reductions deformations resolutions in physics > cq-arr

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reductions deformations resolutions in physics > ln-arr

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reductions deformations resolutions in physics > pg-arr

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reductive algebraic group

## Properties

### Group cohomology

* [[Friedlander-Milnor isomorphism conjecture]]

## Related concepts

* [[reductive group]]

* [[relation between compact Lie groups and reductive algebraic groups]]

## References

* [[Kevin Buzzard]], _Forms of reductive groups_ ([[BuzzardReductive.pdf:file]])

[[!redirects reductive algebraic groups]]


reductive dual pair

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## Idea

(...)

## Related concepts

* [[reductive Lie algebra]]

## References

The original articles:

* [[Roger Howe]], *Remarks on classical invariant theory*, Trans. Amer. Math. Soc. **313** (1989) 539-570 [[doi:10.2307%2F2001418](https://doi.org/10.2307%2F2001418)]

* [[Roger Howe]], *Transcending classical invariant theory*, J. Amer. Math. Soc. **2** (1989) 535-552 [[doi:10.2307%2F1990942](https://doi.org/10.2307%2F1990942)]

See also:

* Wikipedia, *[Reductive dual pair](https://en.wikipedia.org/wiki/Reductive_dual_pair)*

Generalization to [[super Lie algebras]]:

* [[Allan Merino]], Hadi Salmasian, *Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup* [[arXiv:2208.09746](https://arxiv.org/abs/2208.09746)]

Analogues of Howe dual pairs in [[vertex operator algebra]] theory

* B. Lian, [[Andrew Linshaw]], _Howe pairs in the theory of vertex algebras_, J. Algebra __317__ (2007) 111-152 arXiv:[math.RT/0605174](https://arxiv.org/abs/math/0605174) [doi](https://doi.org/10.1016/j.jalgebra.2007.07.002)

[[!redirects reductive dual pairs]]

[[!redirects Howe duality]]
[[!redirects Howe dualities]]


reductive group

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## Definition

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

A [[Lie group]] $G$ is called _reductive_ if its [[Lie algebra]] $\mathfrak{g}$ is *reductive*, i.e., a [[direct sum]] of an abelian and a [[semisimple Lie algebra]].

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

A Lie algebra is reductive if and only if its [[adjoint representation]] is [[completely reducible]], but this does not imply that all of its finite dimensional representations are completely reducible.

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

The concept of reductive is not quite the same for [[Lie groups]] as it is for [[algebraic groups]] (see at _[[reductive algebraic group]]_) because a reductive Lie group can be the group of real points of a unipotent algebraic group.

=--

## Related concepts

* [[semisimple Lie group]]

* [[relation between compact Lie groups and reductive algebraic groups]]

* [[Friedlander-Milnor isomorphism conjecture]]

## References

Review includes

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

Discussion of the "basic" multiplicative [[holomorphic line 2-bundle]] ([[Chern-Simons line 3-bundle]]) on complex reductive groups is in

* [[Jean-Luc Brylinski]], around theorem 5.4.10 (p. 226-227) of _Loop spaces and characteristic classes_, Birkhäuser

* {#Brylinski00} [[Jean-Luc Brylinski]], _Gerbes on complex reductive Lie groups_ ([arXiv:math/0002158](http://arxiv.org/abs/math/0002158))

[[!redirects reductive groups]]

[[!redirects reductive Lie group]]
[[!redirects reductive Lie groups]]


reductive Lie algebra

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## Definition

A [[Lie algebra]] $\mathfrak{g}$ is called _reductive_ if the following equivalent conditions hold:

1. it is the [[direct sum]] $\mathfrak{g} \simeq \mathfrak{h} \oplus \mathfrak{a}$ of a [[semisimple Lie algebra]] $\mathfrak{h}$ and an abelian Lie algebra $\mathfrak{a}$;

1. its [[adjoint representation]] is completely [[reducible representation|reducible]]: every invariant subspace has an invariant complement.

{#OverAFieldOfCharacteristicZero} Over a [[field]] of [[characteristic zero]], the following conditions on $\mathfrak{g}$ are also equivalent to $\mathfrak{g}$ being reductive:

1. the radical of $\mathfrak{g}$ is equal to the centre of $\mathfrak{g}$ (in general, the radical is only contained inside the centre);

1. $\mathfrak{g}$ is a direct sum of its centre with a semisimple ideal;

1. $\mathfrak{g}$ is a direct sum of prime ideals.

More generally:

+-- {: .num_defn #LieAlgebraReductiveInAmbientLieAlgebra}
###### Definition
**([[reductive Lie algebra|Lie algebra reductive]] in ambient Lie algebra)**

A sub-[[Lie algebra]]

$$
  \mathfrak{h} \hookrightarrow \mathfrak{g}
$$

is called _reductive_ if the [[adjoint representation|adjoint]] [[Lie algebra representation]] of $\mathfrak{h}$ on $\mathfrak{g}$ is [[reducible representation|reducible]].

=--

([Koszul 50](#Koszul50), recalled in e.g. [Solleveld 02, def. 2.27](#Solleveld02))

## Properties

* The [[Lie algebra]] of a [[compact space|compact]] and [[connected]] [[Lie group]] is reductive. ([GHV, vol III, section 4.4.](#GHV)).

* The graded algebra of [[invariant polynomial]]s on a reductive Lie algebra is the [[free construction|free]] graded algebra on the [[graded vector space]] of indecomposable invariant polynomials, and via [[transgression]] there generators are in bijection with the odd generators of the [[Lie algebra cohomology]].

## Related concepts

* [[reductive dual pair]]

## References

* {#Koszul50} [[Jean-Louis Koszul]], _Homologie et cohomologie des algèbres de Lie_, Bull. Soc. Math. France 78 (1950), 65-127

* {#GHV} [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], volume III _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

* {#Solleveld02} [[Maarten Solleveld]], _Lie algebra cohomology and Macdonald’s conjectures_, 2002 ([pdf](https://www.math.ru.nl/~solleveld/scrip.pdf))

[[!redirects reductive Lie algebras]]


Reeb graph

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## Definition

Given a [[topological space]] $X$ equipped with a [[continuous function]] $f \,\colon\, X \longrightarrow \mathbb{R}^1$ to the [[Euclidean space]] of [[real numbers]], it's *Reeb graph* is the [[quotient topological space]] of $X$ by the [[equivalence relation]] which regards two points $x, y \,\in\, X$ as equivalent iff they are in the same [[connected component]] of the same [[level set]]. Under good conditions this is a [[CW complex]], necessarily 1-[[dimension of a cell complex|dimensional]], and as such an [[undirected graph]].

## Related concepts

* [[Morse theory]]

* [[level set]]

* [[topological data analysis]]

## References

Original article:

* [[Georges Reeb]], *Sur les points singuliers d'une forme de Pfaff completement integrable ou d'une fonction numerique*, Comptes Rendus Acad. Sciences Paris **222** (1946) 847-849 $[$[crid:1571417125676878592](https://cir.nii.ac.jp/crid/1571417125676878592)$]$

Further developments:

* Łukasz Patryk Michalak, *Realization of a graph as the Reeb graph of a Morse function on a manifold*, Topol. Methods Nonlinear Anal. **52** 2 (2018)  749-762 $[$[arXiv:1805.06727](https://arxiv.org/abs/1805.06727), [doi:10.12775/TMNA.2018.029](https://doi.org/10.12775/TMNA.2018.029)$]$

See also:

* Wikipedia, *[Reeb graph](https://en.wikipedia.org/wiki/Reeb_graph)*

[[!redirects Reeb graphs]]


Reeb sphere theorem

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## Idea

The _Reeb sphere theorem_ in [[differential topology]] says that:

If a [[compact topological space|compact]] [[differentiable manifold]] $X$ admits a [[differentiable function]] $X \to \mathbb{R}$ with exactly two [[critical points]] which are non-degenerate, then $X$ is [[homeomorphism|homeomorphic]] to an [[n-sphere]] (with its [[Euclidean space|Euclidean]] [[metric topology]]).

In fact this holds true even if the two critical points happen to be degenerate ([Milnor 64, theorem 1' on p. 166](#Milnor64))

## Proof

Let $x_{min}$ be the critical point at which $f$ attains its [[minimum]] and $f_{max}$ the critical point at which it attains its maximum.

By the fact that $X$ is a [[differential manifold]] we find an [[closed neighbourhood]] $B \subset X$ of $x_{min}$ which is [[diffeomorphic]] to an [[closed  ball]].

By the assumption that there are no other critical points, $f$ is monotonically increasing along the [[flow lines]] $\gamma$ of $\nabla f$ on $X \setminus \{x_{min}, x_{max}\}$, in that the function $ \nabla_f(d \gamma_t) = \nabla f(\nabla f) =  {\vert \nabla f\vert^2 }$ is strictly positive:

$$
  \vert\nabla f\vert^2 \colon X \setminus \{x_{min}, x_{max}\} \to (0,\infty) \subset \mathbb{R}
  \,.
$$

Hence for $C \subset X$ a [[compact topological space|compact]], the [[extreme value theorem]] implies that this function attains its [[minimum]].

This implies that for every $y \in \mathbb{R}$ with $f(x_{min}) \lt y \lt f(x_{max})$ there exists $t \in \mathbb{R}$ such that the [[flow of a vector field|flow]] $\phi_{\nabla f}$ along $\nabla f$ satisfies $\phi_{\nabla f}(x,t) \geq y$ for all $x \in \partial B$. In particular this is the case for $y$ the maximum of $f|_{C}$. This implies that $C$ is contained in the image of $B$ under some flow of $\nabla f$. But this image is, by the nature of the flow, diffeomorphic to a Euclidean ball.

In conclusion this shows that every compact subspace of $X$ is contained in an open subspace diffeomorphic to a Euclidean ball. With this the claim follows by the [[Brown-Stallings lemma]].

## Applications

Notice that the Reeb sphere theorem does _not_ speak of [[diffeomorphism]] to an [[n-sphere]], just about [[homeomorphism]]. Indeed, the theorem may be used as an ingredient in the construction of [[exotic smooth structures]] on some $n$-spheres.

## References

Due to:

* [[Georges Reeb]], *Sur les points singuliers d'une forme de Pfaff completement integrable ou d'une fonction numerique*, Comptes Rendus Acad. Sciences Paris **222** (1946) 847-849 $[$[crid:1571417125676878592](https://cir.nii.ac.jp/crid/1571417125676878592)$]$

Review:

* [[John Milnor]], theorem 4.1 in _Morse theory_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/milnmors.pdf))

* {#Milnor64} [[John Milnor]], _Differential topology_, chapter 6 in T. L. Saaty (ed.) _Lectures On Modern Mathematic II_ 1964 ([web](https://archive.org/details/LecturesOnModernMathematicsIi), [pdf](https://ia801700.us.archive.org/6/items/LecturesOnModernMathematicsIi/Saaty-LecturesOnModernMathematicsIi.pdf))

See also

* Wikipedia, *[Reeb sphere theorem](https://en.wikipedia.org/wiki/Reeb_sphere_theorem)*


Reeb theorem

See

* [[Reeb sphere theorem]]

* [[Reeb stability theorem]]

[[!redirects Reeb's theorem]]


Reeb vector field

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## Idea

In [[contact geometry]] given a [[contact 1-form]] $\alpha$, then a *Reeb vector field* $R$ is a [[vector field]] satisfying

1. $\alpha(R) = 1$,

1. $d\alpha(R,-) = 0$.

## Related concepts

* [[Hamiltonian vector field]]

## References

Review:

* [[Hansjörg Geiges]], Def. 2.5 in: _Contact geometry_, in F. J. E. Dillen and L. C. A. Verstraelen, (eds.), *Handbook of Differential Geometry* vol. 2  North-Holland, Amsterdam (2006) 315-382 [[arXiv:math/0307242](http://arxiv.org/abs/math/0307242), [ISBN:978-0-444-52052-4](https://www.sciencedirect.com/handbook/handbook-of-differential-geometry/vol/2/suppl/C)]

See also:

* Wikipedia, *[Reeb vector field](https://en.wikipedia.org/wiki/Reeb_vector_field)*

[[!redirects Reeb vector fields]]


Reed Mullanix

* [personal page](https://reedmullanix.com/)

* [blog](https://totbwf.github.io/about.html)

## Selected writings

On [[universe polymorphism]] in [[dependent type theory]] via [[monads]]:

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

## Related pages

* [[1lab]]

category: people


Reedy category

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## Idea

A _Reedy category_ is a [[category]] $R$ equipped with a structure enabling the inductive construction of [[diagrams]] and [[natural transformations]] of shape $R$. It is named after [[Christopher Reedy]].

The most important consequence of a Reedy structure on $R$ is the existence of a certain [[model category|model structure]] on the [[functor category]] $M^R$ whenever $M$ is a [[model category]] (no extra hypotheses on $M$ are required): the _[[Reedy model structure]]_.

## Definition

A **Reedy category** is a [[category]] $R$ equipped with two [[wide subcategories]] $R_+$ and $R_-$ and a [[total ordering]] on the objects of $R$, defined by a _degree_ function $d:Ob(R) \to \alpha$, where $\alpha$ is an [[ordinal number]], such that:

* Every nonidentity morphism in $R_+$ raises degree,
* Every nonidentity morphism in $R_-$ lowers degree, and
* Every morphism $f$ in $R$ factors uniquely as a map in $R_-$ followed by a map in $R_+$.  (Thus $(R_-,R_+)$ is a [[strict factorization system]].)

## Examples

### The simplex category

The prototypical examples of Reedy categories are the [[simplex category]] $\Delta$ and its [[opposite category|opposite]] $\Delta^{op}$.  (More generally, for any [[simplicial set]] $X$, its [[category of simplices]] $\Delta/X$ is a Reedy category.)

\begin{definition}\label{ReedyStructureOnSimplexCategory}
**(Reedy structure on the simplex category)**
\linebreak
The Reedy category structure on $\Delta$ is defined by:

* The degree function $d \colon Ob(\Delta) \to \mathbb{N}$ is defined by $[k] \mapsto k$.

* a map $[k] \to [n]$ is in $\Delta_+$ precisely if it is [[injective]];

* a map $[n] \to [k]$ is in $\Delta_-$ precisely if it is [[surjective]].

And the Reedy category structure on $\Delta^{op}$ is defined by switching $\Delta_+$ and $\Delta_-$.
\end{definition}

### Further examples

* Any ordinal $\alpha$, considered as a [[poset]] and hence a category, is a Reedy category with $\alpha_+=\alpha$, $\alpha_-$ the [[discrete category]] on $Ob(\alpha)$, and $d$ the identity.

* The [[opposite category|opposite]] of any Reedy category is a Reedy category: use the same degree function, and exchange $R_+$ and $R_-$.

* The integers regarded as a poset is NOT a Reedy category, since it is not well-founded in either direction.

* [[Theta category|Joyal's category]] $\Theta$ is also a Reedy category.

* Many very small categories of diagram shapes are Reedy categories, such as $(\cdot\to\cdot\to \dots)$, or $(\cdot\leftarrow \cdot\rightarrow\cdot)$, or $(\cdot\rightrightarrows\cdot)$.  This is of importance for the construction of [[homotopy limits]] and colimits over such diagram shapes.

## Related notions

### Direct and inverse categories

A Reedy category in which $R_-$ contains only identities is called a [[direct category]]; the factorization axiom then says simply that $R=R_+$.  Similarly, if $R_+$ contains only identities it is said to be an [[inverse category]].

Any ordinal is of course a direct category, and so is the subcategory $R_+$ of any Reedy category considered as a category in its own right.  This amounts to "discarding the degeneracies" in a shape category.  In some examples there are no degeneracies to begin with, such as the category of [[opetopes]]; thus these are naturally direct categories.

Dually, so is the subcategory $R_-$ of any Reedy category considered as a category in its own right.  This amounts to "discarding the faces" in a shape category.

### Generalized Reedy categories

One problem with the notion of Reedy category is that it violates the [[principle of equivalence]]: it is not invariant under [[equivalence of categories]].  It's not hard to see that any Reedy category is necessarily [[skeletal category|skeletal]].  In fact, it's even worse: no Reedy category can have _any_ [[identity|nonidentity]] [[isomorphisms]]!

*Proof:* Take any isomorphism $f$, let $f = g h$ and $h f^{-1} = g' h'$ be the unique factorizations. Then $id = g h f^{-1} = (g g') h'$, so $h' = id$ and $g g' = id$, whence $g = id$ and $g' = id$ since $g, g' \in R_+$. Thus $f = h \in R_-$. The same argument applied to $f^{-1}$ shows that $f$ preserves the degree, hence $f = id$. $\qed$

This is problematic for many $\Delta$-like categories such as the [[category of cycles]], Segal's category $\Gamma$, the [[tree category]] $\Omega$, and so on.  The concept of

* [[generalized Reedy category]],

due to [[Clemens Berger]] and [[Ieke Moerdijk]], avoids these problems.  There is a similar notion (which however does not comprise all Reedy categories) due to [[Denis-Charles Cisinski]].  A further generalization which allows noninvertible level morphisms is a

* [[c-Reedy category]].

### Elegant Reedy categories

The notion of [[elegant Reedy category]], introduced by [[Julie Bergner]] and [[Charles Rezk]], is a *restriction* of the notion which captures the property that the Reedy model structure and injective model structure coincide.  Several important Reedy categories are elegant, such as the $\Delta$ and $\Theta$.

### Eilenberg-Zilber categories

[[Eilenberg-Zilber categories]] are a special sort of generalized Reedy category that behave rather like an elegant strict Reedy category.

### Enriched Reedy categories

There is also a generalization of the notion of Reedy category to the context of [[enriched category theory]]: this is an [[enriched Reedy category]].

### Reedy categories with fibrant constants.

If $R$ is a [[direct category]], then for any [[model category]] $M$ the colimit functor $\colim_R \colon M^R \to M$ is a [[Quillen adjunction|left Quillen functor]]. However, there are non-direct Reedy categories with the same property, they are called [[Reedy category with fibrant constants|Reedy categories with fibrant constants]].

## References

See the references at _[[Reedy model structure]]_

[[!redirects Reedy categories]]


Reedy category with fibrant constants

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#### Model category theory
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

A **Reedy category with fibrant constants** is a [[Reedy category]] $R$ such that for every [[model category]] $M$ and every [[fibrant object]] $X \in M$ the constant $R$-[[diagram]] at $X$ is Reedy fibrant.

## Properties

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

For a Reedy category $R$ the following are equivalent.

1. $R$ has fibrant constants.
1. For any model category $M$ the [[colimit]] functor $\colim_R \colon M^R \to M$ is a [[Quillen adjunction|left Quillen functor]].
1. All matching categories of $R$ are [[connected category|connected]] or empty.

=--

This theorem is proven in [Hirschhorn,Proposition 15.10.2 and Theorem 15.10.8](#Hirshhorn).

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

1. The product of two Reedy categories with fibrant constants is a Reedy category with fibrant constants.
1. If $R$ is an [[elegant Reedy category]] and $X$ is a presheaf on $R$, then the [[category of elements]] of $X$ is a Reedy category with fibrant constants.

=--

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

1. This follows by combining point 3. of the theorem above with the fact that the matching categories satisfy the "Leibniz rule", see [Riehl, Verity, Observation 4.2](#RiehlVerity).
1. This is proven in [Hirschhorn, Proposition 15.10.4](#Hirschhorn) in the case of $R = \Delta$, but the argument given there applies to any elegant Reedy category.

=--

## Examples

* The [[simplex category]] $\Delta$ is a Reedy category with fibrant constants.

* More generally, Joyal's [[Theta category|disk categories]] $\Theta_n$ are Reedy categories with fibrant constants.

* Every [[direct category]] is a Reedy category with fibrant constants.

## References

* [[Philip Hirschhorn]],  _Model categories and their localizations_, volume 99 of Mathematical Surveys and Monographs, American Mathematical Society, 2009, {#Hirschhorn}

* [[Emily Riehl]], [[Dominic Verity]] _The theory and practice of Reedy categories_ [arXiv](http://arxiv.org/abs/1304.6871v1) {#RiehlVerity}


Reedy model structure

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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

A _Reedy model structure_ is a [[global model structure on functors]]:

given a [[Reedy category]] $R$ and a [[model category]] $C$ the **Reedy model structure** is a [[model category]] structure on the [[functor category]] $[R,C] = Func(R,C)$.

As opposed to the [[global model structure on functors|projective and injective model structure]] on functors this does not require any further structure on $C$, but instead makes a strong assumption on $R$.

If all three exist, then, in a precise sense, the Reedy model structure sits _in between_ the injective and the projective model structure. As such, it has the advantage that _both_ the cofibrations as well as the fibrations can be fairly explicitly described and detected in terms of cofibrations and fibrations in $C$.

## Definition
 {#Definition}

### Plain version
 {#PlainVersion}

For

* $\mathcal{R}$ a [[Reedy category]]

* $\mathcal{C}$ a [[model category]].

consider

* the [[functor category]] $\mathcal{C}^\mathcal{R}$

  whose [[objects]] $X \colon \mathcal{R} \to \mathcal{C}$ we refer to as ($\mathcal{R}$-shaped) *[[diagrams]]* in $\mathcal{C}$.

\begin{definition}\label{LatchingAndMatchingObjects}
**(latching and matching objects)**
\linebreak
Given

1. a [[diagram]] $X \colon \mathcal{R} \to \mathcal{C}$,

2. an [[object]] $ r \in \mathcal{R}$

then:

The **latching object** of $X$ at stage $r$ is the [[colimit]] in $\mathcal{C}$

\[
  \label{LatchingObject}
  L_r X
  \;\coloneqq\;
  \underset{
    s \underset{+}{\to} r
  }{colim}
  X_s
\]

over the [[full subcategory]] of the [[slice category]] $R_+/r$ containing all objects except the [[identity morphism|identity]] $id_r$.

[[formal duality|Dually]]:

The **matching object** of $X$ at stage $r$ is the [[limit]] in $\mathcal{C}$

\[
  \label{MatchingObject}
  M_r X
  \;\coloneqq\;
  \underset{
    r \underset{-}{\to} s }
  {\lim}
  X_s
  \,,
\]

over the [[full subcategory]] of the [[coslice category]] $r/R_-$ containing all objects except $id_r$.

By the [[universal property]] of ([[colimit|co]])[[limits]] there are evident [[natural transformations]]:

\[
  \label{ComparisonMapsFromLatchingToMatchingObject}
  L_r X
  \longrightarrow
  X_r
  \longrightarrow
  M_r X
  \,.
\]
\end{definition}
(e.g. [Hovey (1999), Def. 5.2.2](#Hovey99); [Hirschhorn (2002), Def. 15.2.5](#Hirschhorn02))

\begin{example}\label{LatchingOfSimplicialObjects}
**(latching of simplicial objects)**
\linebreak
Consider the case that $\mathcal{R} = \Delta^{op}$ is the [[opposite category|opposite]] [[simplex category]] with its canonical Reedy structure ([here](Reedy+category#ReedyStructureOnSimplexCategory)), where $(\Delta^{op})_+ \,=\, (\Delta_-)^{op}$ is the opposite of the [[surjections]] $[r] \to [r-k]$.

For $X \colon \Delta^{op} \to \mathcal{C}$ a [[simplicial object]], its value on such a morphism is the corresponding [[degeneracy map]].

This means that the [[cocone]] morphism (eq:ComparisonMapsFromLatchingToMatchingObject) out of the the latching object (eq:LatchingObject) has as components in degree $r$ all [inclusions of degenerate simplices](simplicial+identities#DegeneracyMapsAreSplitMono)
$$
  X_{r-k} \overset{\phantom{--}}{\hookrightarrow} X_r
$$
which $X_r$ receives.

Hence one may often think of the latching object $L_r X$ of a [[simplicial object]] $X$ as the object of _degenerate_ [[n-simplex|$r$-simplices]] sitting inside the object $X_r$ of all $r$-simplices. Cf. Prop. \ref{EveryBisimplicialSetIsReedyCofibrant} and Prop. \ref{LatchingInAbelianCategoryIsDegeneracySubobject} below.
\end{example}

\begin{remark}
Notice that $L_0 X = 0$ is the [[initial object]] and $M_0 X$ is the [[terminal object]], since there are no objects of degree $\lt 0$.
\end{remark}

\begin{example}
When $\mathcal{R} = \alpha$ is an [[ordinal]] (regarded as a [[poset]] and thus as a [[category]]), then $L_{n+1} X = X_n$ and $M_n X = 1$; and dually for $\mathcal{R}=\alpha^{op}$.
\end{example}

\begin{definition}\label{ReedyModelStructure}
A [[morphism]] $f \colon X \to Y$ in the [[functor category]] $\mathcal{C}^{\mathcal{R}}$ (hence a [[natural transformation]] between [[diagrams]]) is called:

* a *Reedy equivalence* if each component morphism $f_r$ is a [[weak equivalence]] in $\mathcal{C}$;

* a *Reedy-cofibration* or *acyclic Reedy-cofibration* if for all $r \in \mathcal{R}$, the map

  \[
    \label{CharacterizingReedyCofibrations}
    L_r Y
     \overset
       {L_r X}
       {\amalg}
    X_r
      \longrightarrow
    Y_r
  \]

  is a [[cofibration]] or [[acyclic cofibration]] in $\mathcal{C}$, respectively;

* a *Reedy-fibration* or *acyclic Reedy-fibration* if for all $r \in \mathcal{R}$, the map

  \[
    \label{CharacterizingReedyFibrations}
    X_r
     \longrightarrow
    M_r X
     \underset
     {M_r Y}
     {\times}
    Y_r
  \]

  is a [[fibration]] or [[acyclic fibration]] in $\mathcal{C}$, respectively.

\end{definition}
(e.g. [Hirschhorn (2002), Def. 15.3.3](#Hirschhorn02))

\begin{remark}\label{ReedyCofibrantObjects}
In particular this means that:

1. An object $X$ is Reedy-cofibrant iff the comparison maps $L_r \to X_r$ (eq:ComparisonMapsFromLatchingToMatchingObject) from its latching object are cofibrations.

1. An object $X$ is Reedy-fibrant iff the comparison maps $X_r \to M_r X$ (eq:ComparisonMapsFromLatchingToMatchingObject) to its matching object are fibrations.

\end{remark}

\begin{proposition}\label{model}
**(Reedy model structure)**
\linebreak
The classes of morphisms in Def. \ref{ReedyModelStructure} constitute a [[model category]]-[[structure]] on the [[functor category]] $\mathcal{C}^{\mathcal{R}}$, called the *Reedy model structure*.
\end{proposition}
(e.g. [Hovey (1999), Thm. 5.2.5](#Hovey99); [Hirschhorn (2002), Thm 15.3.4](#Hirschhorn02))

\begin{remark}
To see the factorization of a morphism $f \colon X\to Y$ in either of the two necessary ways, construct the factorization $f_r = g_r h_r$ inductively on $r$, by factoring the induced morphism
$$ L_r Z \amalg_{L_r X} X_r \to M_r Z \times_{M_r Y} Y_r$$
in the appropriate way in $\mathcal{C}$.
\end{remark}

### Enriched version

For $V$ a [[cosmos|suitable enriching category]], there is a refinement of the notion of Reedy category to a notion of _$V$-[[enriched Reedy category]]_ such that if $C$ is a $V$-[[enriched model category]] -- in particular when it is a [[simplicial model category]] for $V = $ [[SSet]] -- the [[enriched functor category]] $[R,C]$ is itself a $V$-[[enriched model category]] (see [Angeltveit](#Angeltveit)).

In the case that we do have extra assumptions on the codomain in that

* $C$ is a [[combinatorial simplicial model category]]

* with  $C^\circ$ the [[(∞,1)-category]] [[presentable (∞,1)-category|presented]] by $C$

* and with the [[Reedy category]] $R$ an ordinary category regarded as a [[SSet]]-[[enriched category]],

the Reedy model structure, having the same weak equivalences as the [[global model structure on functors]], presents similarly the [[(∞,1)-category of (∞,1)-functors]] $Func_\infty(R,C^\circ)$, from $R$ into the [[(∞,1)-category]] [[presentable (∞,1)-category|presented by]] $C$.

## Remarks

* Any Reedy cofibration or fibration is, in particular, an objectwise one, but the converse does not generally hold.

* An object $X$ is Reedy cofibrant if and only if each map $L_r X \to X_r$ is a cofibration in $M$.  In particular, this implies that each $X_r$ is cofibrant in $M$.

* For some $M$, $M^R$ also admits a [[projective model structure|projective]] or [[injective model structure|injective]] model structures.  For instance for $M = $ [[SSet]] this is the [[global model structure on simplicial presheaves]].

  In general the Reedy structure will not be the same as either, but will be a kind of mixture of both. If $R = R_+$ then the Reedy model structure coincides with the [[projective model structure]], if $R = R_-$ it coincides with the [[injective model structure]].

  For a detailed comparison of Reedy and global projective/injective model structures see around example A.2.9.22 in [[Higher Topos Theory|HTT]].

* In addition to its existing for all $C$, another advantage of the Reedy structure is the explicit nature of its cofibrations, fibrations, and factorizations.

* If $R$ admits more than one structure of Reedy category, then $C^R$ will have more than one Reedy model structure.  For instance, if $R = (\cdot\to\cdot)$ is the [[walking arrow]], then we can regard it as either the ordinal $2$ or its opposite $2^{op}$, resulting in two different Reedy model structures on $C^2$.

* For a general Reedy category $R$, the diagonal functor $\Delta : C\to C^R$ need not be either a right or a left [[Quillen adjunction|Quillen functor]] (although, of course, it has left and right adjoints given by colimits and limits over $R$).  $\Delta$ is a left Quillen functor if and only if for all objects $r$, the category $r / R_-$ is either connected or empty.  Dually, $\Delta$ is a right Quillen functor if and only if for all objects $r$, the category $R_+ / r$ is empty or connected.  In these cases, one can construct [[homotopy limit|homotopy limits]] and colimits using the derived functors of the Quillen adjunctions $\mathrm{colim} \dashv \Delta \dashv \mathrm{lim}$.

## Properties
  {#Properties}

### Properness

\begin{proposition}\label{ReedyStructureInheritsProperness}
  For $\mathcal{R}$ a [[Reedy category]] and $\mathcal{C}$ a [[model category]] which is left or right [[proper model category|proper]], then also the Reedy model structure on $Func(\mathcal{R}, \mathcal{C})$ (Prop. \ref{model}) is left or right [[proper model category|proper]], respectively.
\end{proposition}
This appears as [Hirschorn (2002), Thm. 15.3.4 (2)](#Hirschorn02), there attributed to [[Daniel Kan]].

### Combinatorial structure
 {#CombinatorialStructure}

\begin{proposition}
\label{ReedyStructureCofibrantlyGenerated}
**(cofibrant generation)**
\linebreak
  If $\mathcal{C}$ is a [[cofibrantly generated model category]] (and $\mathcal{R}$ is a [[Reedy category]]) then also the Reedy model structure (Def. \ref{ReedyModelStructure}) on the [[functor category]] $\mathcal{C}^{\mathcal{R}}$ is cofibrantly generated.
\end{proposition}
This is due to [Hirschhorn (2002), Theorem 15.6.27](#Hirschhorn02), assuming here that the [[domains]] of generating (acyclic) cofibrations are relatively [[small objects]]. In the generality of [[generalized Reedy categories]] cf. [Berger & Moerdijk (2011), Thm. 7.5](generalized+Reedy+category#BergerMoerdijk11)

The [[formal duality|dual]] statement concerning *fibrant* generation is in [BHKKRS15, Thm. 5.9](#BHKKRS15).

\begin{corollary}
\label{ReedyStructureCombinatorial}
**(combinatoriality)**
\linebreak
  If $\mathcal{C}$ is a [[combinatorial model category]] (and $\mathcal{R}$ is a [[small category|small]] [[Reedy category]]) then also the Reedy model structure (Def. \ref{ReedyModelStructure}) on the [[functor category]] $\mathcal{C}^{\mathcal{R}}$ is combinatorial.
\end{corollary}
\begin{proof}
  Recall that "combinatorial" means "locally presentable and cofibrantly generated". Prop. \ref{ReedyStructureCofibrantlyGenerated} gives the cofibrant generation, and local presentability follows from general results on locally presentable functor categories (see [there](locally+presentable+category#PresheavesWithValuesInLocPresAreLocPres)).
\end{proof}

\begin{proposition}\label{LeftProperness}
**(left properness)**
\linebreak
  If $\mathcal{C}$ is

* [[combinatorial model category|combinatorial]] and [[left proper model category|left proper]]

or

* [[cellular model category|cellular]] and [[left proper model category|left proper]]

then also the Reedy model structure on $\mathcal{C}^{\mathcal{R}}$ is (combinatorial, by Cor. \ref{ReedyStructureCombinatorial}, or cellular, respectively, and) left proper.
\end{proposition}
This is mentioned in [Dugger 2001, item 3. on p. 5008 (6 of 25)](#Dugger01).

### Enriched model structure
 {#EnrichedModelStructure}

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

For $C$ a [[Reedy category]] and $A$ a [[symmetric monoidal category|symmetric]] [[monoidal model category]], the Reedy model structure on $[C,A]_{Reedy}$ is naturally an $A$-[[enriched model category]].

If in addition $A$ is a $V$-[[enriched model category]] for some symmetric monoidal model category $V$, then so is $[C,A]_{Reedy}$

=--

This appears as ([Barwick, lemma 4.2, corollary 4.3](#Barwick)).

(...check assumptions...)

### Relation to other model structures

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

Let $C$ be a [[combinatorial model category]] and $R$ a [[Reedy category]].

The identity functors provide left [[Quillen equivalences]]

$$
  [R,C]_{proj} \stackrel{\simeq_{Quillen}}{\to}
  [R,C]_{Reedy}
  \stackrel{\simeq_{Quillen}}{\to}
  [R,C]_{inj}
$$

from the projective [[model structure on functors]] to the injective one.

=--

See also [[Higher Topos Theory|HTT, remark A.2.9.23]]

## Examples

### Over the arrow category

The simplest nontrivial example is obtained for

$$
  R = I = \{1 \to 0\}
$$

the [[interval category]].

In this case the [[functor category]] $[I,C]$ is the [[arrow category]] $C$.

We take the degree on the objects to be as indicated. Then $R_- = R$ and $R_+$ contains only the identity morphisms.

For $F : I \to C$ a functor, i.e. a morphism $F(1) \to F(0)$ in $C$, we find

* the latching object $latch_0 F = colim_{(s \stackrel{+}{\to} 0)} F(s) = \emptyset$;

* the latching object
  $latch_1 F = colim_{(s\stackrel{+}{\to}1)} F(s) = \emptyset$;

* the matching object $match_0 F = lim_{(0 \stackrel{-}{\to}s)} F(s) = {*}$;

* the matching object $match_1 F = lim_{(1 \stackrel{-}{\to}s)} F(s) = F(0)$

where $\emptyset$ denotes the [[initial object]] and ${*}$ the [[terminal object]] (being the [[colimit]] and [[limit]] over the empty [[diagram]],  respectively).

From this we find that for a [[natural transformation]] $\eta : F \to G$

  $$
    \array{
      F(1) &\stackrel{\eta_1}{\to}& G(1)
      \\
      \downarrow && \downarrow
      \\
      F(0) &\stackrel{\eta_0}{\to}& G(0)
    }
  $$

that

* it is a Reedy cofibration in $[I,C]$ if

  * $\eta_0 : F(0) \coprod_{\emptyset} \emptyset = F(0) \to G(0)$ is a cofibration

  and

  * $\eta_1 : F(1) \coprod_{\emptyset} \emptyset = F(1) \to G(1)$ is a cofibration

* it is a Reedy fibration in $[I,C]$ if

  * $\eta_0 : F(0) \to G(0) \times_{*} {*} = G(0)$ is a fibration

  * the universal morphism $F(1) \to G(1) \times_{G(0)} F(0)$

    $$
    \array{
      F(1)
      \\
      & \searrow
      \\
      && F(0) \times_{G(0)} G(1)
      &\to& G(1)
      \\
      && \downarrow && \downarrow
      \\
      && F(0) &\stackrel{\eta_0}{\to}& G(0)
    }
    $$

    is a fibration.

  Notice that since fibrations are preserved by pullbacks and under composition with themselves, it follows that also $\eta_1 : F(1) \to G(1)$ is a fibration.

* The cofibrant objects in $[I,C]$ are those arrows $F(1) \to F(0)$ in $C$ for which $F(1)$ and $F(0)$ are cofibrant;

* The fibrant objects in $[I,C]$ are those arrows $F(1) \to F(0)$ in $C$ that are fibrations between fibrant objects in $C$.

So in accord with the proposition above one finds that this Reedy model structure on $[I,C]$ coincides with the _injective_ [[global model structure on functors]] on $I$.

### Over the tower category
 {#OverTheTowerCategory}

Let $R = \mathbb{N}^{op} = \{\cdots \to 2 \to 1 \to 0\}$ be the natural numbers regarded as a [[poset]] using the greater-than  relation.

With the degree as indicated, this is a Reedy category with $R_- = R$ and $R_+$ containing only identity morphisms.

Now the [[functor category]] $[R,C]$ is the category of _towers_ of morphisms in $C$.

The analysis of the Reedy model structure on this involves just a repetition of the steps involved in the analysis of the arrow category in the above example. One finds:

* a natural transformation $\eta : F \to G$ is a fibration precisely if

  * the component $\eta_0 : F(0) \to G(0)$ is a fibration

  * all universal morphisms $F(n) \to F(n-1) \times_{G(n-1)} G(n)$ are fibrations.

* the fibrant objects are the towers of fibrations on fibrant objects in $C$.

A detailed discussion of the model structure on towers is for instance in ([GoerssJardine, chapter 6](#GoerssJardine))

By duality it follows that analogously there is a model structure on co-towers

$$
  X_0 \to X_1 \to X_2 \to \cdots
$$

in a model category $C$, whose fibrations and weak equivalences are the degreewise ones, and whose cofibrations are those transformations that are a cofibration in degree 0 and where the canonical pushout-morphisms in each square are cofibrations.

### Over the simplex category
  {#SimplexCategory}

The motivating and central example of [[Reedy categories]] is the
[[simplex category]] $\Delta$.

Recall that

* a map $[k] \to [n]$ is in $\Delta_+$ precisely if it is an injection;

* a map $[n] \to [k]$ is in $\Delta_-$ precisely if it is a surjection.

Dually, for the [[opposite category]] $\Delta^{op}$

* a morphism $[n] \leftarrow [k]$ is in $(\Delta^{op})_-$ precisely if the map underlying it is an injection;

* a morphism $[k] \leftarrow [n]$ is in $(\Delta^{op})_+$ precisely if the map underlying it is a surjection.

Let $C = sSet_{Quillen}$ be the category [[sSet]] equipped with the standard [[model structure on simplicial sets]] and consider the Reedy model structures on $[\Delta^{op}, sSet_{Quillen}]$ $[\Delta,sSet_{Quillen}]$.

#### With values in simplicial sets
  {#OverDeltaWithValuesInSimplicialSets}

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

For $X  \in [\Delta^{op},Set]$ a [[simplicial object]]
and $n \in \mathbb{N}$,

* the [[latching object]] $L_n X$ is the [[union]] of all degenerate $n$-cells;

* the [[matching object]] is $M_n X \simeq X^{\partial \Delta[n]}$, the [[powering]] of the [[boundary of a simplex|boundary of the n-simplex]] into $X$, hence the $n$-cells of the $(n-1)$-[[skeleton]] of $X$.

=--

More details on this are currently at _[[generalized Reedy model structure]]_.

\begin{proposition}\label{EveryBisimplicialSetIsReedyCofibrant}
If $\mathcal{C} = sSet$ is the [[classical model structure on simplicial sets]], then every object in $[\Delta^{op}, \mathcal{C}]_{Reedy}$ (i.e. every [[bisimplicial sets]]) is Reedy-cofibrant.
\end{proposition}

([Hirschhorn (2002), corollary 15.8.8](#Hirschhorn02))

The idea is to observe that the latching objects are the [[sub-objects]] of degeenrate cells, so that the comparison morphism (eq:ComparisonMapsFromLatchingToMatchingObject) is a [[monomorphism]] and hence a [[cofibration]] in the [[classical model structure on simplicial sets]].

Similarly:

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

The canonical cosimplicial simplicial set

$$
  \Delta[-] : \Delta \to sSet
$$

is Reedy cofibrant in $[\Delta,sSet_{Quillen}]$.

=--

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

The latching object at $n$ is

$$
  L_n(\Delta[-]) =
  \lim_\to
  \left(
    ([k] \to [n]\; inj.\in\;\Delta) \mapsto
    \Delta[k]
  \right)
  \,.
$$

This is $\partial \Delta[n]$. The inclusion $\partial \Delta[n] \to \Delta[n]$ is a monomorphism, hence a cofibration in $sSet_{Quillen}$ (in fact these are the generating cofibrations).

=--

+-- {: .num_cor}
###### Corollary

Every [[simplicial set]] is the [[homotopy colimit]] over its diagram of simplices (with values in the constant simplicial set on the sets of simplicies $X_n$):

$$
  X \simeq hocolim ( [n] \mapsto const X_n)
  \,.
$$

=--

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

Because $X_{(-)} : \Delta^{op} \to sSet$ is Reedy cofibrant by the above, by the discussion at [[homotopy colimit]] we can compute the hocolim by the [[coend]]

$$
  \int^{[n]} Q(*)_n \cdot X_n
  \,,
$$

where $Q(*) : \Delta \to sSet$ is a cofibrant [[resolution]] of the point in $[\Delta, sSet_{Quillen}]_{Reedy}$. Using the above observation, we can take this to be $\Delta[-]$ since this is cofibrant by the above observation and clearly $\Delta[-] \to *$ is objectwise a weak equivalence in $sSet_{Quillen}$.

Therefore the hocolim is (up to equivalence) represented by the simplicial set

$$
  \int^{[n]} \Delta[n] \cdot X_n
  \,.
$$

But by the [[co-Yoneda lemma]] this is in fact [[isomorphism|isomorphic]] to $X$, hence in particular weakly equivalent to $X$.

=--

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

This kind of argument has many immediate generalizations. For instance for $C = [K^{op}, sSet_{Quillen}]_{inj}$ the injective [[model structure on simplicial presheaves]] over any small category $K$, or any of its left [[Bousfield localization of model categories|Bousfield localizations]], we have that the cofibrations are objectwise those of simplicial sets, hence objectwise monomorphisms, hence it follows that every simplicial presheaf $X$ is the hocolim over its simplicial diagram of component presheaves.

=--

For the following write $\mathbf{\Delta} : \Delta \to sSet$ for the [[fat simplex]].

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

The fat simplex is Reedy cofibrant.

=--

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

By the discussion at [[homotopy colimit]], the fat simplex is cofibrant in the projective [[model structure on functors]] $[\Delta, sSet_{Quillen}]_{proj}$. By the [general properties](#Properties) of Reedy model structures, the identity functor $[\Delta, sSet_{Quillen}]_{proj} \to [\Delta, sSet_{Quillen}]_{Reedy}$ is a [[Quillen adjunction|left Quillen functor]], hence preserves cofibrant objects.

=--

#### With values in an addive model category
 {#WithValuesInAnAbelianCategory}

In this section, let $\mathcal{A}$ be an [[additive category|additive]] [[model category]] in which all [[retractions]] exhibit [[direct sums]] (such as in an [[abelian category]], such as in a [[model structure on chain complexes]]).

In this situaiton the [[Dold-Kan correspondence]] applies (see [there](Dold-Kan+correspondence#StatementAdditiveACase)) and can be used to get further information about the Reedy model structure on the category $s\mathcal{A} \coloneqq \mathcal{A}^{(\Delta^{op})}$ of [[simplicial objects]] in $\mathcal{A}$.

\begin{proposition}
  \label{LatchingInAbelianCategoryIsDegeneracySubobject}
Under the above assumptions, for any [[simplicial object]] $X_\bullet \in s\mathcal{A}$ and all $n \in \mathbb{N}$, the comparison morphism (eq:ComparisonMapsFromLatchingToMatchingObject) from the $n$th latching object (Def. \ref{LatchingAndMatchingObjects}) of $X_\bullet$ is a ([[split monomorphism|split]]) [[monomorphism]]:

$$
  L_n X \xhookrightarrow{\phantom{---}} X_n
  \,.
$$
\end{proposition}
This statement and the following proof spells out a suggestion by [[Charles Rezk]] in [MO:a/300657](https://mathoverflow.net/a/300657/381).
\begin{proof}
  Under the given assumption on $\mathcal{A}$ the [[Dold-Kan correspondence]] applies (see [there](Dold-Kan+correspondence#StatementAdditiveACase)) to [[simplicial objects]] in $\mathcal{A}$ and says that, up to [[isomorphism]], $X_\bullet$ is of the form $X_\bullet \simeq \Gamma(V_\bullet)$ for some [[connective chain complex]] $V_\bullet$ (in fact $V_\bullet \simeq N X_\bullet$ is the [[normalized chain complex]] of $X_\bullet$), in that each $X_r$ is the [[direct sum]] of one copy of $V_s$ for each [[surjection]] $[r] \twoheadrightarrow [s]$ in [[simplex category|$\Delta$]] (by [this Prop.](Dold-Kan+correspondence#ExplicitFormOfGamma)):

$$
  X_r
  \;\simeq\;
  \underset
    {[r] \twoheadrightarrow [s]}
    {\bigoplus}
  V_s
  \;\simeq\;
  \underset{
    (D X)_r
  }{
  \underbrace{
  \left(
  \underset
    {
      {[r] \overset{p}{\twoheadrightarrow} [s]}
      \atop
      p \neq id_{[r]}
    }
    {\bigoplus}
  V_s
  \right)
  }
  }
  \,\oplus\,
  \underset{
    (N X)_r
  }{
  \underbrace{
    V_r
  }
  }
  \,.
$$
(On the right we have split off the summand corresponding to the [[identity morphism]], just for emphasis, since this is what matters in a moment).

Moreover (still by [this Prop.](Dold-Kan+correspondence#ExplicitFormOfGamma)), under this identification the [[degeneracy maps]] $X_{[r'] \overset{p}{\twoheadrightarrow} [r]}$ of $X_\bullet$ are simply given on these [[direct sum|direct summands]] by precomposition of their labels with $p$:
$$
  \array{
    X_{r}
    &
    \overset{X_{[r'] \overset{p}{\twoheadrightarrow} [r]} }{\longrightarrow}
    &
    X_{r'}
    \\
    \big(
      [r] \overset{q}{\twoheadrightarrow} [s]
      ,\,
      v \in V_s
    \big)
    &\mapsto&
    \big(
      [r'] \overset{q \circ p}{\twoheadrightarrow} [s]
      ,\,
      v \in V_s
    \big)
  }
$$

We claim (argument below) that this implies that the colimit defining $L_r X$ consists of [[equivalence classes]] of triples of the form
$
\Big(
    [r]
      \overset{p}{\twoheadrightarrow}
    [r']
      \overset{q}{\twoheadrightarrow}
    [s]
   ,\,
  v \in V_s
  \Big)
$
for non-identity $p$,
under the [[equivalence relation]] which identifies $(p,q,v)$ with $(p'q',v')$ iff $p \circ q \,=\, p' \circ q'$ and $v = v'$. This will mean that the equivalence classes are simply labeled by pairs $\big([r] {\twoheadrightarrow} [s] ,\,  v \in V_s\big)$ where the surjection is non-identity, hence that the colimit is
$$
  L_r X
  \;\coloneqq\;
  \underset{
    \underset
      {
        { p \colon [r] \twoheadrightarrow [s] }
        \atop
        { p \neq id }
      }
      {\longrightarrow}
  }{\lim}
  X_s
  \;\simeq\;
  \underset{
      {
        { p \colon [r] \twoheadrightarrow [s] }
        \atop
        { p \neq id }
      }
  }{\bigoplus}
  V_s
$$
and that the comparison map $L_r X \to X_r$ (eq:ComparisonMapsFromLatchingToMatchingObject) is the canonical inclusion of all the non-identity [[direct summands]]:
$$
  L_r X
  \xhookrightarrow{\phantom{--}}
  L_r X \,\oplus\, V_r
  \;\simeq\;
  X_\bullet
  \,,
$$
which proves the proposition.

Hence it remains to show the claim about the colimit. Since the degeneracy maps on the triples $\big( [r] \overset{p}{\twoheadrightarrow} [r'] \overset{q}{\twoheadrightarrow} [s], \, v \in V_2 \big)$ all act trivially on the $v$-component, we may regard this as a colimit in [[Set]] over the $(p,q)$-pairs. This may be computed as the set of all such pairs quotiented by the [[equivalence relation]] which is generated from the [[relation]] imposed by the morphisms in the diagram, which, unwinding the definitions, is:
$$
  (p,q) \simeq (p',q')
  \;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;
  \exists
  \;\;\;
  \array{
   && [r]
   \\
   &
   \mathllap{^p}\swarrow
   &&
   \searrow\mathrlap{^{p'}}
   \\
   [s]
   &&
   \longrightarrow
   &&
   [s']
   \\
   &
   \mathllap{_q}
   \searrow
   &&
   \swarrow\mathrlap{_{q'}}
   \\
   &&
   [t]
   \mathrlap{\,.}
  }
$$
This already shows the claim for the case that the horizontal [[lift]] seen in the diagram on the right exist.

In general, given a [[commuting diagram]] in the [[simplex category|$\Delta$]] of the form

\begin{tikzcd}[sep=20pt]
    &
    {[r]}
    \ar[dl,->>, "{p}"{swap}]
    \ar[dr,->>, "{p'}"]
    \\
    {[s]}
    \ar[dr,->>, "{q}"{swap}]
    &
    &
    {[s']}
    \ar[dl,->>, "{q'}"]
    \\
    &
    {[t]}
\end{tikzcd}

(without necessarily a horizontal lift), then we may equivalently factor it as a [[zig-zag]] of diagrams

\begin{tikzcd}[sep=20pt]
    &
    {[r]}
    \ar[dl,->>, "{p}"{swap}]
    \ar[dr,->>, "{p'}"]
    \ar[d, ->>]
    \\
    {[s]}
    \ar[dr,->>, "{q}"{swap}]
    &
    {[t]}
    \ar[d, equals]
    \ar[from=r, ->>]
    \ar[from=l, ->>]
    &
    {[s']}
    \ar[dl,->>, "{q'}"]
    \\
    &
    {[t]}
\end{tikzcd}

where the left half of this zig-zag is

\begin{tikzcd}[sep=20pt]
    &
    {[r]}
    \ar[dl,->>, "{ p  }"{swap}]
    \ar[dr,->>, "{ q \,\circ\, p }"]
    \\
    {[s]}
    \ar[dr,->>, "{q}"{swap}]
    \ar[rr, ->>, "{q}"{description}]
    &
    &
    {[t]}
    \ar[dl,->>, "{\mathrm{id}}"]
    \\
    &
    {[t]}
\end{tikzcd}

and exhibits that, in the colimit, $(p,q,v)$ is identified with $(q\circ p, id, v) \,=\, (q' \circ p', id, v)$; while, analogously, the right half exhibits that this in turn is identified with $(p',q',v)$.
\end{proof}

\begin{corollary}
\label{SimplicialReedyCofibsInAbelianCatsUnderDoldKan}
  Let $\mathcal{A}$ be a [[model category]] whose [[underlying]] category is as above, then a morphism $f_\bullet \,\colon\, X_\bullet \to Y_\bullet$ in $s\mathcal{A}$ is a Reedy cofibration (Def. \ref{ReedyModelStructure}) if and only if the [[Dold-Kan correspondence|Dold-Kan-corresponding]] [[chain map]] $N f_\bullet$ of [[normalized chain complexes]] is degreewise a cofibration in $\mathcal{A}$.
\end{corollary}
\begin{proof}
  By Prop. \ref{LatchingInAbelianCategoryIsDegeneracySubobject} the maps (eq:CharacterizingReedyCofibrations) are of this form
$$
  \array{
    L_r Y \overset{L_r X}{\amalg} X_r
    &\longrightarrow&
    Y_r
    \\
    (N X)_r \oplus (D Y)_r
    &\overset{(N f)_r \oplus id}{\longrightarrow}&
    (N Y)_r \oplus (D Y)_r
  }
$$
But $(N f)_r \oplus id$ is clearly a cofibration iff $(N f)_r$ is.
\end{proof}
\begin{remark}
  One direction of the implication in Cor. \ref{SimplicialReedyCofibsInAbelianCatsUnderDoldKan} holds generally without any assumptions on $\mathcal{A}$ or $\mathcal{R}$:

If a [[morphisms]] $f_\bullet \colon X_\bullet \to Y_\bullet$ of functors $\mathcal{R} \to \mathcal{C}$ is Reedy cofibrant (Def. \ref{ReedyModelStructure}) then for each $r \in \mathcal{R}$ the induced morphism
$$
  X_r/L_r X
   \overset
     {\;\; f_r/L_r f \;\;}
     {\longrightarrow}
  Y_r/L_r Y
$$
is a cofibration in $\mathcal{C}$.
\end{remark}
\begin{proof}
Consider the following [[commuting diagram]] in $\mathcal{C}$:
$$
  \array{
    L_r X
      &\longrightarrow&
    X_r
    \\
    \big\downarrow
      &&
    \big\downarrow
      &
    \searrow\mathrlap{{}^{f_r}}
    \\
    L_r Y
      & \longrightarrow &
    L_r Y \overset{L_r X}{\sqcup} X_r
      & \underset{\in Cof}{\longrightarrow} &
    Y_r
    \\
    \big\downarrow
      &&
    \big\downarrow
      &&
    \big\downarrow
    \\
    \ast
      &\longrightarrow&
    X_r / L_r X
      &\underset{f_r / L_r f}{\longrightarrow}&
    Y_r/L_r Y
  }
$$
Here the top square, the left rectangle and the bottom rectagle are [[pushouts]] by definition of the latching object (Def. \ref{LatchingAndMatchingObjects}) and by definition of the quotients $X_r/L_r X \,\coloneqq\, cofib(L_r X \to X_r)$.
Therefore the [[pasting law]] implies first that the left bottom square and then that the right bottom square is a pushout.
This exhibits $f_r/L_r f$ as the pushout of a cofibration, and hence as a cofibration itself.
\end{proof}

\linebreak

#### With values in an arbitrary model category

Let $C$ be a [[model category]].

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

For $X \in [\Delta^{op}, C]$ a Reedy cofibrant object, the [[Bousfield-Kan map]]

$$
  \int^{[n]} \mathbf{\Delta}[n] \cdot X_n
  \to
  \int^{[n]} \Delta[n] \cdot X_n
$$

is a weak equivalence in $C$.

=--

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

The [[coend]] over the [[copower|tensor]] is a left [[Quillen bifunctor]]

$$
  \int (-)\cdot (-) : [\Delta,sSet_{Quillen}]_{Reedy} \times
    [\Delta^{op}, C]_{Reedy}
$$

(as discussed there). Therefore with its second argument fixed and cofibrant it is a [[Quillen adjunction|left Quillen functor]] in the remaining argument. As such, it preserves weak equivalences between cofibrant objects (by the [[factorization lemma]]). By the [above discussion](#OverDeltaWithValuesInSimplicialSets), both $\mathbf{\Delta}[n]$ and $\Delta[-]$ are indeed cofibrant in $[\Delta,sSet_{Quillen}]_{Reedy}$. Clearly the functor $\mathbf{\Delta}[-] \to \Delta[-]$ is objectwise a weak equivalence in $sSet_{Quillen}$, hence is a weak equivalence.

=--

#### Enrichment
 {#EnrichmentOverTheSimplexCategory}

The following proposition should be read as a **warning** that
an obvious idea about simplicial enrichment of Reedy model
structures over the simplex category does _not_ work.

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

For $C$ a [[model category]], the [[category of simplicial objects]] $[\Delta^{op}, C]$ in $C$ is canonically an [[sSet]]-[[enriched category]].

However, this does **not** in general harmonize with the Reedy model structure to make $[\Delta^{op}, C]_{Reedy}$ a [[simplicial model category]].

More precisely the following parts of the [[pushout-product axiom]] for the $sSet$-[[tensoring]] hold. Let $f : A \to B$ be a cofibration in $[\Delta^{op}, C]_{Reedy}$ and $s : S \to T$ be a cofibration in $sSet_{Quillen}$.

1. the [[pushout-product]] $f \bar \otimes g$ is a cofibration in $[\Delta^{op}, C]_{Reedy}$ ;

1. and it is an acyclic cofibration if $f$ is;

1. it is **not necessarily** acyclic if $s$ is.

=--

That the first two items do hold is discussed for instance as [Dugger 2001, prop. 4.4](#Dugger01). A counterexample for the third item is in [Dugger 2001, remark 4.6](#Dugger01).

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

However, there are [[Bousfield localization of model categories|left Bousfield localizations]] of $[\Delta^{op}, sSet]_{Reedy}$ for which

1. the above $sSet$-enrichment does constitute an $sSet$-[[enriched model category]];

1. the result model structure is Quillen equivalent to $C$ itself.

This is in fact a useful technique for replacing $C$ by a Quillen equivalent
and $sSet$-enriched model structure. More discussion of this point is
at _[[simplicial model category]]_ in the section
_[Simplicial Quillen equivalent models](simplicial+model+category#SimpEquivMods)_.

=--

## Related concepts

* [[generalized Reedy model structure]]

* [[Joyal-Tierney calculus]]

## References

The original text is

* [[Chris Reedy]], _Homotopy Theory of Model Categories_  ([retyped pdf](http://www-math.mit.edu/~psh/reedy.pdf))

Textbook accounts:

* {#Hovey99} [[Mark Hovey]], Section 5.2 of: *[[Model Categories]]*, Mathematical Surveys and Monographs, **63** AMS (1999) [[ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover)]

* {#Hirschhorn02} [[Philip Hirschhorn]], Chapter 15 of: _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs Vol 99 (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

* {#Lurie} [[Jacob Lurie]], Section A.2.9 in _[[Higher Topos Theory]]_  (2009)

* [[Emily Riehl]], Chapter 14 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

An overview stressing the role of [[weighted colimit|weighted colimits]] is in

* [[Emily Riehl]], [[Dominic Verity]], *The theory and practice of Reedy categories*, Theory and Applications of Categories **29** 9 (2014) 256-301 [[tac:29-09](http://www.tac.mta.ca/tac/volumes/29/9/29-09abs.html),[arXiv:1304.6871](https://arxiv.org/abs/1304.6871)]

Discussion of functoriality of Reedy model structures:

* {#Barwick} [[Clark Barwick]], _On Reedy Model Categories_ ([arXiv:0708.2832](http://arxiv.org/abs/0708.2832))

On [[enriched model category|enriched]]

* {#Angeltveit} [[Vigleik Angeltveit]], *Enriched Reedy categories*, Proceedings of the American Mathematical Society **136** 7 (2008)  [[arXiv:math/0612137](http://arxiv.org/abs/math/0612137), [AMS:S0002-9939-08-09185-5](https://www.ams.org/journals/proc/2008-136-07/S0002-9939-08-09185-5/home.html), [jstor:20535419](https://www.jstor.org/stable/20535419)]

and [[monoidal model category|monoidal]] Reedy model structures:

* Moncef Ghazel, Fethi Kadhi, _Reedy diagrams in V-model categories_, Applied Categorical Structures **27** (2019) 549–566 [[arXiv:1610.06535](https://arxiv.org/abs/1610.06535), [doi:10.1007/s10485-019-09566-w](https://doi.org/10.1007/s10485-019-09566-w)]

Discussion of fibrant generation of Reedy model structure:

* {#BHKKRS15} [[Marzieh Bayeh]], [[Kathryn Hess]], [[Varvara Karpova]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], Thm. 5.9 in:  *Left-induced model structures and diagram categories*, in: *Women in Topology: Collaborations in Homotopy Theory*, Contemporary Mathematics **641** American Mathematical Society (2015) [[arXiv:1401.3651](http://arxiv.org/abs/1401.3651), [ISBN:978-1-4704-2495-4](https://bookstore.ams.org/conm-641)]

The main statement is theorem 4.7 there.

The Reedy model structure on towers is discussed for instance in chapter 6 of

* {#GoerssJardine} [[Paul Goerss]], [[Rick Jardine]], _[[Simplicial homotopy theory]]_ ([dvi](http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html) [PDF](http://dodo.pdmi.ras.ru/~topology/books/goerss-jardine.pdf))

The Reedy model structure on categories of simplicial objects is discussed in more detail for instance in

* {#Dugger01} [[Dan Dugger]], _Replacing model categories with simplicial ones_, Trans. Amer. Math. Soc. **353** 12 (2001) 5003-5027 [[ams:S0002-9947-01-02661-7](https://www.ams.org/journals/tran/2001-353-12/S0002-9947-01-02661-7), [pdf](https://www.ams.org/journals/tran/2001-353-12/S0002-9947-01-02661-7/S0002-9947-01-02661-7.pdf)]

[[!redirects Reedy model category]]
[[!redirects Reedy model categories]]

[[!redirects latching object]]
[[!redirects latching objects]]
[[!redirects matching object]]
[[!redirects matching objects]]
[[!redirects latching map]]
[[!redirects latching maps]]
[[!redirects matching map]]
[[!redirects matching maps]]

[[!redirects Reedy model structures]]


Reeh-Schlieder theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The *Reeh-Schlieder theorem* is a [[theorem]] about [[local nets]] in the [[Haag-Kastler approach]] to [[quantum field theory]]. It states that for certain nets the set of vectors $\mathcal{M}(\mathcal{O}) \Omega$ (local algebra of a bounded open set applied to the vacuum vector, see [[Haag-Kastler vacuum representation]] for definitions and details) is dense in the given [[state]] space, a [[Hilbert space]] $\mathcal{H}$.

The Reeh-Schlieder theorem can be proven to be valid in the [[Haag-Kastler vacuum representation]], but the statement itself is sometimes used as an axiom and called the **Reeh-Schlieder property** in this case.

The Reeh-Schlieder theorem is of central importance for the mathematical structure theory of the [[Haag-Kastler approach]]. The _physical interpretation_ is counterintuitive and therefore to a certain degree controversial: Intuitively one might expect that for a localized observable $A \in \mathcal{M}(\mathcal{O})$ the vector $A \Omega$ should be localized in $\mathcal{O}$, that is the state should look like the vacuum in the causal complement of $\mathcal{O}$. But the Reeh-Schlieder theorem says that every, arbitrary state can be approximated by states of the kind $A \Omega$. This shows that the concept of _localized states_ is nontrivial in [[AQFT]] and needs to be handled with care.

Complementary statements about the asymptotic "vacuum-like appearance" of localized observables exist, too, and are commonly called [[cluster theorem]]s.

In other parts of the physics literature, notably in the context of [[CFT]], the Reeh-Schlieder property is also called the **state-operator correspondence**. See also the remark in ([Schroer, footnote 14, page 34](#Schroer)).

## Abstract ##

We state the theorem for a vacuum representation on [[Minkowski spacetime]]. Since the theorem has some controverse consequences for the physical interpretation of the theory, some of which will be mentioned in the corresponding paragraph, we take a quick look at which axioms really enter into the proof of the theorem. The references include work on the generalization of the theorem to more general [[spacetimes]] as well as work concerned with dodging the theorem and its consequences by choosing an alternative set of axioms.

## Definition ##
All necessary definitions can be found at [[Haag-Kastler vacuum representation]].

## the Theorem ##
Both the statement and the proof we mention here refer to the [[Haag-Kastler vacuum representation]], but both the statement and the proof can be generalized resp. translated to different contexts, like e.g. to a [[Wightman theory]], to more general spacetimes and to other states than the vacuum vector.

+-- {: .un_theorem}
###### Reeh-Schlieder
In the [[Haag-Kastler vacuum representation]] the vacuum vector $\Omega$ is [[cyclic vector|cyclic]] and [[separating vector|separating]] for every local algebra $\mathcal{M}(\mathcal{O})$.

=--

+-- {: .proof}
###### Proof
This proof does not strive for maximal generality, for example we specialize to $d = 4$ dimensions. The generalization to other dimensions can be done by the reader. We will comment on which part of the axioms of the vacuum representation are actually used below.

First some preliminary observations:

The covariance axiom says that we have a strongly continuous representation $U$ of the Poincare group, the SNAG-Theorem (see [[spectral measure]]) therefore provides us with a spectral measure for the translation subgroup $\mathcal{T}$, that we identify with $\mathbb{R}^4$

$$
\mathcal{U}(x) = \int_{k\in \mathbb{R}^4} e^{i \langle x, k\rangle} \mathcal{P}(k) \qquad \forall x \in \mathcal{T} \cong \mathbb{R}^4
$$

We may restrict the domain of integration to the closure of the forward lightcone $cloV_+$ by the spectrum condition. The spectral measure can be used to define operators for $z = x + i y \in \mathcal{R}^4 + i V_+$ via

$$
U(z) := \int_{cloV_+} e^{i \langle z, k\rangle} \mathcal{P}(k)
$$

The dominated convergence theorem for spectral integrals (see [[Banach space]]) tells us that the strong limit of the left side of the following equation exists and is equal to the right side:
$$
\lim_{y \downarrow 0} U(x + i y) = U (x)
$$

Next we choose a vector $u \in \mathcal{H}$ from our Hilbert space, a bounded operator $A$ on $\mathcal{H}$ and form the complex-valued function
$$
f_{u, A}(z) := \langle u, U(z) A \Omega \rangle
$$

which is holomorphic on $\mathcal{R}^4 + i V_+$ by definition and continuous on $\mathcal{R}^4 + i (V_+ \cup \{0 \}) $ by our last observation.

We can define a second such function $g$ on $\mathcal{R}^4 - i (V_+ \cup \{0 \}) $ by complex conjugation:
$$
g_{u, A}(z) = \overline{f_{u, A}(\overline z)}
$$

If there is a $U \subset \mathbb{R}^4$ open such that $f_{u, A}$ is real valued on $U$, then $f_{u, A}$ and $g_{u, A}$ coincide on $U$ and we can invoke a suitable version of the [[edge-of-the-wedge theorem]] as stated on [[analytic geometry]] to conclude that $f_{u, A}$ and $g_{u, A}$ are the branches of a unique holomorphic function, that is holomorphic at least on a complex neighborhood of $U$.

Now the proof that $\Omega$ is [[cyclic vector|cyclic]] for $\mathcal{M}(\mathcal{O})$:

Choose a bounded open $\mathcal{O}_0$ and a neighborhood of zero $V \subset \mathcal{R}^4$ such that $\mathcal{O}_0 + V \subset \mathcal{O}$.

Suppose that $\Omega$ is not cyclic for $\mathcal{M}(\mathcal{O})$, then $\mathcal{M}(\mathcal{O}) \Omega$ is not dense in $\mathcal{H}$ and we can choose a vector $v \in \mathcal{H}, v \neq 0$ such that

$$
 \langle v, U(z) A \Omega \rangle = 0
$$
for all $A \in \mathcal{M}(\mathcal{O})$. In particular we have for all $A_0 \in \mathcal{M}(\mathcal{O}_0)$ and $x \in V$
$$
f_{v, A_0} (x) = \langle v, U(x) A_0 \Omega \rangle = \langle v, U(x) A_0 U(x)^{-1} \Omega \rangle = 0
$$

Now we see from our previous considerations that there is a function holomorphic in a complex neighborhood of $V$ that restricts to $f_{v, A_0}$ on $V$, vanishes on $V$ and has therefore to vanish everywhere. That means the previous equality holds for arbitrary translations $U(x), x \in \mathbb{R}^4$.

Recall that weak additivity holds in the vacuum representation. This together with the previous statement implies that  $\langle v, R \Omega \rangle = 0$ for all $R \in \mathcal{R}$, the global algebra. But since $\mathcal{R} \Omega$ is dense in $\mathcal{H}$ by assumption (see the axiom about the existence of a vacuum vector), we get that $v$ must be zero, contradiction: $\Omega$ has to be cyclic for $\mathcal{M}(\mathcal{O})$.

Now the proof that $\Omega$ is [[separating vector|separating]] for $\mathcal{M}(\mathcal{O})$:

Choose a bounded open set $\mathcal{O}_2$ such that $\mathcal{O} \perp \mathcal{O}_2$, then by locality we have $\mathcal{M}(\mathcal{O}) \subseteq  (\mathcal{M}(\mathcal{O}_2))'$. We know already that $\Omega$ is cyclic for $\mathcal{M}(\mathcal{O}_2)$, therefore it is separating for  $\mathcal{M}(\mathcal{O})$.

=--

## Preconditions Needed for the Proof ##
Since the theorem and its consequences are counterintuitive for the physical interpretation of the theory, it is worthwhile to take a look which axioms enter the proof: Is the theorem a true feature of [[quantum field theory]] or is it an artifact caused by ill chosen assumptions?

The proof that the vacuum vector is cyclic given in the previous paragraph makes use of:

1. Strong representation of the translation subgroup and the spectrum condition.

Note that we do not need the representation of the whole Poincare group, but really only that of the translations.

2. isotony,

3. weak additivity and

4. of course the existence of the vacuum vector itself and that $\mathcal{R} \Omega$ is dense in $\mathcal{H}$.

Now weak additivity can be proven to hold using isotony and additiviy.

It is sometimes argued that the Reeh-Schlieder theorem - that is the part that the vacuum vector is cyclic for every local algebra - is somehow a contradiction to _locality_ . The remarkable fact in this context is that  **the locality axiom does not enter the proof** that the vacuum vector is cyclic. (It was used for the proof that the vacuum vector is also separating, though).

For some further elaboration of this point see the paper by Halvorson cited in the references.

## Consequences ##
The fact that the vacuum vector is separating implies that there cannot be a number operator $N$ associated to any bounded open set $\mathcal{O}$, because $N$ would annihilate the vacuum vector. This implies that the notion of _particles_ in relativistic quantum physics cannot be quite as simple as in classical physics or in nonrelativistic quantum physics.

More generally there cannot be a nonzero localized observable that annihilates the vacuum. This implies that the stress-energy tensor, if localized to a bounded region, cannot be a positive operator, since its expectation value in the vacuum is zero, in fact it cannot be bounded from below. This is sometimes mentioned as a pathology occuring in general spacetimes that do not have a global timelike [[Killing vector]] field, but in our context it actually holds in [[Minkowski spacetime]] in the vacuum state.

The fact that the vacuum vector is cyclic means that any arbitrary state in the vacuum representation can be approximated by measurements in an arbitrary small bounded open region applied to the vacuum vector. This fact is sometimes referred to as the existence of [[vacuum fluctuations]]. A direct consequence of the Reeh-Schlieder theorem is therefore that to any regions $\mathcal{O}_1, \mathcal{O}_2$, no matter how far apart, there are many projections in the corresponding local algebras that are positively correlated in the vacuum state.

## Related theorems

* [[spin-statistics theorem]]

* [[Osterwalder-Schrader theorem]]

* [[state-field correspondence]]

## References ##

* Stephen J. Summers: _Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State_ ([arXiv:0802.1854] (http://de.arxiv.org/abs/0802.1854))

* [[Edward Witten]], section 2 of _Notes on Some Entanglement Properties of Quantum Field Theory_ ([arXiv:1803.04993](https://arxiv.org/abs/1803.04993))

See also

* Wikipedia on the [Reeh-Schlieder Theorem](http://en.wikipedia.org/wiki/Reeh%E2%80%93Schlieder_theorem)

The Reeh-Schlieder theorem can be generalized from [[Minkowski spacetime]] to more general [[spacetimes]], see for example:

* Alexander Strohmaier, Rainer Verch, Manfred Wollenberg: _Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems_ ([arXiv](http://xxx.uni-augsburg.de/abs/math-ph/0202003))

About work to avoid the Reeh-Schlieder theorem and its counter intuitive implications see this:

* [[Hans Halvorson]], _Reeh-Schlieder Defeats Newton-Wigner: On alternative localization schemes in relativistic quantum field theory_ ([arXiv](http://xxx.uni-augsburg.de/abs/quant-ph/0007060))

The observation that the Reeh-Schlieder property describes what elsewhere is called the _operator-state correspondence_ is made explicit for instance in footnote 14 on page 34 of

* [[Bert Schroer]], _Particle Physics and QFT at the Turn of the Century:
Old principles with new concepts ([pdf](http://cds.cern.ch/record/367824/files/9810080.pdf))
{#Schroer}

[[!redirects Reeh-Schlieder Theorem]]
[[!redirects Reeh-Schlieder property]]
[[!redirects Reeh-Schlieder Property]]

[[!redirects operator-state correspondence]]
[[!redirects state-operator correspondence]]


Rees matrix semigroup

Let $G$ be a group and $0$, a separate symbol. Then the set $G\cup 0$
is a semigroup with $0$ with respect to the operation extending the multiplication in $G$ by the rule $0 g = g 0 = 0 0 = 0$ for all $g\in G$. The same is actually true even if $G$ is a semigroup.

In his 1940 article, Rees introduced matrix semigroups starting with
a semigroup with $0$ two sets $I$ and $\Lambda$ and a matrix $p$ with labels in $\Lambda\times I$ and values $p_{\lambda i}$ in $G \cup 0$. A Rees matrix is a rectangular matrix with values in $G$ which has at most only one nonzero entry.
The elements of the Rees matrix semigroup $M(G, I,\Lambda,p)$ are all possible Rees $I\times \Lambda$ matrices with values in $G$ with multiplication $a \circ b = a p b$ (with usual multiplication of matrices, where obvious auxiliary addition is taken, where never more than one element is nonzero).

In the same article he characterized a class of 0-simple regular semigroups and made steps useful for classification of some types of simple semigroups.

Important role in the theory of Rees matrix semigroups is played by primitive idempotents. A theorem of Rees in that context is an analogue of the second Wedderburn theorem in the theory of simple associative algebras where [[primitive idempotent]]s play a major role.

Brandt groupoids (whose data are equiavlent to the connected groupoids in the modern sense) are very simple subcase of Rees matrix semigroups. In that case $p$ is the Kronecker matrix.

* the current wikipedia article has errors (for example no condition on zeros in $p$ is made) [Rees matrix semigroup](https://en.wikipedia.org/wiki/Rees_matrix_semigroup)
* A. H. Clifford, G. B. Preston, _The algebraic theory of semigroups_, vol. 1
* [[David Rees]], On semi-groups 3, Proc. Cambridge. Math. Soc. 387–400 (1940)
* John M. Howie, _Fundamentals of semigroup theory_, Clarendon Press 1995

category: algebra
[[!redirects Rees matrix semigroups]]


Reese Harvey

* [webpage](http://math.rice.edu/People/Homepages/harvey.html)

## selected writings

* {#HarveyLawson82} [[Reese Harvey]], [[H. Blaine Lawson]], _Calibrated geometries_, Acta Math. Volume 148 (1982), 47-157 ([Euclid:1485890157](https://projecteuclid.org/euclid.acta/1485890157))

## related $n$Lab entries

* [[calibrated geometry]]

* [[Spin(4).Spin(3)]]

category: people

[[!redirects F. Reese Harvey]]


REF MakkaiReyes77

[[!redirects REF First Order Categorical Logic]]

* {#MakkaiReyes77} [[Michael Makkai]] and [[Gonzalo Reyes]], *First Order Categorical Logic:  Model-theoretical methods in the theory of topoi and related categories*, Lecture Notes in Math. __611__,  Springer-Verlag, 1977. re-written by Francisco Marmolejo in 2018 ([web announcement](https://reyes-reyes.com/2018/04/02/first-order-categorical-logic-2/), [pdf](https://marieetgonzalo.files.wordpress.com/2018/04/makkai-reyes-book.pdf))


refinement

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

#Contents#
* table of contents
{:toc}

## Idea

In general, let $U = (U_i)_{i: I}$ and $V = (V_j)_{j: J}$ be two families of [[objects]] of some [[category]] $C$.  We say that $V$ is a __refinement__ of $U$ if there is a [[function]] $f: J \to I$ of indices and a [[morphism]] $V_j \to U_{f(j)}$ for each $j \in J$.

A common special case is the concept of refinement of [[open covers]], example \ref{RefinementOfOpenCovers} below.

## Examples ##

We state a list of examples, beginning with general cases and then consecutively making them more specific.

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

Very often we do this in the [[slice category]] $C/X$ for some object $X$.  If you spell this out, then you have families $(U_i \to X)_i$ and $(V_j \to X)_j$ of morphisms to $X$; $V$ is a refinement of $U$ if there are a function $f: J \to I$ and a commutative diagram
\[ \label{slice}
  \array{
    V_j &&\to&& U_{f(j)}
    \\
    & \searrow && \swarrow
    \\
    && X
  }
\]
for each $j$.

=--

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

More specifically, apply this to the poset of [[subobjects]] of $X$.  Then you have families $(U_i \hookrightarrow X)_i$ and $(V_j \hookrightarrow X)_j$ of subobjects of $X$; $V$ is a refinement of $U$ if there are a function $f: J \to I$ and a commutative diagram (eq:slice) for each $j$.

=--

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

Yet more specifically, apply this to the lattice of [[subsets]] of some set $X$.  Then you have families $(U_i \subseteq X)_i$ and $(V_j \subseteq X)_j$ of subsets of $X$; $V$ is a refinement of $U$ if there is a function $f: J \to I$ such that each $V_j$ is contained in $U_{f(j)}$.

Yet more specifically, let the families of subsets be indexed by themselves.  Then have collections $U \subseteq \mathcal{P}X$ and $V \subseteq \mathcal{P}X$ of subsets of $X$; $V$ is a refinement of $U$ if for each $j$ there is an $i$ such that $V_j$ is contained in $U_i$.

=--

+-- {: .query}
Actually, this definition is slightly weaker than the previous one in the absence of the [[axiom of choice]].  Perhaps in that case the general definition should say that for each $j$ there is an $i$ and a morphism $V_j \to U_i$.
=--

+-- {: .num_example #RefinementOfOpenCovers}
###### Example
**(refinement of [[open covers]])

Special cases of example \ref{RefinementOfSystemsOfSubsets} include refinement of [[filters]] and refinement of [[open covers]] of [[topological spaces]].

Let $(X,\tau)$ be a [[topological space]], and let $\{U_i \subset X\}_{i \in I}$ be a set of [[open subsets]] which [[cover|covers]] $X$ in that $\underset{i \in I}{\cup} U_i \;= \;X$.

Then a _refinement_ of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an [[open cover]] in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.

=--

On the other hand, you might want to generalise the case of open covers to [[covers]] or covering [[sieves]] on a [[site]].  In that case, the general definition still applies; you have covering families $(U_i \to X)_i$ and $(V_j \to X)_j$ of some object $X$; $V$ is a refinement of $U$ if there are a map $f: J \to I$ and a commutative diagram (eq:slice) for each $j$.

## Examples

Refinement of open covers is a concept appearing in the definition of

* [[compact topological space]], [[paracompact topological space]], etc.

## Related concepts

* [[Čech homology]]

[[!redirects refinements]]

[[!redirects refinement of a cover]]
[[!redirects refinements of a cover]]

[[!redirects refinement of covers]]
[[!redirects refinements of covers]]

[[!redirects refinement of an open cover]]
[[!redirects refinements of an open cover]]

[[!redirects refinement of open covers]]
[[!redirects refinements of open covers]]


refl

In [[homotopy type theory]], the [[introduction rule]] is a [[dependent function]]
$$\operatorname{refl} : \prod_{a : A} \big( a =_A a \big)$$
called **reflexivity**, which says that every element of $A$ is equal to itself (in a specified way). We regard $\operatorname{refl}_a$ as being the constant path at the point $a$.

In particular, this means that if $a$ and $b$ are [[judgmentally equal]], $a \equiv b$, then we also have an element $\operatorname{refl}_a : a =_A b$. This is well-typed because $a \equiv b$ means that also the type $a =_A b$ is judgmentally equal to $a =_A a$, which is the type of $\operatorname{refl}_a$.

* From page 63 of the book, Homotopy Type Theory.


reflected limit

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

# Reflection of limits
* table of contents
{: toc}

## Idea

A [[functor]] is said to _reflect_ [[limits]] ([[colimits]]) of a given shape if a [[cone]] ([[cocone]]) is (co-)limiting whenever its image under $F$ is.

## Definition

+-- {: .num_prop #ReflectedLimit}
###### Definition

Let $F\colon C\to D$ be a [[functor]] and $J\colon I\to C$ a [[diagram]].  We say that $F$ **reflects** [[limits]] of $J$ if whenever we have a [[cone]] $\eta\colon const^I_x \to J$ over $J$ in $C$ such that $F(\eta)$ is a [[limit]] of $F\circ J$ in $D$, then $\eta$ was already a limit of $J$ in $C$.

=--

Of course, a functor $F$ reflects a colimit if $F^{op}$ reflects the corresponding limit.

If $F$ reflects all limits or colimits of a given type (i.e. over a given category $I$), we simply say that $F$ reflects that sort of limit (e.g. $F$ reflects products, $F$ reflects equalizers, etc.).

A functor which both reflects *and* preserves limits, and such that limits exist in its domain whenever they do in its codomain, is said to [[created limit|create]] them.

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

Reflection of limits is distinct from [[preservation of limits]], although there are relationships, e.g prop. \ref{ConservativeFunctors}.

=--

## Examples
 {#Examples}

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

A [[faithful functor]] reflects [[epimorphisms]] and [[monomorphisms]].

=--

(The simple proof is spelled out [here](epimorphism#EpimorphismsAreReflectedByFaithfulFunctors) at _[[epimorphism]]_.)

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

A [[fully faithful functor]] (hence a [[full subcategory]] inclusion) reflects all limits and colimits.

=--

This is evident from inspection of the defining [[universal property]].

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

A [[conservative functor]] reflects any limits which exist in its domain and that it [[preserved limit|preserves]].

=--

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

If $J$ in def. \ref{ReflectedLimit} has some limit $\theta$ which is preserved by $F$, then there is a unique induced map $\eta\to\theta$ by the universal property of a limit, which becomes an isomorphism in $D$ since $F(\eta)$ and $F(\theta)$ are both limits of $F\circ J$; hence if $F$ is conservative then it must already have been an isomorphism in $C$, and so $\eta$ was already also a limit of $J$.

=--

## Related pages

* [[preserved limit]]

* **reflected limit**

* [[created limit]]

* [[lifted limit]]

## References

* [[Emily Riehl]], §3.3 in: *[[Category Theory in Context]]*, Dover Publications (2017) [[pdf](http://www.math.jhu.edu/~eriehl/context.pdf), [book website](http://www.math.jhu.edu/~eriehl/context/)]

[[!redirects reflection of limits]]
[[!redirects reflected limits]]
[[!redirects reflected colimit]]
[[!redirects reflected colimits]]
[[!redirects reflection of colimits]]

[[!redirects reflects]]
[[!redirects reflected]]


reflection

## Disambiguation

There are many different notions of "reflection" in mathematics.

* For the notion of reflection across a hyperplane in [[geometry]], see *[[reflection at a hyperplane]]*.

* For the notion of reflection through a point in [[geometry]], see *[[point inversion]]*.

* For the notion of reflection of [[waves]] at an interface between two different mediums, see *[[reflection of waves]]*.

* For the notion of reflection for a [[function]] $f$ in [[number theory]], [[real analysis]], [[functional analysis]], see *[[reflection relation]]*.

* For the notion of reflection in [[category theory]], see *[[reflective subcategory]]* and *[[idempotent monad]]*.

* For the notion of reflection in [[modal logic]] and [[modal type theory]], see *[[modal operator]]*, *[[reflective subuniverse]]*.

* For the notion of equality reflection in [[dependent type theory]], see *[[extensional type theory]]*.

* For the notion of reflection in [[set theory]], see *[[reflection principle]]*.

* For the notion of reflection in [[computer science]], see *[[reflective programming]]*.

category: disambiguation

[[!redirects reflections]]


reflection at a hyperplane

> This entry is about the concept in [[geometry]]/[[physics]]/[[linear algebra]]. For other notions of reflection in mathematics, see [[reflection]].

---

#Contents#
* table of contents
{:toc}

## Idea

Given a [[hyperplane]] in a [[Euclidean space]], then _reflection_ at that hyperplane is the operation that sends each point to its mirror image at that hyperplane, the result of translating the point in the [[direction of a vector|direction]] [[perpendicular]] to the hyperplane by twice its [[distance]] to the hyperplane.

### Related concepts

* [[rotation]], [[translation]], [[boost]]

* [[space group]], [[crystallographic group]]


reflection equation algebra

## Reflection Equation

Given a solution $R$ of the [[quantum Yang-Baxter matrix|Yang-Baxter]] equation, the reflection equation (with additive spectral parameter) reads

$$
R(u-v) K_1 (u) R(u+v) K_2 (v) = K_2 (v) R(u+v) K_1 (u) R(u-v)
$$

## References

* Dimitri Gurevich, Pavel Pyatov, Pavel Saponov, _Reflection equation algebra in braided geometry_, Journal of Generalized Lie Theory and Applications __2__ (2008), No. 3, 162–174 [pdf](http://www.ashdin.com/journals/jglta/2008/3/v2_n3_13.pdf)
* P. P. Kulish, _Reflection equation algebras and quantum groups_, in:
Quantum and Non-Commutative Analysis, Mathematical Physics Studies __16__ (1993) 207-220 [doi](http://dx.doi.org/10.1007/978-94-017-2823-2_16)
* A. Mudrov, _Characters of $U_q(gl(n))$-Reflection Equation Algebra_, Lett. Math. Phys. 60:3, 283-291 (2002) [doi](http://dx.doi.org/10.1023/A:1016283527111)
* Christian Schwiebert, _Extended reflection equation algebras, the braid group on a handlebody and associated link polynomials_, J. Math. Physics 02/1994; [doi](http://dx,doi.org/10.1063/1.530751) [hep-th/9402051](http://arxiv.org/abs/hep-th/9402051); _Reflection equation and link polynomials for arbitrary genus solid tori_, [hep-th/9301023](http://arxiv.org/abs/hep-th/9301023)

[[!redirects reflection equation]]
[[!redirects reflection equation algebras]]


reflection positivity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] _reflection positivity_ is the incarnation of unitarity under [[Wick rotation]].

## References

* [[Arthur Jaffe]], _Reflection Positivity Then and Now_ ([arXiv:1802.07880](https://arxiv.org/abs/1802.07880))

Discussion with tools from [[extended TQFT]] and [[stable homotopy theory]] (aimed at [[invertible TQFTs]]):

* {#FreedHopkins21} [[Daniel Freed]], [[Michael Hopkins]], *Reflection positivity and invertible topological phases*, Geometry & Topology **25** (2021) 1165–1330  [[arXiv:1604.06527](https://arxiv.org/abs/1604.06527), [doi:10.2140/gt.2021.25.1165](https://doi.org/10.2140/gt.2021.25.1165)]

In relation to the [[conformal bootstrap]]:

* Leandro Lanosa, Mauricio Leston, Mario Passaglia, *Interplay between reflection positivity and crossing symmetry in the bootstrap approach to CFT* ([arXiv:2112.00232](https://arxiv.org/abs/2112.00232))


reflection principle

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

#Contents#
* table of contents
{:toc}

## Idea

A **reflection principle** is roughly a statement or formula (scheme) that expresses the idea that a certain logical object contains copies of itself. They play an important role in different guises in several fields of [[mathematical logic]] e.g. in [[set theory]], [[proof theory]], [[type theory]] and, recently, [[constructive analysis]].

Gödel's encoding of an unprovability predicate into [[Peano arithmetic]] $PA$ can be viewed as an _'anti-reflection'_ principle for $PA$. Conversely, this suggests that the validity of reflection principles for some theory $T$ can be informally understood as expressing or approximating the (internal) _consistency_ and _completeness_ of $T$.

### Set-theoretic reflection principles

A _reflection principle_ in [[set theory]] states that it is possible to find [[sets]] that resemble the [[class]] of all sets that are constructed in some ways; in other words, any statement true of the universe $V$ already holds at some initial segment $V_\alpha$. This can be interpreted as saying that no first-order property can distinguish the set-theoretic universe from the extensions of all its member sets thereby suggesting the _expansiveness_ or strong infinity of the universe: $V$ gets 'arbitrarily large'.

The first reflection principles go back to [[Richard Montague|R. Montague]] and A. Lévy around 1960. It emerged from their work that reflection principles can supplant the traditional axioms of [[Zermelo-Fraenkel set theory]], e.g. the [[axiom of replacement]] (cf. [Bell-Machover 1977](#BellMach77), p.495). This capacity and the intuition that they express 'completeness' make them interesting candidates in the quest for new axioms of set theory (cf. [Koellner 2009](#Koell09)).

### Universes in type theory

In [[type theory]] this principle is embodied by a [[type of types]] ([Martin-Löf 74, p. 6](#MartinLoef74)).

## Related entries

* [[universe]]

* [[Russell universe]], [[Coquand universe]]

* [[cumulative hierarchy]]

* [[continuous truth]]

## Related page

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

## References

* {#BellMach77} [[John Lane Bell]], M. Machover, _A Course in Mathematical Logic_, North-Holland Amsterdam 1977. (ch. 10,§5) ([ISBN:9780720428445](https://www.elsevier.com/books/a-course-in-mathematical-logic/bell/978-0-7204-2844-5))

* [[Michael Fourman|M. P. Fourman]], _Continuous Truth II: Reflections_ , LNCS **8071** (2013) pp.153-167. ([preprint](http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2013-continuous-truth-II.pdf))

* {#Koell09}P. Koellner, _On Reflection Principles_ , APAL **157** (2009) pp.209-219. ([preprint](http://logic.harvard.edu/koellner/ORP_final.pdf))

* [[Georg Kreisel]], Mathematical Logic , pp. 95-195 in Saaty (ed.) , Lectures on Modern Mathematics III , Wiley New York 1965.

* [[Georg Kreisel]], A. Lévy, _Reflection Principles and their Use for Establishing the Complexity of Axiomatic Systems_, Mathematical Logic Quarterly **14** (7-12) pp.97–142, 1968.

* A. Lévy, _Axiom Schemata of Strong Infinity in Axiomatic Set Theory_ , Pacific J. Math. **10** (1960) pp.223-238. ([pdf](http://projecteuclid.org/download/pdf_1/euclid.pjm/1103038638))

* {#MartinLoef74} [[Per Martin-Löf]], section 1.9, p. 7 of  _An intuitionistic theory of types: predicative part_, In Logic Colloquium (1973), ed. H. E. Rose and J. C. Shepherdson (North-Holland, 1974), pp.73-118. ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926))


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

This entry collects links related to the book

* [[Stefania Centrone]], [[Deborah Kant]], [[Deniz Sarikaya]] (editors):

  \linebreak

  **Reflections on the Foundations of Mathematics**

  \linebreak

  Synthese Library, vol 407.

  Springer, Cham. 12 November 2019.

  [ISBN:978-3-030-15654-1, [doi:10.1007/978-3-030-15655-8](https://doi.org/10.1007/978-3-030-15655-8)]

## Contents

* Part I: Current Challenges for the Set-Theoretic
Foundations

  * [[Mirna Džamonja]], [[Deborah Kant]], *Interview With a Set Theorist* [[pdf](https://hal.archives-ouvertes.fr/hal-01858063/document), [pdf](https://deborahkant.org/wp-content/uploads/2018/07/PhilTalk0716.pdf)]

  * [[Laura Fontanella]], *How to Choose New Axioms for Set Theory?* [[pdf](https://lacl.fr/~lfontanella/papers/paper%20philo%202016.pdf)]

  * [[Claudio Ternullo]], *Maddy On The Multiverse* [[pdf](https://www.claudioternullo.com/_files/ugd/8787da_7474d586e9034716b2eea201b3e00191.pdf)]

  * [[Philip Welch]], *Proving Theorems from Reflection* [[pdf](https://people.maths.bris.ac.uk/~mapdw/Welch%20Proving%20Theorems%20from%20Reflection-revised%2014-08-18.pdf)]

* Part II: What Are Homotopy Type Theory and the Univalent Foundations?

  * [[Thorsten Altenkirch]], *Naïve Type Theory* [[pdf](https://www.cs.nott.ac.uk/~psztxa/publ/fomus19.pdf)]

  * [[Benedikt Ahrens]], [[Paige Randall North]], *Univalent Foundations and the Equivalence Principle* [[arXiv:2202.01892](https://arxiv.org/abs/2202.01892)]

  * [[Ulrik Buchholtz]], *Higher Structures in Homotopy Type Theory* [[arXiv:1807.02177](https://arxiv.org/abs/1807.02177)]

  * [[Anthony Bordg]], *Univalent Foundations and the UniMath Library* [[arXiv:1710.02723](https://arxiv.org/abs/1710.02723)]

  * [[Andrei Rodin]], *Models of HoTT and the Constructive View of Theories* [[pdf](https://philomatica.org/wp-content/uploads/2013/01/MH1.pdf)]

* Part III: Comparing Set Theory, Category Theory, and Type Theory

  * [[Neil Barton]], [[Sy-David Friedman]], *Set Theory and Structures* [[pdf](https://philsci-archive.pitt.edu/15908/7/Set_theory_structures_2.pdf)]

  * [[Mirna Džamonja]], *A New Foundational Crisis in Mathematics, Is It Really Happening?* [[arXiv:1802.06221](https://arxiv.org/abs/1802.06221)]

  * [[Ansten Klev]], *A Comparison of Type Theory with Set Theory* [[pdf](https://drive.google.com/file/d/1FIfAvSkt9uJh6R2uS7gz-fWh5hLKVx5w/view)]

  * [[Penelope Maddy]], *What Do We Want a Foundation to Do? Comparing Set-Theoretic, Category-Theoretic, and Univalent Approaches* [[pdf](https://bpb-us-e2.wpmucdn.com/faculty.sites.uci.edu/dist/3/399/files/2023/09/What-do-we-want-a-foundation-to-do.pdf)]

* Part IV: Philosophical Thoughts on the Foundations of Mathematics

  * [[Merlin Carl]], *Formal and Natural Proof: A Phenomenological Approach* [[draft](https://kops.uni-konstanz.de/server/api/core/bitstreams/96c96cc1-322e-430c-b4d0-63f2a7abd7fe/content), [pdf](https://www.math.uni-konstanz.de/~carl/Paper/FormalNaturalProofFinal2015.pdf)]

  * [[Michèle Friend]], *Varieties of Pluralism and Objectivity in Mathematics* [[Varieties_of_Pluralism_and_Objectivity_in_Mathematics.pdf:file]]

category: reference


reflective (infinity,1)-subcategory > history

< [[reflective (infinity,1)-subcategory]]