regular category

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### Context
#### Regular and Exact categories
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[[!include regular and exact categories - contents]]
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#### Category Theory
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[[!include category theory - contents]]
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#Contents#
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## Idea

A _regular category_ is a [[finitely complete category]] which admits a good notion of [[image]] factorization. A primary _raison d'être_ behind regular categories $C$ is to have a decently behaved _calculus of [[relations]]_ in $C$ (see [[Rel]]).

Regular categories also provide a natural [[semantics|semantic]] environment to [[interpretation|interpret]] a particularly well behaved positive [[fragment]] of [[first order logic]] having connectives $\top$, ${\wedge}$, $\exists$; in other words, their [[internal logic]] is [[regular logic]].

## Definition

\begin{definition}\label{RegularCategory}
**(regular category)**
\linebreak

A [[category]] $C$ is called **regular** if

1. it is [[finitely complete category|finitely complete]];

1. the [[kernel pair]]

   $$
     \array{
       d\times_c d
       &\stackrel{p_1}{\longrightarrow}&
       d
       \\
       {{}^\mathllap{p_2}}
       \big\downarrow
       &&
       \big\downarrow{{}^\mathrlap{f}}
       \\
       d &\stackrel{f}{\longrightarrow}& c
     }
   $$

   of any [[morphism]] $f: d \to c$ admits a [[coequalizer]] $d \times_c d \,\rightrightarrows\, d \to coeq(p_1,p_2)$;

1. the [[pullback]] of a [[regular epimorphism]] along any morphism is again a regular epimorphism.

\end{definition}

\begin{remark}
The kernel pair is always a [[congruence]] on $d$ in $C$; informally, $\ker(f) = d\times_c d$ is the [[subobject]] of $d \times d$ consisting of pairs of elements which have the same value under $f$ (sometimes called the "kernel" of a function in [[Set]]). The coequalizer above is supposed to be the "object of equivalence classes" of $\ker(f)$ as an internal [[equivalence relation]].
\end{remark}

\begin{remark}
A map which is the coequalizer of a parallel pair of morphisms is called a _[[regular epimorphism]]_. In fact, in any category satisfying the first two conditions in Def. \ref{RegularCategory}, every regular epimorphism is the coequalizer of its kernel pair. (See for instance Lemma 5.6.6 in _[[Practical Foundations]]_.)
\end{remark}

\begin{remark}
The last condition in Def. \ref{RegularCategory} may equivalently be stated in the form "coequalizers of kernel pairs are [[pullback-stable colimit|stable under pullback]]". However, it is not generally true in a regular category that the pullback of a general coequalizer diagram

  $$e \;\rightrightarrows\; d \to c$$

  along a morphism $c' \to c$ is again a coequalizer diagram (nor need a regular category have coequalizers of all parallel pairs).
\end{remark}

In fact, a characterization equivalent to Def. \ref{RegularCategory} is:

\begin{proposition}
\label{CharacterizationViaImageFactorization}
A **regular category** is a [[finitely complete category]] with [[pullback]]-[[stable factorization system|stable]] [[image]] [[factorization system|factorizations]] in that

1. every morphism factors as a [[regular epimorphism]] followed  by a [[monomorphism]]

1. these factorizations are preserved under [[pullback]].

\end{proposition}
(eg. [Gran 2020, Thm. 1.14](#Gran20), cf. also [MO:a/252842](https://mathoverflow.net/a/252842/381))

See also *[[familial regularity and exactness]]* for a generalization of this approach to include [[coherent category|coherent]] categories as well.

## Examples

Examples of regular categories include the following:

\begin{example}
[[Set]] is a regular category.
\end{example}

\begin{example}\label{ToposesAreRegular}
  More generally, any [[topos]] is regular.
\end{example}
([Barr, Grillet & von Osdol 1971, p. 17](#BarrGrilletvonOsdol), [Borceux 1994III, Prop. 3.4.14](#Borceux94III), [Johnstone 2002, p. 92](#Johnstone02))

\begin{example}
Even more generally, a [[locally cartesian closed category]] with [[coequalizers]] is regular, and so any [[quasitopos]] is regular.
\end{example}

\begin{example}
The category of [[algebra over a Lawvere theory|models]] of any finitary [[algebraic theory]] (i.e., [[Lawvere theory]]) $T$ is regular. This applies in particular to the category [[Ab]] of [[abelian groups]].
\end{example}

\begin{example}\label{CategoryOfGroupsIsRegular}
  The category [[Grp]] of all [[groups]] (including [[non-abelian groups]]) is regular.
\end{example}
(e.g. [Borceux 1994 II, Ex. 2.4.3](#Borceux94II))

\begin{example}
Actually, any category that is [[monadic functor|monadic]] over [[Set]] is regular. For example, the category of [[frames]] $Frm \simeq Loc^{op}$ is regular, and the category of [[compact Hausdorff spaces]] is regular. A proof may be found [here](/nlab/show/colimits+in+categories+of+algebras#exact).
\end{example}

\begin{example}
Any [[abelian category]] is regular.
\end{example}

\begin{example}
If $C$ is regular, then so is the [[functor category]] $C^D$ for any category $D$.
\end{example}

\begin{example}
If $C$ is regular and $T$ is a [[Lawvere theory]], then the category $Mod(T, C)$ of $T$-models in $C$ is also regular. See Theorem 5.11 in Barr's [Exact Categories](#Barr).
\end{example}

\begin{example}
A [[slice]] of a regular category is also regular; cf. [[locally regular category]]. So is any [[co-slice]]. (Source: [[Borceux-Bourn]], Appendix section 5.)
\end{example}

\begin{example}
If $Q$ is a quasitopos, then $Q^{op}$ is regular. Source: A2.6.3(i) in the [[Elephant]].
\end{example}

\begin{example}
The [[opposite category]] [[Top]]$^{op}$ is regular. The key facts are that [[regular monomorphisms]] in $Top$ are the same as [[subspace]] inclusions, and that the [[pushout]] of a subspace inclusion is a subspace inclusion as proven [there](subspace+topology#pushout).
\end{example}

\begin{example}
The category of ([[Hausdorff space|Hausdorff]]) [[Kelley spaces]] is regular (however, is not locally cartesian closed, nor is it [[exact category|exact]]) ([Cagliari-Matovani-Vitale 95](#CagliariMatovaniVitale95))
\end{example}

\begin{example}
**(counter-examples)**
Examples of categories which are **not regular** include

* [[Cat]], [[Pos]], and [[Top]].

The following example proves failure of regularity in all three cases:

Let $A$ be the poset $\{a, b\} \times (0 \to 1)$; let $B$ be the poset $(0 \to 1 \to 2)$, and let $C$ be the poset $(0 \to 2)$. There is a regular epi $p: A \to B$ obtained by identifying $(a, 1)$ with $(b, 0)$, and there is the evident inclusion $i: C \to B$. The pullback of $p$ along $i$ is the inclusion $\{0, 2\} \to (0 \to 2)$, which is certainly an epi but not a regular epi. Hence regular epis in $Pos$ are not stable under pullback.

Interpreting the posets as categories, the same example works for $Cat$, and also for preorders. On the other hand, the category of finite preorders is equivalent to the category of [[finite topological spaces]], so this example serves to show also that the category [[Top]] of *all* topological spaces is not regular.
\end{example}

However:

\begin{example}
\label{CompactlyGeneratedHausdorffSpacesAreRegular}
**(compactly generated Hausdorff spaces form a regular category)**
\linebreak
  The [[convenient category of topological spaces|convenient]] [[full subcategories]]
$$
  kTop
    \xhookrightarrow{\;}
  kHaus
    \xhookrightarrow{\;}
  Top
$$
of
[[compactly generated Hausdorff spaces]]
and of
[[compactly generated weakly Hausdorff spaces]]
are regular
([Cagliari, Matovani & Vitale 1995, p. 3](#CagliariMatovaniVitale95)).
\end{example}

\begin{example}
\label{CompactlyGeneratedGSpacesFormARegularCategory}
**(compactly generated $G$-spaces form a regular category)**
For $G \,\in\, Grp(kTop)$ a [[compactly generated weakly Hausdorff space|compactly generated weak Hausdorff]] [[topological group]], its category of [[internal actions]], hence the category of CGWH-[[topological G-spaces]] $G Act(kTop)$ is a regular category.
\end{example}
\begin{proof}
  The [[forgetful functor]] $G Act(kTop) \xrightarrow{\;} kTop$
[[created limit|creates]] all [[limits]] and [[colimits]] ([this Prop.](topological+G-space#ForgetfulFunctorCreatesLimitsAndColimits)). Since regularity is entirely a condition on limits and colimits ([this Def.](regular+category#RegularCategory)) the statement follows from Ex. \ref{CompactlyGeneratedHausdorffSpacesAreRegular}.
\end{proof}

\begin{example}
If $T$ is a [[Mal'cev theory]] (e.g., the theory of groups), then the category $Top^T$ of $T$-models in [[Top]] is regular. This is because for $T$ Mal'cev, coequalizer maps in $Top^T$ are necessarily open surjections, and open surjections are stable under pullback.
\end{example}

## Properties

### Epimorphisms
 {#Epimorphisms}

\begin{prop}
For a [[morphism]] $f$ in a [[regular category]], the following conditions are all equivalent:

1. $f$ is an [[effective epimorphism]];

1. $f$ is a [[regular epimorphism]];

1. $f$ is a [[strong epimorphism]];

1. $f$ is an [[extremal epimorphism]].

\end{prop}

### Pullback and pasting law

In a [[regular category]], the usual [[pasting law]] implication also applies along [[regular epimorphisms]] also in the reverse direction, :

\begin{proposition}
\label{WrongWayPastingLawInRegularCategory}
  In a [[regular category]], consider a [[commuting diagram]] of the form
\begin{tikzcd}
  {}
  \ar[r]
  \ar[d]
  \ar[dr,phantom,"\mbox{\tiny(pb)}"]
  &
  {}
  \ar[r]
  \ar[d]
  &
  {}
  \ar[d]
  \\
  {}
  \ar[r, ->>]
  &
  {}
  \ar[r]
  &
  {}
\end{tikzcd}
where

1. the left square is a [[pullback]];

1. the bottom left morphism is an [[regular epimorphism]].

Then right right square is a pullback iff the total rectangle is.
\end{proposition}
(e.g. [Gran 2020, Lem. 1.15](regular+category#Gran20))

### Factorization properties

+-- {: .num_prop}
###### Proposition
**image factorization**

In a regular category, every morphism $f : x\to y$ can be factored -- uniquely up to [[isomorphism]] -- through its [[coimage]] $coim(f)$ as

$$
  f : x \stackrel{e}{\to} coim(f) \stackrel{i}{\to} y
  \,,
$$

where $e$ is a [[regular epimorphism]] and $i$ a [[monomorphism]].

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###### Proof

Let $e : x \to coim(f)$ be the [[coequalizer]] of the [[kernel pair]] of $f$. Since $f$ coequalizes its kernel pair, there is a unique map $i: coim(f) \to y$ such that $f = i e$. It may be shown from the regular category axioms that $i$ is monic and in fact represents the [[coimage]] of $f$, i.e., the smallest subobject through which $f$ factors.

This is the mere definition of first isomorphism theorem.

A proof is spelled out on p. 30 of ([vanOosten](#vanOosten)).

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###### Proposition

The classes of [[regular epimorphism]], [[monomorphism]]s in a regular category $C$ form a [[orthogonal factorization system|factorization system]].

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### Embedding properties

\begin{remark}
\label{barrembeddingtheorem}
If a regular category is small, it admits particularly nice embeddings into presheaf categories. See *[[Barr embedding theorem]]* for more.
\end{remark}

### Axiomatizability properties

Roughly speaking, regular categories tend to be relatively well-behaved when it comes to desribing them in formalized logics.

### Regular functors over a small regular category

\begin{proposition}
**(Makkai)**
\linebreak
If a regular category $\mathcal{R}$ is small, then the full subcategory of the functor category $[\mathcal{R},\mathsf{Set}]$ consisting of the [[regular functors]] only is an [[elementary class]] w.r.t. the signature given by (the underlying graph) of $\mathcal{R}$.

\end{proposition}

## Stronger conditions

### Exactness

If a regular category additionally has the property that every [[congruence]] is a kernel pair (and hence has a quotient), then it is called a (Barr-) [[exact category]].  Note that while regularity implies the existence of some coequalizers, and exactness implies the existence of more, an exact category need not have all coequalizers (only coequalizers of congruences), whereas a regular category can be [[cocomplete category|cocomplete]] without being exact.

Regularity and exactness can also be phrased in the language of [[Galois connection]]s, as a special case of the notion of [[generalized kernels]].

### Higher arity

As exactness properties go, the ones possessed by general regular categories are fairly moderate; the main condition is of course stability of regular epis under pullback.  A natural generalization is to include (finite or infinite) unions of subobjects, or equivalently images of (finite or infinite) families as well as of single morphisms.  This leads to the notion of [[coherent category]].

Just as regularity implies the existence of certain coequalizers, coherence implies the existence of certain [[coproducts]] and [[pushouts]], but not all.  A [[lextensive category]] has all (finite or infinite) coproducts that are disjoint and stable under pullback.  It is easy to see that a lextensive regular category must actually be coherent.

## The regular topology

Any regular category $C$ admits a [[subcanonical site|subcanonical]] [[Grothendieck topology]] whose covering families are generated by single [[regular epimorphisms]]: the [[regular coverage]].  If $C$ is [[exact category|exact]] or has pullback-stable [[reflexive coequalizer]]s, then its [[codomain fibration]] is a [[stack]] for this topology (the necessary and sufficient condition is that any pullback of a kernel pair is again a kernel pair).

## Making categories regular

Any category $C$ with [[finite limits]] has a **reg/lex completion** $C_{reg/lex}$ with the following properties:

* There is a [[full and faithful functor]] $C\hookrightarrow C_{reg/lex}$
* Each object of $C$ becomes [[projective object|projective]] in $C_{reg/lex}$
* Each left-exact functor $C\to D$, where $D$ is regular, extends to an essentially unique [[regular functor]] $C_{reg/lex}\to D$.

In particular, the reg/lex completion is a left adjoint to the [[forgetful functor]] from regular categories to lex categories (categories with finite limits).  The reg/lex completion can be obtained by "formally adding images" for all morphisms in $C$, or by "closing up" $C$ under images in its [[presheaf category]] $[C^{op},Set]$; see [[regular and exact completions]].  In general, even if $C$ is regular, $C_{reg/lex}$ is larger than $C$ (that is, it is a [[free cocompletion]] rather than merely a [[completion]]), although if $C$ satisfies the [[axiom of choice]] (in the sense that all [[regular epimorphisms]] are split), then $C\simeq C_{reg/lex}$.

Regular categories of the form $C_{reg/lex}$ for a lex category $C$ can be characterized as those regular categories in which every object admits both a regular epi from a projective object and a monomorphism into a projective object, and the projective objects are closed under finite limits.  In this case $C$ can be recovered as the subcategory of projective objects.  In fact, the construction of $C_{reg/lex}$ can be extended to categories having only [[weak finite limits]], and the regular categories of the form $C_{reg/lex}$ for a "weakly lex" category $C$ are those satisfying the first two conditions but not the third.

When the reg/lex completion is followed by the [[ex/reg completion]] which completes a regular category into an [[exact category|exact one]], the result is unsurprisingly the [[ex/lex completion]].  See [[regular and exact completions]] for more about all of these operations.

## Related entries

* [[exact category]], [[pretopos]]

* [[regular 2-category]], [[regular derivator]], [[regular (∞,1)-category]]

* [[Barr embedding theorem]]

* [[allegory]]

* [[Rel]]

[[!include categories and logic - table]]

## References

The notion of regular categories was introduced in:

* {#BarrGrilletvonOsdol} [[Michael Barr]],  [[Pierre Grillet]], [[Donovan van Osdol]], *Exact Categories and Categories of Sheaves*,  Lec. Notes in Math. __236__, Springer 1971, ([doi:10.1007/BFb0058579](https://link.springer.com/book/10.1007/BFb0058579))

Historical context:

* [[Peter Johnstone]], Introduction of: _[[Topos Theory]]_ (1977)

Introduction:

* [[Carsten Butz]], _Regular Categories and Regular Logic_ , BRICS LS-98-2 Aarhus 1998. ([brics](http://www.brics.dk/LS/98/2/))

* {#Gran20} [[Marino Gran]], *An introduction to regular categories*, in: New Perspectives in Algebra, Topology and Categories,  Coimbra Mathematical Texts, Springer (2021) [[arXiv:2004.08964](https://arxiv.org/abs/2004.08964), [ISBN:978-3-030-84319-9](https://www.springerprofessional.de/en/new-perspectives-in-algebra-topology-and-categories/19764096)]

Textbook accounts:

* [[Peter Freyd]], [[Andre Scedrov]], Chapter 1.5 of: _[[Categories, Allegories]]_ , North-Holland Amsterdam 1990. (chap. 1.5. pp.68ff)

* {#Borceux94II} [[Francis Borceux]], Chapter 2 of: *[[Handbook of Categorical Algebra]]*, Vol. 2: *Categories and Structures*, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]

* {#Borceux94III} [[Francis Borceux]], Section 3.3.4 of: *[[Handbook of Categorical Algebra]]*. Vol. 3. *Categories of Sheaves*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) [[doi:10.1017/CBO9780511525872](https://doi.org/10.1017/CBO9780511525872)]

* {#Johnstone02} [[Peter Johnstone]], Section A1.3. pp.18ff of _[[Sketches of an Elephant]] I_ , Oxford UP 2002.

* [[Dominique Bourn]], [[Marino Gran]], _Regular, Protomodular, and Abelian Categories_, Chapter IV, pp.165-211 in: Maria Pedicchio, [[Walter Tholen]] (eds.), _Categorical Foundations_, Cambridge University Press 2004 ([doi:10.1017/CBO9781107340985.007](https://doi.org/10.1017/CBO9781107340985.007))

The following set of course notes has a section on regular categories

* {#vanOosten}[[Jaap van Oosten]], _Basic category theory_ , BRICS LS-95-1 Aarhus 1995. ([Section 4.1](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf#page=30)) ([pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf))

An application of the regularity condition[^knop] is found in:

* F. Knop, _Tensor envelopes of regular categories_, ([arXiv:math/0610552v2](http://arxiv.org/abs/math/0610552))

[^knop]: Knop's condition for regularity is slightly different from that presented here; he works with categories that when augmented by an absolutely initial object are regular in the terminology here.  In the paper, Knop generalizes a construction of Deligne by showing how to construct a symmetric pseudo-abelian [[tensor category]] out of a regular category through the calculus of relations.

Generalization of the notion of regular categories to [[enriched category theory]]:

* [[Brian Day]], [[Ross Street]], _Localisation of locally presentable categories_, J. Pure and Appl. Algebra __58__ (1989) 227-233.

* Dimitri Chikhladze, _Barr's embedding theorem for enriched categories_, J. Pure Appl. Alg. __215__, n. 9 (2011) 2148-2153, [arxiv/0903.1173](http://arxiv.org/abs/0903.1173), [doi](http://dx.doi.org/10.1016/j.jpaa.2010.12.004)

* [[Stephen Lack]], [[Giacomo Tendas]], _Enriched regular theories_, J. Pure Appl. Alg. __226__, n. 6 (2020) 106268, [arXiv:1907.02301](https://arxiv.org/abs/1907.02301), [doi](https://doi.org/10.1016/j.jpaa.2019.106268)

Regularity of (Hausdorff) [[compactly generated topological spaces]]:

* {#CagliariMatovaniVitale95} F. Cagliari, S. Mantovani, [[Enrico Vitale]], *Regularity of the category of Kelley spaces*, Applied Categorical Structures volume 3, pages 357–361 (1995) ([doi:10.1007/BF00872904](https://link.springer.com/article/10.1007/BF00872904), [pdf](http://www.dm.unibo.it/~cagliari/articoli/Regularkelley.pdf))

[[!redirects regular categories]]


regular coverage

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

For a [[regular category]] $C$, the _regular coverage_ on $C$ is the [[coverage]] whose each [[covering]] family has one element which is a [[regular epimorphism]].

The [[Grothendieck topology]] generated from a regular coverage is called the _regular topology_.

[[subcanonical site|subcanonical]] [[Grothendieck topology]] whose covering families are generated by single [[regular epimorphisms]]: the [[regular coverage]].

## Properties

* If $C$ is [[exact category|exact]] or has pullback-stable [[reflexive coequalizer]]s, then its [[codomain fibration]] is a [[stack]] for this topology (the necessary and sufficient condition is that any pullback of a kernel pair is again a kernel pair).

* For $\mathcal{C}_{\mathbb{T}}$ the [[syntactic category]] of a [[regular theory]], the regular coverage makes it the [[syntactic site]], which is a site of defininition for the [[classifying topos]] of $\mathbb{T}$.

[[!redirects regular coverages]]


regular differential operator

Grothendieck has developed a deep version of _differential calculus_, based on a linearization of $O_X$-bimodules. It is also related to the (de Rham) descent data for the stack of $O_X$-modules over the simplicial scheme resolving the diagonal of $X$. As abstract [[descent]] data correspond to the flat [[connections]] for the corresponding [[monad]], this was historically the first case in which this correspondence was noted; in positive characteristics Grothendieck called the corresponding descent data for the de Rham site "costratifications", see

* P. Berthelot,  A. Ogus, _Notes on crystalline cohomology_, Princeton Univ.P. 1978. vi+243, ISBN0-691-08218-9

This corresponds to looking at a sequence of infinitesimal neighborhoods of the diagonal. This geometrical principle can be applied to other categories; it is the basis of the study of [[jet schemes]] and close in spirit to some constructions in synthetic differential geometry.

Given a commutative unital ring $R$, a filtration $M_n$ ($n\geq -1$) on a $R$-$R$-bimodule $M$ is a __differential filtration__ if the commutator $[r,P]$ for any $P$ in $M_n$ and $r$ in $R$ is in $M_{n-1}$, and $M_{-1} = 0$. A bimodule is differential if it has an exhaustive ($\cup_N M_n = M$) differential filtration. Every $R$-$R$-bimodule has a __differential part__, i.e. the maximal differential submodule of $M$.

__Regular differential operators__, as defined by Grothendieck, are the elements of the differential part $Diff(R,R)$ of $Hom(R,R)$ i.e. a maximal differential subbimodule in  $Hom(R,R)$. The operators in $Diff(R,R)_n$ are called the differential operators of degree $\leq n$.
If $R\to B$ is a ring morphism, then the differential part of $B$ via its natural $R$-$R$-bimodule structure is also an object of $R\backslash \mathrm{Ring}$; in particular $Diff(R,R)$ is a ring and $R\hookrightarrow Diff(R,R)$ is an embedding of rings.

More generally (and in the affine case equivalently), for a $S$-scheme $X$, let $P^n_{X/S}$ denote the sheaf $(O_X\otimes_{f^{-1}(O_S)} O_X)/I^{n+1}$, where $I$ is the ideal of the diagonal (this makes sense since the diagonal morphism is an immersion, cf. EGAI, 5.3.9.), and $f:X\rightarrow S$ the structure morphism. Consider $P^n_{X/S}$ as $O_X$-module via the morphism $O_X\rightarrow O_X\otimes_{f^{-1}(O_S)} O_X$, $a\mapsto a\otimes 1$.

For $O_X$-modules $E,F$,  $Diff_S(F,E)_n$ is defined to be $Hom_{O_X}(P_{X/S}^n\otimes_{O_X} F, E)$. Note that $P^n_{X/S}$ has two canonical structures as $O_X$-module given by the projections $p_i: X \times_S X\rightarrow X$. The tensor product $P_{X/S}^n \otimes_{O_X} F$ is understood to be constructed via $p_1$ and considered as an $O_X$-module via $p_0$.

In the affine case, and in characteristics zero, the sheaf of regular differential operators is locally isomorphic to the [[Weyl algebra]]. For that simple case, a good reference is

* S. C. Coutinho, A primer of algebraic $D$-modules,
London Math. Soc. Stud. Texts, 33, Cambridge University Press, Cambridge, 1995. xii+207 pp.

Regular differential operators have been nontrivially generalized to noncommutative rings (and schemes) by V. Lunts and [[Alexander Roseberg|A. L. Rosenberg]], as well as to the setting of braided monoidal categories. Their motivation is an analogue of a Beilinson-Bernstein localization theorem for quantum groups. The category of differential bimodules is categorically characterized in their work as the minimal [[coreflective subcategory|coreflective]] [[topologizing subcategory| topologizing]] [[monoidal category|monoidal]] subcategory of the abelian monoidal category of $R$-$R$-bimodules which is containing $R$. In the case of noncommutative rings, Lunts-Rosenberg definition of differential operators has been recovered from a different perspective in the setup of [[noncommutative algebraic geometry]] represented by monoidal categories; the emphasis is on the duality between infinitesimals and differential operators:

 * [[Tomasz Maszczyk]], Noncommutative geometry through monoidal categories, [arXiv:0611806](http://arxiv.org/abs/math/0611806)

See also [[regular differential operator in noncommutative geometry]].

MathOverflow: [Equivalence of "Weyl Algebra" and "Crystalline" definitions of rings of differential operators between modules?](http://mathoverflow.net/questions/210891/equivalence-of-weyl-algebra-and-crystalline-definitions-of-rings-of-differen), [Ring of differential operators of a quotient ring](http://mathoverflow.net/questions/194218/ring-of-differential-operators-of-a-quotient-ring)

category: algebraic geometry
[[!redirects regular differential operators]]


regular differential operator in noncommutative geometry

[[regular differential operator|Regular differential operators]] have been nontrivially generalized to noncommutative rings (and schemes) by [[Valery Lunts|V. Lunts]] and [[Alexander Rosenberg|A. L. Rosenberg]], as well as to the setting of braided monoidal categories. As in the commutative case, regular differential operators on a $k$-algebra $R$ form the differential part of the bimodule of the $k$-endomorphisms. The differential part is geometrically defined as the $\Delta$-part where $\Delta$ is the so-called diagonal [[topologizing subcategory]] of the abelian category of endofunctors $End_c A$ of the category $A$ of quasicoherent sheaves ($R$-modules in affine case) having right adjoint. The diagonal is by the definition the smallest coreflective topologizing subcategory in $End_c A$ containing the identity functor. For every coreflective topologizing subcategory $\mathbb{T}$ in the abelian category satisfying the property sup one defines the notions of $\mathbb{T}$-torsion and $\mathbb{T}$-part of any object $M$, see [[differential monad]].

The following two papers dwell mainly on the affine and projective cases

* V. A. Lunts, A. L. Rosenberg, _Differential operators on noncommutative rings_, Selecta Math. (N.S.)  __3__  (1997),  no. 3, 335--359 ([doi](http://dx.doi.org/10.1007/s000290050014)); sequel: _Localization for quantum groups_, Selecta Math. (N.S.) __5__ (1999), no. 1, pp. 123--159 ([doi](http://dx.doi.org/10.1007/s000290050044)).

and the following two unpublished preprints outline a more general categorical and geometric picture including the [[Sasha Beilinson|Beilinson]]'s notion of [[D-affinity]] generalized to (co)monads

* V. A. Lunts, A. L. Rosenberg, _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3894), II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3916)

Their motivation is an analogue of a Beilinson-Bernstein localization theorem for [[quantum groups]]. The category of differential bimodules is categorically characterized in their work as the minimal [[coreflective subcategory|coreflective]] [[topologizing subcategory| topologizing]] [[monoidal category|monoidal]] subcategory of the abelian monoidal category of $R$-$R$-bimodules which is containing $R$. In the case of noncommutative rings, Lunts-Rosenberg definition of differential operators has been recovered from a different perspective in the setup of [[noncommutative algebraic geometry]] represented by monoidal categories; the emphasis is on the duality between infinitesimals and differential operators:

 * [[Tomasz Maszczyk]], _Noncommutative geometry through monoidal categories_, [arXiv:0611806](http://arxiv.org/abs/math/0611806)

There are some other approaches to rings of differential operators in noncommutative geometry. In easy semicommutative cases (like nilpotent thickenings of commutative schemes) one can use the standard Grothendieck definition without change. On the other hand, there is an approahc by generators and relations in affine case, corresponding to the recipe for preprojective algebras of quivers. It has some nice localization properties and relations to double derivations and double Poisson geometry. See papers by Yuri Berest and

* [[V. Ginzburg]], [[T. Schedler]], _Differential operators and BV structures in noncommutative geometry_, Sel. Math. New Ser. 16, 673–730 (2010) [doi](https://doi.org/10.1007/s00029-010-0029-8) [arXiv:0710.3392](https://arxiv.org/abs/0710.3392)

and a sequence of article by Yuri Berest and various colaborators including

* Yuri Berest, Oleg Chalykh, Farkhod Eshmatov, _Recollement of deformed preprojective algebras and the Calogero-Moser correspondence_, [arxiv/0710.3392](http://arxiv.org/abs/0710.3392)

category: noncommutative geometry


regular element

> This entry is about regular elements in [[formal logic]] and [[topology]]. For regular elements in [[physics]]/[[quantum field theory]] see at _[[regularization (physics)]]_. For regular elements in [[ring theory]] and [[commutative algebra]], see [[cancellative element]].

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A __regular element__ of a [[Heyting algebra]] is an element $x$ such that $\neg{\neg{x}} = x$.

Thus a [[Boolean algebra]] is precisely a Heyting algebra in which every element is regular.

As a special case, a __regular open__ in a [[locale]] $S$ is a regular element of $S$ as a [[frame]].  These define the __regular [[open sublocales]]__ of $S$.  We may also say __regular open [[subspace]]__ for this (or the following concept).

An analogous kind of regular open subspace is a __regular open set__ in a [[topological space]] $X$, which is a regular element of the [[frame of open sets]] of $X$.  Equivalently, this is an [[open set]] in $X$ that equals the interior of its closure, or equivalently the exterior of its exterior.  (This is the origin of the term, related to a [[regular space]].)

The __regularization__ of $x$ is $\neg{\neg{x}}$; note that this is regular.  In fact, any element of the form $\neg{y}$ is regular.  Note that $x \leq \neg{\neg{x}}$; in logic, this means that a double negation is a weaker statement.

In a topological space, the regularization of an open set $G$ can be constructed as $Int(Cl(G))$, or equivalently as $Ext(Ext(G))$.  Sometimes one performs this operation to an arbitrary set (in the space) to produce a regular open set.  But note that while $G \subseteq Int(Cl(G))$ when $G$ is open, this does not hold for an arbitrary set.

[[!redirects regular element]]
[[!redirects regular elements]]

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regular epi > history

< [[regular epi]]
[[!redirects regular epi -- history]]


regular epimorphism

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#### Higher category theory
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#Contents#
* table of contents
{:toc}

## Idea

A _regular epimorphism_ is a [[morphism]] $c\to d$ in a [[category]] which behaves in the way that a [[covering]] is expected to behave, in the sense that "$d$ is the union of the parts of $c$, identified with each other in some specified way".

In a category with [[pullbacks]], regular epimorphisms are the same as [[effective epimorphisms]] and [[strict epimorphisms]].  Every regular epimorphism is an [[epimorphism]], and every [[split epimorphism]] is regular, but the converses frequently fail.

## Definition

\begin{definition}
\label{RegularEpimorphism}
A **regular epimorphism** is a [[morphism]] $f \colon c \to d$ (in a given [[category]]) that is the [[coequalizer]] of _some_ [[parallel pair]] of morphisms, i.e. if there exists _some_ [[colimit]] diagram of the form
$$
  a
    \;\rightrightarrows\;
  c
    \xrightarrow{\;f\;}
  d.
$$
\end{definition}

\begin{remark}
The [[formal duality|dual]] concept to Def. \ref{RegularEpimorphism} is that of *[[regular monomorphism]]*.
\end{remark}

## Properties

### Epimorphy

Although Def. \ref{RegularEpimorphism} does not state it explicitly, it is true (and easy to prove) that:

\begin{proposition}
Every regular epimorphism (Def. \ref{RegularEpimorphism}) is an [[epimorphism]].
\end{proposition}
In fact, every regular epimorphism is a [[strong epimorphism]], hence an [[extremal epimorphism]].  In particular, this implies that a regular epimorphism which is also a [[monomorphism]] must in fact be an [[isomorphism]].

Frequently (such as in a [[regular category]]), every strong or extremal epimorphism is regular.  Moreover, in a regular category, every regular epimorphism is [[pullback-stability|pullback-stable]], and therefore a [[descent morphism]].  If the category is moreover [[exact category|exact]], or has stable [[reflexive coequalizers]], then every regular epimorphism is an [[effective descent morphism]].

On the other hand, every [[split epimorphism]] is regular, but the converse holds only rarely (it is an [[internalization|internal]] form of the [[axiom of choice]]).

### Effective epimorphisms

Def. \ref{RegularEpimorphism} refers only to *some* parallel pair, but often there is a canonical choice: the [[kernel pair]] of the morphism in question:

\begin{prop}
\label{MorphismsWithKernelPairsAreRegularIffEffective}
A morphism having a [[kernel pair]] (such as any morphism in a category with [[pullbacks]]) is a regular epimorphism (Def. \ref{RegularEpimorphism}) if and only if it is the [[coequalizer]] of its kernel pair.
\end{prop}
E.g. [Borceux 1994 I, Prop. 2.5.7](#Borceux94I), [Taylor 1999, Lemma 5.6.6](#Taylor99).
This also follows from the theory of [[generalized kernels]].

\begin{definition}
\label{EffectiveEpimorphism}
A regular epimorphism
that satisfies the following two conditions (which are equivalent, by Prop. \ref{MorphismsWithKernelPairsAreRegularIffEffective})

1. it has a [[kernel pair]],

1. it is the [[coequalizer]] of its kernel pair

is called an _[[effective epimorphism]]_.
\end{definition}

### Pullbacks and pushouts

\begin{proposition}
\label{InRegularCategoryRegularEpisAreStableUnderPullback}
In a [[regular category]], [[regular epimorphisms]] are [[preserved limit|preserved]] by [[pullback]].
\end{proposition}
This is by definition of "[[regular category]]".

\begin{proposition}
\label{InRegularCategoryEffectiveEpisAreStableUnderPullback}
**(in regular category pullback preserves effective epimorphisms)**
\linebreak
  In a [[regular category]], [[effective epimorphisms]] (Def. \ref{EffectiveEpimorphism}) are preserved by [[pullback]]:

Given a [[commuting diagram]] of the form

\begin{tikzcd}
  R'
  \ar[d, shift left=4pt]
  \ar[d, shift right=4pt]
  \ar[r]
  \ar[dr, phantom, "\mbox{\tiny(pb)}"]
  &
  R
  \ar[d, shift left=4pt]
  \ar[d, shift right=4pt]
  \\
  X'
  \ar[d, ->>]
  \ar[r]
  \ar[dr, phantom, "\mbox{\tiny(pb)}"]
  &
  X
  \ar[d, ->>]
  \\
  Y'
  \ar[r]
  &
  Y
\end{tikzcd}

where

* the right vertical column is both a [[kernel pair]] and a [[coequalizer]], hence the kernel pair of a regular epimorphism,

* all parallel squares are [[pullbacks]]

then also the left vertical column is both a kernel pair and a coequalizer, hence the kernel pair of a regular epimorphism.
\end{proposition}
([Borceux 1994 II, Prop. 2.3.3](#Borceux94II))
\begin{proof}
  The left column is again a kernel pair since [[limits commute with limits]] and the left bottom vertical morphism is again a regular epimorphism, by Prop. \ref{InRegularCategoryRegularEpisAreStableUnderPullback}.

Therefore the left vertical column is also a coequalizer by Prop. \ref{MorphismsWithKernelPairsAreRegularIffEffective} and hence an effective epimorphism by Def. \ref{EffectiveEpimorphism}.
\end{proof}

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

In a [[regular category]], if in a [[pullback]] [[diagram]]

$$
  \array{
     & \swarrow & \searrow
     \\
     & \searrow & \swarrow
  }
$$

the two bottom morphisms are regular epis (hence by prop. \ref{InRegularCategoryRegularEpisAreStableUnderPullback} all four morphisms are), then the diagram is also a [[pushout]].

=--

(e.g. [Johnstone, section A, prop. 1.4.3](#Johnstone))

+-- {: .num_prop}
###### Proposition
In a regular category $C$, regular epimorphisms and [[strong epimorphisms]] (= [[extremal epimorphisms]] since $C$ admits pullbacks) coincide; hence regular epimorphisms are closed under composition.
=--

(See [Johnstone, A.1.3.4](#Johnstone).)

+-- {: .num_remark}
###### Remark
Regular epimorphisms are not generally closed under composition. [Taylor 1999 p. 289](#Taylor99) gives the following example in [[Cat]]. Let $\mathbb{2}$ be the [[interval category]]; then the coequalizer of the two object inclusions $1 \rightrightarrows \mathbb{2}$ is a regular epi $\mathbb{2} \to B\mathbb{N}$ (the codomain is the additive monoid $\mathbb{N}$ made into a 1-object category; the epi sends the non-identity arrow of $\mathbb{2}$ to $1 \in \mathbb{N}$). There is an evident regular epi $e: \mathbb{N} \to \mathbb{Z}/3$ of monoids. But the composite

$$\mathbb{2} \to B\mathbb{N} \stackrel{B e}{\to} B\mathbb{Z}/3$$

is not a regular epi in $Cat$. A very similar example is found at [Bednarczyk et al.](#Bed), Example 4.4.
=--

## Examples

* In [[Set|the category of sets]], every epimorphism is regular. Thus, it can be hard to know, when generalising concepts from $\Set$ to other categories, what kind of epimorphism to use.  Frequently, regular epimorphisms are a good choice.  In particular, one may define a [[projective object]] (and hence the [[axiom of choice]]) using regular epimorphisms.

* More generally, in every [[Grothendieck topos]] every epimorphism is regular (e.g. [Borceux 1994III, Prop. 3.4.13 on p. 218](#Borceux94III) and in fact [[effective epimorphism|effective]], see there).

* In [[Grp|the category of groups]], every epimorphism is regular. A number of proofs can be found in the literature; one proof that avoids case analysis is given [here](http://ncatlab.org/toddtrimble/published/epimorphisms+in+the+category+of+groups).

* In the category of [[monoid]]s, the inclusion $\mathbb{N}\hookrightarrow\mathbb{Z}$ is an epimorphism, even though it is far from a [[surjection]].  But in this or any other [[algebraic category]] (a category of models of an [[algebraic theory]]), the morphisms whose underlying function is surjective are precisely the regular epimorphisms.  Thus $\mathbb{N}\hookrightarrow\mathbb{Z}$ is not a regular epimorphism.

* Similarly, in the category of rings, not every epimorphism is regular, as shown by the example $\mathbb{Z} \hookrightarrow \mathbb{Q}$.

* In [[Diff]], the category of smooth (paracompact) manifolds, regular epimorphisms are not as useful as in other settings. As any [[split epimorphism]] is regular, and split epimorphisms are badly behaved in $\Diff$ (for example, pullbacks of split epis do not necessarily exist), the usual procedure is to consider the smallest class of arrows inside regular epis of which all pullbacks exist, namely the surjective [[submersions]]. In the setting of [[differentiable stacks]] and [[Lie groupoids]] it is surjective submersions that play the role of regular epimorphisms.

## Related pages

* The page [[epimorphism]] has a list of many types of epimorphism and their relationships.

* The dual concept is [[regular monomorphism]].  In particular, in the presence of pullbacks, [[effective epimorphisms]] and [[strict epimorphisms]] are the same as regular ones.

* In higher category theory:

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

## References

Textbook accounts:

* [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]], Def. 7.71 in: *[[Abstract and Concrete Categories -- The Joy of Cats]]* John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories **17** (2006) 1-507 ([tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html), [book webpage](http://katmat.math.uni-bremen.de/acc/))

* {#Borceux94I} [[Francis Borceux]], Def. 4.3.1 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

* {#Borceux94II} [[Francis Borceux]], Chapter 2 of: *[[Handbook of Categorical Algebra]]*, Vol. 2: *Categories and Structures*, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) ([doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865))

* {#Borceux94III} [[Francis Borceux]], Section 3.3.4 of: *[[Handbook of Categorical Algebra]]*. Vol. 3. *Categories of Sheaves*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) [[doi:10.1017/CBO9780511525872](https://doi.org/10.1017/CBO9780511525872)]

Exposition and examples:

* [[Qiaochu Yuan]], *[Regular and effective monomorphisms and epimorphisms](https://qchu.wordpress.com/2012/11/03/regular-and-effective-monomorphisms-and-epimorphisms/)*, 2012

See also:

* {#Johnstone} [[Peter Johnstone]], Section A of: _[[Sketches of an Elephant]]_

* {#Bed} [[Marek A. Bednarczyk]], [[Andrzej M. Borzyszkowski]], [[Wieslaw Pawlowski]], *Generalized congruences – epimorphisms in $\mathcal{C}a t$*, Theory and Applications of Categories **5** 11 (1999) 266-280 [[tac:5-11](http://www.tac.mta.ca/tac/volumes/1999/n11/5-11abs.html), [dml:120226](https://eudml.org/doc/120226?lang=fr&limit=5)]

* {#Taylor99} [[Paul Taylor]], *[[Practical Foundations of Mathematics]]*, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press 1999 ([webpage](http://www.paultaylor.eu/~pt/prafm/index.html))

[[!redirects regular epimorphisms]]
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regular epimorphism in an (infinity,1)-category

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## Definition

By comparing the notions of [[regular epimorphism]] and [[effective epimorphism]] with that of [[effective epimorphism in an (∞,1)-category]], we propose to call a morphism $f : c \to d$ in an [[(∞,1)-category]] a **regular epimorphism** if it is the [[colimit]] of some simplicial diagram, i.e. if there exists a functor $c : \Delta^{op} \to C$, such that $f$  is the [[colimit in a quasi-category|colimiting cocone]]

$$
  \cdots c_2 \stackrel{\to}{\stackrel{\to}{\to}} c_1 \stackrel{\to}{\to} c \stackrel{f}{\to} d
$$

over this diagram.

Equivalently, this is a morphism such that for all objects $e \in C$ the induced morphism $f^* : C(d,e) \to C(c,e)$ is a [[regular monomorphism in an (∞,1)-category]] in the [[(∞,1)-category]] [[∞Grpd]].

**Warning**. Such a morphism may not be an [[epimorphism in an (∞,1)-category]] (i.e. a [[monomorphism in an (∞,1)-category|monomorphism]] in the opposite category).

## Questions

* If $f$ has a [[Cech nerve]] and is a regular epimorphism above, does it follow that it is the colimit of its Cech nerve (that is, that it is an [[effective epimorphism in an (∞,1)-category]])?

* Is there a notion of a [[strict epimorphism]] in an [[(∞,1)-category]]?

[[!redirects regular epimorphism in an (∞,1)-category]]
[[!redirects regular epimorphisms in an (infinity,1)-category]]
[[!redirects regular epimorphisms in an (∞,1)-category]]


regular extension axiom

# The regular extension axiom
* table of contents
{: toc}

## Idea

The **Regular Extension Axiom** (REA) is a foundational axiom which asserts the existence of arbitrarily large [[regular cardinal]]-like sets.  It has several variants, some of which are provable in [[ZF]], some of which are provable  from the [[axiom of choice]] or weaker variants thereof such as [[SVC]], and some of which are not even provable in ZFC. REA is usually considered in the context of [[CZF]].

## Variants

There is some discussion [here](http://golem.ph.utexas.edu/category/2012/04/pssl_93_trip_report.html#c041293).

uREA is REA for union closed regular sets. In CZF it implies the set generated axiom (SGA):

 For each set S and each subset Z of Fin(S) × Pow(Pow(S)), the class

M(Z) = {α ∈ Pow(S) | ∀(σ, Γ) ∈ Z[σ ⊆ α ⇒ ∃U ∈ Γ(U ⊆ α)]}

is set-generated. This axiom is also implies by [[relativised dependent choice]].

## References

* [[Michael Rathjen]] and [[Robert Lubarsky]], *On the regular extension axiom and its variants*, [PDF](http://www1.maths.leeds.ac.uk/~rathjen/REA.pdf)

* Peter Aczel, Hajime Ishihara, Takako Nemoto and Yasushi Sangu, *Generalized geometric theories and set-generated classes*, [PDF](http://www.newton.ac.uk/preprints/NI12032.pdf)

category: foundational axiom

[[!redirects regular extension axiom]]
[[!redirects Regular Extension Axiom]]
[[!redirects REA]]


regular field extension

#Contents#
* table of contents
{:toc}

## Idea

An algebraic [[field extension]] $k\subset L$ is __regular__ if it is separable and $k$ is [[algebraically closed field|algebraically closed]] in $L$.

## Related entries

* [[ramified extension]]

## References

* [wikipedia](http://en.wikipedia.org/wiki/Regular_extension)

category: algebra


regular functor

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#Contents#
* table of contents
{:toc}

## Definition

A [[functor]] $F : C \to D$ between [[regular categories]] is called **regular** if it preserves [[finite limit]]s and the canonical [[cover]]s: [[regular epimorphism]]s.

## Properties

* For $\mathcal{T}$ a [[regular theory]] and $\mathcal{C}_{\mathbb{T}}$ its [[syntactic category]], regular functors $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ into some [[topos]] $\mathcal{E}$ are precisely [[model]]s of the theory in $\mathcal{E}$.

  For $\mathcal{C}_{\mathbb{T}}$ equipped with the structure of the [[syntactic site]] (the [[regular coverage]]), this is in turn equivalent to [[geometric morphism]]s $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$ into the [[sheaf topos]] over $\mathcal{C}_{\mathbb{T}}$ (the [[classifying topos]] for the theory).

## Related concepts

* [[cartesian functor]]

* **regular functor**

* [[coherent functor]]

* [[geometric functor]]

[[!redirects regular functors]]


regular homotopy

Let $M, N$ be manifolds, and let $Imm(M, N)$ denote the space of [[immersions]] of $M$ in $N$, with the [[compact-open topology]]. A **regular homotopy** between immersions $f, g: M \to N$ is a [[path]] $\alpha: I \to Imm(M, N)$ where $\alpha(0) = f$ and $\alpha(1) = g$.


regular hyperdoctrine

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#Contents#
* table of contents
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## Idea

A **regular hyperdoctrine**, also called an **elementary existential doctrine** or a **regular fibration**, is a version of a [[first-order hyperdoctrine]] that is appropriate for [[regular logic]].

## Definition

Let $C$ be a [[category]] with [[finite limits]]. A **[[regular logic|regular]] [[hyperdoctrine]]**, or **elementary existential doctrine**, over $C$ is a [[functor]]

$$
  P \;\colon\; C^{op} \to InfSemiLattice
$$

from the [[opposite category]] of $C$ to the category of inf-[[semilattices]], such that for every [[morphism]] $f : A \to B$ in $C$, the functor $P(A) \to P(B)$ has a [[left adjoint]] $\exists_f$ satisfying

1. [[Frobenius reciprocity]];

1. [[Beck-Chevalley condition]].

\begin{remark}
The original definition of Lawvere did not include the last two conditions and allowed the values of the functor to be arbitrary categories with finite products.  Other references sometimes require the adjoints $\exists_f$ to be defined only along projections (corresponding to ordinary [[existential quantification]] in logic) and diagonals (corresponding to [[equality]]); Lawvere showed that in the presence of Frobenius and Beck-Chevalley more general quantifiers can be constructed in terms of these.
\end{remark}

## Related concepts

[[!include categories and logic - table]]

## References

* [[William Lawvere]], *Equality in hyperdoctrines and comprehension schema as an adjoint functor*

* Davide Trotta, *The existential completion*, 2021 ([arxiv:2108.03416](https://arxiv.org/abs/2108.03416))

* Davide Trotta, *An algebraic approach to the completions of elementary doctrines*, 2021 ([arxiv:2108.03415](https://arxiv.org/abs/2108.03415))

* [[Bart Jacobs]], *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf), [webpage](http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html)]

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regular language

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# Contents
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## Idea

Regular languages are a simple kind of [[formal grammar|formal languages]], the least expressive in the [[Chomsky hierarchy]].

## Definition
 {#Definition}

Fix a [[finite set]] $V$ (serving as the [[alphabet]]) and denote its [[free monoid]] by $V^\star$. A [[language]] $L \subseteq V^\star$ is *regular* whenever there is a [[formal grammar]] $G = (V, X, R, s)$ that generates it (i.e. $L(G) = L$) such that all rules in $R$ are of the form $x \to w x'$ for a terminal $w \in V$ and non-terminal symbols $x, x' \in X$.

## Properties

Regular languages can be characterised by _regular expressions_. Regular expressions are inductively generated by the following rules:

1. For each terminal $w \in V$, $w$
2. Basic expressions $\emptyset$, $\epsilon$
3. For each pair of regular expressions $R_1,R_2$, $R_1|R_2$ and $R_1 \cdot R_2$
4. For each regular expression $R$ the Kleene star $R^\star$.

These can be given a semantics in the [[poset]] of languages $P(V)$ by [[universal properties]].

1. $L(\emptyset) = \emptyset$ and $L(R_1 | R_2) = L(R_1) \cup L(R_2)$. These are nullary and binary joins.
2. $L(w) = \{ w \}$. These are [[representable]] viewing languages as de-categorified [[presheaves]].
3. $L(\epsilon)$ consists of only the empty string, $L(R_1\cdot R_2) = \{\alpha \in V^\star \vert \exists \beta \in L(R_1), \gamma \in L(R_2) \cdot \alpha = \beta \gamma \}$. This is a de-categorified version of the [[Day convolution]] monoidal structure.
4. $L(R^\star) = \{ \alpha \in V^\star \vert \exists n \in \mathbb{N}. \exists\beta_0,\ldots,\beta_{n-1} \in L(R). \alpha = \beta_0\cdots\beta_{n-1} \}$. In [[category theory|category theoretic]] terms this is the least solution to the recurrence equation $L(R^\star) = L(\epsilon| (R\cdot R^\star))$, and can be constructed by [[Kleene's fixed point theorem]] as an $\omega$-[[colimit]].

Every regular language can is denoted by some regular expression in this way.

Equivalently, a language is regular if it can be recognised by a finite-state [[automaton]].

## Related entries

* [[context-free grammar]]

[[!redirects regular languages]]


regular Lindelöf spaces are paracompact

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### Context###
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

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

Let $X$ be a [[topological space]] which is

1. [[Lindelöf space|Lindelöf]] (every [[open cover]] has a [[countable cover|countable]] subcover);

1. [[regular topological space|regular]].

Then $X$ is [[paracompact topological space]].

=--

## Related statements

* [[paracompact Hausdorff spaces are normal]]

* [[second-countable regular spaces are paracompact]]

* [[second-countable regular spaces are paracompact]]


regular local ring

A __regular local ring__ $R$ is a Noetherian commutative unital [[local ring]] whose [[Krull dimension]] agrees with the minimal number of generators of its maximal ideal $I$. Equivalently, the Krull dimension equals $dim_k (I/I^2)$ where $k = R/I$ is the residue field of $R$. A Noetherian local ring is regular iff its global dimension is finite; it follows that its global and Krull dimension coincide.

The notion should not be confused with that of a [[von Neumann regular ring]] in noncommutative algebra.

* [wikipedia](http://en.wikipedia.org/wiki/Regular_local_ring)
* [[eom]]: [regular ring](https://encyclopediaofmath.org/wiki/Regular_ring_(in_commutative_algebra)) (in commutative algebra)

[[!redirects regular ring]]


regular logic

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### Context
#### Type theory
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[[!include type theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

Regular logic is the [[internal logic]] of [[regular categories]].  Its logical operations consist only of [[truth]], [[conjunction]], and [[existential quantification]], which makes it a superset of [[finite-limit logic]] and a subset of [[coherent logic]] and [[geometric logic]].

## Related concepts

[[!include categories and logic - table]]

## References

* [[Carsten Butz]], _Regular Categories and Regular Logic_ , BRICS LS-98-2 Aarhus 1998. ([brics](http://www.brics.dk/LS/98/2/))

* [[Brendan Fong|B. Fong]], [[David Spivak|D. Spivak]], _Graphical Regular Logic_ , arXiv:1812.05765 (2018). ([abstract](https://arxiv.org/abs/1812.05765))

* [[Peter Johnstone]], _[[Sketches of an Elephant]] II_ , Oxford UP 2002. (Section D1.2)

* [[Jaap van Oosten]], _Basic category theory_ , BRICS LS-95-1 Aarhus 1995. (Section 4.2) ([pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf))

[[!redirects logic of regular categories]]


regular monomorphism

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#### Higher category theory
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# Contents
* table of contents
{:toc}

## Idea

A [[monomorphism]] is _regular_ if it behaves like an [[embedding]].

* [[effective epimorphism]] $\Rightarrow$ [[regular epimorphism]] $\Leftrightarrow$ [[covering]]

* [[effective monomorphism]] $\Rightarrow$ [[regular monomorphism]] $\Leftrightarrow$ [[embedding]] .

The universal factorization through an embedding is the [[image]].

## Definition

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

A **regular monomorphism** is a [[morphism]] $f : c \to d$ in some [[category]] which occurs as the [[equalizer]] of _some_ [[pair of parallel morphisms]] $d \stackrel{\to}{\to} e$, i.e. for which a [[limit]] [[diagram]] of the form

$$
  c \stackrel{f}{\to} d \stackrel{\longrightarrow}{\longrightarrow} e
$$

exists.

=--

From the defining [[universal property]] of the [[limit]] it follows directly that a regular monomorphism is in particular a [[monomorphism]].

The dual concept is that of a [[regular epimorphism]].

\begin{rmk} Beware that ([CassidyHebertKelly](#CassidyHebertKelly)) use 'regular monomorphism' in a more general way: for them, a regular monomorphism is by definition the joint equalizer of an arbitrary family of parallel pairs of morphisms with common domain.
This concept is sometimes called [[strict monomorphism]], dual to the more commonly used [[strict epimorphism]]. \end{rmk}

## Properties

### Relation to effective monomorphisms

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

A monomorphism $i: A \to B$ is an [[effective monomorphism]] if it is the [[equalizer]] of its [[cokernel pair]]: if the [[pushout]]

$$\array{
A & \stackrel{i}{\to} & B \\
i \downarrow & & \downarrow i_1 \\
B & \underset{i_2}{\to} & B +_A B
}$$

exists and $i$ is the equalizer of the pair of coprojections $i_1, i_2: B \stackrel{\to}{\to} B +_A B$. Obviously effective monomorphisms are regular.

=--

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

In a [[category]] with [[equalizers]] and [[cokernel pairs]], the class of  regular monomorphism coincides with that of [[effective monomorphism]] (def. \ref{EffectiveMonomorphism}).

=--

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

It is clear that every effective monomorphism is regular, we need to show the converse.

Suppose $i \colon A \to B$ is the equalizer of a pair of morphisms $f, g: B \to C$, and with notation as in def. \ref{EffectiveMonomorphism}, let $j: E \to B$ be the equalizer of the pair of coprojections $i_1, i_2$. Since $f \circ i = g \circ i$, there exists a unique map $\phi: B +_A B \to C$ such that $\phi \circ i_1 = f$ and $\phi \circ i_2 = g$. Then, since
$$f j = \phi i_1 j = \phi i_2 j = g j$$
and since $i: A \to B$ is the equalizer of the pair $(f, g)$, there is a unique map $k: E \to A$ such that $j = i k$. Since $i_1 i = i_2 i$, there is a unique map $l: A \to E$ such that $i = j l$. The maps $k$, $l$ are mutually inverse.

=--

## Examples {#Examples}

* In [[Set]], or more generally in any [[pretopos]], every [[monomorphism]] is regular. More generally, this holds in every [[adhesive category]].

* Similarly, in [[Ab]], and more generally any [[abelian category]], every monomorphism is regular.

+-- {: .num_prop #RegularMonomorphismsOfTopologicalSpaces}
###### Proposition
**(regular monomorphisms of [[topological spaces]])**

In the [[category]] [[Top]] of [[topological space]],

1. the [[monomorphisms]] are the those [[continuous functions]] which are [[injective functions]];

1. the regular monomorphisms are the [[topological embeddings]] (that is, the injective continuous functions whose sources have the [[induced topology|topologies induced]] from their targets); these are in fact all of the [[extremal monomorphisms]].

=--

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

Regarding the first statement: an injective continuous function $f \colon X \to Y$ clearly has the cancellation property that defines monomorphisms: for parallel continuous functions $g_1,g_2 \colon Z \to X$: if $f \circ g_1 = f \circ g_1$, then $g_1 = g_2$ because continuous functions are equal precisely if their underlying functions of sets are equal. Conversely, if $f$ has the cancellation property, then testing on points $g_1, g_2 \colon \ast \to X$ gives that $f$ is injective.

Regarding the second statement: from the construction of [[equalizers]] in [[Top]] ([this example](Top#EqualizerInTop)) we have that these are topological subspace inclusions.

Conversely, let  $i \colon X \to Y$ be a [[topological subspace embedding]]. We need to show that this is the equalizer of some pair of parallel morphisms.

To that end, form the [[cokernel pair]] $(i_1, i_2)$ by taking the [[pushout]] of $i$ against itself (in the category of sets, and using the [[quotient topology]] on a [[disjoint union space]]). By prop. \ref{RegEquEff}, the equalizer of that pair is the set-theoretic equalizer of that pair of functions endowed with the [[subspace topology]]. Since monomorphisms in [[Set]] are regular, we get the function $i$ back and (again by [this example](Top#EqualizerInTop)) it is equipped with the subspace topology.

=--

+-- {: .num_prop #reg}
###### Proposition
In [[Grp]], the monics are (up to [[isomorphism]]) the inclusions of [[subgroup]]s, and every monomorphism is regular.
=--

In contrast, the [[normal monomorphisms]] (where one of the morphisms $d \to e$ is required to be the [[zero morphism]]) are the inclusions of [[normal subgroups]].

+-- {: .proof}
###### Proof
The elementary proof we give follows [exercise 7H](http://katmat.math.uni-bremen.de/acc/acc.pdf#page=129) of ([AdamekHerrlichStrecker](#AdamekHerrlichStrecker)). It is however nonconstructive (because it contains if-then-else lines); for a constructive proof, see [here](/nlab/show/Grp#eq).

Let $K \hookrightarrow H$ be a subgroup. We
need to define another group $G$ and group homomorphisms
$f_1, f_2 : H \to G$ such that

$$
  K = \{h \in H  | f_1(h) = f_2(h)\}
  \,.
$$

To that end, let

$$
  X  := H/K  \coprod  \{\hat K\}  := \{ h K | h \in H \}  \coprod  \{\hat K\}
$$

be the set of [[coset]]s together with one more element $\hat K$.

Let then

$$
  G = Aut_{Set}(X)
$$

be the [[permutation group]] on $X$.

Define $\rho \in G$ to be the permutation that exchanges the coset $e K$ with the extra element $\hat K$ and is the identity on all other elements.

Finally define group homomorphism $f_1,f_2 : H \to G$ by

$$
  f_1(h) : x \mapsto
  \left\{
     \array{
        h h' K & if x = h' K
        \\
        \hat K & if x = \hat K
     }
  \right.
$$

and

$$
  f_2(h) = \rho \circ f_1(h) \circ \rho^{-1}
  \,.
$$

It is clear that these maps are indeed group homomorphisms.

So for $h \in H$ we have that

$$
  f_1(h) : \hat K \mapsto \hat K
  \,,
$$

and

$$
  f_1(h) : e K \mapsto h K
$$

and

$$
  f_2(h)
  :
  \hat K
   \mapsto
  e K
    \mapsto
   h K
   \mapsto
  \left\{
    \array{
     \hat K & if h \in K
     \\
     h K & otherwise
   }
   \right.
  \,.
$$

$$
  f_2(h) : e K \mapsto \hat K \mapsto \hat K \mapsto e K
  \,.
$$

So we have $f_1(h) = f_2(h)$ precisely if $h \in K$.

=--

## In an $(\infty,1)$-category {#Infty1Version}

In the context of [[higher category theory]]
the ordinary [[limit]] diagram $c \stackrel{f}{\to} d \stackrel{\to}{\to} e $ may be thought of as the beginning of a [[homotopy limit]] diagram over a [[cosimplicial object|cosimplicial]] diagram

$$
  c \stackrel{f}{\to} d_0 \stackrel{\to}{\to} d_1
  \stackrel{\to}{\stackrel{\to}{\to}}
  d_2
  \cdots
  \,.
$$

Accordingly, it is not unreasonable to define a
[[regular monomorphism in an (∞,1)-category]], to be a morphism which is the [[limit in a quasi-category]] of a cosimplicial diagram.

In practice this is of particular relevance for the $\infty$-version of [[regular epimorphism]]s: with the analogous definition as described there, a morphism $f : c \to d$ is a [[regular epimorphism]] in an [[(∞,1)-category]] $C$ if for all objects $e \in C$ the induced morphism $f^* : C(d,e) \to C(c,e)$ is a [[regular monomorphism]] in [[∞Grpd]] (for instance [[model structure on simplicial sets|modeled]] by a [[homotopy limit]] over a cosimplicial diagram in [[SSet]]).

**Warning.** The same warning as at [[regular epimorphism]] applies: with this definition of regular monomorphism in an [[(∞,1)-category]] these may fail to satisfy various definitions of plain monomorphisms that one might think of.

## Related pages

* [[regular epimorphism]] (containing more results which of course have duals that could be added here)

## References

Textbook accounts:

* {#AdamekHerrlichStrecker} [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]], Def. 7.56 in: *[[Abstract and Concrete Categories -- The Joy of Cats]]* John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories **17** (2006) 1-507 ([tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html), [book webpage](http://katmat.math.uni-bremen.de/acc/), [pdf](http://katmat.math.uni-bremen.de/acc/acc.pdf))

See also:

* {#CassidyHebertKelly} C. Cassidy, M. Hébert, [[Max Kelly]], _Reflective subcategories, localizations, and factorization systems_,  J. Austral. Math Soc. (Series A) 38 (1985), 287--329 ([doi:10.1017/S1446788700023624](https://doi.org/10.1017/S1446788700023624))

[[!redirects regular monomorphism]]
[[!redirects regular monomorphisms]]
[[!redirects regular mono]]
[[!redirects regular monos]]
[[!redirects regular monic]]
[[!redirects regular monics]]

[[!redirects regular subobject]]
[[!redirects regular subobjects]]


regular monomorphism in an (infinity,1)-category

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#### $(\infty,1)$-Category theory
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#Contents#
* table of contents
{:toc}

## Idea

 A [[regular monomorphism]] in a [[category]] is a morphism that is the [[equalizer]] of some pair of morphisms. A _regular monomorphism in an $(\infty,1)$-category_ is its analog in an [[(∞,1)-category theory]].

Beware that this need not be a [[monomorphism in an (∞,1)-category]].

## Definition

Let $C$ be an [[(∞,1)-category]]. A [[morphism]] $f : x \to y$ in $C$ is a **regular monomorphism** if there exists a [[cosimplicial object|cosimplicial]] [[diagram]] $D : \Delta \to C$ with $D[0] = y$ such that $f$ is the [[(∞,1)-limit]] over this diagram.

$$
  x \stackrel{f}{\to} y \stackrel{\to}{\to} y_1 \stackrel{\to}{\stackrel{\to}{\to}} y_2 \cdots
$$

[[!redirects regular monomorphism in an (∞,1)-category]]

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


regular polygon

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#### Geometry
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#Contents#
* table of contents
{:toc}

## Definition ##

In [[Euclidean geometry]], a **regular polygon** is a (simple, non self-intersecting) [[polygon]] in a [[Euclidean space]] $\mathbb{R}^n$ such that every [[segment]] of the [[boundary]] of the polygon has the same [[length]] and every [[angle]] between two segments of the boundary of the polygon has the same angle [[measure]]. (The boundary of a regular polygon is sometimes called called a **regular polygonal line**)

## Perimeter and area ##

We are using the [[circle constant]] $\tau = 2 \pi$.

Every regular polygon is the union of $n$ congruent [[triangles]] each with segments of length $r$ and $r$ respectively and an angle of $\frac{\tau}{n}$ between the two segments of length $r$. As a result, the triangles are isosceles and the altitude from the center of the regular polygon to the third segment of the triangles bisects the angle: such that the angles between the [[circumradius]] and the altitude is $\frac{\tau}{2n}$. The length of the third segment is thus given by

$$b \coloneqq r \sin\left(\frac{\tau}{2n}\right) + r \sin\left(\frac{\tau}{2n}\right) = 2 r \sin\left(\frac{\tau}{2n}\right)$$

The perimeter of a regular polygon $\mathcal{P}_n$ (strictly speaking, a regular polygonal line) with $n$ sides and [[circumradius]] $r$ is given by the sequence of functions $P_\mathcal{P}:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$

$$P_\mathcal{P}(n)(r) = n b = r (2 n) \sin\left(\frac{\tau}{2n}\right)$$

The area of a regular polygon $\mathcal{P}_n$ with $n$ sides and [[circumradius]] $r$ is given by the sequence of functions $P:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$

$$A_\mathcal{P}(n)(r) = n \left(\frac{1}{2} r b\right) = n \frac{1}{2} r (2 r) \sin\left(\frac{\tau}{2n}\right) = \frac{1}{2} r^2 (2 n) \sin\left(\frac{\tau}{2n}\right)$$

## See also ##

* [[polygon]]

* [[center of a regular polygon]]

* [[circumference of a circle]]

* [[area of a circle]]

## References

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

[[!redirects regular polygons]]

[[!redirects regular polygonal line]]
[[!redirects regular polygonal lines]]


regular polyhedron

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#### Group Theory
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#Contents#
* table of contents
{:toc}

## Idea

...regular [[polyhedron]]...

## Examples

* [[tetrahedron]]

* [[Platonic solid]]

## Related concepts

* [[regular polytope]]

* [[finite rotation group]]

## References

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

[[!redirects regular polyhedra]]


regular polytope

#Contents#
* table of contents
{:toc}

## Idea

...a regular [[polytope]]...

... the higher [[dimension|dimensional]] analog of a [[regular polyhedron]]...

## Examples

### In 3 dimensions

* [[Platonic solids]]

### In 4 dimensions

* [[24-cell]], [[120-cell]], [[600-cell]], ...

## Related concepts

* [[finite rotation group]]

## References

See also

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

* Wikipedia, _[List of regular polytopes and compunds](https://en.wikipedia.org/wiki/List_of_regular_polytopes_and_compounds#Convex_4)_

[[!redirects regular polytopes]]


regular representation

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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
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=--
=--

# Contents
* table of contents
{: toc}

## Idea

Many algebraic objects have a [[representation theory]] in which they act on other things; for example, [[groups]] often act on sets and [[algebras]] act on [[modules]].
In these situations, the acting object can often be viewed as an example of the type of thing acted upon: a group has an underlying set and an algebra has an underlying module.
The multiplication in the acting object then defines an action of the object on this unstructured copy of itself.
This is called the **regular representation** and is an extremely useful representation to study as it only involves the object itself, whence is in a sense *canonical*, but contains a lot of information, as opposed to, say, the [[trivial representation]] (which is also canonical).

## Definition

The definition is the same in each case, but we shall give the actual definitions for the usual suspects.

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

Let $G$ be a [[group]] with multiplication $\mu$.  Let us write ${|G|}$ for the underlying set of $G$.  The **left regular representation** of $G$ is

1. as a [[permutation representation]]: the action $G \times {|G|} \to {|G|}$ defined by $g \cdot h = \mu(g,h)$;

1. as a [[linear representation]]: the corresponding representation on the [[linear span]] of $G$.

The **right regular representation** is defined analogously.
=--

+-- {: .num_defn #algreg}
###### Definition
Let $A$ be an [[associative unital algebra]] with multiplication $\mu$.  Let us write ${|A|}$ for the underlying module of $A$.  The **left regular representation** of $A$ is the action $A \otimes {|A|} \to {|A|}$ defined by $a \cdot m = \mu(a,m)$.

The **right regular representation** is defined analogously.
=--

These can be seen as examples of a more general concept.

+-- {: .num_defn #monrep}
###### Definition
Let $(C,\otimes,I)$ be a [[monoidal category]].  Let $M = ({|M|},\mu,\eta)$ be a [[monoid]] in $C$, where ${|M|}$ is the underlying object of $M$ in $C$.
The **regular representation** of $M$ is the action of $M$ on ${|M|}$ induced by the product $\mu$.
=--

## Properties
 {#Properties}

* The regular representation of $G$ as a linear representation is the [[induced representation]] $Ind_{1}^G 1$ of the trivial representation along the inclusion of the trivial [[subgroup]].

\begin{prop}\label{RegularRepDecomposedIntoIrreps}
Over the [[complex numbers]], the regular representation of a [[finite group]] is a [[direct sum]] that contains each [[irreducible representation]] $\rho_i$ with multiplicity its dimension

$$
  \mathbb{C}[G]
  \;\simeq\;
  \underset{i}{\sum}
  dim_{\mathbb{C}}(\rho_i)
    \cdot
  \rho_i
  \,.
$$
\end{prop}
(e.g. [tom Dieck 09, Thm. (1.10.2)](#tomDieck09) and using [[Schur's lemma]] for the complex case)

## Related concepts

* [[trivial representation]]

* [[alternating representation]]

## References

Lecture notes include

* {#tomDieck09} [[Tammo tom Dieck]], section 1.10 of _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

[[!redirects regular representation]]
[[!redirects regular representations]]


regular scheme

#Contents#
* table of contents
{:toc}

## Idea

A [[regular scheme]] $(X,\mathcal{O}_X)$ is an [[algebraic scheme]] such that for every $x\in X$ there exists an affine open neighbourhood $U\subset X$ of $x$ such that the ring $\mathcal{O}_X(U)$ is regular (many authors also require Noetherian). Equivalently, [[stalk]] $\mathcal{O}_{X,x}$ of the structure sheaf $\mathcal{O}_X$ at any point $x\in X$ (as a locally ringed space) is a [[regular local ring]].

## References

* stacks project: [28.9 Regular schemes](https://stacks.math.columbia.edu/tag/02IR)
* [[eom]]: *[regular scheme](http://www.encyclopediaofmath.org/index.php/Regular_scheme)*
* wikipedia: [regular scheme](https://en.wikipedia.org/wiki/Regular_scheme)

On pro-algebraic resolutions of regular schemes over algebraically complete fields:

* [[Adrian Clough]], *Pro-algebraic resolutions of regular schemes*, MSc thesis, ETC Zürich (2014) [[pdf](https://web.ma.utexas.edu/users/adrian.clough/Master_thesis.pdf)]

category: algebraic geometry
[[!redirects regular schemes]]


regular semicategory

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#Contents#
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## Idea
The notion of a **regular semicategory** generalizes the notion of a _regular algebra_ from ring theory to (semi-)category theory. Just like regular algebras are (in general) non-unital algebras that nevertheless "behave" in many respects like unital ones, regular semicategories are semicategories on the verge of being categories.

>Note that regular semicategories that are categories **are not** (necessarily) [[regular category|regular categories]] in the usual sense. In this case there is a **clash of terminology** between category theory and algebra.

## Preliminaries

For convenience let us first recall a couple of concepts

+-- {: .num_defn #semifunctor}
###### Definition
Let $\mathcal{G}$, $\mathcal{H}$ be [[semicategories]]. A _morphism $F:\mathcal{G}\to\mathcal{H}$ of semicategories_ assigns to all objects $X\in\mathcal{G}$ an object $F(X)\in\mathcal{H}$ and to every morphism $f:X\to Y$ in $\mathcal{G}$ a morphism $F(f):F(X)\to F(Y)$ in $\mathcal{H}$ such that $F(f\circ g)=F(f)\circ F(g)$.

A _natural transformation_ $\alpha:\mathcal{G}\Rightarrow\mathcal{H}$ consists of a family $\{\alpha_X:F(X)\to G(X)\}_{X\in|\mathcal{G}|}$ in $\mathcal{H}$ indexed by the objects of $\mathcal{G}$ such that for all $f:X\to Y$ in $\mathcal{G}$ the following diagram commutes:

$$
  \array{
     F(X)& \overset{F(f)}{\longrightarrow} & F(Y)
     \\
     {}_{\alpha_X}\downarrow & & \downarrow _{\alpha_Y}
     \\
     G(X)& \overset{G(f)}{\longrightarrow} & G(Y)
  }
$$

=--

+-- {: .num_defn #semipresheaf}
###### Definition
A [[semipresheaf|_presheaf_ on a semicategory]] $\mathcal{G}$ is a morphism of semicategories $F:\mathcal{G}^{op}\to Set$. The category $Prsh(\mathcal{G})$ has objects presheaves on $\mathcal{G}$ and morphisms the natural transformations and is the called _the category of presheaves of the semicategory $\mathcal{G}$_.
=--

$Prsh(\mathcal{G})$ is indeed a _category_! Denoting $\overline{\mathcal{G}}$ the category resulting from $\mathcal{G}$ by adding the missing identity morphisms, it is easy to check that $Prsh(\mathcal{G})\simeq PrSh(\overline{\mathcal{G}})$ and that the latter coincides with the usual [[presheaf category]] hence $Prsh(\mathcal{G})$ is even a [[Grothendieck topos]].

Given $\mathcal{G}$ there is also a **Yoneda morphism** $Y_\mathcal{G}:\mathcal{G}\to Prsh(\mathcal{G})$ defined on objects as usual by $X\mapsto Hom_\mathcal{G}({}_-,X)$. Since semicategories embed into categories only iff they are categories themselves it follows that $Y_\mathcal{G}$ is fully-faithful iff $\mathcal{G}$ is a category!

## Definition

...

## Example

...

## Properties

The most striking result is that although for a general regular semicategory $\mathcal{G}$ regular presheaves will not be Yoneda presheaves and vice versa nevertheless the subcategories $RegPsh(\mathcal{G})$ and $YonPsh(\mathcal{G})$ are _identical_ in an [[adjoint cylinder|identity-and-unity of opposites]] in the sense of [[William Lawvere|Lawvere]] i.e. both are equivalent and occur in an [[level|essential localization]] of $Prsh(\mathcal{G})$.

+-- {: .num_prop #unity_of_opposites}
###### Theorem
Let $\mathcal{G}$ be a regular semicategory. The functor $k:RegPsh(\mathcal{G})\to Prsh(\mathcal{G})$ defined on objects by $F\mapsto Nat(Y_\mathcal{G}({}_-),F)$ is fully-faithful and part of an adjoint string

$$
i\dashv j\dashv k:RegPsh(\mathcal{G})\hookrightarrow Prsh(\mathcal{G})
$$

with $k$ identifying the regular presheaves with the Yoneda presheaves and $i$ identifying them with the presheaves that are colimits of representables.
=--

For the proof see Moens et al. ([2002](#MBB02), p.179).

## Related entries

* [[semicategory]]

* [[semipresheaf]]

* [[adjoint cylinder|unity-and-identity of opposites]]

* [[Yoneda lemma]]

## References

Regular semicategories were introduced in

* {#MBB02}M.-A. Moens, U. Bernani-Canani, [[Francis Borceux|F. Borceux]], _On regular presheaves and regular semi-categories_ , Cah. Top. Géom. Diff. Cat. **XLIII** no.3 (2002) pp.163-190. ([numdam](http://www.numdam.org/item?id=CTGDC_2002__43_3_163_0))

Their quantaloid-enriched theory is studied in

* {#Stubbe05}[[Isar Stubbe]], _Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories_ , Cah. Top. Géom. Diff. Cat. **XLVI** no.2 (2005) pp.99-121. ([numdam](http://www.numdam.org/item/CTGDC_2005__46_2_99_0))

For the origins in algebra of the concept see

* F. Grandjean, [[Enrico Vitale|E. M. Vitale]], _Morita equivalence for regular algebras_ , Cah. Top. Géom. Diff. Cat. **XXXIX** (1998) pp.137-153. ([numdam](http://www.numdam.org/item/CTGDC_1998__39_2_137_0))

[[!redirects regular semicategories]]
[[!redirects regular semi-category]]
[[!redirects regular semi-categories]]
[[!redirects regular presheaf]]
[[!redirects regular presheaves]]


regular sequence

## References

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

[[!redirects regular sequences]]


regular singular point

Let a [[meromorphic map|meromorphic]] linear [[ordinary differential equation]]

$$
  y^{(n)}+h_1 y^{(n-1)}+ \ldots + h_n y = 0\,\,\,\,\,\,\,\,\,\, (E)
$$

defined everywhere in the complex plane (in dimension 1) outside of a finite set of singular points (where some of the meromorphic coefficients have singularity), is given. Equivalently, we can instead of equation (E) of higher order, look at a system

$$
\frac{\mathrm{d}u}{\mathrm{d}z} = A u \,\,\,\,\,\,\,\,\,\,(S)
$$

of meromorphic first order differential equations where $A$ is the matrix of coefficients and $z\in\mathbb{C}^n$ is a column vector.

A singular point $z_0$ is **regular singular** if the module of solutions of the equation in a neighborhood of that point the solutions make a module over the algebra of (convergent) meromorphic functions possibly with pole at $z_0$ which is generated by functions of the form $z^a (log z)^b$ where $a\in \mathbb{C}$ and $b\in \mathbb{N}_0$. If a singular point is not regular it is called **irregular singular point**.

It can be shown that $z_0$ is regular singular if a formal Laurent power expansion of a formal solution around $z_0$ every small sector around $z_0$ has a branch bounded by $|z|^{-N}$ for some $N\geq 0$ as $z\to 0$.

Fuchs criterium: (E) has a regular singular point at $0$ iff $ord(h_i)\geq -i$ for $1\leq i\leq n$ where $ord$ is the order function at $0$. There is no similar iff criterium for the coefficients of the linear system. However there is a sufficient condition due Savage: (S) has a regular singular point at $0$ if $)$ is a simple singularity of all entries of the connection matrix. This condition is clearly not invariant under the meromorphic changes of coordinates.

This notion goes back to Riemann and Fuchs. Equations whose all singular points are regular singular are called [[Fuchsian equation|Fuchsian]].

* wikipedia: [regular singular point](http://en.wikipedia.org/wiki/Regular_singular_point)

* Yu. S. Il'yashenko, _[Regular singular point](http://eom.springer.de/r/r080870.htm)_, Springer Enc. of Math. 2001

* V.S. Varadarajan, _Linear meromorphic differential equation: a modern point of view_, Bull. AMS __33__, n. 1, 1996, [pdf](http://www.ams.org/bull/1996-33-01/S0273-0979-96-00624-6/S0273-0979-96-00624-6.pdf)

* P. [[Deligne]], _Équations différentielles à points singuliers réguliers_, Lect. Notes in Math. __163__, Springer-Verlag (1970)

[[!redirects irregular singular point]]


regular space

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

# Regular spaces
* table of contents
{: toc}

## Idea

A _regular space_ is a [[topological space]] (or variation, such as a [[locale]]) that has, in a certain sense, enough [[regular open subspaces]].  The condition of regularity is one of the [[separation axioms]] satsified by every [[metric space]] (and in this case, by every pseudometric space).

## Definitions

### Classical formulation
 {#ClassicalFormulation}

\begin{definition}\label{classical}
**(regular topological spaces)**
\linebreak
A [[topological space]] $X$ is called *regular* (or $R_2$) if for
$C \,\subset\, X$ any [[closed subset]]
and
$x \in X$ any point not contained in it, $x \,\notin\,C$,
then there exist [[neighbourhoods]] $U_x ,\, U_C \,\subset X\,$ of $x$ and of $C$, respectively,
which are [[disjoint subsets|disjoint]]: $U_x \cap U_C \,=\, \varnothing$.
\end{definition}

\begin{definition}\label{RegularHausdorffSpace}
**(regular Hausdorff space)**
\linebreak
A [[topological space]] is called a *regular Hausdorff space* (or *$T_3$*) if

1. it is regular according to Def. \ref{classical}

1. it is a [[T0-space|$T_0$-space]]

   in that for any [[pair]] of distinct points, *at least one* point has a [[neighbourhood]] that does not contain the other point.

\end{definition}
\begin{proposition}\label{RegularT0SpacesAreHausdorff}
  Regular Hausdorff spaces according to Def. \ref{RegularHausdorffSpace} are indeed [[Hausdorff spaces]] (or $T_2$), in that for any pair of distinct points, there is a pair of [[disjoint subsets|disjoint]] [[neighbourhoods]].
\end{proposition}
\begin{proof}
  Let $x,y \,\in\,X$ be two distinct points in a regular $T_0$-space. By the [[T0-space|T0-condition]], one of them, say $x$, is not in the [[topological closure]] of the other: $x \,\notin\, cl(y)$ (using [this Prop.](closed+subspace#UnionOfOpensGivesClosure)). Therefore, by regularity, $\{x\}$ and $cl(y)$ have disjoint neighbourhoods, but these are then also disjoint neighbourhoods of $\{x\}$ and $\{y\}$.
\end{proof}

\begin{example}\label{RegularSpacesWhichAreNotHausdorff}
**(regular spaces which is not Hausdorff)**
\linebreak
  There are regular spaces (Def. \ref{classical}) which are not regular Hausdorff (Def. \ref{RegularHausdorffSpace}).

The simplest example is the [[codiscrete space]] $CoDisc\big(\{0,1\}\big)$ with two points, whose set of [[open subsets]] $\big\{ \varnothing, \, \{0,1\} \big\}$ consists of just the [[empty set]] $\varnothing$ and the total set.
This is not a [[T0-space|T0-space]], since the only open subset containing one of the two points is $\{0,1\}$, which hence also contains the other.
However, $\varnothing$ and $\{0,1\}$ are also the only two [[closed subsets]], so that the regularity condition just asks for a pair of disjoint neighbourhoods around $\varnothing$ and any one of the two points. But this is provided by $\varnothing$ and $\{0,1\}$.
\end{example}

\begin{remark}
  Beware that some authors use "$T_3$-space" to refer to all regular spaces according to Def. \ref{classical}, instead of just the regular Hausdorff spaces according to Def. \ref{RegularHausdorffSpace}.
\end{remark}

### Alternative formulations
 {#AlternativeFormulations}

In many contexts, such as in [[constructive mathematics]], it is more helpful to change perspective in the definition of regular spaces, from a closed set that $a$ does *not* belong to, to an open set that $a$ *does* belong to.  Then the definition reads, equivalently:
+-- {: .num_defn #constructive}
###### Definition
A [[topological space]] $X$ is *regular* if
given any point $a$ and neighbourhood $U$ of $a$, there exist a neighbourhood $V$ of $a$ and an open set $G$ such that $V \cap G = \empty$ but $U \cup G = X$.
=--
You can think of $V$ as being half the size of $U$, with $G$ the exterior of $V$.  (In a [[metric space]], or even in a [[uniform space]], this can be made into a proof.)

If we apply the regularity condition twice, then we get what at first might appear to be a stronger result:
+-- {: .num_defn #closed}
###### Definition
Given any point $a$ and neighbourhood $U$ of $a$, there exist a neighbourhood $W$ of $a$ and an open set $G$ such that $Cl(W) \cap Cl(G) = \empty$ but $U \cup G = X$ (where $Cl$ indicates topological closure).
=--
+-- {: .proof}
###### Proof of equivalence
Find $V$ and $G$ as above.  Now apply the regularity axiom to $a$ and the interior $Int(V)$ of $V$ to get $W$ (and $H$).
=--
In terms of the classical language of separation axioms, this says that $a$ and $F$ are separated by *closed* neighbourhoods.

It is possible to describe the regularity condition fairly simply entirely in terms of the algebra of open sets.  First notice the relevance above of the condition that $Cl(V) \subset U$; we write $V \subset\!\!\!\!\subset U$ in that case and say that $V$ is __well inside__ $U$.  We now rewrite this condition in terms of open sets and regularity in terms of this condition.
+-- {: .num_defn #locale}
###### Definition
Given sets $U, V$, then $V \subset\!\!\!\!\subset U$ iff there exists an open set $G$ such that $V \cap G = \empty$ but $U \cup G = X$.  Then $X$ is regular iff, given any open set $U$, $U$ is the union of all of the open sets that are well inside $U$.
=--
This definition is suitable for [[locales]].  As the definition of a Hausdroff locale is rather more complicated, one often speaks of compact regular locales where classically one would have spoken of [[compact Hausdorff space|compact Hausdorff spaces]]. (The theorem that compact regular $T_0$ spaces and compact Hausdorff spaces are the same works also for locales, and every locale is $T_0$, so compact regular locales and compact Hausdorff locales are the same.)

The condition that a space $X$ be regular is related to the _[[regular open sets]]_ in $X$, that is those open sets $G$ such that $G$ is the interior of its own closure.  (In the [[Heyting algebra]] of open subsets of $X$, this means precisely that $G$ is its own [[double negation]]; this immediately generalises the concept to locales.)  Basically, we start with a neighbourhood $U$ of $x$ and reduce that to a closed neighbourhood $Cl(V)$ of $x$.  Then $Int(Cl(V))$ is a regular open neighbourhood of $x$.

This gives us another way to characterise regular spaces, as follows:
+-- {: .num_defn #closed_basis}
###### Definition
Given a neighbourhood $U$ of $x$,
there is a closed neighbourhood of $x$ that is contained in $U$.
(Equivalently, $x$ has a regular open neighbourhood, or indeed any neighbourhood, well inside $U$.)
In other words, the closed neighbourhoods of $x$
form a local base (a base of the [[neighbourhood filter]]) at $x$.
=--

\begin{remark}\label{regularOpenWarning}
**(Warning)**
\linebreak
It is *not* sufficient that the regular open neighbourhoods themselves form a local base of each point; see Counterexample \ref{dppesFriendCounterexample}. It\'s the *closures* of the regular open neighbourhoods (which are arbitrary closed neighbourhoods) that form the basis. But compare semiregular spaces [below](#semiregular).
\end{remark}

In [[constructive mathematics]], Definition \ref{constructive} is good; then everything else follows without change, except for the equivalence with \ref{classical}. Even then, the classical separation axioms hold for a regular space; they just are not sufficient.

## Variations

Following up on Definition \ref{closed_basis}, we have:
+-- {: .num_cor #regular_basis}
###### Corollary
For any regular space $X$,
the regular open sets form a basis for the topology of $X$.
=--
+-- {: .proof}
###### Proof
For any closed neighbourhood $V$ of $x \in X$,
the interior $Int(V)$ is a regular open neighbourhood of $x$.
Using Definition \ref{closed_basis} finishes the proof.
=--
This suggests a slightly weaker condition, that of a __semiregular space__:
+-- {: .un_defn #semiregular}
###### Definition (of semiregular)
The regular open sets form a basis for the topology of $X$.
=--

As we\'ve seen above, a regular $T_0$ space ($T_3$) is [[Hausdorff space|Hausdorff]] ($T_2$); we can also remove the $T_0$ condition from the latter to get $R_1$:
+-- {: .un_defn #R1}
###### Definition (of R₁)
Given points $a$ and $b$, if every neighbourhood of $a$ meets every neighbourhood of $b$, then every neighbourhood of $a$ is a neighbourhood of $b$.
=--
It is immediate that $T_2 \equiv R_1 \wedge T_0$, and the proof above that $T_3 \Rightarrow T_2$ becomes a proof that $R_2 \Rightarrow R_1$; that is, every regular space is $R_1$.  An $R_1$ space is also called _preregular_ (in _[[HAF]]_) or _reciprocal_ (in [[convergence space]] theory).

A bit stronger than regularity is _complete regularity_; a bit stronger than $T_3$ is $T_{3\frac{1}{2}}$.  The difference here is that for a [[completely regular space]] we require that $a$ and $F$ be separated *by a function*, that is by a continuous real-valued function.  See [[Tychonoff space]] for more. This strengthening implies (Tychonoff Embedding Theorem) that the space embeds into a product of [[metric space]]s.

For locales, there is also a weaker notion called [[weakly regular locale|weak regularity]], which uses the notion of [[fiberwise closed sublocale]] instead of ordinary closed [[sublocales]].

## Examples
 {#Examples}

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

Let $(X,d)$ be a [[metric space]] regarded as a [[topological space]]
via its [[metric topology]]. Then this is a [[normal Hausdorff space]], in particular hence a regular Hausdorff space.

=--

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

We need to show is that given two [[disjoint subsets|disjoint]] [[closed subsets]] $C_1, C_2 \subset X$
then their exists [[disjoint subset|disjoint]] [[open neighbourhoods]] $U_{C_1} \subset C_1$
and $U_{C_2} \supset C_2$.

Consider the function

$$
  d(S,-) \colon X \to \mathbb{R}
$$

which computes [[distances]] from a subset $S \subset X$, by forming the [[infimum]] of the distances to all its points:

$$
  d(S,x) \coloneqq inf\left\{ d(s,x) \vert s \in S \right\}
  \,.
$$

Then the
[[unions]] of [[open balls]]

$$
  U_{C_1}
    \coloneqq
  \underset{x_1 \in C_1}{\cup} B^\circ_{x_1}( d(C_2,x_1) )
$$

and

$$
  U_{C_2}
    \coloneqq
  \underset{x_2 \in C_2}{\cup} B^\circ_{x_2}( d(C_1,x_2) )
  \,.
$$

have the required properties.

=--

\begin{example}\label{KtopCounterexample}
**(counter example)**
\linebreak
The [[real numbers]] equipped with their [[K-topology]] $\mathbb{R}_K$ are a [[Hausdorff topological space]] which is not a [[regular Hausdorff space]] (hence in particular not a [[normal Hausdorff space]]).
\end{example}

\begin{proof}
By construction the K-topology is [[finer topology|finer]] than the usual [[Euclidean space|euclidean]] [[metric topology]]. Since the latter is Hausdorff, so is $\mathbb{R}_K$. It remains to see that $\mathbb{R}_K$ contains a point and a [[disjoint subset|disjoint]] closed subset such that they do not have disjoint [[open neighbourhoods]].

But this is the case essentially by construction: Observe that

$$
  \mathbb{R} \backslash K
  \;=\;
  (-\infty,-1/2)
    \cup
  \left(
    (-1,1) \backslash K
  \right)
    \cup
  (1/2, \infty)
$$

is an open subset in $\mathbb{R}_K$, whence

$$
  K = \mathbb{R} \backslash ( \mathbb{R} \backslash K  )
$$

is a [[closed subset]] of $\mathbb{R}_K$.

But every [[open neighbourhood]] of $\{0\}$ contains at least $(-\epsilon, \epsilon) \backslash K$ for some positive real number $\epsilon$. There exists then $n \in \mathbb{N}_{\geq 0}$ with $1/n \lt \epsilon$ and $1/n \in K$. An open neighbourhood of $K$ needs to contain an open interval around $1/n$, and hence will have non-trivial intersection with $(-\epsilon, \epsilon)$. Therefore $\{0\}$ and $K$ may not be separated by disjoint open neighbourhoods, and so $\mathbb{R}_K$ is not normal.
\end{proof}

\begin{example}\label{dppesFriendCounterexample}
**(counter example)**
\linebreak
Let $\bigl((0, 1)\times(0, 1)\bigr)\cup\{0\}$ be equipped with the Euclidean topology on $(0, 1)\times(0,1)$ and have the sets of the form $(0, 1/2)\times(0, \varepsilon)\cup\{0\}$ (for $\varepsilon \in (0, 1)$) as a basis of open neighbourhoods for the point $0$.

* This space is not regular since we cannot separate $0$ from $[1/2, 1)\times(0,1)$
* Every point $p = (p_1, p_2) \neq 0$ has the euclidean balls of centre $p$ and radius $\varepsilon \in (0, p_1)$ as regular neighbourhood basis.
* The provided basis for the neighbourhoods of $0$ already is a system of regular open sets.

Therefore, this space has the property that every point has a neighbourhood basis of regular open sets (and consequently, the space is semiregular, an even weaker property), but $0$ does not have a neighbourhood basis of closed sets (and consequently, the space is *not* regular).  The problem is that, while every basic neighbourhood of $0$ (and therefore every neighbourhood of $0$ whatsoever) contains a regular open neighbourhood of $0$, none of these basic neighbourhoods contains the closure of any of these regular open neighbourhoods (or any other closed neighbourhood of $0$).
\end{example}

\begin{example}\label{LocallyCompactHausdorffSpacesAreCompletelyRegular}
  [[locally compact Hausdorff space|Locally compact Hausdorff spaces]] are completely regular.
\end{example}
(e.g. [Engelking 1989, Thm. 3.3.1](#Engelking89))
\begin{remark}
  Example \ref{LocallyCompactHausdorffSpacesAreCompletelyRegular} plays a key role in discussion of [[slice theorem|slice theorems]], see there for more.
\end{remark}

## Properties
 {#Properties}

[[!include main separation axioms -- as lifting properties]]

### Further properties

* [[second-countable regular spaces are paracompact]]

* [[paracompact Hausdorff spaces are normal|paracompact Hausdorff spaces are regular]]

## Related concepts

[[!include main separation axioms -- table]]

A [[uniform space]] is automatically regular and even completely regular, at least in [[classical mathematics]].  In [[constructive mathematics]] this may not be true, and there is an intermediate notion of interest called [[uniform regularity]].

Every regular space comes with a naturally defined (point-point) [[apartness relation]]: we say $x # y$ if there is an open set containing $x$ but not $y$.  This can be defined for any topological space and is obviously irreflexive, but in a regular space it is symmetric and a [[comparison]], hence an apartness.  For symmetry, if $x\in U$ and $y\notin U$, let $V$ be an open set containing $x$ and $G$ an open set such that $V\cap G = \emptyset$ and $G\cup U = X$; then $y\in G$ (since $y\notin U$) while $x\notin G$ (since $x\in V$).  With the same notation, to prove comparison, for any $z$ we have either $z\in G$, in which case $z # x$, or $z\in U$, in which case $z # y$.  Note that this argument is valid constructively; indeed, classically, the much weaker $R_0$ [[separation axiom]] is enough to make this relation symmetric, and it is a comparison on any topological space whatsoever.

Note that if a space is [[Hausdorff space|localically strongly Hausdorff]] (a weaker condition than regularity), then it has an apartness relation defined by $x \# y$ if there are disjoint open sets containing $x$ and $y$.  If $X$ is regular, then this coincides with the above-defined apartness.

## References

Textbook accounts:

* {#Engelking89} [[Ryszard Engelking]], *General Topology*, Sigma series in pure mathematics **6**, Heldermann 1989 ([ ISBN 388538-006-4](https://www.heldermann.de/SSPM/SSPM06/sspm06.htm))

See also:

* Wikipedia, *[Regular space](https://en.wikipedia.org/wiki/Regular_space)*

[[!redirects completely regular]]
[[!redirects completely regular space]]

[[!redirects regular space]]
[[!redirects regular spaces]]
[[!redirects R2 space]]
[[!redirects R2 spaces]]
[[!redirects T3 space]]
[[!redirects T3 spaces]]

[[!redirects T3]]

[[!redirects regular Hausdorff space]]
[[!redirects regular Hausdorff spaces]]
[[!redirects regular Hausdorff topological space]]
[[!redirects regular Hausdorff topological spaces]]

[[!redirects regular topological space]]
[[!redirects regular topological spaces]]
[[!redirects regular topology]]
[[!redirects regular topologies]]
[[!redirects R2 topology]]
[[!redirects R2 topologies]]
[[!redirects T3 topology]]
[[!redirects T3 topologies]]
[[!redirects T3 axiom]]

[[!redirects regular locale]]
[[!redirects regular locales]]

[[!redirects semiregular space]]
[[!redirects semiregular spaces]]


regular summation method

A definition of a sum for a class of possibly divergent series is a __regular summation method__ if all [[convergent series]] are in the class and the sum assigned to every convergent series is the usual sum of the series.

* I. I. Volkov (at [[eom]]), [Regular summation methods](http://www.encyclopediaofmath.org/index.php?title=Regular_summation_methods&oldid=26222)

Related entries: [[series]], [[Borel summability]]

category: analysis

[[!redirects regular summation]]