reflexive coequalizer

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# Reflexive coequalisers
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## Definitions

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

A **reflexive pair** is a [[parallel pair]] $f,g\colon A\rightrightarrows B$ having a common [[section]], i.e. a map $s\colon B\to A$ such that $f \circ s = g \circ s = 1_B$.  A **reflexive coequalizer** is a [[coequalizer]] of a reflexive pair.  A category **has reflexive coequalizers** if it has coequalizers of all reflexive pairs.

Dually, a reflexive coequalizer in the [[opposite category]] $C^{op}$ is called a __coreflexive equalizer__ in $C$.

=--

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

Reflexive coequalizers should not be confused with [[split coequalizers]], a distinct concept.

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

Any [[congruence]] is a reflexive pair, so in particular any [[quotient object|quotient]] of a congruence is a reflexive coequalizer.

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## Properties

+-- {: .num_theorem #LintonTheorem}
###### Theorem

If $T$ is a [[monad]] on a [[cocomplete category]] $C$, then the category $C^T$ of [[Eilenberg-Moore category|Eilenberg Moore algebras]] is cocomplete if and only if it has reflexive coequalizers.  This is the case particularly if $T$ preserves reflexive coequalizers.

=--

This is due to ([Linton](#Linton)).

+-- {: .proof}
###### Proof
Suppose $C^T$ has reflexive coequalizers. Then $C^T$ certainly has coproducts, because if $A_i$ is a collection of $T$-algebras, then we can form the coequalizer in $C^T$ of the reflexive pair

$$
  \sum_i F U F U A_i
   \underoverset
     {\sum_i F U \varepsilon A_i}
     {\sum_i \varepsilon  F U A_i}
     {\rightrightarrows}
  \sum_i F U A_i$$

using the fact that the displayed coproducts exist because, for example,

$$\sum_i F U A_i \cong F(\sum_i U A_i)$$

since the left adjoint $F$ preserves coproducts, assumed to exist in $C$. That this reflexive coequalizer is the coproduct $\sum_i A_i$ in $C^T$ is routine.

Finally, a category with coproducts and reflexive coequalizers is cocomplete. It suffices that general coequalizers exist, but it is easily seen that if

$$f, g \colon A \stackrel{\to}{\to} B$$

is a parallel pair, then the coequalizer of the reflexive pair

$$A + B \stackrel{\overset{(f, 1_B)}{\to}}{\underset{(g, 1_B)}{\to}} B$$

(note both maps are retracts of the inclusion $B \to A + B$) also exists, and gives the coequalizer of the first pair. This completes the proof.
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+-- {: .num_prop #PreservingReflectiveCoequalizersInTwoVariables}
###### Proposition

If $F\colon C\times D\to E$ is a [[functor]] of two variables which preserves reflexive coequalizers in each variable separately (that is, $F(c,-)$ and $F(-,d)$ preserve reflexive coequalizers for all $c\in C$ and $d\in D$), then $F$ preserves reflexive coequalizers in both variables together.

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

This is emphatically *not* the case for arbitrary coequalizers.

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+-- {: .proof}
###### Proof
**of proposition \ref{PreservingReflectiveCoequalizersInTwoVariables}**
Suppose given two reflexive coequalizers

$$c_0 \stackrel{\to}{\to} c_1 \to c_2$$

$$\,$$

$$d_0 \stackrel{\to}{\to} d_1 \to d_2$$

and let $c_{i j}$ denote $F(c_i, d_j)$ so that we have a diagram

$$\array{
c_{0 0} & \stackrel{\to}{\to} & c_{0 1} & \to & c_{0 2} \\
\downarrow \downarrow & & \downarrow \downarrow & & \downarrow \downarrow \\
c_{1 0} & \stackrel{\to}{\to} & c_{1 1} & \to & c_{1 2} \\
\downarrow & & \downarrow & & \downarrow \\
c_{2 0} & \stackrel{\to}{\to} & c_{2 1} & \to & c_{2 2}
}$$

in which all rows and columns are reflexive coequalizers (using preservation of reflexive coequalizers in separate variables), and all squares are _serially_ commutative. According to [Toposes, Triples, Theories](#BarrWells), lemma 4.2 page 248, the diagonal is also a (reflexive) coequalizer, as claimed. (See also the lemma on page 1 of Johnstone's [Topos Theory](#Johnstone).)
=--

Proposition \ref{PreservingReflectiveCoequalizersInTwoVariables} is particularly interesting when $F$ is the [[tensor product]] of a cocomplete [[closed monoidal category]] $C$.  In this case it implies that the [[free monoid monad]] on such a category preserves reflexive coequalizers, and thus (by [Linton's theorem](#LintonTheorem)) the category of [[monoid objects]] in $C$ is cocomplete.

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

Reflexive coequalizers in [[Set]] [[limits commuting with colimits|commute]] with [[finite products]]:

the $n$-fold product functors $Set^n \stackrel{\prod}{\to} Set$ preserve reflexive coequalizers.

=--

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

This follows from prop. \ref{PreservingReflectiveCoequalizersInTwoVariables} as well as from the fact that the [[diagram]] category $\{ 0 \stackrel{\overset{d_0}{\to}}{\stackrel{\overset{s_0}{\leftarrow}}{\underset{d_1}{\to}}} 1\}$ with $d_0 \circ s_0 = d_1 \circ s_0 = id$ is a [[sifted category]].

=--

Of course, the diagonal functor $\Delta: Set \to Set^n$, being [[left adjoint]] to the product functor, preserves reflexive coequalizers; therefore the composite

$$Set \stackrel{\prod \Delta}{\to} Set: x \mapsto \hom(n, x)$$

also preserves reflexive coequalizers.

This has a further consequence which is technically very convenient:

+-- {: .num_theorem}
###### Theorem

If $T$ is a [[finitary monad]] on $Set$, then $T$ preserves reflexive coequalizers.

=--

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

We have a [[coend]] formula for $T$:

$$T(-) \cong \int^{n \in Fin} T(n) \times \hom(n, -)$$

and since this is a colimit of functors $\hom(n, -)$ which preserve reflexive coequalizers, $T$ must also preserve reflexive coequalizers.

=--

Since finitary monads $T$ preserve reflexive coequalizers, it follows that the monadic functor $U \colon Set^T \to Set$ reflects reflexive coequalizers, and so since $Set$ has reflexive coequalizers, $Set^T$ must as well. Therefore, by proposition \ref{LintonTheorem}, $Set^T$ is cocomplete. This is actually true for infinitary monads $T$ on $Set$ as well, at least if we assume the axiom of choice (see [here](http://ncatlab.org/nlab/show/colimits+in+categories+of+algebras#reflexive_coequalizers_and_cocompleteness_16) for a proof), but the argument just given is a choice-free proof for the case of finitary monads.

## Applications

* Reflexive coequalizers figure in the [[crude monadicity theorem]].

## See also

* [[local reflexive coequalizer]]

## References

* {#Linton} [[Fred Linton]], *Coequalizers in categories of algebras*, in: *Seminar on Triples and Categorical Homology Theory*, Lecture Notes in Mathematics **80** (1969) 75-90 [[doi:10.1007/BFb0083082](https://doi.org/10.1007/BFb0083082)]

* [[Michael Barr]], [[Charles Wells]], *[[Toposes, Triples, and Theories]]*, Grundlehren der math. Wissenschaften **278** Springer (1983), republished in: Reprints in Theory and Applications of Categories **12** (2005) 1-287 [[tac:tr12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html) in [[TAC reprints series]]]

* {#Johnstone} [[Peter Johnstone]], *Topos Theory*, London Mathematical Society Monographs **10**, Academic Press (1977)

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reflexive completion

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## Definition

The **reflexive completion** (also called the **Isbell completion**) of a category $A$ is the [[fixed point of an adjunction|fixed point]] of the [[Isbell duality]] adjunction for $A$.

## Properties

* The reflexive completion is idempotent: $\mathcal{R}(\mathcal{R}(A)) \simeq \mathcal{R}(A)$.

* The reflexive completion is [[Cauchy complete]]. The Cauchy completion $\bar A$ of a small category $A$ is a full subcategory of the reflexive completion $\mathcal{R}(A)$. We have that $\mathcal{R}(\bar A) \simeq \mathcal{R}(A)$. Hence [[Morita equivalent]] categories have equivalent reflexive completions. See §10 of [Avery and Leinster](#AveryLeinster).

* $A \to \mathcal{R}(A)$ [[created limit|creates]] limits and colimits. $\mathcal{R}(A) \to \hat{A}$ preserves limits and [[reflected limit|reflects]] limits and colimits. See §11 of [Avery and Leinster](#AveryLeinster).

* The reflexive completion of a small category has an [[initial object]] and a [[terminal object]].

* The reflexive completion is a full subcategory of the [[Isbell envelope]], and can be concretely described as a [[2-pullback]]:
\begin{tikzcd}
	{\mathcal R(A)} & {[A^{\mathrm{op}}, \mathrm{Set}]} \\
	{[A, \mathrm{Set}]^{\mathrm{op}}} & {\mathcal I(A)}
	\arrow[hook, from=1-1, to=1-2]
	\arrow[hook, from=1-1, to=2-1]
	\arrow[hook, from=1-2, to=2-2]
	\arrow[hook, from=2-1, to=2-2]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=2-2]
\end{tikzcd}
See [[envelope of an adjunction]].

## Related concepts

* [[Isbell duality]]

* [[Isbell envelope]]

* [[fixed point of an adjunction]]

* [[MacNeille completion]]

## References

* {#AveryLeinster} [[Tom Avery]], [[Tom Leinster]]. _Isbell conjugacy and the reflexive completion_. Theory and Applications of Categories, **36** 12 (2021) 306-347 [[tac:36-12](http://www.tac.mta.ca/tac/volumes/36/12/36-12abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf)]

* [[Richard Garner]], *Topological functors as total categories*, [TAC](http://www.tac.mta.ca/tac/volumes/29/15/29-15abs.html)

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reflexive graph

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## Idea

A [[graph]] is _reflexive_ if for each [[vertex]] $v$ there is a (specified) [[edge]] $v \to v$.

A **reflexive [[quiver]]** has a specified identity edge $i_X: X \to X$ on each object (vertex) $X$.
The [[free category]] on a reflexive quiver has the same objects,
 [[identity morphism]]s corresponding to the identity edges,
 and non-identity morphisms consisting of paths of non-identity edges.

## Properties

+-- {: .num_prop}
###### Proposition
The category of reflexive directed graphs $RefGph$, i.e., reflexive [[quivers]], equipped with the functor $U: RefGph \to Set$ which sends a graph to its set of edges, is [[monadic functor|monadic]] over $Set$.
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+-- {: .proof}
###### Proof
$RefGph$ is the category of functors $R \to Set$ where $R$ is the [[walking structure|walking]] [[reflexive fork]], consisting of two objects $0, 1$ and generated by arrows $i: 0 \to 1$ and $s, t: 1 \to 0$ subject to $s i = 1_0 = t i$ and no other relations. This $R$ is in turn the [[Cauchy completion]] of a monoid $M$ consisting of two elements $e_0 = i s, e_1 = i t$ and an identity, with multiplication $e_0 e_0 = e_0 = e_1 e_0$ and $e_1 e_1 = e_1 = e_0 e_1$, and therefore $RefGph$ is equivalent to the category of functors $M \to Set$, i.e., the category of $M$-sets $Set^M$. This is the category of algebras of the monad $M \times -$ whose monad structure is induced from the monoid structure of $M$.
=--

$Cat$ is monadic over $RefGph$ and $RefGph$ is monadic over $Set$ but $Cat$ is not monadic over $Set$; this is a nice example of how the relation 'being monadic over' is not transitive.

## In dependent type theory

In [[dependent type theory]], a graph $(A; x:A, y:A \vdash R(x, y))$ is **reflexive** if

* it comes with a family of elements $x:A \vdash \mathrm{refl}_A^R(x):R(x, x)$.
* it comes with a family of functions $\mathrm{idtofam}(x, y):(x =_A y) \to R(x, y)$.

These two definitions are the same. From $\mathrm{refl}_A^R(x)$, we can inductively define $\mathrm{idtofam}(x, y)$ on reflexivity:

$$\mathrm{idtofam}(x, x)(\mathrm{refl}_A(x)) \coloneqq \mathrm{refl}_A^R(x)$$

From $\mathrm{idtofam}(x, y)$ we define $\mathrm{refl}_A^R(x)$ to be
$$\mathrm{refl}_A^R(x) \coloneqq \mathrm{idtofam}(x, x)(\mathrm{refl}_A(x))$$

## Related concepts

* [[reflexive relation]]
* [[univalent reflexive graph]]

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reflexive object

> This entry is about a notion of reflexivity in [[cartesian closed categories]] of relevance in the [[categorical semantics]] of [[lambda calculus]]. For the notion of *reflexive [[weak dual objects]]* in [[closed monoidal categories]] see [there](dualizable+object#ReflexiveDualizableObject).

***

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## Idea

A _reflexive object_ is a model of the pure (untyped) [[lambda calculus]] validating the $\beta$ law, and sometimes also the $\eta$ law.

## Definition

A **reflexive object** in a [[cartesian closed category]] is an [[object]] $U$ equipped with a pair of [[morphisms]]

$$
\array{U & \overset{app}{\underset{lam}{\rightleftarrows}} & U^U}
$$

(where $(-)^{(-)}$ denotes the [[exponential object]])
such that $app \circ lam = 1$.

In other words, a reflexive object is an object $U$ together with data $U^U \lhd U$ exhibiting $U^U$ as a [[retract]] of $U$.

A reflexive object is said to be **extensional** (or "strict") when also $lam \circ app = 1$, so that there is an isomorphism $U^U \cong U$.

Viewed as a model of lambda calculus, the equation $app \circ lam = 1$ of a reflexive object represents $\beta$-equality $(\lambda x.t)(u) = t[u/x]$, while the equation $lam \circ app = 1$ of an extensional reflexive object represents $\eta$-equality $\lambda x.t(x) = t$.

### In a closed (symmetric) monoidal category

The definition has a straightforward generalization to any [[symmetric monoidal closed category]] (i.e., not necessarily [[cartesian monoidal category|cartesian monoidal]]), where we just replace the [[exponential object]] $U^U$ by the [[internal hom]] $[U,U]$.  Indeed, the definition also makes sense in any left-closed or right-closed monoidal category.

A reflexive object in a symmetric monoidal closed category provides a model of [[linear lambda calculus]].

### In a closed monoidal 2-category

The definition of reflexive object also has a natural generalization to any closed monoidal [[2-category]] (or cartesian closed 2-category), where the retraction should be replaced by an [[adjunction]] $app \dashv lam : [U,U] \to U$.  Then the [[counit]] of the adjunction $app \circ lam \Rightarrow 1_{[U,U]}$ models [[beta-reduction]] $(\lambda x.t)(u) \to t[u/x]$, while the [[unit]] $1_U \Rightarrow lam \circ app$ models [[eta-expansion]] $t \to \lambda x.t(x)$.

## Examples

### Terminal object

The [[terminal object]] of a ccc provides a degenerate example of a (extensional) reflexive object, and for cardinality reasons, this is the only reflexive object in [[Set]].

### Scott's $D_\infty$ construction

The first non-degenerate model of untyped lambda calculus was described by [[Dana Scott]] (see [Scott (1970)](#Scott70) and [Scott (1972)](#Scott72)), who solved the cardinality issue by replacing sets with [[algebraic lattices]], and arbitrary set-theoretic functions by [[Scott topology|Scott-continuous]] functions.  The so-called "$D_\infty$" model is built by a limit construction, starting from $D_0 = D$ an arbitrary algebraic lattice, and taking $D_{n+1} = D_n \to D_n$.

### Enumeration operator model

Scott later gave a more concrete model of lambda calculus in [Scott (1976)](#Scott76), defining a reflexive object with carrier the lattice $P\omega$ of all [[subsets]] of the non-negative integers, and with the maps $app$ and $lam$ (there called "fun" and "graph") defined as follows:

$$app(u)(x) = \{m \mid \exists e_n \subseteq x. (n,m) \in u\}$$

$$lam(f) = \{(n,m) \mid m \in f(e_n)\}$$

Here $e_n$ stands for the set whose elements are the exponents in the binary expansion of $n$ (thus $e_n$ is the $n$th subset in the standard enumeration of finite subsets of $\omega$), while "$(n,m)$" stands for the standard enumeration of pairs of integers

$$(n,m) = \frac{1}{2}(n+m)(n+m+1)+m$$

Note that the enumeration operator model is not "extensional" in the sense that it only validates $\eta$-conversion as an inclusion $u \subseteq \ell(a(u))$, which is not an equality in general.

### Scott's representation theorem

[Scott (1980)](#Scott80) proved a sort of representation theorem for pure lambda calculus, showing that _any_ "type-free theory" (a.k.a., [[lambda theory]]) can be realized as a reflexive object.  He began by constructing a category whose objects are the closed lambda terms $A$ such that one can prove $A = \lambda x.A(A(x))$, and whose morphisms $f : A \to B$ are (equivalence classes of) closed terms $f$ such that one can prove $f = \lambda x.B(f(A(x)))$ (cf. [[Karoubi envelope]]), with identity and composition defined by
$$
1_A = A
\qquad
f \circ g = \lambda x.f(g(x))
$$
This category is cartesian closed, where products and exponentials are defined (on objects) by
$$
A \times B = \lambda u\lambda z.z(A(u(\lambda x\lambda y.x)))(B(u(\lambda x\lambda y.y)))
\qquad
A \to B = \lambda f.B \circ f \circ A
$$
Now, the identity term $U = \lambda x.x$ lives inside this category since obviously $U = U\circ U$, and moreover, every object $A$ is a retract of $U$, since $A : A \to U$ and $A : U \to A$ and $A \circ A = A = 1_A$. Thus, $U$ equipped with its retraction from $U \to U$ is a reflexive object.

A more abstract analysis of Scott's representation theorem appears in [Hyland (2013)](#Hyland13).

### Higher-order abstract syntax

Another way of looking at the _free_ cartesian closed category containing a reflexive object is as a representation of pure lambda terms (up to $\beta$-equality) in the style of so-called [[higher-order abstract syntax]].  The operation $app : U \to U^U$ is the [[currying|curried]] form of application, while the operation $lam : U^U \to U$ represents lambda abstraction.  The [[global elements]] of $U$ can be interpreted as closed lambda terms, and more generally, morphisms $U^n \to U$ can be interpreted as terms with $n$ free variables.

## References

Dana Scott's original work on lattice models of the lambda calculus can be found in:

* {#Scott70} [[Dana S. Scott]], *Outline of a mathematical theory of computation*, in: Proceedings of the *Fourth Annual Princeton Conference on Information Sciences and Systems* (1970) 169–176. [[pdf](https://ropas.snu.ac.kr/~kwang/520/readings/sco70.pdf), [[Scott-TheoryOfComputation.pdf:file]]]

* {#Scott72} [[Dana Scott]]. Continuous Lattices. _Toposes, Algebraic Geometry and Logic_ (ed.: E. Lawvere), pages 97-136, Lecture Notes in Mathematics, vol 274. 1972. doi:[10.1007/BFb0073967](https://doi.org/10.1007/BFb0073967)  ([pdf](https://www.cs.ox.ac.uk/files/3229/PRG07.pdf))

* {#Scott76} [[Dana Scott]], *Data types as lattices*. SIAM Journal of Computing **5** 3 (1976) 522--587 [[doi:10.1137/0205037](https://doi.org/10.1137/0205037), [pdf](https://www.cs.ox.ac.uk/files/3287/PRG05.pdf)]

Scott made the underlying categorical structure explicit in a later paper, where he introduced the definition of a reflexive object in a ccc, and proved a representation theorem for theories of pure lambda calculus:

* {#Scott80} Dana Scott. Relating theories of the $\lambda$-calculus. In _To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism_ (eds. Hindley and Seldin), Academic Press, 403--450, 1980.

For a modern analysis of Scott's representation theorem, see:

* {#Hyland13} [[Martin Hyland]]. _Classical lambda calculus in modern dress_ Mathematical Structures in Computer Science, 27(5) (2017) 762-781. doi:[10.1017/S0960129515000377](https://doi.org/10.1017/S0960129515000377). [arxiv:1211.5762](http://arxiv.org/abs/1211.5762)

* {#Hyland14} [[Martin Hyland]], _Towards a Notion of Lambda Monoid_ , Electronic Notes in Theoretical Computer Science **303** (2014) pp.59-77, doi:[10.1016/j.entcs.2014.02.004](https://doi.org/10.1016/j.entcs.2014.02.004)

[[2-category|Bicategorical]] and [[monoidal category|monoidal]] generalizations of the notion of reflexive object are discussed in:

* [[R. A. G. Seely]]. _Modelling Computations : a 2-categorical Framework_. LICS 1987. ([pdf](http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf))

* [[Bart Jacobs]]. *Semantics of lambda-I and of other substructure lambda calculi*, in:  M. Bezem and J.F. Groote (eds.) _Typed Lambda Calculi and Applications_, Springer
LNCS 664, 1993, p. 195-208 ([doi:10.1007/BFb0037107](https://doi.org/10.1007/BFb0037107))

* [[Noam Zeilberger]], *Linear lambda terms as invariants of rooted trivalent maps*, Journal of Functional Programming, 26, e21  2016 ([doi:10.1017/S095679681600023X](https://doi.org/10.1017/S095679681600023X), [arXiv:1512.06751](http://arxiv.org/abs/1512.06751))

## Related concepts

* [[lambda calculus]]
* [[domain theory]]
* [[lambda theory]]


reflexive relation

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## Definition

A (binary) [[relation]] $\sim$ on a set $A$ is __reflexive__ if every element of $A$ is related to itself:
$$\forall (x: A),\; x \sim x$$

In the language of the $2$-poset [[Rel]] of sets and relations, a relation $R: A \to A$ is __reflexive__ if it contains the identity relation on $A$:
$$\id_A \subseteq R$$

## Relation to graphs

A set with a reflexive relation is the same as a [[loop digraph object|loop]] [[digraph]] $(V, E, s:E \to V, t:E \to V)$ with function $refl:V \to E$ such that

* for every $a \in V$, $s(refl(a)) =_V a$
* for every $a \in V$, $t(refl(a)) =_V a$

## Related concepts

* [[relation]]

  * **reflexive relation**, [[irreflexive relation]]

  * [[symmetric relation]]

* [[first law of thought]]

* [[equivalence relation]]

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reflexive set

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## Definition

### In unsorted set theory

In [[unsorted set theory]], a **reflexive set** is a set that belongs to itself:
$$ X \in X .$$
Equivalently, $X$ is reflexive iff it equals its [[successor]] $X \cup \{X\}$.  Compare [[transitive sets]].

If the [[axiom of foundation]] holds in a set theory, then there are no reflexive sets.  In non-well-founded set theory, however, there may be many reflexive sets.

A __Quine atom__ is a minimally reflexive set:
$$ X = \{X\} .$$

### In two-sorted set theory

In [[two-sorted set theory]] with [[element reflection]], a **reflexive set** is a set whose element reflection belongs to itself:
$$\mathrm{asElem}(X) \in X$$
Similarly, in a [[two-sorted set theory]] with [[set reflection]], a **reflexive element** is an element which belongs to its set reflection:
$$X \in \mathrm{asSet}(X)$$

## Examples

In [[Peter Aczel]]\'s ill-founded set theory, there is a unique Quine atom.  On the other hand, by exempting Quine atoms (and only Quine atoms) from the axiom of foundation, one obtains a theory of pure sets equivalent to well-founded material sets with [[urelements]].

### Fully formal ETCS

Take any [[single-sorted definition of a category|single-sorted definition]] of a [[well-pointed topos]] $\mathcal{E}$, such as [[fully formal ETCS]], which by definition has a [[morphism]] $1$ representing the [[terminal object]], the [[identity morphism]] of the terminal object, and the single [[global element]] of the terminal object. Sets are represented by morphisms with [[codomain]] $1$, and elements are represented by morphisms with [[domain]] $1$. Thus, we define the [[predicates]] $\mathrm{set}(f) \coloneqq \mathrm{codom}(f) = 1$ and $\mathrm{element}(f) \coloneqq \mathrm{dom}(f) = 1$. The codomain and domain of general functions are then defined as usual to be sets, $\mathrm{set}(\mathrm{codom}(f))$ and $\mathrm{set}(\mathrm{dom}(f))$. We define the membership relation $a \in b$ as requiring the morphism $a$ to be an [[element]], the morphism $b$ to be a [[set]], and the codomain of $a$ to be $b$:

$$a \in b \coloneqq \mathrm{element}(a) \wedge \mathrm{set}(b) \wedge \mathrm{codom}(a) = b$$

The terminal object $1$ is then a [[Quine atom]] with respect to $\in$.

## See also

* [[unsorted set theory]]
* [[fully formal ETCS]]

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reflexive symmetric relation

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#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

\tableofcontents

## Definition

An **reflexive symmetric relation** is a [[binary relation]] that is [[reflexive relation|reflexive]] and [[symmetric relation|symmetric]].

## Examples

* Every [[equivalence relation]] is an reflexive symmetric relation which is also a [[transitive relation]].

* The negation of a [[irreflexive symmetric relation]] is a reflexive symmetric relation.

* The [[double negation]] of every [[equivalence relation]] is an irreflexive symmetric relation. In [[constructive mathematics]], this is not guaranteed to be transitive.

## In constructive mathematics
 {#ConstructiveMathematics}

In constructive mathematics, the [[double negation]] of [[equality]] cannot in general be proven to be an [[equivalence relation]], because the [[denial inequality]] cannot be proved to be an [[apartness relation]]. Instead, the double negation of equality is only a stable reflexive symmetric relation. Indeed, since excluded middle cannot be proven, we could distinguish between the following reflexive symmetric relations in constructive mathematics:

* [[equality]]

* [[equivalence relations]], a transitive reflexive symmetric relation

* [[double negation]] of [[equality]]

* [[stable relation|stable]] reflexive symmetric relation, a reflexive symmetric relation $\approx$ which additionally satisfies $\neg \neg (a \approx b)$ implies $a \approx b$

* [[decidable relation|decidable]] reflexive symmetric relation, an reflexive symmetric relation $\approx$ which additionally satisfies $(a \approx b)$ or $\neg (a \approx b)$

## See also

* [[double negation]]
* [[irreflexive symmetric relation]]
* [[equivalence relation]]

[[!redirects reflexive symmetric relation]]
[[!redirects reflexive symmetric relations]]

[[!redirects stable reflexive symmetric relation]]
[[!redirects stable reflexive symmetric relations]]

[[!redirects decidable reflexive symmetric relation]]
[[!redirects decidable reflexive symmetric relations]]


refutation by contradiction

# Refutation by contradiction
* table of contents
{:toc}

## Idea

*Refutation by contradiction* is the "basic" method of [[formal proof]] for proving a [[negation]]: if assuming $P$ leads to a [[contradiction]], then we conclude $\neg P$.  It is the [[introduction rule]] for the [[connective]] of [[negation]] in [[intuitionistic logic]]; if $\neg P$ is defined as $P\to \bot$ (the [[function type]] from $P$ to [[false]]), then it follows from the introduction rule for [[implication]].

Refutation by contradiction is also called *proof of negation*, but this can be confusing because there may be other "indirect" ways to prove a negation.

Rarely one finds logics that have a basic notion of "refutation of $A$" that is distinct from (though generally closely related to) a "proof of $\neg A$" [(Nelson 1949)](#Nelson49).  In that case, refutation by contradiction is more precisely a method of refuting $A$ by deriving a contradiction from an assumption of $A$, which might be distinguished from other more "direct" ways of refuting $A$ (for example, a refutation of a [[universal quantifier|universally quantified]] statement $\forall x.B(x)$ that exhibits an explicit counterexample, i.e., a term $t$ together with a refutation of $B(t)$).

## Related concepts

* [[proof by contradiction]]

## References

* {#bauer} [[Andrej Bauer]], _[Proof of negation and proof by contradiction](http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/)_ 2010

* {#Lopez-Escobar72} E. G. K. López-Escobar, _Refutability and Elementary Number Theory_ , Indag. Math. **34** 1972 pp.362-374.

* {#Nelson49} David Nelson, Constructible Falsity, _The Journal of Symbolic Logic_, Vol. 14, No. 1 (Mar., 1949), pp. 16-26. ([jstor](http://www.jstor.org/stable/2268973))

[[!redirects proof of negation]]


Regge trajectory

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

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], specifically in [[quantum hadrodynamics]] and in [[string theory]], a _Regge trajectory_ is, for any given type of [[physical system]] but specifically for [[hadrons]] and for [[strings]], the [[graph of a function|graph]] of the [[mass]]-squared as a [[function]] of total [[angular momentum]] (with all other properties held fixed).

## Properties

### Hadron supersymmetry

The Regge trajectories of [[mesons]] and of [[hadrons]] are observed to have essentially the same slope.

\begin{imagefromfile}
    "file_name": "MesonBaryonReggeTrajectoriesKlemptMetsch.jpg",
    "width": 420,
    "unit": "px",
    "caption": "from [Klempt-Metsch 12](#KlemptMetsch12)"
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "MesonBaryonReggeTrajectories.jpg",
    "width": 600,
    "unit": "px",
    "caption": "from [BTDL 16](hadron+supersymmetry#BTDL16)"
\end{imagefromfile}

This phenomenon is explained by [[hadron supersymmetry]].

## Related concepts

* [[string tension]]

## References

* {#KlemptMetsch12} E. Klempt, B. C. Metsch,  _Multiplet classification of light-quark baryons_, Eur. Phys. J. A 48, 127 (2012) ([doi:10.1140/epja/i2012-12127-1](https://doi.org/10.1140/epja/i2012-12127-1))

See also:

* Wikipedia, _[Regge theory](https://en.m.wikipedia.org/wiki/Regge_theory)

[[!redirects Regge trajectories]]

[[!redirects Regge slope]]
[[!redirects Regge slopes]]


Regina Tix

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=113662)

## Selected writings

On [[category theory|category theoretic]] [[probability theory]] (cf. *[[convex powerset of distributions monad]]*):

* [[Regina Tix]], [[Klaus Keimel]], [[Gordon D. Plotkin]]: *Semantic domains for combining probability and non-determinism*, Electronic Notes in Theoretical Computer Science, **222** (2009) 3–99 [[doi:10.1016/j.entcs.2009.01.002](https://doi.org/10.1016/j.entcs.2009.01.002)]

category: people


regular

* [[regular ring]]
* [[regular scheme]]


regular (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _regular $(\infty,1)$-category_ is the analog of a [[regular category]] for [[(∞,1)-category]] theory.

## Definition

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

Let $\mathcal{C}$ be an [[(∞,1)-category]]. This is called an _exact $(\infty,1)$-category_ if

1. $\mathcal{C}$ has a [[terminal object in an (∞,1)-category|terminal object]] and [[homotopy fiber products]];

1. $\mathcal{C}$ admits a [[factorization system]] $(S_L,S_R)$, where $S_L$ is the collection of [[regular epimorphism|regular]] [[n-connected]] morphisms and $S_R$ is the collection of [[n-truncated]] morphisms.

=--

\begin{definition}
An [[(∞,1)-category]] $C$ is __regular__ if it admits finite [[(∞,1)-limits]], every morphism in $C$ has an [[image]], i.e., can be written as a composition of an [[(∞,1)-quotient morphism]] and an (∞,1)-monomorphism, and the collection of [[(∞,1)-quotient morphisms]] is closed under base changes.
\end{definition}

## Related concepts

* [[regular category]]

* [[exact (infinity,1)-category]], [[coherent (infinity,1)-category]], [[(infinity,1)-pretopos]]

* [[infinity-allegory]]

## References

* [[Jacob Lurie]], [[Kerodon]], [Tag 04TM](https://kerodon.net/tag/04TM).

[[!redirects regular (infinity,1)-categories]]

[[!redirects regular (∞,1)-category]]

[[!redirects regular (∞,1)-categories]]


regular 2-category

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _regular 2-category_ is the analog in [[2-category theory]] of the notion of _[[regular category]]_ in [[category theory]].

## Definition

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

A [[2-category]] $K$ is called **regular**  if

1. It is finitely complete (has all [[finite limit|finite]] [[2-limits]] );

1. [[eso morphism|esos]] are stable under [[2-pullback]];

1. Every [[2-congruence]] which is a kernel can be completed to an
   exact [[2-fork]].

=--

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

In particular, the last condition implies that every 2-congruence which is a kernel has a [[quotient]].

=--

## Examples

* [[nLab:Cat|Cat]] is regular.

* A [[1-category]] is regular as a 2-category iff it is [[regular category|regular]] as a 1-category, since the esos in a 1-category are precisely the [[strong epimorphism|strong epis]].

* Every finitely complete] [[(0,1)-category]] (that is, every [[meet-semilattice]]) is regular.

## Properties

### Factorizations

In [[StreetCBS]] the last condition is replaced by

* Every morphism $f$ factors as $f\cong m e$ where $m$ is [[ff morphism|ff]] and $e$ is eso.

We now show that this follows from our definition.  First we need:

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

**(Street's Lemma)**  Let $K$ be a finitely complete 2-category where esos are stable under pullback, let $e:A\to B$ be eso, and let $n:B\to C$ be a map.
1. If the induced morphism $ker(e) \to ker(n e)$ is ff, then $n$ is faithful.
1. If $ker(e) \to ker(n e)$ is an equivalence, then $n$ is ff.

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

First note that $ker(e)\to ker(n e)$ being ff means that if $a_1,a_2: Y \rightrightarrows A$ and $\delta_1,\delta_2 : e a_1 \;\rightrightarrows\; e a_2$ are such that $n \delta_1 = n \delta_2$, then $\delta_1=\delta_2$.  Likewise, $ker(e)\to ker(n e)$ being an equivalence means that given any $\alpha: n e a_1 \to n e a_2$, there exists a unique $\delta: e a_1 \to e a_2$ such that $n \delta = \alpha$.

We first show that $n$ is faithful under the first hypothesis.  Suppose we have $b_1,b_2:X \rightrightarrows B$ and $\beta_1,\beta_2:b_1\to b_2$ with $n \beta_1 = n \beta_2$.  Take the pullback
$$\array{&Y & \overset{r}{\to} & X \\
  (a_1,a_2) & \downarrow && \downarrow & (b_1,b_2)\\
  &  A\times A & \overset{e\times e}{\to} & B\times B}$$
Then we have two 2-cells
$$\beta_1 r, \beta_2 r: b_1 r \;\rightrightarrows\; b_2 r$$
such that the composites
$$n e a_1 \cong n b_1 r \overset{n \beta_1 r = n \beta_2 r}{\to}
n b_2 r \cong n e a_2$$
are equal.  By the hypothesis, $n \beta_1 r = n \beta_2 r$ implies $\beta_1 r = \beta_2 r$.  But $r$ is eso, since it is a pullback of the eso $e\times e$, so this implies $\beta_1=\beta_2$.  Thus, $n$ is faithful.

Now suppose the (stronger) second hypothesis, and form the pair of pullbacks:
$$\array{(n e / n e) & \overset{g}{\to} & n / n & \to & C^{\mathbf{2}}\\
  \downarrow && \downarrow && \downarrow \\
  A\times A & \overset{e\times e}{\to} & B\times B & \overset{n\times
n}{\to} & C\times C}$$
Then $g$, being a pullback of $e\times e$, is eso.  We also have a commutative square
$$\array{(e/e) & \to & (n e / n e)\\ \downarrow && \downarrow g \\
  B^{\mathbf{2}} & \to & (n/n).}$$
By assumption, $(e/e) \to (n e / n e)$ is an equivalence.  Since we have shown that $n$ is faithful, the bottom map $B^{\mathbf{2}} \to (n/n)$ is ff, so since the eso $g$ factors through it, it must be an equivalence as well.  But this says precisely that $n$ is ff.

=--

+--{: .num_theorem #StreetDefn}
###### Theorem

A 2-category is regular if and only if
1. it has finite limits,
1. esos are stable under pullback,
1. every morphism $f$ factors as $f\cong m e$ where $m$ is ff and $e$ is eso, and
1. every eso is the quotient of its kernel.

=--

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

First suppose $K$ is regular; we must show the last two conditions.  Let $f:A\to B$ be any morphism.  By assumption, the kernel $ker(f)$ can be completed to an exact 2-fork $ker(f) \rightrightarrows A \overset{e}{\to} C$.  Since $e$ is the quotient of the 2-congruence $ker(f)$, it is eso, and since $f$ comes with an action by $ker(f)$, we have an induced map $m:C\to B$ with $f\cong m e$.  But since the 2-fork is exact, we also have $ker(f)\simeq ker(e)$, so by Street's Lemma, $m$ is ff.

Now suppose that in the previous paragraph $f$ were already eso.  Then since it factors through the ff $m$, $m$ must be an equivalence; thus $f$ is equivalent to $e$ and hence is a quotient of its kernel.

Now suppose $K$ satisfies the conditions in the lemma.  Let $f:A\to B$ be any morphism; we must show that $ker(f)$ can be completed to an exact 2-fork.  Factor $f = m e$ where $m$ is ff and $e$ is eso.  Since $m$ is ff, we have $ker(f)\simeq ker(e)$.  But every eso is the quotient of its kernel, so the fork $ker(f) \rightrightarrows A \overset{e} \to C$ is exact.

=--

In [[StreetCBS]] it is claimed that the final condition in Theorem \ref{StreetDefn} follows from the other three, but there is a flaw in the proof.

### Subobjects

In a regular 2-category $K$, we call a ff $m:A\to X$ with codomain $X$ a **subobject** of $X$.  We write $Sub(X)$ for the [[nLab:preorder|preorder]] of subobjects of $X$, as a full sub-2-category of the [[slice 2-category]] $K/X$.  Since $K$ is finitely complete and pullbacks preserve ffs, we have pullback functors $f^*:Sub(Y)\to Sub(X)$ for any $f:X\to Y$.

If $g \cong m e$ where $m$ is ff and $e$ is eso, we call $m$ the **image** of $g$.  Taking images defines a left adjoint $\exists_f:Sub(X)\to Sub(Y)$ to $f^*$ in any regular 2-category, and the [[nLab:Beck-Chevalley condition|Beck-Chevalley condition]] is satisfied for any pullback square, because esos are stable under pullback.

### Preservation

It is easy to check that if $K$ is regular, so are:

* its [[nLab:opposite 2-category|2-cell dual]] $K^{co}$ (by the remarks about opposite [[2-congruence]]s).
* the (2,1)-category $gpd(K)$ of [[groupoidal object]]s in $K$.
* the (1,2)-category $pos(K)$ of [[posetal object]]s in $K$.
* the 1-category $disc(K)$ of [[discrete object]]s in $K$.
* and more generally the $n$-category $trunc_n(K)$ of $n$-[[truncated object|truncated objects]] in $K$.

The [[slice 2-category]] $K/X$ does not, in general, inherit regularity, but we have:

+--{: .num_theorem}
###### Theorem

If $K$ is regular, so are the [[fibrational slice]]s $Opf(X)$ and $Fib(X)$.

=--

### Regular completion

See at _[[2-congruence]]_ the section _[Regularity](http://ncatlab.org/nlab/show/2-congruence#Regularity)_.

## Related concepts

* [[regular category]]

* [[regular completion]]

## References

The above definitions and observations are originally due to

* [[Michael Shulman]], _[[michaelshulman:regular 2-category]]_ (see [[StreetCBS]])

[[!redirects regular 2-categories]]

[[!redirects regular (2,1)-category]]
[[!redirects regular (2,1)-categories]]


regular action

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

#Contents#
* table of contents
{:toc}

## Definition

An [[action]]

$$
  (-)\cdot(-) \;\colon\; G \times X \to X
$$

of a [[group]] $G$ on an [[inhabited set]] $X$ is called _regular_ if it is both [[transitive action|transitive]] and [[free action|free]].

In terms of [[elements]] this means that if for any pair of elements $x,y \in X$, there is _exactly one_ group element $g \in G$ such that $g \cdot x = y$.

More abstractly this means that the [[shear map]]

$$
  \array{
    G \times X
    &
    \overset
      { (pr_2, \cdot) }
      {\longrightarrow}&
    X \times X
    \\
    (g, x)
    &
    \mapsto
    &
    \big( x, g \cdot x \big)
    \,.
  }
$$

is an [[isomorphism]].

In this case $X$ is also called a *$G$-[[torsor]]*. If the condition is dropped that $X$ be [[inhabited object|inhabited]] it is still called a *[[pseudo-torsor]]*.

## Properties

+-- {: .num_prop}
###### Proposition
Suppose $G$ acts transitively on $X$ by $* : G \times X \to X$, and suppose moreover that this action is [[faithful action|faithful]].  Then $G$ acts freely (and hence regularly) on $X$ if and only if the group $Aut_G(X)$ of $G$-equivariant automorphisms (i.e., [[bijections]] $\phi : X \to X$ commuting with the action of $G$) acts transitively (and hence regularly) on $X$.
=--

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

First we show that $Aut_G(X)$ acts freely on $X$. Suppose $\phi \in Aut_G(X)$ is such that $\phi(x) = x$ for some $x\in X$, and let $y\in X$ be arbitrary.  By the assumption that $G$ acts transitively, there is a $g \in G$ such that $y = g*x$. But then $G$-equivariance implies that
$$\phi(y) = \phi(g*x) = g*\phi(x) = g*x = y.$$
Since this holds for all $y\in Y$, $\phi$ must be equal to the identity $\phi = id_X$, and therefore $Aut_G(X)$ acts freely on $X$.

Next, suppose that $G$ also acts freely on $X$, and let $x,y \in X$ be arbitrary.  Then we can define a $G$-equivariant automorphism $\phi$ such that $\phi(x) = y$ by
$$\phi = z \mapsto g_z*y,$$
where for each $z$, $g_z$ is the unique group element such that $z = g_z*x$.  Conversely, suppose that $Aut_G(X)$ acts transitively on $X$, and let $x\in X$, $g\in G$ such that $g*x = x$.  By the assumption, for any $y \in X$, there exists $\phi \in Aut_G(X)$ such that $\phi(x) = y$, from which it follows that
$$g*y = g*\phi(x) = \phi(g*x) = \phi(x) = y.$$
Since $g*y = y$ for all $y \in X$, therefore $g = 1$ by the assumption that $G$ acts faithfully on $X$.

=--

## Examples

* The action of $G$ on itself by multiplication $\cdot : G \times G \to G$ (on the left or on the right) is a regular action, called the (left or right) [[regular representation]] of $G$.

* If one views a [[combinatorial map]] $M$ as the transitive action of a certain group of permutations, then $M$ represents a _regular map_ ([Siran 2006](#SiranSurvey)) just in case this action is regular.  For example, the five [[Platonic solids]] may be represented as regular combinatorial maps.

##In homotopy type theory
 {#InHomotopyTypeTheory}

We discuss regular actions via [[homotopy type theory]].

Since doing group [representation theory in homotopy type theory](representation+theory#InHomotopyTypeTheory) corresponds to working in the [[context]] of a [[delooped group]] in [[homotopy type theory]], the regularity of an action is naturally expressed there. Transitivity ensures that all the points in the [[homotopy quotient]] are connected by equivalences, while freeness means that the space of equivalences between two points is itself [[contractible space|contractible]]. Hence if $\ast \colon \mathbf{B} G \vdash X(\ast): Type$ corresponds to a regular action, then the quotient $\sum_{\ast: \mathbf{B} G} X(\ast)$ is contractible.

Restriction to 1-groups is unnecessary here, and we say

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

An [[∞-action]] of an [[∞-group]] is a **regular $\infty$-action** if its [[homotopy quotient]] is [[contractible type|contractible]].

=--

For any $G$-action ([[∞-action]]) $X \colon BG \to U$, its [[automorphism group]] is (see at [automorphism ∞-group in HoTT](automorphism+infinity-group#InHomotopyTypeTheory))

$$
  B Aut_G(X) \coloneqq \sum_{(P:BG \to U)} \|P=X\|
$$

and

$$
   \tilde{X} \colon B Aut_G(X) \to U
$$

by $\tilde{X}(P,-) \coloneqq P(\ast)$.

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

If $X$ is regular, then $\tilde{X}$ is regular.

=--

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

First, we need to argue that $X(\ast)$ is [[mere proposition|merely]] [[inhabited type|inhabited]]. Since $X$ is regular, we have $\sum_{(b:BG)} X(b)$ [[contractible type|contractible]]. This gives a center of contraction $(b,x)$. Now, since $B G$ is connected, it follows that $\|b=\ast\|$. Since we are proving the [[mere proposition]] $\|X(\ast)\|$, we get to use $b=\ast$. Now we obtain $\|X(\ast)\|$.

Next, to show that $\tilde{X}$ is regular we need to show that $\tilde{X}$ has a contractible total space. The [[dependent sum]] type $\sum_{(b : BAut_G(X))} \tilde{X}(b)$ is equivalent to $\sum_{(P:BG \to U)} \|P=X\| \times P(\ast)$. Contractibility is a [[mere proposition]], and we have $\|X(\ast)\|$, so we get to use a point $x:X(\ast)$. This gives us a center of contraction $(X,refl,x)$ of the total space of $\tilde{X}$.

Now let $P : BG \to U$, let $ \| P = X \|$, let $p_0 : P(\ast)$. To show regularity, it suffices to find a [[term]] of type

$$
  \sum_{\alpha : P = X} \mathrm{trans}(\alpha)(p_0) = x
  \,.
$$

This type is equivalent to showing that there are

$$
  K : \prod_{b:BG} P(b) \simeq X(b)
  \;\;\;\;
  \text{and}
  \;\;\;\;\;
  K(*,p_0) = x_0
  \,.
$$

Now we use that $X(b)$ is equivalent to $b=\ast$ (we get this fact from regularity, together with a point $x:X(\ast)$).
Since we need this particular fiberwise [[equivalence in homotopy type theory|equivalence]], it suffices to show that

$\sum_{b:BG} P(b)$

is [[contractible type|contractible]]. Now this is a [[mere proposition]], so we can eliminate $t : \| P = X \|$ to obtain the proof.

=--

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

If $X$ is a principal homogeneous space on $G$, in the sense that the type $\sum_{(g:G)} g_\ast(x)=y$ is contractible for all $x,y:X(\ast)$, and $\tilde{X}$ is regular, then $X$ is regular.

=--

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

Again, we first show that $X(\ast)$ is [[mere proposition|merely]] [[inhabited type|inhabited]]. The total space of $\tilde{X}$ has center of contraction $(P,p_0)$. Since $\|P=X\|$ and since we are proving a [[mere proposition]], we get to use $P=X$. Now $\|X(\ast)\|$ follows from $p_0:P(\ast)$. The regularity of $X$ is a mere proposition, so we get to use $x_0:X(\ast)$. This gives us the center of contraction $(\ast,x_0)$. It remains to show that

$\prod_{(b:BG)} \prod_{(x:X(b))} \sum_{(\alpha : b=\ast)} \mathrm{trans}(\alpha,x) = x_0$.

Of course, it would suffice to prove the stronger statement

$\prod_{(b:BG)} \prod_{(x:X(b))} \mathrm{isContr} (\sum_{(\alpha : b=\ast)} \mathrm{trans}(\alpha,x) = x_0)$.

However, now we get to use that $BG$ is connected. Therefore it suffices to show that

$\prod_{x:X(*)} isContr (\sum_{\alpha : G} \mathrm{trans}(\alpha,x) = x_0)$

This holds by assumption.
=--

## Related concepts

* [[transitive action]], [[free action]], [[semi-free action]], [[regular action]],

* [[proper action]], [[properly discontinuous action]]

* [[torsor]]

## References

* {#SiranSurvey} Jozef Siran, "Regular Maps on a Given Surface: A Survey", _Topics in Discrete Mathematics_, 2006. ([pdf](http://dfgm.math.msu.su/files/papers-sym/regular%20maps%20on%20a%20given%20surface.pdf))

* Group Properties Wiki: [Regular group action](http://groupprops.subwiki.org/wiki/Regular_group_action).

[[!redirects regular actions]]
[[!redirects regular group action]]
[[!redirects regular group actions]]

[[!redirects regular infinity-action]]
[[!redirects regular ∞-action]]
[[!redirects regular infinity-actions]]
[[!redirects regular ∞-actions]]


regular and exact categories - contents

**[[∞-ary regular and exact categories]]**

* [[exactness property]]

* [[arity class]]: unary, finitary, infinitary

**regularity**

* [[regular category]] = unary regular

* [[coherent category]] = finitary regular

  * [[Heyting category]]/[[logos]] = finitary regular + [[universal quantifier]]

* [[geometric category]] = infinitary regular

**exactness**

* [[exact category]] = unary exact

* [[pretopos]] = finitary exact = [[exact category|exact]] + [[extensive category|extensive]]


regular and exact completions

+-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

* table of contents
{: toc}

## Idea

The [[forgetful functor|forgetful]] [[2-functors]]

* from [[regular categories]] to [[lex categories]],
* from [[exact categories]] to [[lex categories]], and
* from [[exact categories]] to [[regular categories]]

have [[left adjoints]], and in fact are [[monadic adjunction|(2-)monadic]].  Their left adjoints are called ([[free construction|free]]) regular or exact completions.

In the third case the [[2-monad]] is [[idempotent monad|idempotent]], so the left adjoint can properly be called a [[completion]], while in the first two cases, the 2-monad is only [[lax-idempotent monad|lax-idempotent]], so the left adjoint should technically be called a [[free completion]].  However, the phrases *regular completion* and *exact completion* are also commonly used for the first two cases.  To disambiguate the second and third cases the phrases *ex/lex completion* and *ex/reg completion* are also used, and so by analogy the first case is called *reg/lex completion*.

In fact, the reg/lex and ex/lex completion can be applied to a category that merely has [[weak finite limits]], although in this case they are not left adjoint to the obvious forgetful functor.  A general context which includes *all* of these types of these completions is the 2-category of [[unary sites]], in which the categories of regular and exact categories form reflective sub-2-categories.

## Constructions

### The ex/lex completion
 {#TheExLexCompletion}

There are several constructions of the ex/lex completion.  Perhaps the quickest one to state (Hu-Tholen 1996) is that if $C$ is [[small category|small]], then $C_{ex/lex}$ is the full subcategory of its [[presheaf category]] $Set^{C^{op}}$ spanned by those presheaves $F$ such that

* $F$ admits a [[regular epimorphism]] $y(X)\twoheadrightarrow F$ from a [[representable presheaf]],
* with the additional property that if $K\rightrightarrows y(X)$ is the [[kernel pair]] of $y(X)\twoheadrightarrow F$, then $K$ also admits a regular epi $y(Z)\to K$ from a representable presheaf.

A more explicit construction is as follows.  Let us think, informally, of the objects of $C$ as [[presets]] and the morphisms of $C$ as "proofs".  An object of $C_{ex} = C_{ex/lex}$ will then be a "set" or [[setoid]] constructed from $C$.  Precisely, we take the objects of $C_{ex}$ to be the [[pseudo-equivalence relations]] in $C$: a pseudo-equivalence relation consists of:

* an object $X\in C$ and
* a parallel pair $s,t\colon R\rightrightarrows X$, such that
* there exists an arrow $i\colon X\to R$ with $s i = t i = 1_X$,
* there exists an arrow $v\colon R\to R$ with $s v = t$ and $t v = s$, and
* there exists an arrow $c\colon R\times_X R \to R$ with $s c = s \pi_1$ and $t c = t \pi_2$.  (If $C$ has merely weak finite limits, we assert this for some, and hence every, weak pullback $R\times^w_X R$.)

If $(s,t)\colon R\to X\times X$ is a [[monomorphism]], then these conditions make it precisely a [[congruence]] or internal [[equivalence relation]] in $C$.  In general, we can think of the fiber of $R$ over $(x_1,x_2)$ as giving a collection of "reasons" or "proofs" that $x_1 \mathrel{R} x_2$.  Then $i$ supplies a uniform proof that $x \mathrel{R} x$ for every $x$, while $v$ supplies a uniform proof that $x \mathrel{R} y$ implies $y \mathrel{R} x$, and $c$ supplies a uniform proof that $x \mathrel{R} y$ and $y \mathrel{R} z$ imply $x \mathrel{R} z$.

If $R\rightrightarrows X$ and $S\rightrightarrows Y$ are two pseudo-equivalence relations, a morphism between them in $C_{ex}$ is defined to be a morphism $f\colon X\to Y$ in $C$, such that there exists a morphism $f_1\colon R\to S$ with $s f_1 = f s$ and $t f_1 = f t$.  That is, $f_1$ supplies a uniform proof that if $x \mathrel{R} y$ then $f(x) \mathrel{S} f(y)$.  Moreover, we declare two such morphisms $f,g\colon X\to Y$ to be *equal* if there exists a morphism $h\colon X\to S$ such that $s h = f$ and $t h = g$ (that is, a uniform proof that $f(x) \mathrel{S} g(x)$).  Because $S\rightrightarrows Y$ is a pseudo-equivalence relation, this defines an actual equivalence relation on the morphisms $f\colon X\to Y$, which is compatible with composition; thus we have a well-defined category $C_{ex}$.

We have a [[full and faithful functor]] $C\to C_{ex}$ sending an object $X$ to the pseudo-equivalence relation $X\rightrightarrows X$.  One can then verify directly that $C_{ex}$ is exact, that this embedding preserves finite limits, and that it is universal with respect to lex functors from $C$ into exact categories.

There are also other constructions.  Of course, the ex/lex completion can also be obtained by composing (any construction of) the reg/lex completion with (any construction of) the ex/reg completion.

### The reg/lex completion

The reg/lex completion $C_{reg}= C_{reg/lex}$ of a lex category $C$ is perhaps most succinctly described as the subcategory of $C_{ex}$ consisting of those objects which admit monomorphisms into objects of $C$.  That is, instead of adding *all* quotients of pseudo-equivalence relations in $C$, we only add those quotients which are necessary in order to be able to construct [[images]] of morphisms in $C$.  For many construction of $C_{ex}$, this idea can then be made more explicit and sometimes simplified.

For instance, if we regard $C_{ex}$ as a full subcategory of $Set^{C^{op}}$ as above, then we can likewise regard $C_{reg}$ as the full subcategory of $Set^{C^{op}}$ determined by those presheaves $F$ such that

* $F$ admits a regular epimorphism $y(X) \twoheadrightarrow F$ from a representable presheaf, and
* $F$ admits a monomorphism $F\rightarrowtail y(Z)$ into a representable presheaf.

If we construct $C_{ex}$ using pseudo-equivalence relations, as above, then we can characterize the pseudo-equivalence relations which we need to form $C_{reg}$ as precisely the [[kernel pairs]] of morphisms of $C$ (or finite families of such).  Therefore, we obtain an equivalent definition of $C_{reg}$ as follows.  Its objects are morphisms of $C$ (regarded as stand-ins for their formally added images).  A morphism from $p\colon X\to Y$ to $q\colon Z\to W$ should be a morphism $f\colon X\to Z$ for which there exists an $f_1$ relating the kernel of $p$ to the kernel of $q$, modulo an equivalence relation generated by maps from $X$ to the kernel of $q$.  But by definition of kernel pairs, two morphisms will be identified under this latter equivalence relation if and only if they have the same composite with $q$, so it makes sense to define the morphisms of $C_{reg}$ from $p\colon X\to Y$ to $q\colon Z\to W$ to be certain morphisms $\overline{f}\colon X\to W$ which factor through $q$ (non-uniquely).  We still have to impose the condition that $f$ should preserve the kernel pairs, but in terms of $\overline{f}$ this is simply the statement that $\overline{f} r = \overline{f} s$, where
$(r,s)$ is the kernel pair of $p$.  This is the definition of $C_{reg}$ given in the [[Elephant]].

We can then verify that $C_{reg}$ is regular, that we have a full and faithful functor $C\to C_{reg}$, which preserves finite limits, and is universal among lex functors from $C$ to regular categories.  Again, there are also other constructions.

Also, just as for the free exact completion, the construction works essentially the same if $C$ has only weak finite limits.  In this case, instead of the objects of $C_{ex}$ admitting a monomorphism to a single object of $C$, we have to consider those admitting a jointly-monic finite family of morphisms into objects of $C$, with similar modifications for the other descriptions.

### The ex/reg completion

If $C$ is regular, a quick definition of $C_{ex/reg}$ is as the full subcategory of the category $Sh(C)$ of [[sheaves]] for the [[regular coverage]] on $C$ spanned by those sheaves which are quotients of [[congruences]] in $C$.  (Lack 1999)

A more explicit description can be obtained by first passing from $C$ to its [[allegory]] of internal relations, then [[split idempotent|splitting]] the idempotents which are equivalence relations in $C$, and finally reconstructing a regular category from the resulting allegory.  Yet more explicitly, this means that the objects of $C_{ex/reg}$ are congruences in $C$, and the morphisms are relations which are [[entire relation|entire]] and [[functional relation|functional]] relative to the given congruences.

### The higher categorical approach

A somewhat more unified approach to all these completions can be obtained as follows.  Observe that in the classical situation (that is, in the presence of choice), *sets* can be identified with all of the following:

* 0-trivial [[groupoids]], i.e. groupoids in which any two parallel morphisms are equal, i.e. [[equivalence relations]].
* 0-trivial [[2-groupoids]], i.e. 2-groupoids in which any two parallel 2-morphisms are equal and any two parallel 1-morphisms are isomorphic.
* and so on...
* 0-trivial [[n-groupoids]] for any $0\le n \le \infty$.

In the absence of choice, this is still true as long as the morphisms between 0-trivial n-groupoids are $n$-[[anafunctors]].  If instead we consider only actual functors, however, in the absence of choice what we obtain are various completions of $Set$.  Specifically:

* $Set_{reg/lex}$ can be identified with the category whose objects are 0-trivial groupoids, and whose morphisms are natural isomorphism classes of functors.
* $Set_{ex/lex}$ can be identified with the category whose objects are 0-trivial 2-groupoids, and whose morphisms are pseudonatural equivalence classes of 2-functors.  In the notion of 2-groupoid here we also demand that each 1-cell be equipped with a *specified* inverse [[equivalence]].

This idea can be generalized to provide alternate constructions of the completions for an arbitrary $C$ with finite limits.  The notions of [[internal category|internal]] $n$-category and internal $n$-functor in such a $C$ make perfect sense for any $n$.  The same is true of the notion of $n$-groupoid, as long as we interpret this to mean the *structure* of "inverse-assigning" morphisms in $C$.  The statement "any two parallel $n$-cells are equal" also makes sense in any lex category, since it demands that a certain specified morphism is monic.  Finally, we can also interpret "any two parallel $k$-cells are equivalent" algebraically by specifying a particular equivalence between any such pair.  (Note that for $k=(n-1)$, since parallel $n$-cells are equal there is a unique way to do this.)  We thereby obtain a notion of internal 0-trivial $n$-groupoid in any lex category, and we write $0 triv n Gpd(C)$ for the category of such things and internal $n$-natural equivalence classes of functors.  We then have:

* It is fairly clear from the above explicit description that $C_{reg/lex}$ is the full subcategory of $0 triv 1 Gpd(C)$ determined by the [[kernel pairs]] (which are [[congruences]], i.e. internal 0-trivial 1-groupoids).  If, like $Set$, $C$ is already exact, so that every congruence is a kernel pair, then $C_{reg/lex}\simeq 0 triv 1 Gpd(C)$.

* $C_{ex/lex}$ is always equivalent to $0 triv 2 Gpd(C)$.  To see this, note that a pseudo-equivalence relation (together with chosen maps $i$, $c$, and $v$) can be regarded as the 1-skeleton of an internal [[bicategory]] in $C$ with specified inverse equivalences for every 1-cell.  There is then a unique way to add 2-cells to make it a 0-trivial bigroupoid.

It is not clear how 0-trivial $n$-groupoids fit into this picture for $n\gt 2$, although it seems likely that the objects of *iterated* reg/lex and ex/lex completions can be identified with some type of internal [[n-fold category]].

Now if $C$ is already regular, then we can define a notion of internal [[anafunctor]] between internal $n$-categories. It is then easily seen that

* $C_{ex/reg}$ is equivalent to the category of 0-trivial 1-groupoids, and natural isomorphism classes of internal anafunctors between them.

Again, it is not entirely clear how the 0-trivial $n$-groupoids and anafunctors behave for $n\gt 1$, although it seems fairly likely (to [[Mike Shulman|me]]) that in this case the process will stabilize at $n=1$, i.e. 0-trivial $n$-groupoids with equivalence classes of ana-$n$-functors will give $C_{ex/reg}$ for all $n\ge 1$.

### Completions of unary sites

The descriptions of the ex/lex and ex/reg completions in terms of pseudo-equivalence relations and equivalence relations, respectively, have a common generalization.  Let $C$ be a [[unary site]], so that it has a notion of "covering morphism" and admits "finite local unary prelimits".  In particular, any cospan $X\to Z\leftarrow Y$ has a *local unary pre-pullback*, which is a commutative square
$$ \array{ P & \to & Y \\ \downarrow && \downarrow\\ X & \to & Z } $$
such that for any other commutative square
$$ \array{ V & \to & Y \\ \downarrow && \downarrow\\ X & \to & Z } $$
there is a cover $U\to V$ and a map $U\to P$ such that the induced composites $U\to X$ and $U\to Y$ are equal.

Now we can define a **unary congruence** in $C$ to consist of:

* An object $X\in C$

* A parallel pair $s,t:R\rightrightarrows X$, such that

* There exists a cover $p:Y\to X$ and a map $i:Y\to R$ with $s i = t i = p$,

* There exists a cover $q:S\to R$ and a map $v:S\to R$ with $s v = t q$ and $t v = s q$, and

* There exists a local unary pre-pullback $T$ of the cospan $R \xrightarrow{t} X \xleftarrow{s} R$ and an arrow $c:T\to R$ such that $s c = s \pi_1$ and $t c = t \pi_2$, where $\pi_1$ and $\pi_2$ are the projections of $T$ to $R$.

If $C$ has a trivial topology, then local unary prelimits are simply weak limits, and this reduces to the definition of pseudo-equivalence relation.  On the other hand, if $C$ is regular with its regular topology, then these conditions ensure exactly that the [[image]] of $R\to X\times X$ is an internal equivalence relation on $X$.

Now we can define morphisms between unary congruences using a suitable kind of either entire and functional relations or anafunctors, and obtain the exact completion of the unary site $C$.  This construction exhibits the 2-category of exact categories as a reflective sub-2-category of the 2-category of unary sites, and restricts to the ex/wlex and ex/reg completions on the sub-2-categories of categories with weak finite limits and trivial topologies and of regular categories with regular topologies, respectively.  It can also be modified to construct regular completions.  See ([Shulman](#Shulman)) for details.

## Generalizations to higher arity

More generally, any [[∞-ary site]] has a $\kappa$-ary exact completion, which is a [[∞-ary exact category]].  This exhibits the 2-category of $\kappa$-ary exact categories as a reflective sub-2-category of that of $\kappa$-ary sites.  See ([Shulman](#Shulman)) for details.  In particular:

* When $\kappa=\omega$, this is called the [[pretopos completion]], and applies in particular to [[coherent categories]].

* When $\kappa$ is the size of the [[universe]], this applies to any [[small category|small]] [[site]] and constructs its [[topos of sheaves]].

## Properties of regular and exact completions

Many categorical properties of interest are preserved by one or more of the regular and exact completions.  That is, if $C$ has these properties, then so does the completion, and the inclusion functor preserves them.  Note that frequently, for a completion to have some structure, it suffices for $C$ to have a "weak" version of that structure.

* Of course, [[finite limits]] are preserved by all three completions.  In fact, as we have remarked, for the ex/lex and reg/lex completions, $C$ need only have weak finite limits.

* $C$ is [[extensive category|lextensive]] if and only if $C_{ex/lex}$ is, and if and only if $C_{reg/lex}$ is, and in this case the embeddings preserve [[coproducts]] (Menni 2000).  It follows that if $C$ is a [[pretopos]], then so is $C_{ex/lex}$, although the inclusion $C\to C_{ex/lex}$ is not a "pretopos functor" as it does not preserve [[regular epis]].

* If $C$ is lextensive and has [[coequalizers]] (and hence has finite colimits), then so do $C_{ex/lex}$ and $C_{reg/lex}$ (Menni 2000).  However, the inclusion functors do not preserve coequalizers.  In fact, it suffices for $C$ to be lextensive with *quasi-coequalizers*, meaning that for every $f,g\colon Y\rightrightarrows X$ there exists $q\colon X\to Q$ with $q f = q g$, such that for any $h\colon X\to Z$ with $h f = h g$, $h$ coequalizes the [[kernel pair]] of $q$.

* The categories $C_{reg/lex}$ and $C_{ex/lex}$ always have [[projective object|enough (regular) projectives]].  In fact, the objects of $C$ are precisely the projective objects of these categories.  Moreover, an exact category $D$ is of the form $C_{ex/lex}$ for some $C$ (with weak finite limits) if and only if it has enough projectives, in which case of course $C$ can be taken to be the subcategory of projectives (Carboni--Vitale 1998).  Note that if $D$ has enough projectives, then its subcategory of projectives always has weak finite limits.  Similarly, a regular category $D$ is of the form $C_{reg/lex}$ for some $C$ (with weak finite limits) if and only if it has enough projectives and every object can be embedded in a projective one.

* If $C$ is a regular category satisfying the "regular" [[axiom of choice]] (i.e. every regular epi splits), then it is *equivalent* to $C_{reg/lex}$, and hence the latter also satisfies the axiom of choice.  Similarly, if $C$ is exact and satisfies choice, then it is equivalent to $C_{ex/lex}$.  Conversely, if the inclusion $C\to C_{reg/lex}$ or $C\to C_{ex/lex}$ is an equivalence, then since the objects of $C$ are projective in these completions, $C$ must satisfy the axiom of choice.

  In fact, if we assume merely that $C_{ex/lex} \to (C_{ex/lex})_{ex/lex}$ is a equivalence, then since the objects of $C_{ex/lex}$ are projective in $(C_{ex/lex})_{ex/lex}$, they must also all be projective in $C_{ex/lex}$, and therefore $C\to C_{ex/lex}$ is also an equivalence.  It follows by induction that if the sequence of iterations of $(-)_{ex/lex}$ stabilizes at any finite stage, it must in fact stabilize at the very beginning and $C$ must satisfy the axiom of choice.  A similar argument applies to the reg/lex completion.  (The ex/reg completion, of course, always stabilizes after one application.)

* [[cartesian closed category|Cartesian closure]] is preserved by the ex/lex completion (Carboni--Rosolini 2000).  In fact, $C_{ex/lex}$ is cartesian closed if and only if $C$ has *weak simple products*, meaning [[weak dependent product]]s along [[product]] projections.

* [[locally cartesian closed category|Local cartesian closure]] is also preserved by the ex/lex completion (Carboni--Rosolini 2000).  In fact, $C_{ex/lex}$ is locally cartesian closed if and only if $C$ is *weakly locally cartesian closed*, meaning that each [[slice category]] has weak dependent products.  It follows in particular that if $C$ is a [[Π-pretopos]], then so is $C_{ex/lex}$.  For each $\Pi$-pretopos $C$ we thus obtain a sequence $C$, $C_{ex}$, $(C_{ex})_{ex}$, ... of $\Pi$-pretopoi, which in general does not stabilize.

* (Local) cartesian closure is seemingly not always preserved by the reg/lex completion, but it is under certain hypotheses.  Recalling that $C_{reg/lex}$ is the full subcategory of $C_{ex/lex}$ consisting of the kernel pairs, suppose that $C$ has pullback-stable (epi,regular mono) factorizations and that every [[regular monomorphism|regular]] congruence is a kernel pair.  Then $C_{reg/lex}$ is [[reflective subcategory|reflective]] in $C_{ex/lex}$ (BCRS 1998, Menni 2000): the reflection of a pseudo-equivalence relation $R\rightrightarrows X$ is its (epi,regular mono) factorization.  Moreover, the reflection preserves products, and also pullbacks along maps in $C_{reg/lex}$, from which it follows that if $C_{ex/lex}$ is cartesian closed or locally cartesian closed, so is $C_{reg/lex}$.

  Thus, if $C$ is weakly cartesian closed (resp. weakly locally cartesian closed), has pullback-stable (epi,regular mono) factorizations, and every regular congruence is a kernel pair, then $C_{reg/lex}$ is cartesian closed (resp. locally cartesian closed).  In particular, the local versions of these hypotheses apply in particular to [[Top]] and to any [[quasitopos]].  Note that $Top_{reg/lex}$ is called the category of [[equilogical space]]s.

* If $C$ is lextensive with coequalizers (or "quasi-coequalizers") and a [[strong-subobject classifier]], then so is $C_{reg/lex}$ (Menni 2000).  It follows that if $C$ is a lextensive [[quasitopos]], then so is $C_{reg/lex}$.  For each lextensive quasitopos $C$ we thus obtain a sequence $C$, $C_{reg}$, $(C_{reg})_{reg}$, ... of lextensive quasitopoi, which in general does not stabilize.

* If $C$ has a [[natural numbers object]], then so do $C_{reg/lex}$ and $C_{ex/lex}$.  (Does $C_{ex/reg}$?  What about more general [[W-type]]s?)

* If $C$ is a [[ΠW-pretopos]], then so is $C_{ex/lex}$ (see [vandenBerg](#vdB), Theorem 1.1).

* $C_{ex/lex}$ is an [[elementary topos]] iff $C$ has weak dependent products and a [[generic proof]] ([Menni2000](#Menni)).  Note that if $C$ is a topos satisfying the axiom of choice, then its subobject classifier is a generic proof.  It follows that in this case $C_{ex/lex}$ is a topos---but we already knew that, because $C_{ex/lex}$ is equivalent to $C$ for such a $C$.

* Expanding on the last point, for a [[presheaf topos]] $C = Set^{D^{op}}$, the category $C_{ex/lex}$ is a topos iff $D$ is a [[groupoid]] ([Menni2000](#Menni)).

* If $C$ is regular, locally cartesian closed, and has a *generic mono*, i.e. a monomorphism $\tau\colon \Upsilon\to \Lambda$ such that every monomorphism is a pullback of $\tau$ (not necessarily uniquely), then $C_{ex/reg}$ is a topos ([Menni2000](#Menni)).

* If $C$ is an [[additive category]], then $C_{ex/lex}$ is an [[abelian category]].

On the other hand, some properties are *not* preserved by the completions.

* Of course, all the completions are regular categories, but the inclusions are not [[regular functor]]s, since they do not preserve regular epis.

* We have seen that the existence of a [[subobject classifier]] or [[power objects]] is not, in general, preserved by the completions (although if $C$ is a topos, then of course so is $C_{ex/reg}$, since it is equivalent to $C$).

* Similarly, if $C$ is [[well-powered category|well-powered]], it does not follow that $C_{reg/lex}$ or $C_{ex/lex}$ are.  In particular, for $X\in C$, the subobject preorders $Sub_{C_{reg/lex}}(X)$ and $Sub_{C_{ex/lex}}(X)$ are equivalent to the preorder reflection of the slice category $C/X$, and it is easy to construct examples in which this is not essentially small[^fine].

* If $C$ is a [[coherent category]], it does not follow that $C_{ex/lex}$ or $C_{reg/lex}$ is.  However, if $C$ is additionally [[extensive category|lextensive]], we have seen above that so are these completions, and hence in particular also coherent (any extensive regular category is coherent).  One can also write down the "free coherent completion" and the "free pretopos completion" of a lex category, and the "pretopos completion" of a coherent category; see [[familial regularity and exactness]] for some clues on how to proceed.

* If $C$ is a [[Heyting category]], it does not follow that $C_{ex/lex}$ or $C_{reg/lex}$ is.  However, if $C$ is additionally lextensive and locally cartesian closed, we have seen above that so are these completions, and hence Heyting (any lextensive locally cartesian closed regular category is Heyting).

* Unsurprisingly, if $C$ is a [[Boolean category]], it does not follow that $C_{ex/lex}$ or $C_{reg/lex}$ is, even if $C$ is lextensive and LCC so that its completions are Heyting.

  In fact, a stronger statement is true: if $C$ is lextensive and regular, then $C_{reg/lex}$ and $C_{ex/lex}$ are Boolean if and only if $C$ *satisfies the axiom of choice* (in which case they are of course equivalent to $C$).  More precisely, if $X\in C$ is such that every subobject of $X$ in $C_{reg/lex}$ is complemented, then $X$ is projective in $C$.  (The same argument applies to $C_{ex/lex}$.)  For suppose that $p\colon Y\to X$ is a regular epi in $C$.  Recall that $Sub_{C_{reg/lex}}(X)$ is the preorder reflection of $C/X$.  Thus $p$, considered as an object of $C/X$, defines a monomorphism in $C_{reg/lex}$.  By assumption, this monic is complemented; let its complement be represented by $q\colon Z\to X$.  Since complements are disjoint, and meets in $Sub_{C_{reg/lex}}(X)$ are given by pullbacks in $C/X$, the pullback of $p$ and $q$ admits a morphism to the initial object $0$, and hence is itself initial since $C$ is extensive.  Now $p$ is regular epi, hence so is its pullback $0\to Z$.  But in a lextensive regular category, disjointness of the coproduct $1+1$ implies that $0\to 1$ is the equalizer of of the coprojections $1\rightrightarrows 1+1$, and therefore any epimorphism with domain $0$ is an isomorphism; thus $Z$ is also initial.  Now since joins in $Sub_{C_{reg/lex}}(X)$ are given by coproducts in $C/X$, the induced map $Y+Z \to X$ must become an isomorphism in $Sub_{C_{reg/lex}}(X)$, which means that it must admit a section; but since $Z$ is initial this means that $p$ itself has a section.

* If $C$ is [[well-pointed topos|well-pointed]], it does not follow that $C_{ex/lex}$ or $C_{reg/lex}$ are (in the stronger sense appropriate for non-toposes).  It is of course always true that $1$ is projective in the completions.  And if $C$ is lextensive, so that its completions are coherent, then $1$ is indecomposable in them as soon as it is so in $C$.  However, it does not follow that $1$ is a *strong* generator in the completions even if it is so in $C$, since the completions have (in general) many more monomorphisms than $C$ does.

  If $C$ is a well-pointed topos such that $C_{ex/lex}$ is also a topos, then the latter cannot be well-pointed unless $C$ satisfies AC, because the completion would then be Boolean, and hence AC holds by the proof above.

## Examples

(to be written...)

* The category [[Set]] of [[Bishop set|Bishop sets]] is the ex/lex completion of the category of [[hSet|hSets]] in a theory without [[quotient sets|quotients]].

* [[Realizability toposes]] arise as ex/lex completions of categories of partitioned assemblies based on a [[partial combinatory algebra]] $A$. In fact the reg/lex completion gives, as an interesting intermediate step, the category of assemblies based on $A$ (which turns out to be the [[quasitopos]] of $\neg\neg$-separated objects inside the realizability topos). This is discussed in [Menni](#Menni).

* The category $TF$ of [[torsion|torsion-free]] [[abelian groups]] is regular, but not exact.  For instance, the [[congruence]] $\{ (a,b) | a \equiv b \mod 2 \} \subseteq \mathbb{Z}\times\mathbb{Z}$ is not a kernel in $TF$.  Unsurprisingly, the ex/reg completion of $TF$ is equivalent to the category $Ab$ of all abelian groups.

  Note that although $TF$ is not exact, its inclusion into $Ab$ does have a left adjoint (quotient by torsion), and thus $TF$ is cocomplete.  Herein lies a subtle trap for the unwary: since the ex/reg completion monad is idempotent, it is in particular lax-idempotent, which means that any left adjoint to the unit $C \hookrightarrow C_{ex/reg}$ is in fact a (pseudo) algebra structure; but since the monad is actually idempotent, any algebra structure is an equivalence.  Of course, the reflection $Ab \to TF$ is *not* an equivalence, which doesn't contract the general facts because this left adjoint is not a regular functor, and hence not an adjunction in the 2-category on which the monad $(-)_{ex/reg}$ lives.  In fact, it is not hard to check that $C \hookrightarrow C_{ex/reg}$ has a left adjoint in $Cat$ if and only if $C$ has [[coequalizers]] of [[congruences]] (while if it has a left adjoint in $Reg$ then it must be an equivalence).

* A [[free cocompletion]] of a (possibly large) finitely complete category, i.e., the category of [[small presheaves]] on $C$, is the ex/lex completion of the [[free cartesian category|free coproduct completion]] of $C$. This appears as Lemma 3 [here](#JR).

## Related concepts

* [[homotopy exact completion]]

## References

* Carboni and Celia Magno, "The free exact category on a left exact one", J. Austral. Math. Soc. (Ser. A), 1982.

* {#JR} Jiří Rosický, _Cartesian closed exact completions_, JPAA 142 no. 3 (October 1999), 261-270. ([doi](https://doi.org/10.1016/S0022-4049(98)00146-7))

* Hu and Tholen, "A note on free regular and exact completions and their infinitary generalizations", [TAC](http://www.tac.mta.ca/tac/volumes/1996/n10/2-10abs.html) 1996.

* [[Aurelio Carboni]], [[Enrico Vitale]], *Regular and exact completions*, Journal of Pure and Applied Algebra **125** 1–3  (1998) 79-116 (<a href="https://doi.org/10.1016/S0022-4049(96)00115-6">doi:10.1016/S0022-4049(96)00115-6</a>)

* Birkedal and Carboni and Rosolini and Scott, "Type Theory via Exact Categories," 1998

* Stephen Lack, "A note on the exact completion of a regular
 category, and its infinitary generalizations" [TAC](http://www.tac.mta.ca/tac/volumes/1999/n3/5-03abs.html) 1999.

* The [[Elephant]], Sections A1.3 and A3.

* Carboni and Rosolini, "Locally cartesian closed exact completions", JPAA 2000

* Matías Menni, "Exact completions and toposes," Ph.D. Thesis, University of Edinburgh, 2000. ([web](http://www.lfcs.inf.ed.ac.uk/reports/00/ECS-LFCS-00-424/))
{#Menni}

* [[Michael Shulman]], "Exact completions and small sheaves".  *Theory and Applications of Categories*, Vol. 27, 2012, No. 7, pp 97-173.  [Free online](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html)
{#Shulman}

* [[Benno van den Berg]], _Inductive types and exact completion_, Ann. Pure Appl. Logic, 134 (2005), 95–121. ([online .pdf file](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.18.5456&rep=rep1&type=pdf))
 {#vdB}

[^fine]: For example, let $C = Set^{\bullet \rightrightarrows \bullet}$ be the topos of [[quiver|directed graphs]]. For each ordinal $\alpha$, let $G_\alpha$ be the directed graph whose nodes are elements of $\alpha$ and with a directed edge from $\beta$ to $\gamma$ if $\beta \lt \gamma$ in $\alpha$. Then in the poset reflection $Pos(C)$, we have a class of proper monomorphisms, e.g., $[G_\alpha] \lt [G_{\alpha'}]$ whenever $\alpha \lt \alpha'$. Thus $Pos(C)$ is a large poset. This example also shows that $Pos(C)$ need not be a [[total category]] even if $C$ is.

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Regular axiom is stronger than hausdroff axiom with example? Pls

> Reply:

> An example is spelled out [here](https://ncatlab.org/nlab/show/regular+space#RegularSpacesWhichAreNotHausdorff) at *[[regular topological space]]*.

> But for asking questions you should please turn to the [nForum](https://nforum.ncatlab.org/) instead; the $n$Lab itself is for recording mathematical content.

> Moreover, for asking basic maths questions there are other sites to turn to, such as [math.stackexchange](https://math.stackexchange.com/).


regular cardinal

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### Context
#### Foundations
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[[!include foundations - contents]]
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# Contents
* table of contents
{: toc}

## Idea

A regular cardinal is a [[cardinal number]] that is 'closed under [[union]]'.  The [[category]] of [[sets]] bounded by a regular cardinal has several nice properties, making it a [[universe]] that is handy for some purposes but falls short of being a [[Grothendieck universe]].  Unlike Grothendieck universes (which are based on [[inaccessible cardinals]] rather than regular cardinals), it is easy to prove that (even [[uncountable set|uncountable]]) regular cardinals exist.

## Definitions

An [[infinite set|infinite]] [[cardinal]] $\kappa$ is a **regular cardinal** if it satisfies the following equivalent properties:

* no set (in a [[material set theory]]) of cardinality $\kappa$ is the [[union]] of fewer than $\kappa$ sets of cardinality less than $\kappa$.

* no set (in a [[structural set theory]]) of cardinality $\kappa$ is the [[disjoint union]] of fewer than $\kappa$ sets of cardinality less than $\kappa$.

* given a function $P \to X$ (regarded as a [[family of sets]] $\{P_x\}_{x\in X}$) such that ${|X|} \lt \kappa$ and ${|P_x|} \lt \kappa$ for all $x \in X$, then ${|P|} \lt \kappa$.

* the [[category]] $\Set_{\lt\kappa}$ of sets of cardinality $\lt\kappa$ has all [[colimits]] (or just all [[coproducts]]) of size $\lt\kappa$.

* the [[cofinality]] of $\kappa$ is equal to $\kappa$.

A cardinal that is not regular is called **singular**.

### Finite regular cardinals?
{#finite}

Traditionally, one requires regular cardinals to be infinite.  This clause may be removed, in which case $0$, $1$, and $2$ are all regular cardinals.

Other modifications of the definition which are equivalent for infinite cardinals may include some of $0$, $1$, and $2$ but not all.  For instance, if we regard an indexed [[disjoint union]] $\sum_{i\in\lambda} \alpha_i$ as a binary operation taking as input $\lambda$ and a $\lambda$-indexed family $\alpha$, then closure under this binary operation as in the above definition also entails closure under the ternary version $\sum_{i\in\lambda} \sum_{j\in \mu_i} \alpha_{i,j}$, and so on.  The unary version is simply the identity operation, whereas the nullary version will always output the [[singleton set]] $1$.  (This can be seen by thinking in terms of trees of uniform finite height, or remembering that a [[dependent sum]] includes a binary [[cartesian product]] as a special case, so a nullary dependent sum should at least include a nullary product.)  Thus, from this perspective, $2$ is a regular cardinal, but $0$ and $1$ are not.  In applications for which this perspective is the relevant one, such as [[familial regularity and exactness]], one may more precisely be interested in an [[arity class]] rather than a regular cardinal.

We may rule out all three finite regular cardinals by additionally generalising from indexed disjoint unions to finitary disjoint unions.

Then in terms of $Set_{\lt\kappa}$, the (potential) conditions on a (possibly finite) regular cardinal are as follows:

1.  $Set_{\lt\kappa}$ is closed under iterated disjoint unions ($\biguplus_i A_i$).
2.  $Set_{\lt\kappa}$ is closed under the nullary iterated disjoint union (the [[singleton]]).
3.  $Set_{\lt\kappa}$ is closed under binary disjoint unions ($A \uplus B$).
4.  $Set_{\lt\kappa}$ is closed under the nullary disjoint union (the [[empty set]]).

These are all variations on the theme of closure under disjoint unions.

Clauses (2--4) hold of all infinite cardinals, while clauses (2&3) together force $\kappa$ to be greater than any finite cardinal.  However, if we require only clauses (1&2), then $2$ is a regular cardinal.

### In weak foundations

Thinking of a regular cardinal *as* a cardinal number makes the most sense using the [[axiom of choice]].  Otherwise, we probably want to think of it as a *collection* of cardinals, or equivalently think of it as the category $Set_{\lt\kappa}$.

From this perspective, a regular cardinal is a [[full subcategory]] of $Set$ that is closed under taking [[quotient objects]] and satisfies the condition on $Set_{\lt\kappa}$ above.  We can then recover $\kappa$ as the smallest cardinal number greater than every cardinal in $Set_{\lt\kappa}$, if we accept the axiom of choice.

Note that if we require only conditions (1&2) on $Set_{\lt\kappa}$, then (even classically), $\{1\}$ is an acceptable (and finite) regular collection of cardinals, even though it is not actually of the form $Set_{\lt\kappa}$ for any cardinal number $\kappa$.

In the absence of the axiom of choice, it is not clear that there exist arbitrarily large regular cardinals.  Thus in weaker foundations, regular cardinals (or "regular sets of cardinals") can be regarded as a [[large cardinal]] property.

At least if "regular cardinal" has its classical meaning of a particular ordinal, then the statement that *there exist arbitrarily large regular cardinals* is independent of [[ZF]]; in fact it is consistent with ZF that all uncountable cardinals are singular.  A foundational axiom which is related to the existence of regular cardinals (but considers them as sets with various closure properties, rather than cardinal numbers) is the [[regular extension axiom]].

## Examples

### Regular cardinals

* The first (infinite) regular cardinal is $\aleph_0 = {|\mathbb{N}|}$, because a set with cardinality less than $\aleph_0$ is a [[finite set]], and a finite union of finite sets is still a finite set.

\begin{theorem} Assuming the [[axiom of choice]], the [[successor]] of any infinite cardinal, such as $\aleph_0$, is a regular cardinal.
\end{theorem}

In the case of $\aleph_0$, this means that a countable union of countable sets is countable.  Note that this implies that there exist arbitrarily large regular cardinals: for any cardinal $\lambda$ there is a greater regular cardinal, namely $\lambda^+$.

\begin{proof}
Under the axiom of choice, the successor of a cardinal number is the [[Hartogs number]] (see there): if $\lambda$ is the cardinality of $X$, then $\lambda^+$ is the order type of the well-ordered set $\aleph(X)$. If $(\lambda_\alpha) = (0 = \lambda_0 \lt \lambda_1 \lt \ldots)$ is a set of ordinals with least upper bound $\lambda^+$, and supposing, working toward a contradiction, that this set has cardinality $\leq \lambda$, then the corresponding initial segments $X_\alpha$ of $\aleph(X)$ provide a partition
$$\aleph(X) = \bigcup_\alpha X_{\alpha+1} \setminus X_\alpha$$
and by construction of $\aleph(X)$, each $X_{\alpha+1} \setminus X_\alpha$ has cardinality $\leq \lambda$. Thus the union would have at most $\lambda \cdot \lambda = \lambda$ elements, which is less than the cardinality $\lambda^+$ of $\aleph(X)$, contradiction.
\end{proof}

### Singular cardinals

* $\aleph_\omega = \bigcup_{n\in \mathbb{N}} \aleph_n$ is singular, more or less by definition, since $\aleph_n\lt\aleph_\omega$ and ${|\mathbb{N}|} = \aleph_0 \lt\aleph_\omega$.

* More generally, any limit cardinal that can be "written down by hand" should be singular, since if it were regular then it would be [[weakly inaccessible cardinal|weakly inaccessible]], and the existence of weakly inaccessible cardinals cannot be proven in [[ZFC]] (if $ZFC$ is consistent).  We say 'should' rather than 'must', since there are exceptions, but they are sort of cheating: one (definable with Choice) is 'the smallest limit cardinal that is regular if and only if some weakly innaccessible cardinal exists'.

* Assuming the consistency (with [[ZFC]]) of 'there is a [[proper class]] of [[strongly compact cardinal|strongly compact cardinals]]', it is consistent with $ZF$ that every uncountable cardinal is singular (and in fact every infinite well-orderable cardinal has [[cofinality]] $\aleph_0$), a result due to [[Moti Gitik]].  (Of course this conclusion is inconsistent with $ZFC$, in which many uncountable cardinals, starting with $\aleph_1$, are regular.)

## Relevance for category theory

Regular cardinals $\lambda$ are used in the definitions
of $\lambda$-[[filtered colimits]],
$\lambda$-[[presentable objects]],
$\lambda$-[[accessible categories]],
locally $\lambda$-presentable categories,
$\lambda$-[[ind-completion]],
and many notions derived from these, e.g.,
$\lambda$-[[combinatorial model categories]].

Then notions make sense for all cardinals, not necessarily regular.
However, the relevant concepts reduce to those for regular cardinals.

The relevance of regular cardinals for these concepts
was already pointed out by [[Gabriel]] and [[Ulmer]]
in their original treatise on [[locally presentable categories]], where on page 2 we read:

> Sei $\alpha$ eine Kardinalzahl, wobei $3 \le \alpha \lt \infty$.
Eine geordnete Menge $(N,\le)$ heisst
*$\alpha$-gerichtet*, wenn es für jede Familie $(\nu_i)_{i\in I}$ in $N$
mit $Kard(I) \lt \alpha$ ein $\mu$ gibt
derart, dass $\nu_i\le\mu$.
Sei $\beta$ die kleinste reguläre Kardinalzahl $\ge \alpha$.
Dann ist jede $\alpha$-gerichtete Menge auch $\beta$-gerichtet.
Wir setzen deshalb im folgenden zusatzlich voraus,
dass $\alpha$ regulär ist (vgl. §0).

where the meaning of $\alpha\lt\infty$ is explained on page 13:

> Ausserdem bezeichnen wir mit $\infty$ die kleinste Kardinalzahl, die nicht mehr zu $U$ gehört (die also in unserer Sprache keine Menge ist).

> Dieser Arbeit liegt die Mengenlehre von Zermelo-Fraenkel und ein fest gewähltes
Universum $U$ zugrunde. Wir setzen dabei voraus, dass $U$ die Menge $\mathbf{N}$ der natürlichen Zahlen enthält.

If $\lambda$ is not a [[regular cardinal]],
then a category has $\lambda$-[[filtered colimits]]
if and only if it has $\lambda^+$-[[filtered colimits]],
and $\lambda^+$ is always a [[regular cardinal]] (assuming the [[axiom of choice]]).
In this case,
a category is locally $\lambda$-presentable
if and only if it is locally $\lambda^+$-presentable.

Reference: Proposition 5.6 in [Rezk2021](#Rezk2021).

## Related concepts

* [[large cardinal]]

* [[cofinality]]

* [[arity class]]

* [[product-regular cardinal]]

## References

* [[Peter Gabriel]], [[Friedrich Ulmer]], *Lokal präsentierbare Kategorien*, Lecture Notes in Mathematics 221, Springer 1971 ([doi:10.1007/bfb0059396](http://dx.doi.org/10.1007/bfb0059396), [MR0327863](http://www.ams.org/mathscinet-getitem?mr=0327863)).

* {#Rezk2021} [[Charles Rezk]], *Generalizing accessible ∞-categories*, 2021 ([pdf](https://faculty.math.illinois.edu/~rezk/accessible-cat-thoughts.pdf)).

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