressed in terms of different terminologies:

*  the difference between [[axiom|axiomatic]] and [[definition|defined]] equality;
*  the difference between identity and equality,
*  the difference between intensional and extensional equality,
*  the difference between equality [[judgements]], equality [[propositions]], and equality [[types]]
*  the difference between equality and [[isomorphism]],
*  the difference between equality and [[equivalence]],
*  the possibility of operations that might not preserve equality.

## Notions of equality in type theory

In a formal language such as [[type theory]], one distinguishes various different notions of _equality_ or _equivalence_ of the terms of the language.

In [[type theory]], there are broadly three different notions of equality which could be distinguished: **judgmental equality**, **propositional equality**, and **typal equality**. Judgmental equality is defined as a basic judgment in type theory. Propositional equality is defined as a proposition in any two-layered type theory with a layer of types and a layer of propositions. Typal equality defined as a type in type theory.

### Judgmental equality

The first notion of equality is **[[judgmental equality]]**, where we simply judge two elements to be equal to each other. Judgmental equality is expressed in type theory by a separate [[judgment]]: in addition to typing judgments $\Gamma \vdash (t:A)$, we have equality judgments $\Gamma \vdash (t = t'): A$. The rules for manipulating such judgments then include reflexivity, symmetry, transitivity, making judgmental equality into a judgmental [[equivalence relation]].

However, not all type theories have judgmental equality; most [[first-order theories]] use propositional equality in place of judgmental equality, and [[objective type theories]] use typal equality in place of judgmental equality.

### Propositional equality

In any two-layer type theory with a layer of [[types]] and a layer of [[propositions]], or equivalently a [[first order logic]] over [[type theory]] or a [[first-order theory]], every type $A$ has a binary [[relation]] according to which two elements $x$ and $y$ of $A$ are related if and only if they are equal; in this case we write $x =_A y$. Since relations are propositions in the context of a term variable judgment in the type layer, this is called **[[propositional equality]]**. The formation and introduction rules for propositional equality is as follows

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A, y:A \vdash x =_A y \; \mathrm{prop}} \quad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:A \vdash x =_A x \; \mathrm{true}}$$

Then we have the elimination rules for propositional equality:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A, x =_A y \; \mathrm{true} \vdash P(x, y) \; \mathrm{prop} \quad \Gamma, x:A \vdash P(x, x) \; \mathrm{true}}{\Gamma, x:A, y:A, x =_A y \; \mathrm{true} \vdash P(x, y) \; \mathrm{true}}$$

Propositional equality satisfies the [[principle of substitution]] and the [[identity of indiscernibles]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash P(x) \; \mathrm{prop}}{\Gamma, x:A, y:A \vdash (x =_A y) \implies (P(x) \iff P(y)) \; \mathrm{true}}$$

In addition, one could have propositional equality between the types themselves, with formation and introduction rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A = B \; \mathrm{prop}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A = A \; \mathrm{true}}$$

Notable examples of propositional equality include the equality in all flavors of [[set theory]], such as [[ZFC]] or [[ETCS]].

The presence of propositional equality is part of the axiomatization of first-order theories, although this is usually swept under the rug by including propositional equality by default in all first-order theories.  (It has to be made more explicit in a structural set theory such as [[SEAR]], where there is no propositional equality on sets themselves, only on elements of a particular set.) By contrast, in a first order theory without propositional equality, propositional equality is an [[equivalence relation]] which has to be put onto a [[type]] or [[preset]].

### Typal equality

However, not all type theories are [[first order logic]] over [[type theory]], nor are they even two-layered type theories. In type theories with only one layer for types, equality is not a relation. Instead, we have what are called **[[typal equality]]**, as equality of both terms and types are represented by types.  Typal equality of terms is also called the **equality type**, **identity type**, **path type**, or **identification type**, and the terms of typal equality are called **equalities**, **identities**, **identifications**, **paths**. Notable examples of typal equality of terms include the [[identity type]] in [[Martin-Löf type theory]] and [[higher observational type theory]], and the [[cubical path type]] in [[cubical type theory]]. Typal equality of types is also called **[[equivalence of types]]**, are written as $A \simeq B$ for types $A$ and $B$ and are definable in all type theories with typal equality of terms, [[function types]], [[dependent product types]], and [[dependent sum types]].

Type theories with only typal equality are called [[objective type theories]]. In type theories with both typal equality and either judgmental equality or propositional equality, one could distinguish between [[intensional type theories]] and [[extensional type theories]]: extensional type theories are type theories with an equality reflection rule, where one could derive judgmental or propositional equality of terms from typal equality of terms:

$$\frac{\Gamma \vdash p:a =_A b}{\Gamma \vdash a \equiv b:A} \qquad \frac{\Gamma \vdash p:a =_A b}{\Gamma \vdash a \equiv_A b \; \mathrm{true}}$$

Intensional type theories are then type theories with typal equality and one of either judgmental or propositional equality without an equality reflection rule. Note that this notion of extensionality is different from the [[axiom of extensionality]] or [[function extensionality]].

### Comparison of equalities

This table gives the six different notions of equality found in type theory.

| Judgment | of types | of terms |
|--------|--------|--------|
| Judgmental equality | $\Gamma \vdash A = B \; \mathrm{type}$ | $\Gamma \vdash a = b:A$ |
| Propositional equality | $\Gamma \vdash A = B \; \mathrm{true}$ | $\Gamma \vdash a =_A b \; \mathrm{true}$ |
| Typal equality | $\Gamma \vdash P:A \simeq B$ | $\Gamma \vdash p:a =_A b$ |

## Structural rules of equalities

### Reflexivity

The reflexivity of equality is given by the following judgmental, propositional, and typal rules respectively:

* Reflexivity of judgmental equality

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A = A \; \mathrm{type}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash a = a:A}$$

* Reflexivity of propositional equality
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A = A \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash a =_A a \; \mathrm{true}}$$

* Reflexitivity of typal equality (i.e. [[identity function]] and identity [[path]])
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash id_{A}:A \simeq A}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A}{\Gamma \vdash \mathrm{refl}_{A}(a):a =_A a}$$

### Symmetry

The symmetry of equality is given by the following judgmental, propositional, and typal rules respectively:

* Symmetry of judgmental equality
$$\frac{\Gamma \vdash A = B \; \mathrm{type}}{\Gamma \vdash B = A \; \mathrm{type}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a = b:A}{\Gamma \vdash b = a:A}$$

* Symmetry of propositional equality
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash A = B \; \mathrm{true}}{\Gamma \vdash B = A \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash a =_A b \; \mathrm{true}}{\Gamma \vdash b =_A a \; \mathrm{true}}$$

* Symmetry of typal equality (i.e. [[inverse function]] and inverse path)
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash P:A \simeq B}{\Gamma \vdash P^{-1}:B \simeq A}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:a =_A b}{\Gamma \vdash p^{-1}:b =_A a}$$

### Transitivity

The transitivity of equality is given by the following judgmental, propositional, and typal rules respectively:

* Transitivity of judgmental equality
$$\frac{\Gamma \vdash A = B \; \mathrm{type} \quad \Gamma \vdash B = C \; \mathrm{type} }{\Gamma \vdash A = C \; \mathrm{type}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a = b:A \quad b = c:A }{\Gamma \vdash a = c:A}$$

* Transitivity of propositional equality
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash A = B \; \mathrm{true} \quad \Gamma \vdash B = C \; \mathrm{true}}{\Gamma \vdash A = C \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash c:A \quad \Gamma \vdash a =_A b \; \mathrm{true} \quad \Gamma \vdash b =_A c \; \mathrm{true}}{\Gamma \vdash a =_A c \; \mathrm{true}}$$

* Transitivity of typal equality (i.e. [[composition]] of functions and path concatenation)
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash C \; \mathrm{type} \quad \Gamma \vdash P:A \simeq B \quad \Gamma \vdash Q:B \simeq C}{\Gamma \vdash Q \circ P:A \simeq C}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash c:A \quad \Gamma \vdash p:a =_A b \quad \Gamma \vdash q:b =_A c}{\Gamma \vdash p \bullet q:a =_A c}$$

### Principle of substituting equals for equals

The [[principle of substituting equals for equals]] is given by the following judgmental, propositional, and typal rules respectively:

* Substitution of judgmentally equal terms:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a = b : A \quad \Gamma, x:A, \Delta \vdash B \; \mathrm{type}}{\Gamma, \Delta[b/x] \vdash B[a/x] = B[b/x] \; \mathrm{type}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a = b : A \quad \Gamma, x:A, \Delta \vdash c:B}{\Gamma, \Delta[b/x] \vdash c[a/x] = c[b/x]: B[b/x]}$$

* Substitution of propositionally equal terms:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a : A \quad \Gamma \vdash b : A \quad \Gamma \vdash a =_A b \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash B \; \mathrm{type}}{\Gamma, \Delta[b/x] \vdash B[a/x] = B[b/x] \; \mathrm{true}}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a : A \quad \Gamma \vdash b : A \quad \Gamma \vdash a =_A b \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash c:B}{\Gamma, \Delta[b/x] \vdash c[a/x] =_{B[b/x]} c[b/x] \; \mathrm{true}}$$

* Substitution of typally equal terms (i.e. [[transport]] and [[action on identifications]]):
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:a =_A b \quad \Gamma, x:A, \Delta \vdash B \; \mathrm{type}}{\Gamma, \Delta[b/x] \vdash \mathrm{tr}(x.B, a, b, p):B[a/x] \simeq B[b/x]}$$
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:a =_A b \quad \Gamma, x:A, \Delta \vdash c:B}{\Gamma, \Delta[b/x] \vdash \mathrm{ap}(c, x.B, a, b, p):\mathrm{tr}(x.B, a, b, p)(c[a/x]) =_{B[b/x]} c[b/x]}$$

### Variable conversion

The variable conversion rule is given by the following judgmental, propositional, and typal rules respectively:

* Judgmental variable conversion rule:
$$\frac{\Gamma \vdash A = B \; \mathrm{type} \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}$$

* Propositional variable conversion rule:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash A = B \; \mathrm{true} \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, x:B, \Delta \vdash \mathcal{J}}$$

* Typal variable conversion rule:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash P:A \simeq B \quad \Gamma, x:A, \Delta \vdash \mathcal{J}}{\Gamma, y:B, \Delta[P^{-1}(y)/x] \vdash \mathcal{J}[P^{-1}(y)/x]}$$

$\mathcal{J}$ is a generic judgment thesis.

## Usages of equality

### In definitions of types and terms

The three different notions of equality above could be used in [[definitions]] of [[types]] and [[terms]], as well as in the rules for the [[single assignment operator]] $\coloneqq$. In [[Martin-Löf type theory]] and [[cubical type theory]], judgmental equality is used for definitional equality. In [[ZFC]] and [[ETCS]], propositional equality is used for definitional equality, and in [[objective type theories]], typal equality is used for definitional equality.

### In conversion rules and inductive definitions

The three different notions of equality above could also be used in the [[conversion rules]] for the various type formers, such as [[beta conversion]] or [[eta conversion]], where the equality is called **[[conversional equality]]** or **computational equality**. Computational equality is important because it is the equality used in [[inductive definitions]]. In [[Martin-Löf type theory]] and [[cubical type theory]], judgmental equality is used for computational equality, and in [[objective type theories]], typal equality is used for computational equality.

## Meta-equalities

There are also equalities in [[type theory]] which are not internal to the type theory itself, but rather equalties in the metatheroy. These include **syntactic equality** of expressions, as well as **[[alpha-equivalence]]** and **[[definitional equality]]** of syntactical expressions. When using [[de Bruijn indices]], alpha-equivalence is the same as syntactic equality, but when using [[variables]], alpha-equivalence is different from syntactic equality: rather alpha-equivalence is equality up to the renaming of [[bound variables]]. All type theories, like [[MLTT]], [[cubical type theory]], [[ZFC]], and [[ETCS]], have syntactic equality, alpha-equivalence, and definitional equality of syntactical expressions, including the type theories which lack [[judgmental equality]] like [[objective type theories]].

## Equality in constructive and classical mathematics

Constructive mathematics is mathematics in which the law of excluded middle does not necessarily hold for [[propositions]], [[subsingletons]], or [[h-propositions]]. Classical mathematics is mathematics in which the law of excluded middle does hold for [[propositions]], [[subsingletons]], or [[h-propositions]]. Here, we take equality to mean either typal or propositional equality, so that equality is a relation or a type family on a type or set.

In classical mathematics, equality of sets is a [[stable relation|stable]] [[equivalence relation]], and [[denial inequality]] of sets is a [[tight apartness relation]]. However, in constructive mathematics, equality cannot be proven to be stable for all sets, and denial inequality cannot be proven to be a tight apartness relation for all sets. Instead, one could distinguish between 4 different notions of equality and [[inequality]]:

* [[tight apartness relations]]. However, not all sets have tight apartness relations. The sets which do are called [[inequality spaces]].

* equality, which is an [[equivalence relation]]; [[inequality spaces]] have [[stable equality]].

* [[denial inequality]], which can only be proven to be a [[weakly tight]] [[irreflexive symmetric relation]]. However, all statements in classical mathematics involving only denial inequalities hold in constructive mathematics, by the [[double negation translation]] and the property that for any proposition $P$, $\neg \neg \neg P$ if and only if $\neg P$.

* the [[double negation]] of equality, which can only be proven to be a [[stable relation|stable]] [[reflexive symmetric relation]]. However, all statements in classical mathematics involving only equality hold in constructive mathematics with the equality replaced by its double negation, by the [[double negation translation]].

The sets in which equality and inequality behaves as it does in [[classical mathematics]] are the sets with [[decidable equality]].

As a result, in constructive mathematics, sometimes one takes [[inequality spaces]] instead of general [[sets]] to be the foundational primitive concept. In classical mathematics, this is unnecessary, because every [[set]] is an [[inequality space]].

## Internal equality in set theory

Relations and equality could be [[internal logic of set theory|internalized in any set theory]]: the internal equality in set theory is the smallest internal [[reflexive relation]] on $S$, and it is in fact an internal [[equivalence relation]]; it is the only internal equivalence relation on $S$ that is also a [[partial order]] (although that fact is somewhat circular).  This relation is often called the __identity relation__ on $S$, either because 'identity' can mean 'equality' or because it is the [[identity]] for [[composition]] of relations.

In the [[category]] of reflexive relations on $S$, the [[equality relation]] on $S$ is an [[initial object|initial reflexive relation]]; when the equality relation is a [[type family]] instead of a family of propositions, then the equality relation is the initial type generated by reflexivity or the [[first law of thought]].

As a [[subset]] of $S \times S$, the equality relation is often called the __[[diagonal]]__ and written $\Delta_S$ or similarly.  As an abstract set, this subset is [[isomorphism|isomorphic]] to $S$ itself, so one also sees the diagonal as a map, the [[diagonal function]] $S \overset{\Delta_S}\to S \times S$, which maps $x$ to $(x,x)$; note that $x = x$.  This is in [[Set]]; analogous [[diagonal morphisms]] exist in any [[cartesian monoidal category]].

## Related concepts

* **equality**, [[equation]]

* [[alpha-equivalence]]

* [[definition]], [[assignment]], [[equality]]

* [[identity type]],

* [[identity of indiscernibles]]

* [[inequality]]

  * [[denial inequality]]

  * [[apartness relation]]

* [[isomorphism]]

* [[equivalence]]

* [[weak equivalence]]

* [[homotopy equivalence]], [[weak homotopy equivalence]]

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

* [[equivalence of (∞,1)-categories]]

[[!include logic symbols -- table]]

## References

In

* [[Georg Hegel]], _[[Science of Logic]]_, 1812

equality is the subject of volume 1, book 2 "Die Lehre vom Wesen" (The doctrine of essence). As discussed at _[[Science of Logic]]_, one can roughly identify in Hegel's text there the notion of intensional identity and of the reflector term in [[identity types]].

Texts on [[type theory]] typically deal with the subtleties of the notion of _equality_. For instance

* [[Per Martin-Löf]], _Intuitionistic type theory_, Lecture notes (1980)
  {#PML}

Besides

* [[Kurt Gödel]], _Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes_. Dialectica (1958), pp. 280–287,

which might be the first paper to mention intensional equality (and the fact that it should be decidable), there is

* [[Nicolaas de Bruijn]], _[[Automath]]_,

where de Bruijn makes a distinction between definitional equality and "book" equality.

[[!redirects equal]]
[[!redirects equals]]
[[!redirects equality]]
[[!redirects equalities]]
[[!redirects equality predicate]]
[[!redirects equality predicates]]

[[!redirects equality relation]]
[[!redirects equality relations]]
[[!redirects identity relation]]
[[!redirects identity relations]]
[[!redirects diagonal relation]]
[[!redirects diagonal relations]]

[[!redirects syntactic equality]]
[[!redirects syntactically equal]]


equality and equivalence - contents

**[[equivalence]]**

* [[equality]] ([[definitional equality|definitional]], [[propositional equality|propositional]], [[computational equality|computational]], [[judgemental equality|judgemental]], [[extensional equality|extensional]], [[intensional equality|intensional]], [[decidable equality|decidable]])

* [[identity type]], [[equivalence in homotopy type theory]]

* [[isomorphism]], [[weak equivalence]], [[homotopy equivalence]], [[weak homotopy equivalence]], [[equivalence in an (∞,1)-category]]

* [[natural equivalence]], [[natural isomorphism]]

* [[gauge equivalence]]

* **Examples.**

  * [[equivalence of categories]], [[adjoint equivalence]], [[weak equivalence of internal categories]], [[Morita equivalence]], [[equivalence of 2-categories]], [[equivalence of (∞,1)-categories]]

**[[principle of equivalence]]**

* [[univalence]]

**[[equation]]**

* [[fiber product]], [[pullback]]

* [[homotopy pullback]]

* **Examples.**

  * [[linear equation]], [[differential equation]], [[ordinary differential equation]], [[critical locus]]

  * [[Euler-Lagrange equation]], [[Einstein equation]], [[wave equation]]

  * [[Schrödinger equation]], [[Knizhnik-Zamolodchikov equation]], [[Maurer-Cartan equation]], [[quantum master equation]], [[Euler-Arnold equation]], [[Fuchsian equation]], [[Fokker-Planck equation]], [[Lax equation]]

[Edit this sidebar](/nlab/edit/equality+and+equivalence+-+contents)


This page collects material related to the article

* [[William Lawvere]],

  _Equality in hyperdoctrines and comprehension schema as an adjoint functor_,

  Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

  [pdf](https://ncatlab.org/nlab/files/LawvereComprehension.pdf)

  [GoogleBooks](https://books.google.de/books?hl=en&lr=&id=uuYCCAAAQBAJ&oi=fnd&pg=PA1&dq=Equality+in+hyperdoctrines+and+comprehension+schema+as+an+adjoint+functor&ots=v8gWJvEIyI&sig=pbl-hby1aUC1zJ1sav4flLz-OWM&redir_esc=y#v=onepage&q=Equality%20in%20hyperdoctrines%20and%20comprehension%20schema%20as%20an%20adjoint%20functor&f=false)

on [[hyperdoctrines]] and [[comprehension]] in terms of [[base change]] [[adjoint functors]] (and on the [[Beck-Chevalley condition]]).

## Related articles

* [[William Lawvere]], _[[Adjointness in Foundations]]_, Dialectica 23 (1969), 281-296 ([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html))

* {#Lambek} [[Joachim Lambek]], _The Influence of [[Heraclitus]] on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1981)

category: reference


equalizer

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## Definition

### In category theory

An **equalizer** is a [[limit]]
$$\operatorname{eq}\underset{\quad e \quad}{\to}x\underoverset{\quad g \quad}{f}{\rightrightarrows}y$$
over a parallel pair i.e. of the [[diagram]] of the shape
$$\left\lbrace
      x \underoverset{\quad g \quad}{f}{\rightrightarrows} y
  \right\rbrace.$$
(See also [[fork]] diagram).

This means that for $f : x \to y$ and $g : x \to y$ two [[parallel morphisms]] in a [[category]] $C$, their equalizer is, if it exists

* an object $eq(f,g) \in C$;

* a morphism $eq(f,g) \to x$

* such that

  * pulled back to $eq(f,g)$ both morphisms become [[equality|equal]]: $ (eq(f,g) \to x \stackrel{f}{\to} y) = (eq(f,g) \to x \stackrel{g}{\to} y) $
  * and $eq(f,g)$ is the [[universal property|universal object]] with this property.

The dual concept is that of [[coequalizer]].

### In type theory

In [[type theory]] the [[equalizer]]
$$P \to A \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} B$$
is given by the [[dependent sum]] over the [[dependent type|dependent]] [[equality type]]
$$P \simeq \sum_{a : A} (f(a) = g(a)).$$

## Examples

* In $C =$ [[Set]] the equalizer of two functions of sets is the subset of elements of $c$ on which both functions coincide.
$$eq(f,g)=\left\{
     s \in c |
     f(s) = g(s)
  \right\}.$$

* For $C$ a category with [[zero object]] the equalizer of a morphism $f : c \to d$ with the corresponding [[zero morphism]] is the [[kernel]] of $f$.

## Properties

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

A category has equalizers if it has binary [[product]]s and [[pullback]]s.

=--

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

For $S \stackrel{\overset{g}{\longrightarrow}}{\underset{f}{\longrightarrow}} T$ the given diagram, form the [[pullback]] along the [[diagonal morphism]] of $T$:
$$\array{
     eq(f,g)
       &\longrightarrow&
     S
     \\
     \big\downarrow
       &&
     \big\downarrow {}^{\mathrlap{(f, g)}}
     \\
     T
       &\underset{(id, id)}{\longrightarrow}&
     T \times T
  }.$$

One checks that the horizontal morphism $eq(f,g) \to S$ equalizes $f$ and $g$ and that it does so universally.
=--

+-- {: .num_prop}
###### Proposition
If a category has equalizers and (finite) products, then it has (finite) [[limits]].
=--

For the finite case, we may say equivalently:

+-- {: .num_prop}
###### Proposition
If a category has equalizers, binary products and a terminal object, then it has finite [[limits]].
=--

\begin{prpn}
(Eckmann and Hilton [EH](#EH), Proposition 1.3.)
Let $e: E \rightarrow X$ be an arrow in a category $\mathcal{C}$ which is an equaliser of a pair of arrows of $\mathcal{C}$. Then $e$ is a [[monomorphism]].
\end{prpn}

\begin{proof}
If $g,h : A \rightarrow E$ are arrows of $\mathcal{C}$ such that $e \circ g = e \circ h$, then it follows immediately from the uniqueness part of the universal property of an equaliser that $g = h$.
\end{proof}

## Related concepts

* [[homotopy equalizer]]

## References

Equalizers were defined in the paper

* {#EH} [[Beno Eckmann]], [[Peter J. Hilton]], _Group-like structures in general categories II.  Equalizers, limits, lengths_.  Mathematische Annalen 151:2 (1963), 150–186.  [doi:10.1007/bf01344176](https://doi.org/10.1007/bf01344176).

for any finite collection of parallel morphisms.
The paper refers to them as _left equalizers_, whereas [[coequalizers]] are referred to as _right equalizers_.

Textbook account:

* [[Francis Borceux]], Section 2.4 in Vol. 1: *Basic Category Theory* of: *[[Handbook of Categorical Algebra]]*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

[[!redirects equalizers]]
[[!redirects equaliser]]
[[!redirects equalisers]]


equation

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{:toc}

## Definition

An _equation_ is a [[proposition]] of [[equality]].

For

$$
  x : X \vdash \phi(x) : Z
$$

and

$$
  y : Y \vdash \psi(y) : Z
$$

two [[terms]] of some [[type]] $Z$ [[dependent type|dependent]] on [[variables]] $x$ of type $X$ and $y$ of type $Y$, respectively, the equation asserting that these two formulas are [[equality|equal]] is as a [[proposition]] the [[bracket type]]

$$
 x : X, y : Y \vdash [\phi(x) = \psi(y)] : Type
$$

and as a not-neccessarily [[(-1)-truncated object|(-1)-truncated]] type just the [[identity type]]

$$
 x : X, y : Y \vdash (\phi(x) = \psi(y)) : Type
$$

itself.

The **space of [[solutions]]** of this equation is the [[dependent sum]] over all pairs of terms for which [[equality]] holds

$$
  \vdash \sum_{{x : X} \atop {y : Y}} (\phi(x) = \psi(y)) : Type
  \,.
$$

Hence a particular **[[solution]]** $sol$ is a [[term]] of this type

$$
  \vdash sol : \sum_{{x : X} \atop {y : Y}} (\phi(x) = \psi(y))
  \,,
$$

which means that it is a triple consisting of an $x \in X$, a $y \in Y$ and a witness $eq : (\phi(x) = \psi(y))$ that these indeed solve the equation.

In [[categorical semantics]] this means that the space of solutions to an equation between expression $\phi(x) : Z$ and $\psi(y) : Z$ of [[type]] $Z$ depending on [[variables]] of type $X$ and $Y$, respectively is the  [[pullback]]

$$
  \array{
    \sum_{{x : X} \atop {y : Y}} (\phi(x) = \psi(y))
    &\to& Y
    \\
    \downarrow && \downarrow^{\mathrlap{\psi}}
    \\
    X &\stackrel{\phi}{\to}& Z
  }
  \,,
$$

the [[fiber product]] of $\phi$ with $\psi$. In the context of [[homotopy type theory]] this is the [[homotopy pullback]]/[[homotopy fiber product]].

See at _[homotopy pullback -- concrete constructions -- In homotopy type theory](http://ncatlab.org/nlab/show/homotopy+pullback#ConstructionInHomotopyTypeTheory)_ for more on this.

## Examples

* [[linear equation]]

* [[algebraic equation]]

* [[Diophantine equation]]

* [[trigonometric identity]]

* [[differential equation]]

* [[Euler-Lagrange equation]], [[equation of motion]]

* [[master equation]]

* [[equation of state]]

[[!redirects equations]]


equation of state

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[thermodynamics]] the _equation of state_ of a system is an [[equation]] that relates its state variables to each other. For instance for a [[gas]] the equation of state relates [[pressure]], [[volume]] and [[temperature]].

## Related concepts

* [[QCD cosmology]]

## References

See also:

* Wikipedia, _[Equation of state](https://en.wikipedia.org/wiki/Equation_of_state)_

[[!redirects equations of state]]


equationally presentable category

A category is __equationally presentable__ if it may be described, beginning with the category [[Set]], as the category of [[sets]] equipped with certain operations satisfying certain [[equation]]s involving these operations as [[objects]], and [[functions]] that preserve these operations as [[morphisms]].  A category is __equationally presented__ if it is equipped with such a description.

A more formal definition would be useful here (but perhaps a bit tedious); the point is that an equationally presentable category is given by an [[algebraic theory]] interpreted in [[Set]].

We can generalise to categories given by laws which may be [[implications]] between equations rather than simply equations themselves; call such a category __quasi-equationally presentable__ or __quasi-equationally presented__.

In general, we allow largely many operations and equations (that is a [[proper class]] of them, too many be parametrised by an object of $Set$ itself), although each individual operation must have small arity and each equation must be well-founded.

## Monadic and algebraic categories

Equationally presentable categories capture precisely the categories of [[varieties of algebras]] studied in [[universal algebra]].  However, to study them with [[category theory]], it is somewhat more manageable to use [[monadic categories]] instead.  We have that an algebraic category $C$ is monadic if and only if it has [[free objects]]; that is, the [[forgetful functor]] from $C$ to $Set$ has a [[left adjoint]].

Let a (quasi)-equationally presentable category $C$ be __bounded__ if it may be described using only a small set of operations (and hence a small set of equations); in this case it follows that $C$ is monadic (if it is equationally presentable) or at least [[algebraic category|algebraic]] (if it is only quasi-equationally presentable), although the converse does not hold.  Similarly, let $C$ be __finitary__ if it may be described using only a small set of operations, each of which has finite arity (in which case it must be bounded).  We can characterise boundedness and finite arity nicely in category-theoretic terms; thus one speaks of bounded monadic categories, finitary algebraic categories, etc.  See [[algebraic category]] for the definitions; it is a theorem that the definitions there are equivalent to the definitions here.

+-- {: .query}
Is it true that a quasi-equationally presentable category is algebraic if it has free objects?  Both Johnstone and AHS stop short of this case.
=--

Finitary monadic categories are particularly nice; they are given by [[Lawvere theories]].

Warning:  The meaning of 'algebraic category' varies in the literature; Johnstone in _[[Stone Spaces]]_ defines it to mean monadic, while we follow (here and at [[algebraic category]]) AHS in _The [[Joy of Cats]]_ .  The term 'equationally presentable' is from Johnstone, and the claim that this is equivalent to being given by an [[algebraic theory]] requires using Johnstone\'s definition of the latter from the _[[Elephant]]_.  The term 'quasi-equationally presentable' is [[Toby Bartels|my own]], following a pattern set down by AHS.

## References

Same as [[algebraic category]].

[[!redirects equationally presentable category]]
[[!redirects equationally presented category]]
[[!redirects equationally-presentable category]]
[[!redirects equationally-presented category]]
[[!redirects equationally presentable categories]]
[[!redirects equationally presented categories]]
[[!redirects equationally-presentable categories]]
[[!redirects equationally-presented categories]]

[[!redirects quasi-equationally presentable category]]
[[!redirects quasi-equationally presented category]]
[[!redirects quasiequationally presentable category]]
[[!redirects quasiequationally presented category]]
[[!redirects quasi-equationally-presentable category]]
[[!redirects quasi-equationally-presented category]]
[[!redirects quasiequationally-presentable category]]
[[!redirects quasiequationally-presented category]]
[[!redirects quasi-equationally presentable categories]]
[[!redirects quasi-equationally presented categories]]
[[!redirects quasiequationally presentable categories]]
[[!redirects quasiequationally presented categories]]
[[!redirects quasi-equationally-presentable categories]]
[[!redirects quasi-equationally-presented categories]]
[[!redirects quasiequationally-presentable categories]]
[[!redirects quasiequationally-presented categories]]


equator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given an [[n-sphere]] $S^n$, regarded under its standard [[embedding]] into [[Cartesian space]] $\mathbb{R}^{n+1}$, then its _equator_ is the [[intersection]] with $\mathbb{R}^n \subset \mathbb{R}^{n+1}$.

The equator devides the $n$-sphere into its to [[hemi-n-spheres]].

For $n \geq 1$, the full $n$-sphere is the [[suspension]] of its equator $(n-1)$-sphere, namely the [[union]] of all [[meridians]] passing through the equator.

## Related concepts

* [[meridian]]

* [[hemisphere]], [[antipode]]

* [[stereographic projection]]

* [[clutching construction]]

[[!redirects equators]]


equicontinuous family of functions

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

# Equicontinuous functions
* table of contents
{: toc}

## Idea

A [[function]] $f$ is [[continuous map|continuous]] if, roughly speaking, $f(x)$ is arbitrarily close to $f(y)$ whenever $x$ is sufficiently close to $y$.  However, 'close' is relative, and $f(x)$ may be much closer to $f(y)$ than $g(x)$ is to $g(y)$, even if both $f$ and $g$ are continuous.  Nevertheless, given a [[family]] of functions, we may have that $f(x)$ is arbitrarily close to $f(y)$ for *every* function $f$ in the family at once whenever $x$ is sufficiently close to $y$.  In this case, the family of functions is _equicontinuous_.

## Definitions

Because we are considering the relative degree of closeness between potentially unrelated pairs of points, we need a [[uniform structure]] to define this concept.  So let $X$ and $Y$ be [[uniform spaces]] (although the concept should make sense in somewhat greater generality), and let $\mathcal{F}$ be a [[family]] of [[functions]] from $X$ to $Y$.

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

The family $\mathcal{F}$ is __continuous__ if each member is [[continuous function|continuous]]:  For each [[entourage]] $E$ in $Y$, for each function $f$ in $\mathcal{F}$ and each [[point]] $x \in X$, for some entourage $D$ in $X$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.
=--

In short:
$$ \forall E,\; \forall f,\; \forall x,\; \exists D,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .$$

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

The family $\mathcal{F}$ is __uniformly continuous__ if each member is [[uniformly continuous function|uniformly continuous]]:  For each [[entourage]] $E$ in $Y$, for each function $f$ in $\mathcal{F}$, for some entourage $D$ in $X$, for each [[point]] $x$ in $X$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.
=--

In short:
$$ \forall E,\; \forall f,\; \exists D,\; \forall x,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .$$

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

The family $\mathcal{F}$ is __equicontinuous__ if:  For each [[entourage]] $E$ in $Y$, for each [[point]] $x$ in $X$, for some entourage $D$ in $X$, for each function $f$ in $\mathcal{F}$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.
=--

In short:
$$ \forall E,\; \forall x,\; \exists D,\; \forall f,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .$$

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

The family $\mathcal{F}$ is __uniformly equicontinuous__ if:  For each [[entourage]] $E$ in $Y$, for some entourage $D$ in $X$, for each function $f$ in $\mathcal{F}$ and each [[point]] $x$ in $X$, for each point $y$ in $X$, whenever $x \approx_D y$, we have $f(x) \approx_E f(y)$.
=--

In short:
$$ \forall E,\; \exists D,\; \forall f,\; \forall x,\; \forall y,\; x \approx_D y \;\Rightarrow\; f(x) \approx_E f(y) .$$

All of these definitions are identical except for the placement of the [[quantifiers]] $\forall f$ and $\forall x$ before or after the quantifier $\exists D$.

Just as it is reasonable to speak of a single function $f$ continuous at a single point $x$, so it is reasonable to speak of a family equicontinuous at a single point $x$ (or, for that matter, of a single function $f$ uniformly continuous).  However, it makes no sense to speak of a single function $f$ equicontinuous (or uniformly equicontinuous), nor can we speak of a family of functions uniformly equicontinuous at a single point $x$.

## Properties

### Cauchy sum theorem

The [[Cauchy sum theorem]] holds for an equicontinuous family of functions (from the [[real line]] to itself), *without* the requirement for [[uniform convergence]].

### Arzela-Ascoli theorem

The importance of equicontinuity is perhaps best illustrated by the Arzelà-Ascoli theorem, which gives conditions for a set in a [[exponential law for spaces|function space]] to be compact. A reasonably general version is this:

+-- {: .num_theorem}
###### Theorem
Let $X$ be a [[convergence space]] and $Y$ a [[uniform space]]. Then a subset $F \subseteq C(X, Y)$ is [[relatively compact subspace|relatively compact]] (has compact closure) iff it is equicontinuous and $\{f(x):\; f \in F\}$ is relatively compact in $Y$ for each $x \in X$.
=--

See [BB, corollary 2.4.9](#BB). Here the topology on the space $C(X, Y)$ of continuous functions $X \to Y$ is the so-called _natural topology_, namely the largest topology on $C(X, Y)$ such that for all spaces $A$, the continuity of a map $g: A \times X \to Y$ implies the continuity of its transpose $\hat{g}: A \to C(X, Y)$. This is the same as the exponential in $Top$ whenever the exponential exists. See [Escardó, sections 8.1 and 10.2](#Es); see also [ELS](#ELS) where it is shown that the natural topology on $C(X, Y)$ coincides with the topology of continuous convergence (which is the context for the theorem above).

(Some applications of Arzela-Ascoli should also be given.)

### Representability

It happens that the property of a family of functions being (uniformly) equicontinuous is equivalent to them defining a single function which is (uniformly) continuous.

Recall that for a family of functions $f_i : A \to B$ for $i \in I$, we can define a function $\hat f : A \to B^I$ by $\hat f(a)(i) = f_i(a)$, and that the $f_i$ are all (uniformly) continuous iff $\hat f$ is (uniformly) continuous when $B^I$ is given the product uniformity.

However, we can give $B^I$ another uniformity more analogous to the [[box topology]] of topological spaces. Specifically, the box power $B^{\square I}$ is the uniform space with point set $B^I$, and where $E \subseteq B^I \times B^I$ is an entourage iff there is some entourage $G$ on $B$ such that for all $f \sim_E g$ in $B^I$ and all $i \in I$, $f(i) \sim_G g(i)$. This means that the entoruages of $B^{\square I}$ are generated by $G^I$ for entourages $G$ on $B$.

With this definition, we can see that the family of functions $f_i : A \to B$ is (uniformly) equicontinuous iff $\hat f : A \to B^{\square I}$ is (uniformly) continuous.

This also gives a convenient way to show that an equicontinuous net of continuous functions which converge pointwise converges to a continuous function: a net of equicontinuous functions $f_d : A \to B$ gives a continuous function $A \to B^{\square D}$. By the below proposition, if the $f_d$ converge pointwise, they converge to a continuous function.

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

Let $B$ be a separated uniform space and $D$ a directed set, let $conv(D,B)$ be the subspace of $B^{\square D}$ which contains the convergent nets. Then the map $lim : conv(D,B) \to B$ which takes a net to the point it converges to is continuous.
=--

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

Let $M$ in $conv(D,B)$ and $lim M \in U \subset B$ be open. Then there is an entourage $E$ on $B$ such that $ (E \circ E)[lim M] \subset U $. $E^D[M] := \{N \mid \forall i. M(i) \sim_E N(i) \} \subset conv(D,B)$ is a neighbourhood of $M$. For any $N \in E^D[M]$, we have that $lim N \in \widebar{E[lim M]}$ (as all $N(i)$ are in this closed set), so $lim N \in \widebar{E[x]} \subseteq \(E \circ E)[x] \subseteq U$. This means that $lim$ is continuous at $M$, as required.

=--

## References

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

* R. Beattie and H.-P. Butzmann, _Convergence Structures and Applications to Functional Analysis_, Kluwer Academic Publishers (2002).
 {#BB}

* Martín Escardó, _Synthetic topology of data types and classical spaces_, Electronic Notes in Theor. Comp. Sci. (2004). ([pdf](http://www.cs.bham.ac.uk/~mhe/papers/barbados.pdf))
 {#Es}

* Martín Escardó, Jimmie Lawson, and Alex Simpson, _Comparing cartesian closed categories of (core) compactly generated spaces_, Topology and its Applications Vol. 143 Iss. 1-3 (2004), 105-145. ([web](http://www.sciencedirect.com/science/article/pii/S0166864104000550))
 {#ELS}

[[!redirects equicontinuous family]]
[[!redirects equicontinuous families]]
[[!redirects equicontinuous family of functions]]
[[!redirects equicontinuous families of functions]]
[[!redirects equicontinuous family of maps]]
[[!redirects equicontinuous families of maps]]

[[!redirects equicontinuous collection]]
[[!redirects equicontinuous collections]]
[[!redirects equicontinuous collection of functions]]
[[!redirects equicontinuous collections of functions]]
[[!redirects equicontinuous collection of maps]]
[[!redirects equicontinuous collections of maps]]

[[!redirects equicontinuous sequence]]
[[!redirects equicontinuous sequences]]
[[!redirects equicontinuous sequence of functions]]
[[!redirects equicontinuous sequences of functions]]
[[!redirects equicontinuous sequence of maps]]
[[!redirects equicontinuous sequences of maps]]

[[!redirects equicontinuous function]]
[[!redirects equicontinuous functions]]
[[!redirects equicontinuous map]]
[[!redirects equicontinuous maps]]

[[!redirects uniformly equicontinuous family]]
[[!redirects uniformly equicontinuous families]]
[[!redirects uniformly equicontinuous family of functions]]
[[!redirects uniformly equicontinuous families of functions]]
[[!redirects uniformly equicontinuous family of maps]]
[[!redirects uniformly equicontinuous families of maps]]

[[!redirects uniformly equicontinuous collection]]
[[!redirects uniformly equicontinuous collections]]
[[!redirects uniformly equicontinuous collection of functions]]
[[!redirects uniformly equicontinuous collections of functions]]
[[!redirects uniformly equicontinuous collection of maps]]
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[[!redirects uniformly equicontinuous sequence]]
[[!redirects uniformly equicontinuous sequences]]
[[!redirects uniformly equicontinuous sequence of functions]]
[[!redirects uniformly equicontinuous sequences of functions]]
[[!redirects uniformly equicontinuous sequence of maps]]
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[[!redirects uniformly equicontinuous function]]
[[!redirects uniformly equicontinuous functions]]
[[!redirects uniformly equicontinuous map]]
[[!redirects uniformly equicontinuous maps]]


equifibration

\tableofcontents

\section{Idea}

A _equifibration_ or _equiv-fibration_ is a [[categorification]] of the notion of an [[isofibration]] to the setting of [[2-category|2-categories]]. Roughly speaking, a [[2-functor|functor]] $p:E \rightarrow B$ between 2-categories is an equifibration if equivalences in $B$ can be 'lifted' to equivalences in $E$. However, there are some subtleties to the precise definition, as will be discussed below.

\section{Explicit definition}

The following works for both weak and strict 2-categories and weak and strict 2-functors.

\begin{defn} \label{DefinitionEquifibration} A _equifibration_ is a [[2-functor|functor]] $p: E \rightarrow B$ between [[2-category|2-categories]] such that, for every object $e$ of $E$, and every 1-arrow $f: p(e) \rightarrow b$ of $B$ which is an [[equivalence]], the following hold.

1. There is an object $e'$ of $E$ and an equivalence $g: e \rightarrow e'$ in $E$ such that $p(g) = f$.
1. For every 1-arrow $h: e \rightarrow e'$ of $E$ and every [[2-isomorphism]] $\phi: f \rightarrow p(h)$ in $B$, there is a 2-isomorphism $\psi: g \rightarrow h$ in $E$ such that $p(\psi) = \phi$.

\end{defn}

\begin{rmk} \label{RemarkFirstConditionInLackFibrationAsLiftingCondition} Let $Q$ be the [[free-standing equivalence]]. Then condition 1. in Definition \ref{DefinitionEquifibration} is equivalent to saying that for every (strictly!) commutative diagram of 1-arrows

\begin{tikzcd}
1 \ar[r, "e"] \ar[d, swap, "0"] & E \ar[d, "p"] \\
Q \ar[r, swap, "f"] & B
\end{tikzcd}

in $\mathsf{2-Cat}$, the [[2Cat|2-category of 2-categories]], there is a functor $l: Q \rightarrow E$ such that the following diagram of 1-arrows in $\mathsf{2-Cat}$ (strictly!) commutes.

\begin{tikzcd}
1 \ar[r, "e"] \ar[d, swap, "0"] & E \ar[d, "p"] \\
Q \ar[r, swap, "f"] \ar[ur, "l"] & B
\end{tikzcd}

The same is true if $Q$ is the [[free-standing adjoint equivalence]], due to the fact that any equivalence can be improved to an [[adjoint equivalence]].

\end{rmk}

\begin{rmk} Remark \ref{RemarkFirstConditionInLackFibrationAsLiftingCondition} can be compared with the definition of an [[isofibration]] of 1-categories as expressed by a lifting condition: the condition is exactly the same, with the [[free-standing isomorphism]] replaced by the [[free-standing equivalence]].   \end{rmk}

\begin{rmk} Let us explore the second condition in Definition \ref{DefinitionEquifibration} a little. Note that any 1-arrow which is 2-isomorphic to an equivalence is itself an equivalence. Thus $p(h)$ is an equivalence, and condition 1. then ensures that $p(h)$ lifts to an equivalence $g'$ in $E$ such that $p\left(g'\right) = p(h)$. Condition 2. expresses that $g'$ must be 2-isomorphic to $h$. This implies in particular that $h$ is an equivalence.

Putting everything together, condition 2. is equivalent to: any 1-arrow of $E$ which maps to $f$ under $p$ up to 2-isomorphism is an equivalence, and all equivalences of $E$ which map to $f$ under $p$ up to 2-isomorphism are  2-isomorphic to one another, in such a way that, given 1-arrows $g$ and $g'$ of $E$, a 2-isomorphism $\phi_{g}: f \rightarrow p(g)$ in $B$, and a 2-isomorphism $\phi_{g'}:  f \rightarrow p\left(g'\right)$ in $B$, the 2-isomorphism $\psi : g \rightarrow g'$ in $E$ has the property that $p(\psi) = \phi_{g}^{-1} \circ \phi_{g'}$.

\end{rmk}

\section{References}

The following was where efibrations were introduced.

Equifibrations (there called equiv-fibrations) were introduced in:

* [[Stephen Lack]], _A Quillen model structure for 2-categories_, K-Theory 26, No. 2, 171-205 (2002). [Zentralblatt review](https://zbmath.org/?q=an%3A1017.18005) [author's webpage](http://maths.mq.edu.au/~slack/papers/qmc2cat.html)

However, it contains an error, not pertaining directly to the definition of a equifibration itself, but to obtaining a model structure on the [[2Cat|category of strict 2-categories]] with this definition. The aforementioned error was fixed in the following paper by using the [[free-standing adjoint equivalence]] rather than the free-standing equivalence in the generating acyclic cofibrations.

* [[Stephen Lack]], _A Quillen model structure for bicategories_, K-Theory 33, No. 3, 185-197 (2004). [Zentralblatt review](https://zbmath.org/?q=an%3A1069.18008) [author's webpage](http://maths.mq.edu.au/~slack/papers/qmcbicat.html)

In the paper above, equifibrations were unnamed (and simply referred to as *fibrations*). The term "equifibration" was used in the following paper:

* [[Alexander Campbell]]. _A homotopy coherent cellular nerve for bicategories_, Advances in Mathematics 368 (2020): 107138.

Equifibrations are called **Lack fibrations** in:

* L. Moser, M. Sarazola, P. Verdugo, _A 2Cat-inspired model structure for double categories_, 2020. [arXiv:2004.14233](https://arxiv.org/abs/2004.14233)

[[!redirects equifibrations]]
[[!redirects equiv-fibration]]
[[!redirects equiv-fibrations]]
[[!redirects Lack fibration]]
[[!redirects Lack fibrations]]


equifier

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _equifier_ is a particular kind of [[2-limit]] in a [[2-category]], which universally renders a pair of parallel [[2-morphisms]] equal.

## Definition

Let $f,g\colon A\rightrightarrows B$ be a pair of parallel 1-morphisms in a 2-category and let $\alpha,\beta\colon f\rightrightarrows g$ be a pair of parallel 2-morphisms.  The **equifier** of $\alpha$ is a universal object $V$ equipped with a morphism $v\colon V\to A$ such that $\alpha v = \beta v$.

More precisely, universality means that for any object $X$, the induced functor
$$Hom(X,V) \to Hom(X,A)$$
is fully faithful, and its [[replete image]] consists precisely of those morphisms $u\colon X\to A$ such that $\alpha u=\beta u$.  If the above functor is additionally an *isomorphism* of categories onto the exact subcategory of such $u$, then we say that $V\xrightarrow{v} A$ is a **strict equifier**.

Equifiers and strict equifiers can be described as a certain sort of [[weighted limit|weighted]] 2-limit, where the diagram shape is the [[walking structure|walking]] parallel pair of 2-morphisms $P$, and the weight $P\to Cat$ is the diagram
$$\array{ & \to \\ 1 & \Downarrow\Downarrow & I\\ & \to } $$
where $1$ is the [[terminal category]] and $I$ is the [[interval category]].  Note that this cannot be re-expressed as any sort of [[conical limit|conical]] 2-limit.

An equifier in $K^{op}$ (see [[opposite 2-category]]) is called a **coequifier** in $K$.

## Example: equifiers in the strict 2-category of categories

In the strict 2-category of categories, equifiers can be computed as follows.
The input data is two categories $A$ and $B$,
two functors $F,G\colon A\to B$,
and two natural transformations $p,q\colon F\to G$.
The equifier of $p$ and $q$ is the full subcategory of $A$
consisting of those objects $X\in A$ for which $p_X=q_X$.

## Properties

* The above explicit definition makes it clear that any equifier is a [[fully faithful morphism]].

* Any strict equifier is, in particular, an equifier.  (This is not true for all strict 2-limits.)

* Strict equifiers are, by definition, a particular case of [[PIE-limits]].

[[!redirects equifiers]]
[[!redirects strict equifier]]
[[!redirects strict equifiers]]
[[!redirects coequifier]]
[[!redirects coequifiers]]
[[!redirects strict coequifier]]
[[!redirects strict coequifiers]]


equilibrium

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

## Related concepts

* [[thermodynamics]]

* [[thermal quantum field theory]]

  * [[KMS condition]]

  * [[KMS state]]

## References

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

[[!redirects equilibria]]

[[!redirects thermal equilibrium]]
[[!redirects thermal equilibria]]

[[!redirects thermodynamic equilibrium]]
[[!redirects thermodynamic equilibria]]


equilibrium propagation

## Idea

An alternative [[algorithm]] to [[backpropagation]] used to train [[neural networks]]

## See also

* [[neural networks]]

* [[machine learning]]

* [[backpropagation]]

## References

* Benjamin Scellier and Yoshua Bengio, *Equilibrium Propagation: Bridging the Gap between Energy-Based Models and Backpropagation*, Frontiers in Computational Neuroscience, **11**:24 (2017) $[$[doi:10.3389/fncom.2017.00024](https://doi.org/10.3389/fncom.2017.00024), [arXiv:1602.05179](https://arxiv.org/abs/1602.05179)$]$


equilogical space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

# Equilogical spaces
* table of contents
{: toc}

## Definition

An __equilogical space__ is a [[separation axioms|Kolmogorov]] ($T_0$) [[topological space]] $T$ along with an arbitrary [[equivalence relation]] $\equiv$ on its points (of note, the equivalence relation need not match the topological structure in any way). A morphism between equilogical spaces $(T, {\equiv})$ and $(U, {\cong})$ is a [[continuous function]] $f\colon T \to U$ such that $x \equiv y$ implies $f(x) \cong f(y)$, for all points $x$ and $y$ in $T$. Two such morphisms $f$ and $g$ are considered equal if for all points $x$ in $T$, $f(x) \cong g(x)$.

## Properties

The category __$Equ$__ of equilogical spaces obviously contains the category of $T_0$ topological spaces as a [[full subcategory]] (by using the trivial equivalence relation of [[equality]] on points). Moreover, as opposed to the latter, $Equ$ is in fact [[cartesian closed]]; this can be seen using the [[equivalence of categories|equivalence]] of $Equ$ and the category of [[partial equivalence relation]]s over [[algebraic lattice]]s.

On the other hand, $Equ$ can be identified with a [[reflective subcategory|reflective]] [[exponential ideal]] in the [[ex/lex completion]] of the category $Top_0$ of $T_0$ topological spaces.  This provides an alternative proof of the cartesian closure of $Equ$, since an exponential ideal in a cartesian closed category is cartesian closed, and $(Top_0)_{ex/lex}$ is cartesian closed (in fact, [[locally cartesian closed category|locally cartesian closed]]) since $Top_0$ is weakly locally cartesian closed.

Moreover, in this way we can see that the embedding $Top_0 \to Equ$ preserves all existing [[exponential law for spaces|exponentials]], since the embedding $C \to C_{ex/lex}$ does so, and $Equ$ is closed under exponentials in $(Top_0)_{ex/lex}$ and contains the image of $Top_0$.  This embedding also preserves all [[limits]], but it does not in general preserve [[colimits]].

## Related concepts

* [[effective topological space]]

## References

The concept was originally introduced for [[domain theory]] in [a privately circulated manuscript](#Scott) by [[Dana Scott]].

* {#Scott}[[Dana Scott]], _A New Category? Domains, Spaces, and Equivalence Relations_,

It is then discussed in more detail in

* {#Bauer00} [[Andrej Bauer]], section 4 of _The Realizability Approach to
Computable Analysis and Topology_, PhD thesis CMU (2000) ([pdf](http://andrej.com/thesis/thesis.pdf))

* {#BBS} [[Andrej Bauer]], [[Lars Birkedal]], [[Dana Scott]], _Equilogical Spaces_, 2001  ([ps](http://www.andrej.com/papers/equ-paper.ps), [[BBSEquilogical.pdf:file]])

[[!redirects equilogical space]]
[[!redirects equilogical spaces]]


equivalence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Equivalence
* table of contents
{: toc}

## Idea

In general two objects are considered _equivalent_ if they may be replaced by one another in all contexts under consideration (see also the *[[principle of equivalence]]*).

At the propositional level, the general theory of equivalence is discussed at [[equivalence relation]], but that leaves us with the question of the correct definition of equivalence in various situations.  Furthermore, we can work on a higher level and ask what the equivalences are, not just whether things are equivalent.

## In higher category theory

In [[higher category theory]] an **equivalence** often means a morphism which is invertible in a maximally weak sense (that is, up to all higher equivalences).  Two objects are **equivalent** if there is an equivalence between them.  For example:

* In a [[set]] (a $0$-[[0-category|category]]), equivalence is just [[equality]].

* In a [[category]] (a $1$-[[1-category|category]]), an equivalence is an [[isomorphism]].

* In a (possibly weak) $2$-[[2-category|category]], an equivalence is a morphism $f : x \to y$ such that there exists a morphism $g : y\to x$ and $2$-cell isomorphisms $f g \cong 1_y$ and $g f \cong 1_x$.  In the $2$-category [[Cat]] this reduces to the usual notion of [[equivalence of categories]] (although one must define $\Cat$ carefully using [[anafunctor]]s if one wants to avoid the [[axiom of choice]]).

* In a (possibly weak) $3$-[[3-category|category]], an equivalence is a morphism $f : x \to y$ such that there exists a morphism $g : y\to x$ and 2-cells $f g \to 1_y$ and $g f \to 1_x$ which are equivalences in the appropriate hom-2-categories.  In the (weak) 3-category $Bicat$ of [[bicategory|bicategories]], pseudofunctors, pseudonatural transformations, and modifications, such equivalences are often called _biequivalences_, and that term is sometimes also used for an equivalence in any $3$-category.

* In a (possibly weak) $\infty$-[[infinity-category|category]], an equivalence is a morphism $f : x \to y$ such that there exists a morphism $g : y\to x$ and 2-cells $f g \to 1_y$ and $g f \to 1_x$ which are equivalences in the appropriate hom-$\infty$-categories.  This can be made rigorous as a [[corecursion|corecursive definition]].

There is also the more structured/coherent notion of an [[adjoint equivalence]], but one hopes to have a theorem that any equivalence can be improved to an adjoint one.  This is known in some cases, including 2-categories, 3-categories, strict $\omega$-categories, and some models for $(\infty,1)$-categories.

In addition to this, there is the notion of [[equivalence of categories]] (and higher categories), which is on a separate page.  But it is related: properly done, two $n$-categories are equivalent if and only if they\'re equivalent as objects in the $(n+1)$-category of $n$-categories.

## In homotopy theory

In the category [[Top]] of topological spaces, notions weaker than isomorphism become very important: [[homotopy equivalence]] and [[weak homotopy equivalence]].  The latter notion is generalized to [[weak equivalence]] for objects in any [[model category]].  There is also a notion of equivalence between model categories: [[Quillen equivalence]].

These can be understood to some extent using higher categories.  For example, topological spaces should be weakly homotopy-equivalent if and only if they have equivalent [[fundamental infinity-groupoid|fundamental infinity-groupoids]].  Similarly, Quillen equivalent model categories give rise to equivalent [[(infinity,1)-category|(infinity,1)-categories]].

See [[equivalence in an (infinity,1)-category]].

## In homotopy type theory

In [[homotopy type theory]] equivalences can be axiomatized as those [[terms]] of [[function types]] all whose [[homotopy fibers]] are [[contractible]].

See _[[equivalence in homotopy type theory]]_.

## Related concepts

* [[principle of equivalence]]

* [[identity morphism]]

* [[identity type]]

* [[identity of indiscernibles]]

* [[equality]]

* [[isomorphism]], [[natural isomorphism]]

* **equivalence**, [[natural equivalence]]

* [[weak equivalence]]

* [[logical equivalence]]

* [[equivalence of groupoids]], [[equivalence of categories]]

* [[equivalence in a 2-category]]

* [[equivalence of 2-categories]], [[equivalence of (2,1)-categories]]

* [[homotopy equivalence]], [[weak homotopy equivalence]], [[Quillen equivalence]]

* [[Bousfield equivalence]]

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

* [[equivalence in homotopy type theory]]

* [[equivalence of (∞,1)-categories]]

[[!include logic symbols -- table]]

[[!redirects equivalence]]
[[!redirects equivalences]]
[[!redirects equivalent]]


equivalence class

# Equivalence classes
* table of contents
{: toc}

## Idea

An equivalence class is an [[element]] of a [[quotient set]].

## Definitions

There are a variety of ways to make this precise.

### Axiomatic

Let $S$ be a [[set]], and let ${\sim}$ be an [[equivalence relation]] on $S$.  Then there exists a set $S/{\sim}$, the __quotient set__ of $S$ modulo $\sim$.  Given any element $x$ of $S$, there is an element $[x]_{\sim}$ of $S/{\sim}$, the __equivalence class__ of $x$ modulo ${\sim}$.  Every element of $S/{\sim}$ is of this form.  Furthermore, $[x]_{\sim}$ and $[y]_{\sim}$ are equal in $S/{\sim}$ if and only if $x \sim y$ in $S$.

The [[axiom of quotients]] is an axiom of [[set theory]] which states that the paragraph above is true.  It corresponds to the clause in the definition of a [[pretopos]] (or in [[Giraud's axioms]] for a [[Grothendieck topos]]) that every [[congruence]] has a [[coequaliser]].  In most formulations of set theory, this axiom is not needed; instead, it is a theorem when equivalence classes are defined in one of the ways below.

In [[intensional type theory]] such as [[homotopy type theory]], [[quotient sets]] could be constructed as a [[higher inductive type]], and thus an equivalence class is an element of that higher inductive type.

### As subsets

Again, let $S$ be a set, and let ${\sim}$ be an equivalence relation on $S$.  Let $x$ be an element of $S$.  Then the __equivalence class__ of $x$ modulo ${\sim}$ is the [[subset]] of $S$ consisting of those elements of $S$ that are equivalent to $x$:
$$ [x]_{\sim} \coloneqq \{ y\colon S \;|\; x \sim y \} .$$
Then the __quotient set__ $S/{\sim}$ is the collection of these equivalence classes.

We may construct this collection using the [[power set]] of $S$; therefore, this may be done in any [[elementary topos]] as well as in such diverse set theories as [[ZFC]], [[SEAR]], and [[ETCS]].  This definition of equivalence class is quite natural in [[material set theory]], since it immediately produces a set (assuming that subsets are sets).

Any element $x$ of $S$ is a __representative__ of its equivalence class $[x]$.  Every equivalence class has at least one representative, and its representatives are all equivalent.  The set of representatives *is* the equivalence class in the material set-theoretic sense.

One usually defines properties of equivalence classes and functions on quotient sets by defining them for an arbitrary representative, then proving that the result is independent of the representative chosen.  This does *not* require the [[axiom of choice]].

### Redefined equality

In some [[foundations of mathematics]], sets are not fundamental, but are defined as more basic [[presets]] (sometimes called [[types]] or, confusingly, sets). By definition, a set (sometimes called a [[setoid]]) is a preset equipped with an equivalence (pre)relation.

Once more, let $S$ be a set, and let ${\sim}$ be an equivalence relation on $S$.  Then the quotient set $S/{\sim}$ is the the underlying preset of $S$ equipped with $\sim$ (in place of the original equality on $S$), and the equivalence class $[x]_{\sim}$ is simply $x$.

### As objects of a groupoid

One can also consider sets as [[groupoids]] (or $\infty$-[[infinity-groupoid|groupoids]]) with the property of being [[discrete groupoid|discrete]].

So again, let $S$ be a set, and let ${\sim}$ be an equivalence relation on $S$.  Then the quotient set $S/{\sim}$ is the (higher) groupoid whose objects are the same as those of $S$ and with a single morphism from $x$ to $y$ iff $x \sim y$ (and none otherwise); $[x]_{\sim}$ is simply $x$ again.

## Examples

* [[isomorphism class]]

* [[homotopy class]]

[[!redirects equivalence class]]
[[!redirects equivalence classes]]


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category: svg


equivalence extensionality

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

\tableofcontents

## Definition

In [[dependent type theory]] with [[identity types]], [[equivalence types]], and [[dependent function types]], **equivalence extensionality** characterizes the identity type of equivalence types and is the property that for all [[equivalence in type theory|equivalences]] $f:A \simeq B$ and $g:A \simeq B$, there are equivalences

$$
  \mathrm{equivext}(f, g)
  \;\;\colon\;\;
  \big(f =_{A \simeq B} g\big)
  \;\simeq\;
  \prod_{x \colon A}
  \,
  f(x) =_{B} g(x)
  \,.
$$

While [[function extensionality]] implies equivalence extensionality, it is unknown whether equivalence extensionality implies function extensionality under any conditions or whether equivalence extensionality is strictly weaker than function extensionality.

## See also

* [[equivalence type]]

* [[equivalence of types]]

* [[extensionality]]


equivalence in a 2-category

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#### 2-Category theory
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[[!include 2-category theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

A [[1-morphism]] in a [[2-category]] is an _[[equivalence]]_ if there exists another 1-morphism the other way around, such that both are [[inverses]] up to invertible [[2-morphisms]].

In [[2-category theory]] equivalences in a 2-category are sometimes also called "1-equivalences", to distinguish these weakly invertible [[1-morphisms]] from invertible [[2-morphisms]].

## Examples

* An equivalence in a [[1-category]] regarded as a [[2-category]] with only trivial [[2-morphisms]] is just an _[[isomorphism]]_.

* An equivalence in the 2-category [[Cat]] of all [[categories]] is an _[[equivalence of categories]]_.

* An equivalence in the 2-category of [[algebras]] with [[bimodules]] as 1-morphisms and [[intertwiners]] as 2-morphisms (see [here](bimodule#AsMorphismsInA2Category)) is a _[[Morita equivalence]]_.

## Related concepts

* [[homotopy equivalence]]

* [[equivalence in an (infinity,1)-category]]

[[!redirects equivalences in a 2-category]]
[[!redirects equivalences in 2-categories]]

[[!redirects 1-equivalence]]
[[!redirects 1-equivalences]]


equivalence in an (infinity,1)-category

[[!redirects equivalence in a quasi-category]]

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###Context###
#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#### Equality and Equivalence
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[[!include equality and equivalence - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

For $C$ a [[quasi-category]], a morphism $f : x \to y$ in $C$ (an edge in the underlying [[simplicial set]]) is an **[[equivalence]]** if its image in the [[homotopy category of an (infinity,1)-category|homotopy category]] $Ho(C)$ is an [[isomorphism]].

Equivalently, $f$ is an equivalence if it is the image of a functor of quasi-categories (i.e. a map of simplicial sets) out of the [[nerve]] $N(J)$, where $J$ is the [[interval groupoid]].  This is a quasi-categorical version of the general theorem-schema in [[higher category theory]] that any [[equivalence]] can be improved to an [[adjoint equivalence]].

## Related concepts

* [[equality]]

* [[isomorphism]]

* [[equivalence]]

* [[weak equivalence]]

* [[equivalence in a 2-category]]

* [[homotopy equivalence]], [[weak homotopy equivalence]]

* **equivalence in an (∞,1)-category**

* [[equivalence in homotopy type theory]]

* [[equivalence of (∞,1)-categories]]

[[!redirects equivalence in a quasicategory]]

[[!redirects (infinity,1)-equivalence]]
[[!redirects (∞,1)-equivalence]]
[[!redirects equivalence in an (∞,1)-category]]
[[!redirects equivalence in an (infinity,1)-category]]

[[!redirects equivalences in an (∞,1)-category]]
[[!redirects equivalences in an (infinity,1)-category]]


equivalence of (2,1)-categories

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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _equivalence of (2,1)-categories_ is the concept of [[equivalence of (infinity,1)-categories]] restricted along the inclusion of [[(2,1)-categories]] into all [[(infinity,1)-categories]].

Equivalently, the concept of [[equivalence of 2-categories]] restricted along the inclusion of $(2,1)$-categories into [[2-categories]].

## Properties

Hence in the presence of the [[axiom of choice]], a [[(2,1)-functor]] $f \colon \mathcal{C} \longrightarrow \mathcal{D}$ is an equivalence precisely if

1. it is essentially surjective, hence [[surjection|surjective]] on [[equivalence classes]] of [[objects]];

1. it is fully faithful in that for all objects $c_1,c_2\in \mathcal{C}$ the [[1-functor]] on [[hom-groupoids]] $f_{c_1,c_2}\colon \mathcal{C}(c_1,c_2) \longrightarrow \mathcal{D}(f(c_1),f(c_2))$ is an [[equivalence of groupoids]].

[[!redirects equivalences of (2,1)-categories]]


equivalence of (infinity,1)-categories

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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#### Equality and Equivalence
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[[!include equality and equivalence - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An [[(∞,1)-functor]] between [[(∞,1)-categories]] is an _[[equivalence in an (∞,1)-category|equivalence]]_ in [[(∞,1)Cat]] precisely if it is an [[essentially surjective (∞,1)-functor]] and a [[full and faithful (∞,1)-functor]].

When [[(∞,1)-categories]] are presented by [[quasi-categories]], an equivalence between them is presented by a [[weak equivalence]] in the  [[model structure for quasi-categories]].

## Properties

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

An [[(∞,1)-functor]] $f : C \to D$ is an **equivalence** in [[(∞,1)Cat]] if  the following equivalent conditions hold

* On the underlying [[simplicial sets]] it is a [[weak equivalence]] in Joyal's [[model structure for quasi-categories]] (a "[[weak categorical equivalence]]").

* For every [[simplicial set]] $K$ the induced morphism on [[mapping complexes]] $f_* \colon SSet(K,C) \to SSet(K,D)$ is a [[weak categorical equivalence]].

* For every [[simplicial set]] $K$ the induced morphism $f_* \colon Core(SSet(K,C)) \to Core(SSet(K,D))$ of [[cores]] (the maximal [[Kan complexes]]) is a [[weak equivalence]] in the [[Kan-Quillen model structure]], hence a [[simplicial weak homotopy equivalence]].

* The induced morphism $f_* : Core(SSet(\Delta^1, C)) \to Core(SSet(\Delta^1, D))$ is an equivalence of Kan complexes.

=--

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

The equivalence of the first three points is [[Higher Topos Theory|HTT, lemma 3.1.3.2]].

[Cisinki](#Cisinki), theorem 3.9.2, shows the third point can be weakened to taking just $K = \Delta^0, \Delta^1$, so to show equivalence with the fourth point, it suffices to show the $K = \Delta^1$ case implies the $K = \Delta^0$ case.

Suppose that $f_* : Core(SSet(\Delta^1, C)) \to Core(SSet(\Delta^1, D))$ is an equivalence. For each object $d \in D_0$, there is an edge $\varphi : c \to c'$ in $C_1$ such that $f_*(\varphi) \simeq id_d$. This implies $f(c) \simeq d$ and thus $f_*(id_c) \simeq id_d$, and that $\varphi \simeq id_c$.

Thus, $f_*$ restricts to an equivalence between the subcomplexes consisting of the connected components of the identity morphisms. But for any quasi-category $X$, this subcomplex is $\Core(X)^{\Delta^1} \subseteq \Core(X^{\Delta^1})$, and the degeneracy $\Core(X) \to \Core(X)^{\Delta^1}$ is an equivalence. Thus, the induced map $Core(C) \to Core(D)$ is an equivalence.

=--

## Related concepts

* [[equality]]

* [[isomorphism]]

* [[equivalence]]

* [[weak equivalence]]

* [[homotopy equivalence]], [[weak homotopy equivalence]]

* [[homotopy equivalence of toposes]]

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

* [[equivalence of categories]], [[adjoint functor]]

* [[equivalence of 2-categories]], [[2-adjunction]]

* [[equivalence of (2,1)-categories]]

* **equivalence of (∞,1)-categories**, [[adjoint (∞,1)-functor]]

[[!include properties of functors -- contents]]

[[!redirects equivalence of (∞,1)-categories]]
[[!redirects equivalences of (∞,1)-categories]]

[[!redirects equivalence of (infinity,1)-categories]]
[[!redirects equivalences of (infinity,1)-categories]]

## References

* {#Cisinksi} [[Denis-Charles Cisinski]], "Higher Categories and Homotopical Algebra"

[[!redirects equivalence of quasi-categories]]
[[!redirects equivalences of quasi-categories]]

[[!redirects equivalence of quasicategories]]
[[!redirects equivalences of quasicategories]]


equivalence of 2-categories

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### Context
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
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#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

\tableofcontents

## Idea

The notion of **equivalence of $2$-categories** is the appropriate notion of [[equivalence]] between [[2-categories]], [[categorification|categorifying]] the notion of [[equivalence of categories]]: a [[pair]] of [[2-functors]] back and forth between 2-categories, which are [[inverse]] to each other, up to [[pseudonatural equivalence]].

## Definition

\begin{definition}\label{EquivalenceOf2Categories}
An *equivalence* between [[2-categories]] $C$ and $D$ consists of

1. [[2-functors]]$\;$ $F \,\colon\, \mathcal{C} \to \mathcal{D}$ and $G \,\colon\, \mathcal{D} \to \mathcal{C}$,

1. [[pseudonatural transformations]] ${}\;G \circ F \to Id_{\mathcal{C}}$ and $F \circ G \to Id_{\mathcal{D}}$ which are themselves [[equivalences]],

\end{definition}

\begin{remark}
Def. \ref{EquivalenceOf2Categories} makes sense, and is used, both in the case that $F$ is [[strict 2-functor|strict]], and in the case that it is [[weak 2-functor|weak]]. Note however that in this case $G$ should be allowed to be weak: see  [Lack 2002, Ex, 3.1](#Lack2002).
\end{remark}

\begin{remark}
In the literature this sort of equivalence in Def. \ref{EquivalenceOf2Categories} is often called a **biequivalence**, as it has traditionally been associated with [[bicategories]], the standard algebraic definition of weak $2$-category.

There is a stricter notion of equivalence for strict $2$-categories, which traditionally is called just a *$2$-equivalence* and which on the nLab is called a *[[strict 2-equivalence]]*.
\end{remark}

## Properties

\begin{prop}\label{CharacterizationOfEquivalencesOf2Categories}
**(recognition of [[equivalences of 2-categories]] assuming the [[axiom of choice]])**
\linebreak
  Assuming the [[axiom of choice]], a 2-functor $F \,\colon\, \mathcal{C} \xrightarrow{\;} \mathcal{D}$ is an [[equivalence of 2-categories]] precisely if it is

1. **essentially surjective**:

   [[surjection|surjective]] on [[equivalence in a 2-category|equivalence]] [[equivalence classes|classes]] of [[objects]]: $\pi_0(F) \;\colon\; \pi_0(\mathcal{C}) \twoheadrightarrow \pi_0(\mathcal{D})\;$,

1. **fully faithful** (e.g. [Gabber & Ramero 2004, Def. 2.4.9 (ii)](#GabberRamero04)):

   for each [[pair]] of [[objects]] $X,\, Y \in \mathcal{C}$ the component functor is an [[equivalence of categories|equivalence of]] [[hom-categories]] $F_{X,Y} \,\colon\, \mathcal{C}(X,Y) \xrightarrow{\simeq} \mathcal{D}\big(F(X), F(Y)\big)$,

   which by the analogous theorem for 1-functors ([this Prop.](equivalence+of+categories#ViaEssentiallySurjectiveAndFullyFaithful)) means equivalently that $F$ is (e.g. [Johnson & Yau 2020, Def. 7.0.1](#JohnsonYau20))

   1. **essentially full on 1-cells**:

      namely that each component functor $F_{X,Y}$ is an [[essentially surjective functor]];

   1. **fully faithful on 2-cells**:

      namely that each component functor $F_{X,Y}$ is a [[fully faithful functor]].

\end{prop}

This is classical [[folklore]]. It is made explicit in, e.g. [Gabber & Ramero 2004, Cor. 2.4.30](#GabberRamero04); [Johnson & Yau 2020, Thm. 7.4.1](#JohnsonYau20).

## Internalization

Just as the notion of [[equivalence of categories]] can be [[internalization|internalized]] in any $2$-category, the notion of equivalence for $2$-categories can be internalized in any $3$-category in a straightforward way.  The version above for $2$-categories then results from specializing this general definition to the (weak) $3$-category $2 Cat$ of $2$-categories, (weak) $2$-functors, pseudonatural transformations, and modifications.

There is one warning to keep in mind here.  Every $3$-category is equivalent to a [[semi-strict n-category|semi-strict]] sort of $3$-category called a [[Gray-category]], since it is a category [[enriched category|enriched]] over the monoidal category [[Gray]] of strict $2$-categories and strict $2$-functors.  Of course $Gray$ itself is a Gray-category, but as such it is [not](http://arxiv.org/abs/math/0612299) equivalent to the weak $3$-category $2 Cat$ of weak $2$-categories and *weak* $2$-functors.

In particular, an "internal (bi)equivalence" in $Gray$ consists of *strict* $2$-functors $F,G$ together with pseudonatural equivalences relating $G F$ and $F G$ to identities.  This is a semistrict notion of equivalence, intermediate between the fully weak notion and the fully strict one.

## Related concepts

* [[equality]], [[isomorphism]], [[equivalence]]

* [[weak equivalence]], [[homotopy equivalence]], [[weak homotopy equivalence]]

* [[homotopy equivalence of toposes]]

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

* [[equivalence of categories]], [[adjoint functor]]

* **equivalence of 2-categories**, [[2-adjunction]]

* [[equivalence of (2,1)-categories]]

* [[equivalence of (∞,1)-categories]], [[adjoint (∞,1)-functor]]

[[!include properties of functors -- contents]]

## References

* {#Lack2002} [[Stephen Lack]], _A Quillen model structure for 2-categories_, K-Theory 26, No. 2, 171-205 (2002). [Zentralblatt review](https://zbmath.org/?q=an%3A1017.18005) [author's webpage](http://maths.mq.edu.au/~slack/papers/qmc2cat.html)

* {#GabberRamero04} [[Ofer Gabber]], [[Lorenzo Ramero]], Def. 2.49 with Cor. 2.4.30 in: *Foundations for almost ring theory* ([arXiv:math/0409584](https://arxiv.org/abs/math/0409584))

* {#Zheng12} [[Weizhe Zheng]], Def. 1.6 - Lem. 1.8 of: _Gluing pseudofunctors via $n$-fold categories_, J. Homotopy Relat. Struct. **12** 189–271 (2017) ([arXiv:1211.1877](http://arxiv.org/abs/1211.1877), [doi:10.1007/s40062-016-0126-2](https://doi.org/10.1007/s40062-016-0126-2))

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Section 7 of: _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

[[!redirects equivalence of 2-categories]]
[[!redirects equivalences of 2-categories]]
[[!redirects equivalence of bicategories]]
[[!redirects equivalences of bicategories]]

[[!redirects eqivalence of 2-categories]]
[[!redirects eqivalences of 2-categories]]

[[!redirects equivalence of 2-groupoids]]
[[!redirects equivalences of 2-groupoids]]
[[!redirects equivalence of bigroupoids]]
[[!redirects equivalences of bigroupoids]]

[[!redirects biequivalence]]
[[!redirects biequivalences]]
[[!redirects biequivalent]]