ringed topos

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

#Contents#
* table of contents
{:toc}

## Idea

Since a [[topos]] is a [[cartesian monoidal category]], the notion of a unital [[ring]] and commutative unital ring can be defined [[internalization|internal]] to it.

A _ringed topos_ $(X,\mathcal{O}_{X})$ is a topos $X$ equipped with a choice of ring object $\mathcal{O}$. If $X$ is a [[sheaf topos]] over a [[site]] $C$ then $\mathcal{O}_X$ is a [[sheaf]] of rings on $C$: a [[structure sheaf]].

The notion of ringed topos makes sense for the theory of rings replaced by any [[Lawvere theory]]. Moreover, it makes sense for higher toposes such as [[(∞,1)-topos]]es. This is described at [[structured (∞,1)-topos]].

## Definition

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

A **ringed topos** $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is

* a [[topos]] $\mathcal{X}$

* equipped with a distinguished unital [[ring object]] $\mathcal{O}_{\mathcal{X}} \in \mathcal{X}$: a [[ring]] [[internalization|internal to]] the topos.

If all [[stalk]]s of $\mathcal{O}_{\mathcal{X}}$ are [[local ring]]s, $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is a called a **[[locally ringed topos]]**.

A morphism of ringed toposes $(f, \eta) : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}})$ is

* a [[geometric morphism]]

  $$
    (f^* \dashv f_*) : \mathcal{X} \to \mathcal{Y}
  $$

* and a morphism ("[[comorphism]]") of ring objects in $\mathcal{X}$

  $$
    \eta
      \colon
    f^* \mathcal{O}_{\mathcal{Y}}
      \to
    \mathcal{O}_{\mathcal{X}}
  $$

  which is equivalently, by the $(f^* \dashv f_*)$-[[adjunction]], a morphism of ring objects

  $$
    \tilde \eta : \mathcal{O}_{\mathcal{Y}} \to f_* \mathcal{O}_{\mathcal{X}}
    \,.
  $$

=--

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

The usual variants apply: we can speak of toposes equipped with, specifically, commutative ring objects, unital/nonunital ring objects, ring objects under other ring objects, hence [[associative algebra]] objects.

=--

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

Let $PSh((CRing^{fg})^{op})$ be the [[classifying topos]] for the [[Lawvere theory]] of [[ring]]s.  Then

* a ringed topos $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is a [[geometric morphism]]

  $$
    \mathcal{O}_{\mathcal{X}} : \mathcal{X} \to PSh((CRing^{fg})^{op})
    \,,
  $$

* a morphism $(f,\eta) : (\mathcal{X}, \mathcal{O}_{\mathcal{X}}) \to (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}})$ is a [[diagram]]

  $$
    \array{
      \mathcal{X} &&\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}&& \mathcal{Y}
      \\
      & {}_{\mathllap{}}\searrow \nwarrow^{\mathrlap{\mathcal{O}_{\mathcal{X}}}}
      &\swArrow_{\eta}&
     \swarrow \nearrow_{\mathcal{O}_{\mathcal{Y}}}
      \\
      && PSh((CRing^{fg})^{op})
    }
  $$

  in the [[2-category]] [[Topos]].

So the [[2-category]] of ringed toposes is the [[lax slice 2-category]] $Topos/PSh((CRing^{fp})^{op})$.

=--

More generally:

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

For $T$ a [[Lawvere theory]], a $T$-ringed topos is a [[topos]] $X$ together with a [[product]]-preserving [[functor]] $\mathcal{O}_X : T \to X$.

=--

See [[locally algebra-ed topos]] for more on this.

In order to say what _locally_ $T$-ringed means, one needs the extra structure of a [[geometry (for structured (infinity,1)-toposes)|geometry]] on $T$. See there for details.

## Examples

* A [[ringed space]] $(X,\mathcal{O})$ induces the ringed [[localic topos]] $(Sh(X),\mathcal{O})$ of [[sheaf|sheaves]] on the [[category of open subsets]] of the [[topological space]] $X$.

  Similarly but more generally a  [[ringed site]] $(S, \mathcal{O})$ induces the ringed [[Grothendieck topos]] $(Sh(S), \mathcal{O})$.

* In some applications the ringed topos is refined to a [[lined topos]] when instead of a [[ring]] object an [[algebra]]-object is required.

* For $A \in Alg$ any (possibly non-commutative) algebra, let $\mathcal{C}(A)$ be its [[poset of commutative subalgebras]]. The [[presheaf topos]] $[\mathcal{C}(A), Set]$ naturally carries the commutative ring object $\underline A : (C \in \mathcal{C}(A)) \mapsto C$. This example appears in the description of [[state]]s in [[quantum mechanics]] after "[[Bohrification]]".

## Properties
 {#Properties}

### Limits and colimits
 {#LimitsAndColimits}

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

Let $J \to RingedTopos$ be a [[diagram]] of ringed toposes. Its [[limit]] exists and is given by

* the limiting topos

  $$
    {\lim_\leftarrow}_j (\mathcal{X}_j, \mathcal{O}_{\mathcal{X}_j})
    \stackrel{p_j}{\to}
    (\mathcal{X}_j, \mathcal{O}_{\mathcal{X}_j})
  $$

  of the underlying diagram $J \to RingedTopos \stackrel{}{\to} $ [[Topos]];

* equipped with the [[colimit]]ing [[ring object]] of all the [[inverse image]] rings

  $$
    {\lim_\to}_j p_j^* \mathcal{O}_{\mathcal{X}_j}
    \in
    {\lim_\leftarrow}_j \mathcal{X}_j
    \,.
  $$

=--

In more detail: let

$$
  \array{
    && (\mathcal{Y}, \mathcal{O}_{\mathcal{Y}})
    \\
    & {}^{\mathllap{f_i}}\swarrow &{}^{\mathllap{\simeq}}\swArrow_{\rho}& \searrow^{\mathrlap{f^j}}
    \\
   (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i})
   &&\underset{h_{i j}}{\to}&&
   (\mathcal{X}_j, \mathcal{O}_{\mathcal{X}_j})
  }
$$

be a [[cone]] in $RingedTopos$, then this induces the cocone of ring objects in $\mathcal{Y}$

$$
  \array{
     f_i^* \mathcal{O}_{\mathcal{X}_i}
     &
      \stackrel{f_i^*(h_{i j}^* \mathcal{O}_{\mathcal{X}_j} \to \mathcal{O}_{\mathcal{X}_i} )}{\leftarrow}
     &
     f_j^* h_{i j}^* \mathcal{O}_{\mathcal{X}_j}
     &\underoverset{\simeq}{\rho_{\mathcal{O}_{\mathcal{X}_j}}}{\leftarrow}&
     f_j^* \mathcal{O}_{\mathcal{X}_j}
     \\
     & \searrow && \swarrow
     \\
     && \mathcal{O}_{\mathcal{Y}}
  }
$$

whose commutativity may be understood as being the 2-commutativity of the prism in [[Topos]] over the [[classifying topos]] $PSh(CRing_{fg}^{op})$ with rear side faces $\eta_i$ and $\eta_j$, with front face $\eta_{i j}$ (corresponding to $h_{i j}$) and top face $\rho$.

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

We check the [[universal property]] of the [[limit]]:

for $(\mathcal{Y}, \mathcal{O}_{\mathcal{Y}}) \stackrel{f_i}{\to} (\mathcal{X}_i, \mathcal{O}_{\mathcal{X}_i})$ any [[cone]] over the given diagram, we have by the definition of morphisms of ringed toposes:

1. an essentially unique [[geometric morphism]]

   $$
     h : \mathcal{Y} \to {\lim_\leftarrow}_j (\mathcal{X}_j, \mathcal{O}_{\mathcal{X}_j});
   $$

1. a unique morphism of ring objects

   $$
     h^* {\lim_\to}_j p_j^* \mathcal{O}_{\mathcal{X}_j}
     \simeq
     {\lim_\to}_j h^* p_j^* \mathcal{O}_{\mathcal{X}_j}
     \to
     \mathcal{O}_{\mathcal{Y}}
   $$

   induced from the fact that the [[inverse image]] $h^*$ preserves [[colimit]]s and that the morphisms

   $$
      f_j^* \mathcal{O}_{\mathcal{X}_j} \to \mathcal{O}_{\mathcal{Y}}
   $$

   form a cocone under the diagram of ring objects $f_j^* \mathcal{O}_{\mathcal{X}_j} \in \mathcal{Y}$.

=--

## Related concepts

* [[ringed space]], [[locally ringed space]]

* [[ringed site]], [[locally ringed site]]

* **ringed topos**, [[locally ringed topos]],

* [[locally algebra-ed topos]]

* [[structured (∞,1)-topos]]

## References

Original references:

* [[Alexander Grothendieck]], [[Jean-Louis Verdier]], *Topos annelés, localisation dans les topos annelés*, Section 11 in Exposé iv in: [[M. Artin]] et al., *Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos*, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 ([[SGA 4]]), Lecture Notes in Mathematics __269__, Springer (1972) [[doi:10.1007/BFb0081551](https://link.springer.com/book/10.1007/BFb0081551), [pdf](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf)]

* [[Monique Hakim]], Chapitre III in: *Topos annelés et schémas relatifs*, Ergebnisse der Mathematik und ihrer Grenzgebiete **64**, Springer (1972) [[doi:10.1007/978-3-662-59155-0](https://doi.org/10.1007/978-3-662-59155-0)]

See also:

* {#deJong} [[Aise Johan de Jong]], section 18.7 of: _[[The Stacks Project]]_ ([pdf](http://www.math.columbia.edu/algebraic_geometry/stacks-git/book.pdf)) ([project website](http://www.math.columbia.edu/algebraic_geometry/stacks-git/))

The generalization to [[structured (infinity,1)-toposes]] is due to

* [[Jacob Lurie]], _[[Structured Spaces]]_

See also references at *[[ringed space]]*, such as

* [[eom]], *[Ringed spaces](https://encyclopediaofmath.org/wiki/Ringed_space)*

[[!redirects ringed toposes]]
[[!redirects ringed topoi]]


ringoid > history

< [[ringoid]]
[[!redirects ringoid -- history]]


ringoids > history

< [[ringoids]]


This page collects material related to

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]],

  _Rings, modules and algebras in stable homotopy theory_

  Mathematical Surveys and Monographs Volume 47

  AMS 1997

  [ISBN:978-0-8218-4303-1](https://bookstore.ams.org/surv-47-s)

  [pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf)

on [[higher algebra]] ([[brave new algebra]]) in [[stable homotopy theory]], i.e. on [[ring spectra]], [[module spectra]] etc.

(Sometimes referred to as *EKMM* for short, after the initials of its authors.)

This was the first account to give a [[symmetric monoidal category of spectra]] (here modered as "[[S-modules]]"), in which $A_\infty$ and $E_\infty$-[[ring spectra]] could be defined as ordinary [[monoids]] and their [[modules]].

#Contents#
* table of contents
{:toc}

## Chapter I. Prologue: the category of $\mathbb{L}$-spectra

* [[coordinate-free spectrum]]

## Chapter II. Structured ring and module spectra

* [[ring spectrum]]

* [[module spectrum]]

## Chapter III. The homotopy theory of $R$-Modules

## Chapter IV. The algebraic theory of $R$-modules

## Chapter V. $R$-Ring spectra and the specialization to $MU$

* [[MU]]

## Chapter VI. Algebraic K-theory of $S$-algebras

* [[algebraic K-theory]]

## Chapter VII. $R$-algebras and topological model categories

## Chapter VIII. Bousfield localization of $R$-modules and algebras

## Chapter IX. Topological Hochschild homology and cohomology

* [[topological Hochschild homology]]

## Chapter X. Some basic constructions on spectra

## Chapter XI. Spaces of linear isometries and technical theorems

## Chapter XII. The monadic bar construction

## Chapter XIII. Epilogue. The category of $\mathbb{L}$-spectra under $S$

category: reference

[[!redirects EKMM]]


Rita Fioresi

* [webpage](https://www.dm.unibo.it/~fioresi/)

## Selected writings

On [[supergeometry]] ([[supermanifolds]], [[super Lie groups]]) from the perspective of [[functorial geometry]]:

* [[Claudio Carmeli]], Lauren Caston, [[Rita Fioresi]], _Mathematical Foundations of Supersymmetry_, EMS Series of Lectures in Mathematics Volume: 15; 2011; 263 pp; ( [ISBN:978-3-03719-097-5](https://bookstore.ams.org/emsserlec-15/), [arXiv:0710.5742](https://arxiv.org/abs/0710.5742))

category: people


Rita Jimenez Rolland

* [personal page](https://www.matem.unam.mx/~rita/mainEng.html)

## Selected writings

On [[braid representations]] on the [[ordinary cohomology]] of [[configuration spaces of points]] with focus on the phenomenon of [[representation stability]]:

* [[Rita Jimenez Rolland]], [[Jennifer C. H. Wilson]], *Stability properties of moduli spaces*, Notices of the AMS [**69** 4](https://www.ams.org/notices/202204/202204FullIssue.pdf) (April 2022) [[arXiv:2201.04096](https://arxiv.org/abs/2201.04096), [pdf](https://www.ams.org/journals/notices/202204/rnoti-p522.pdf), [web](https://www.ams.org/journals/notices/202204/noti2452/noti2452.html)]

category: people

[[!redirects Rita J. Rolland]]
[[!redirects Rita Rolland]]


Rng

## Idea

_$Rng$_ is the [[category]] of [[nonunital rings]] and [[homomorphisms]] between them.

## Related categories

* [[Mon]], [[CMon]]

* [[Ring]], [[CRing]]


RO(G)-grading

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

#Contents#
* table of contents
{:toc}

## Idea

In [[equivariant stable homotopy theory]] a _[[genuine G-spectrum]]_  has [[homotopy groups]] [[grading|graded]] not just by the [[integers]], but by the (additive group underlying) the [[representation ring]] of $G$. This is often called _$RO(G)$-grading_.

## Motivation

### Suspension by representation spheres

Since a [[cohomology theory]] is, in particular, a functor

$$
  E^\bullet \colon BasedSpaces^{op} \longrightarrow GradedAbelianGroups
$$

satisfying some axioms, then naively a $G$-equivariant cohomology theory could be defined to be a functor

$$
  E_G^\bullet\colon Based G Spaces^{op} \longrightarrow GradedAbelianGroups
  \,.
$$

This implies in particular the [[suspension isomorphism]]: for  $n\in \mathbb{N}$ there is, for each $G$-space $X$, an identification

$$
   E_G^\bullet(X) \simeq  E_G^{\bullet + n}(S^n \wedge X)
  \,,
$$

where in the smash product $S^n \wedge X$ the group $G$ is to be taken to act trivially on the [[sphere]] $S^n$. From this perspective it is desirable to have an analogous relation also for smashing with spheres on which the group $G$ acts nontrivially. Since we may think of $S^n$ as being the [[representation sphere]] of the trivial action of $G$ on $\mathbb{R}^n$, this leads one to demand that $E^\bullet$ is graded not just by the integers, but by any [[linear representation]] $V$, so that one has _equivariant [[suspension isomorphisms]]_

$$
  E_G^\bullet(X) \simeq E_G^{\bullet + V}(S^V \wedge X)
  \,,
$$

where now $S^V$ denotes the [[representation sphere]] of $V$. This more general grading is, for historical reasons, called "[[RO(G)-grading]]", and an equivariant cohomology theory equipped with this extra structure is called _genuine_ (as opposed to the "naive" case with grading just over the integers).

### Representation by $G$-spectra

Since by the [[Brown representability theorem]], in the absence of a group action a cohomology theory is [[representable functor|represented]] by a [[spectrum]], it is natural to construct a category in which genuine $G$-equivariant cohomology theories also become representables. This is the [[equivariant stable homotopy theory]] of [[genuine G-spectra]].

### Induced transfer and Mackey functors

(...)

## Definition

For $G$ a [[finite group]], $X$ a $G$-[[equivariant spectrum]] modeled as an [[orthogonal spectrum]] equipped with a $G$-[[action]], and for $V$ a linear $G$-[[linear representation|representation]] on a [[real vector space]] of [[dimension]] $n$, the value of $X$ in $RO(G)$-degree $V$ is

$$
  X(V)
  \coloneqq
  \mathbf{L}(\mathbb{R}^n, V)_+ \wedge_{O(n)} X_n
  \,,
$$

where

* $\mathbf{L}(\mathbb{R}^n, V)$ is the linear [[isometries]] $\mathbb{R}^n \to V$ ([[orthogonal bases]]);

(e.g. [Schwede 15 (2.2)](#Schwede15))

The _$RO(G)$-graded [[equivariant homotopy group]]_ of $X$ is (in the notation used there)

$$
  \pi_V^G(X)  \coloneqq
  \underset{\longrightarrow_{\mathrlap{n}}}{\lim}
  [S^{V + n \rho_G}, X(n \rho_G)]_G
  \,.
$$

(e.g. [Schwede 15, p. 40](#Schwede15))

## Properties

### Relation to intrinsic twisting

>  In equivariant cohomology theory, the use of $RO(G)$ is a great convenience, but it is not the thing most intrinsic to the mathematics.  Ignore the multiplication on $RO(G)$,  which is irrelevant to its use for grading theories, and think of it just as an abelian group.  Send a representation $V$ to the isomorphism class of the suspension $G$-spectrum of the one-point compactification $S^V$.  This induces a homomorphism from $RO(G)$ into the Picard group $Pic(Ho G\mathcal{S})$ of the stable homotopy category of  $G$-spectra, namely the abelian group of equivalence classes of
$G$-spectra that are invertible under the smash product.  That homomorphism  is neither a monomorphism nor an epimorphism.  See ([Fausk-Lewis-May 01](#FauskLewisMay01)) for a discussion of that Picard group.  Logically, equivariant cohomology theories really should be graded on $Pic(Ho G\mathcal{S})$, but that group is much less  convenient than $RO(G)$.  ([P. May, comment on MO, Jul 2014](http://mathoverflow.net/a/177154/381))

## Related concepts

* [[G-universe]]

## References

A standard reference is

* {#May96} [[Peter May]], chapters IX.5, X and XIII of _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. ([pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf))

* {#Schwede15} [[Stefan Schwede]], section 4 (from page 40 on) in _[[Lectures on Equivariant Stable Homotopy Theory]]_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/equivariant.pdf))

* {#FauskLewisMay01} [[Halvard Fausk]], [[L. Gaunce Lewis]], [[Peter May]],  _The Picard group of equivariant stable  homotopy theory_, Advances in Mathematics Volume 163, Issue 1, 15 October 2001, Pages 17–33  ([arXiv:2008.05551](https://arxiv.org/abs/2008.05551), [pdf](http://www.math.uchicago.edu/~may/PAPERS/FLMJan01.pdf))

* [[Justin Noel]], _Equivariant cohomology of representation spheres and $Pic(S_G)$-graded homotopy groups_, 2013 ([pdf](http://www.nullplug.org/talks/equivariant.pdf))

[[!redirects RO(G)-gradings]]

[[!redirects RO(G)-degree]]
[[!redirects RO(G)-degrees]]


Rob Kirby > history

see _[[Robion Kirby]]_


Robbert Dijkgraaf

* [website](http://staff.science.uva.nl/~rhd/)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Robbert_Dijkgraaf)

## Selected writings

Review of [[perturbative quantization of 3d Chern-Simons theory]]:

* [[Robbert Dijkgraaf]], _Perturbative topological field theory_, In: *Trieste 1993, Proceedings, String theory, gauge theory and quantum gravity '93* 189-227 ([spire:399223](http://inspirehep.net/record/399223/), [[DijkgraafPerturbativeCS93.pdf:file]])

On [[Dijkgraaf-Witten theory]]:

* {#DijkgraafWitten90} [[Robbert Dijkgraaf]], [[Edward Witten]], _[[DW.pdf:file]]_, Commun. Math. Phys. __129__ (1990), 393

* [[Robbert Dijkgraaf]], V. Pasquier, P. Roche, _QuasiHopf algebras, group cohomology and orbifold models_, Nucl. Phys. B Proc. Suppl. __18B__ (1990), 60-72; _Quasi-quantum groups related to orbifold models_, Modern quantum field theory (Bombay, 1990), 375–383, World Sci. 1991

On [[Drinfeld doubles]] of [[quasi-Hopf algebras]], motivated from [[rational conformal field theory|rational]] [[orbifold]] [[conformal field theories]]:

* [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], *QuasiHopf algebras, group cohomology and orbifold models*, Nucl. Phys. B Proc. Suppl **18** (1990) 60-72 [doi:10.1016/0920-5632(91)90123-V]

* [[Robbert Dijkgraaf]], [[Vincent Pasquier]], [[Philippe Roche]], _Quasi-quantum groups related to orbifold models_, *International Colloquium on Modern Quantum Field Theory* (Bombay, 1990), 375-383, World Sci. (1991) [[cds:206306](https://cds.cern.ch/record/206306?ln=ka), [[DPR-QuasiQuantumGroups.pdf:file]]]

On [[AdS3-CFT2]] for [[D1/D5 brane bound states]] and [[black hole entropy]] [[black holes in string theory|in string theory]]:

* [[Robbert Dijkgraaf]], [[Juan Maldacena]], [[Gregory Moore]], [[Erik Verlinde]], _A Black Hole Farey Tail_ ([arXiv:hep-th/0005003](https://arxiv.org/abs/hep-th/0005003), [spire:526744](http://inspirehep.net/record/526744))

On [[string theory]] [[perturbative string theory vacuum|vacua]] [[KK-compactification|compactified]] on [[orbifolds]] and [[orientifolds]]:

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], section 3 of _Triples, Fluxes, and Strings_, Adv.Theor.Math.Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

On [[topological M-theory]]:

* [[Robbert Dijkgraaf]], [[Sergei Gukov]], [[Andrew Neitzke]], [[Cumrun Vafa]], *Topological M-theory as Unification of Form Theories of Gravity*, Adv. Theor. Math. Phys. **9** (2005) 603-665 [[arXiv:hep-th/0411073](http://arxiv.org/abs/hep-th/0411073), [doi:10.4310/ATMP.2005.v9.n4.a5](https://dx.doi.org/10.4310/ATMP.2005.v9.n4.a5)]

On [[topological string theory]] and [[integrable systems]]:

* [[Mina Aganagic]], [[Robbert Dijkgraaf]], [[Albrecht Klemm]], [[Marcos Marino]], [[Cumrun Vafa]], *Topological Strings and Integrable Hierarchies*, Commun. Math. Phys. **261** (2006) 451-516 [[arXiv:hep-th/0312085](https://arxiv.org/abs/hep-th/0312085), [doi:10.1007/s00220-005-1448-9](https://doi.org/10.1007/s00220-005-1448-9)]

* [[Mina Aganagic]], [[Miranda Cheng]], [[Robbert Dijkgraaf]], Daniel Krefl, [[Cumrun Vafa]], *Quantum Geometry of Refined Topological Strings*, J. High Energ. Phys. **2012** 19 (2012) [[arXiv:1105.0630](https://arxiv.org/abs/1105.0630), doi:10.1007/JHEP11(2012)019]

On [[The Unreasonable Effectiveness of Physics in the Mathematical Sciences]]:

* {#AtiyahDijkgraafHitchin10} [[Michael Atiyah]], [[Robbert Dijkgraaf]], [[Nigel Hitchin]], _Geometry and physics_, Phil. Trans. R. Soc. A 13 March 2010 vol. 368 no. 1914 913-926 ([pdf](http://rsta.royalsocietypublishing.org/content/368/1914/913.full.pdf), [doi;10.1098/rsta.2009.0227](https://royalsocietypublishing.org/doi/10.1098/rsta.2009.0227))

* {#Dijkgraaf14} [[Robbert Dijkgraaf]], _The Unreasonable Effectiveness Of Quantum Physics in Modern Mathematics_, [Perimeter Public Lecture](http://pirsa.org/C04001) 2014 ([web](http://pirsa.org/14030094), [pdf slides](http://pirsa.org/pdf/loadpdf.php?pirsa_number=14030094)).

## Related entries

* [[Dijkgraaf-Witten theory]]

category: people


Robbert Krebbers

* [personal page](https://robbertkrebbers.nl/)

## Selected writings

On [[constructive analysis]] with exact [[real numbers]] via [[type theory]], specifically with [[ufias2012:Type classes]] in [[Coq]]:

* {#KrebbersSpitters11} [[Robbert Krebbers]], [[Bas Spitters]], *Type classes for efficient exact real arithmetic in Coq*, Logical Methods in Computer Science, **9** 1 (2013) lmcs:958 [[arXiv:1106.3448](http://arxiv.org/abs/1106.3448/), doi:10.2168/LMCS-9(1:1)2013]

category: people


Robert A. Wilson

Robert Wilson is a [[finite group|finite ]]-[[group theory|group theorist]]. He is one of the authors of the *[ATLAS of finite groups](https://en.wikipedia.org/wiki/ATLAS_of_Finite_Groups)* and particularly works on [[sporadic finite simple groups]], such as the [[monster group]].

* [website](http://www.maths.qmul.ac.uk/~raw/)

## Selected writings

On [[octonions]] and the [[Leech lattice]]:

* {#Wilson09} [[Robert A. Wilson]], _Octonions and the Leech lattice_, Journal of Algebra, Volume 322, Issue 6, 15 September 2009, Pages 2186-2190, ([pdf](http://www.maths.qmul.ac.uk/~raw/pubs_files/octoLeech1.pdf)), ([slides](http://www.maths.qmul.ac.uk/~raw/talks_files/Cambridge09.pdf))

* {#Wilson09b} [[Robert A. Wilson]], _A new approach to the Leech lattice_, talk at University of Cambridge, 21st October 2009 ([slides pdf](http://www.maths.qmul.ac.uk/~raw/talks_files/Cambridge09.pdf))

On [[subgroups]] of [[Clifford algebras]] reminiscent of [[gauge groups]] of relevance in the [[standard model of particle physics]] and [[grand unified theories]] thereof:

* {#Wilson20} [[Robert A. Wilson]], *A group-theorist's perspective on symmetry groups in physics* ([arXiv:2009.14613](https://arxiv.org/abs/2009.14613))

* {#Wilson21} [[Robert A. Wilson]], *On the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics*, Adv. Appl. Clifford Algebras 31, 59 (2021) ([doi:10.1007/s00006-021-01160-5]( https://doi.org/10.1007/s00006-021-01160-5))

On [[E7]] via [[division algebra]]:

* [[Tevian Dray]], [[Corinne Manogue]], [[Robert Wilson]], *A New Division Algebra Representation of $E_7$* [[arxiv:2401.10534](https://arxiv.org/abs/2401.10534)]

## Related entries

* [[exceptional structure]]

* [[Dwyer-Wilkerson H-space]]

category: people

[[!redirects Robert Wilson]]
[[!redirects R. A. Wilson]]


Robert Atkey

* [webpage](http://bentnib.org/index.html)

## Selected writings

On [[dependent linear type theory]]:

* [[Robert Atkey]], *Syntax and semantics of quantitative type theory*, in *33rd Annual ACM/IEEE Symposium on Logic in Computer Science*, LICS (2018) 56–65 [[doi:10.1145/3209108.3209189](https://dl.acm.org/doi/10.1145/3209108.3209189)]

[[!redirects Bob Atkey]]

category: people


Robert B. Griffiths

Physicist at Carnegie-Mellon University

* [personal page](https://www.cmu.edu/physics/people/emeritus-faculty/griffithsrb.html)

* Wikipedia entry

## Selected writings

Introducing the "[[consistent histories]]"-[[interpretation of quantum mechanics]]:

* [[Robert Griffiths]], *Consistent histories and the interpretation of quantum mechanics*, Journal of Statistical Physics **36** (1984) 219-272 [[doi:10.1007/BF01015734](https://doi.org/10.1007/BF01015734)]

* [[Robert Griffiths]], *Consistent Quantum Theory*, Cambridge University Press (2001) [[doi:10.1017/CBO9780511606052](https://doi.org/10.1017/CBO9780511606052), [webpage](http://quantum.phys.cmu.edu/CQT/)]

On [[quantum physics]] in general:

* [[Robert B. Griffiths]], *Consistent Quantum Theory*, Cambridge University Press (2002) [[doi:10.1017/CBO9780511606052](https://doi.org/10.1017/CBO9780511606052), [webpage](https://quantum.phys.cmu.edu/CQT)]

On [[quantum operations]] ([[Kraus decomposition]], etc.)

* [[Robert B. Griffiths]], *Quantum Channels, Kraus Operators, POVMs* (2012) [[pdf](https://quantum.phys.cmu.edu/QCQI/qitd412.pdf), [[Griffiths-QuantumChannels.pdf:file]]]

## Related entries

* [[consistent histories]]$\;$[[interpretation of quantum mechanics]]

category: people

[[!redirects Robert Griffiths]]


Robert B. Laughlin

* [Institute page](http://large.stanford.edu/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_B._Laughlin)

## Selected writings

On the (fractional) [[quantum Hall effect]]:

* [[Robert B. Laughlin]], *Quantized Hall conductivity in two dimensions*, Phys. Rev. B **23** 5632(R) (1981) [[doi:10.1103/PhysRevB.23.5632](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.23.5632)]

and introducing the [[Laughlin wavefunction]]:

* [[Robert B. Laughlin]], *Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations*, Phys. Rev. Lett. **50** (1983) 1395 $[$[doi:10.1103/PhysRevLett.50.1395](https://doi.org/10.1103/PhysRevLett.50.1395)$]$

On [[quantum spin liquids]] (then called "[[resonating valence bond states]]"):

* V. Kalmeyer, [[Robert B. Laughlin]], *Equivalence of the resonating-valence-bond and fractional quantum Hall states*, Phys. Rev. Lett. **59** (1987) 2095 [[doi:10.1103/PhysRevLett.59.2095](https://doi.org/10.1103/PhysRevLett.59.2095)]

category: people

[[!redirects Robert Laughlin]]


Robert Brandenberger

* [webpage](http://www.physics.mcgill.ca/~rhb/)

## related $n$Lab entries

* [[moduli stabilization]]

category: people


Robert Brandom

**Robert Brandom** is an American philosopher whose work on the philosophy of language, notably the monumental *Making It Explicit* (Harvard UP 1994), combines insights from 'continental' philosophy with ideas of pragmatism and analytic philosophy.

* [homepage](http://www.pitt.edu/~rbrandom/)

## Related entries

* [[Georg Hegel|Georg Wilhelm Friedrich Hegel]]

category:people


Robert Brout

* [Wikipedia entry](http://en.wikipedia.org/wiki/Robert_Brout)

category: people


Robert Bruner

* [webpage](http://www.math.wayne.edu/~rrb/)

## Selected writings

On [[H-infinity ring spectra|$H_\infty$-ring spectra]]:

* {#BrunerMayMcLureSteinberger86} [[Robert Bruner]], [[Peter May]], [[James McClure]], [[Mark Steinberger]], *$H_\infty$-Ring Spectra and their Applications*, Lecture Notes in Mathematics **1176**, Springer (1986) [[doi:10.1007/BFb0075405](https://link.springer.com/book/10.1007/BFb0075405), [pdf](http://www.math.uchicago.edu/~may/BOOKS/h_infty.pdf)]

On the [[topological Hochschild homology]] of [[topological modular forms]]:

* [[Robert Bruner]], [[John Rognes]], *Topological Hochschild homology of topological modular forms*, talk at *Nordic Topology Meeting* NTNU (2008) [[[Bruner-Rognes_THHtmf.pdf:file]]]

On [[equivariant K-theory]] and [[KR-theory]] for [[finite group|finite]] [[equivariance groups]]:

* {#BrunerGreenlees10} [[Robert Bruner]], [[John Greenlees]], *Connective Real K-Theory of Finite Groups*, Mathematical Surveys and Monographs **169** AMS 2010 ([ISBN:978-0-8218-5189-0](https://bookstore.ams.org/surv-169))

## Related $n$Lab entries

* [[Adams spectral sequence]]

* [[Steenrod algebra]]

category: people

[[!redirects Bob Bruner]]


Robert Bryant

* [webpage](http://fds.duke.edu/db/aas/math/faculty/bryant/)

## Selected writings

On [[twistor fibrations]]:

* [[Robert Bryant]], Section 1 of: _Conformal and minimal immersions of compact surfaces into the 4-sphere_, J. Differential Geom. Volume 17, Number 3 (1982), 455-473 ([euclid:jdg/1214437137](https://projecteuclid.org/euclid.jdg/1214437137))

* {#Bryant85} [[Robert Bryant]], _Lie groups and twistor spaces_, Duke Mathematical Journal 52 (1985), pp. 223–261, ([euclid:dmj/1077304286](https://projecteuclid.org/euclid.dmj/1077304286))

On the [[metric cone over complex projective 3-space]] and the [[twistor fibration]]:

* [[Bobby Acharya]], [[Robert Bryant]], [[Simon Salamon]], _A circle quotient of a $G_2$ cone_, Differential Geometry and its Applications Volume 73, December 2020, 101681  ([arXiv:1910.09518](https://arxiv.org/abs/1910.09518), [doi:10.1016/j.difgeo.2020.101681](https://doi.org/10.1016/j.difgeo.2020.101681))

On the construction of [[G2-manifolds]]:

* [[Robert Bryant]], [[Simon Salamon]],  _On the construction of some complete metrics with exceptional holonomy_, Duke Mathematical Journal 58: 829–850 (1989) ([euclid:dmj/1077307681](https://projecteuclid.org/euclid.dmj/1077307681))

On [[special holonomy]] and [[G-structure]]:

* [[Robert Bryant]], _Metrics with Exceptional Holonomy_, Annals of Mathematics Second Series, Vol. 126, No. 3 (Nov., 1987), pp. 525-576 ([jstor:1971360](https://www.jstor.org/stable/1971360))

On [[G2-manifolds]] via [[G-structure]]:

* {#Bryant05} [[Robert Bryant]], _Some remarks on $G_2$-structures_, [Proceedings of the 12th Gökova Geometry-Topology Conference 2005](http://gokovagt.org/proceedings/2005), pp. 75-109 ([arXiv:math/0305124](https://arxiv.org/abs/math/0305124), [webpage](http://gokovagt.org/proceedings/2005/bryant.html), [pdf](http://gokovagt.org/proceedings/2005/ggt05-bryant.pdf))

On [[Spin(8)]]

* [[Robert Bryant]], _Remarks on Spinors in Low Dimension_ ([pdf](https://services.math.duke.edu/~bryant/Spinors.pdf), [[BryantRemarksOnSpinorsInLowDimension.pdf:file]])

category: people


Robert C. Gunning

* [personal page](https://web.math.princeton.edu/~gunning/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_C._Gunning)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=8120)

## Selected writings

On [[Riemann surfaces]]:

* {#Gunning66} [[Robert C. Gunning]], *Lectures on Riemann Surfaces*, Princeton University Press (1966) [[pdf](http://ronan.terpereau.perso.math.cnrs.fr/Master_Class_2023_Dijon/GUNNING_Lectures%20on%20Riemann%20Surfaces.pdf)]

On [[analytic functions]] in [[complex analysis]]:

* {#GunningRossi} [[Robert C. Gunning]], [[Hugo Rossi]], *Analytic functions of several complex variables*, Prentice-Hall Inc., Englewood Cliffs (1965)

## Related entries

* [[fine sheaf]]

* [[Cartan theorem B]]

category: people

[[!redirects Robert Gunning]]


Robert C. Helling

* [institute page](https://www.theorie.physik.uni-muenchen.de/MATH/members/asc/sci_mem/helling_robert/index.html)

* [Google Scholar page](https://scholar.google.com/citations?user=t05gBxwAAAAJ&hl=en)

## Selected writings

On [[quantization]] of the [[M2-brane]] via the [[BFSS matrix model]]:

* {#NicolaiHelling98} [[Hermann Nicolai]], [[Robert C. Helling]], _Supermembranes and M(atrix) Theory_, In _Trieste 1998, Nonperturbative aspects of strings, branes and supersymmetry_ 29-74 ([arXiv:hep-th/9809103](http://arxiv.org/abs/hep-th/9809103), [spire:476366](http://inspirehep.net/record/476366))

On [[graviton]] [[scattering amplitudes]] in [[D=11 supergravity]] via the [[BFSS matrix model]]:

* [[Robert Helling]], [[Jan Plefka]], Marco Serone, Andrew Waldron, _Three-graviton scattering in M-theory_, Nuclear Physics B Volume 559, Issues 1–2, 18 October 1999, Pages 184-204 ([arXiv:hep-th/9905183](https://arxiv.org/abs/hep-th/9905183))

On [[perturbative quantum field theory]] with emphasis on conceptual aspects of [[non-perturbative effects]] and [[renormalization]]:

* {#FloryHellingSluka12} [[Mario Flory]], [[Robert C. Helling]], Constantin Sluka, *How I Learned to Stop Worrying and Love QFT* ([arXiv:1201.2714](https://arxiv.org/abs/1201.2714))

and notes on the perturbation series specifically for [[phi^4 theory]]:

* {#Helling} [[Robert Helling]], p. 4 of _Solving classical field equations_ ([pdf](http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf), [[HellingClassicalQFT.pdf:file]])

category: people

[[!redirects Robert Helling]]


Robert Calderbank

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_Calderbank)

## Selected writings

On [[quantum error correcting codes]]:

* [[Robert Calderbank]], [[Peter W. Shor]], *Good Quantum Error-Correcting Codes Exist*, Phys. Rev. A, Vol. 54, No. 2, pp. 1098-1106, 1996 ([doi:10.1103/PhysRevA.54.1098](https://doi.org/10.1103/PhysRevA.54.1098))

Introducing [[stabilizer codes]]:

* [[Robert Calderbank]], E. M Rains, [[Peter W. Shor]], N. J. A. Sloane, *Quantum Error Correction and Orthogonal Geometry*, Phys. Rev. Lett. 78:405-408, 1997 ([arXiv:quant-ph/9605005](https://arxiv.org/abs/quant-ph/9605005))

category: people


Robert Connelly

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_Connelly)

## Selected writings

* [[Robert Connelly]], _A new proof of Brown's collaring theorem_, Proceedings of the American Mathematical Society 27 (1971), 180 – 182 ([jstor:2037284](https://www.jstor.org/stable/2037284))

  (on the [[collar neighbourhood theorem]])

## Related $n$Lab entries

* [[collar neighbourhood theorem]]

category: people


Robert Constable

* [webpage](http://www.cs.cornell.edu/home/rc/)

## Selected writings

On [[inductive types]]:

* [[Robert L. Constable]], [[N. Paul Francis Mendler]], *Recursive definitions in type theory*, in *Logic of Programs 1985*, Lecture Notes in Computer Science **193** Springer (1985) $[$[doi:10.1007/3-540-15648-8_5](https://doi.org/10.1007/3-540-15648-8_5)$]$

On [[type theory]] in [[mathematics]] and [[computer science]]:

* [[Robert Constable]], *The Triumph of Types: Creating a Logic of Computational Reality*, lecture at *[Types, Semantics and Verification](http://www.cs.uoregon.edu/research/summerschool/summer11/)*, Oregon (2011) [[pdf](http://www.cs.uoregon.edu/research/summerschool/summer11/lectures/Triumph-of-Types-Extended.pdf), [[ConstableTriumphOfTypes.pdf:file]]]

## Related entries

* [[NuPRL]]

* [[computational type theory]]

category: people

[[!redirects Robert L. Constable]]


Robert D. Tennent

* [institute page](https://www.cs.queensu.ca/people/Bob/Tennent)

## Selected writings

On [[denotational semantics]] and [[domain theory]] for [[programming languages]]:

* [[Robert D. Tennent]], *The denotational semantics of programming languages*, Communications of the ACM **19** 8 (1976) 437–453 [[doi:10.1145/360303.360308](https://doi.org/10.1145/360303.360308), [pdf](https://dl.acm.org/doi/pdf/10.1145/360303.360308)]

* [[Robert D. Tennent]], *Denotational semantic*, in: *[[Handbook of Logic in Computer Science]]* **3**, Oxford University Press (1995) [[ISBN:9780198537625](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537625?cc=de&lang=en&)[

On [[bireflective subcategories]] with [[ambidextrous adjoints]]:

* {#FOPTST99} [[Peter Freyd]], [[Peter O’Hearn]],  [[A. John Power]], [[M. Takeyama]], [[Ross Street]], [[Robert D. Tennent]], *Bireflectivity*, Theoretical Computer Science **228** 1–2 (1999) 49-76 [doi:10.1016/S0304-3975(98)00354-5]

category: people

[[!redirects R. D. Tennent]]
[[!redirects Robert Tennent]]
[[!redirects Robert D. Tennent]]

[[!redirects Bob Tennent]]

[[!redirects Robert D. Tennent]]
[[!redirects Robert D. Tennent]]


Robert Dawson

* [webpage](https://www.smu.ca/math-cs/dawson.html)

## Selected Writings

On [[double categories]]:

* [[R. Dawson]] and [[Robert Paré]], _Characterizing tileorders_, Order, Vol. 10, 1993, pp. 111–128. [[doi:10.1007/BF01111295](https://doi.org/10.1007/BF01111295)]

* [[Robert Dawson]] and [[Robert Paré]], _General associativity and general composition for double categories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 34, No. 1, 1993, pp. 57-79. [[cahiers](http://archive.numdam.org/item/CTGDC_1993__34_1_57_0/)]

category: people

[[!redirects R. Dawson]]


Robert Eugene Marshak

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_Marshak)

* [[E. C. G. Sudarshan]] (ed.), _A Gift of Prophecy -- Essays in Celebration of the Life of Robert Eugene Marshak_, World Scientific  (1995)  ([doi:10.1142/2580](https://doi.org/10.1142/2580))

In relation to the [[Connes-Lott-Chamseddine-Barrett models]]:

* {#G-B19} [[José Gracia-Bondia]], _On Marshak's and Connes' views of chirality_, in: [[E. C. G. Sudarshan]] (ed.), _A Gift of Prophecy -- Essays in Celebration of the Life of [[Robert Eugene Marshak]]_, pp. 208-217, World Scientific  (1995)  ([arXiv:hep-th/9706200](https://arxiv.org/abs/hep-th/9706200), [doi:10.1142/9789812831408_0017](https://doi.org/10.1142/9789812831408_0017))

## Selected writings

On [[baryon-lepton symmetry]]:

* A. Gamba, [[Robert Marshak]], S. Okubo, _On a symmetry in weak interactions_, PNAS June 1, 1959 45 (6) 881-885 ([doi:10.1073/pnas.45.6.881](https://doi.org/10.1073/pnas.45.6.881))

* [[Robert Marshak]], _Yukawa Meson, Sakata Model and Baryon-Lepton Symmetry Revisited_, Progress of Theoretical Physics Supplement, Volume 85, May 1985, Pages 61–74 ([dpi:10.1143/PTP.85.61](https://doi.org/10.1143/PTP.85.61))

On [[particle physics]]:

* [[Robert E. Marshak]], *Conceptual Foundations of Modern Particle Physics*, World Scientific 1993 ([doi:10.1142/1767](https://doi.org/10.1142/1767))

category: people

[[!redirects Robert Marshak]]
[[!redirects Robert E. Marshak]]


Robert Friedman

* [webpage](http://www.math.columbia.edu/~rf/)

## Selected writings

On [[complex surfaces]] and their [[elliptic fibrations]]:

* {#FriedmanMorgan94} [[Robert Friedman]], [[John Morgan]], _Smooth Four-Manifolds and Complex Surfaces_, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (1994) ([doi:10.1007/978-3-662-03028-8](https://www.springer.com/gp/book/9783540570585))

category: people


Robert Geroch

* [Wikipedia entry](http://en.wikipedia.org/wiki/Robert_Geroch)

## Selected writings

On [[Cauchy surfaces]] in [[globally hyperbolic spacetimes]]:

* [[Robert Geroch]], §5, Thm. 11 in: *Domain of Dependence*, J. Math. Phys. **11** (1970) 437–449 [[doi:10.1063/1.1665157](https://doi.org/10.1063/1.1665157)]

On [[quantum mechanics]]:

* [[Robert Geroch]], *Geometrical Quantum Mechanics*, University of Chicago (1974) [[pdf](http://strangebeautiful.com/other-texts/geroch-geom-qm.pdf), [[Geroch-GeometricalQuantumMechanics.pdf:file]]]

On [[mathematical physics]], making explicit the [[categories]] of some of its basic ingredients:

* [[Robert Geroch]], _Mathematical Physics_, Chicago 1985 ([ISBN:9780226223063](https://press.uchicago.edu/ucp/books/book/chicago/M/bo4158035.html))

On [[partial differential equations]] appearing in [[physics]]:

* [[Robert Geroch]], _Partial Differential Equations of Physics_, in: _[General Relativity: Proceedings](https://inspirehep.net/literature/328567)_ Edited by G.S. Hall and J.R. Pulham. Edinburgh, IOP Publishing, 1996. p. 19 ([arXiv:gr-qc/9602055](http://arxiv.org/abs/gr-qc/9602055))

## Related entries

* [[general relativity]]

category: people


Robert Ghrist

Robert Ghrist is a mathematician who applies methods from [[algebraic topology]] to problems arising in other areas of study ([[applied topology]]).

* [webpage](http://www.math.upenn.edu/~ghrist/)

## Publications

* [publication list](http://www.math.upenn.edu/~ghrist/preprints.html)

On [[applied topology]]:

* Robert Grist, Abubakr Muhammad, _Coverage and Hole-detection in sensor networks via homology_, in Proc. Information Processing in Sensor Networks, 2005.

* [[Robert Ghrist]], *Elementary Applied Topology* (2014) [[web](https://www2.math.upenn.edu/~ghrist/notes.html), [ISBN:978-1502880857]()]

On [[barcodes]] of [[persistent homology]] in [[topological data analysis]]:

* [[Robert Ghrist]], _Barcodes: The Persistent Topology of Data_, Bull. Amer. Math. Soc. 45 (2008), 61-75 ([doi:10.1090/S0273-0979-07-01191-3](https://doi.org/10.1090/S0273-0979-07-01191-3), [pdf](https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf))

category: people


Robert Goldblatt

Robert Goldblatt is a logician based at Victoria University, Wellington, New Zealand.

* [home page](http://homepages.mcs.vuw.ac.nz/~rob/)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Robert_Goldblatt)

## Selected writings

On [[modal logic]]:

* [[Robert Goldblatt]], *Mathematical modal logic: A view of its evolution*, Journal of Applied Logic **1** 5–6 (2003) 309-392 [doi:10.1016/S1570-8683(03)00008-9]

On [[topos theory]] and [[internal logic|internal]] [[formal logic]]:

* _Topoi: The Categorial Analysis of Logic_, 2nd ed. North-Holland Amsterdam 1984. ([Dover reprint](https://store.doverpublications.com/0486450260.html) New York 2006; [Project Euclid](http://projecteuclid.org/euclid.bia/1403013939))

  > A point of terminology: – I have consistently used the word “categorial” where the literature uniformly employs “categorical”. The reason is that while both can serve as adjectival forms of the noun “category”, the second of them already has a different and long established usage in the domain of logic, one that derives from its ordinary-language meaning of “absolute”. Logicians have known since the work of Gödel that set theory has no categorical axiomatisation. One function of this book will be to explain to them why it does have a categorial one. [preface, p. x]

## Related entries

* [[Goldblatt-Thomason theorem]]

category: people

[[!redirects R. Goldblatt]]


Robert Gompf

* [webpage](http://www.ma.utexas.edu/users/gompf/)

category: people


Robert Gordon

## Selected writings

On [[coherence theorems]] for [[tricategories]]:

* [[Robert Gordon]], [[A. John Power]], [[Ross Street]], _Coherence for tricategories_, Mem. Amer. Math Soc. **117** (1995) no 558 [[ISBN:978-1-4704-0137-5](https://bookstore.ams.org/memo-117-558)]

category: people

[[!redirects R. Gordon]]


Robert Griess

* [Wikipedia entry](http://en.wikipedia.org/wiki/R._L._Griess)

## Selected writings

* {#Griess82} [[Robert Griess]], Jr. _The Friendly Giant_, Invent. Math. 69 (1982) 1-102 ([doi:10.1007/BF01389186](https://doi.org/10.1007/BF01389186))

* {#Griess98} [[Robert Griess]], _Twelve Sporadic Groups_, Springer, 1998 ([doi:10.1007/978-3-662-03516-0](https://link.springer.com/book/10.1007/978-3-662-03516-0))

* {#GriessLam11} [[Robert Griess]] Jr., Ching Hung Lam, _A new existence proof of the Monster by VOA theory_ ([arXiv:1103.1414](https://arxiv.org/abs/1103.1414))

## Related $n$Lab entries

* [[sporadic finite simple group]]

  * [[Monster group]]

  * [[Happy Family]]

  * [[pariah group]]

* [[Moonshine]]

  * [[automorphism of a vertex operator algebra]]

category: people


Robert Griffiths > history

see *[[Robert B. Griffiths]]*


Robert H. Szczarba

* [institute page](https://fas.yale.edu/book/faculty-retirement-tributes-2000/robert-h-szczarba)

* [obituary](https://news.yale.edu/2011/10/27/memoriam-robert-h-szczarba)

* [mathematics genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6499)

## Selected writings

On [[rational homotopy theory]], the [[fundamental theorem of dg-algebraic rational homotopy theory]] and its generalization both to [[Borel-equivariant rational homotopy theory]] (for [[covering spaces]] of non-[[nilpotent spaces]]) and to [[real homotopy theory]] (using [[continuous cohomology|continuous]] [[real cohomology]] and [[topological algebra|topological]] [[dgc-algebras]]):

* {#BrownSzczarba95} [[Edgar Brown]], [[Robert H. Szczarba]], _Real and Rational Homotopy Theory_, Chapter 17 in: *[[Handbook of Algebraic Topology]]*,  Elsevier, 1995 869-915 ([doi:10.1016/B978-044481779-2/50018-3](https://doi.org/10.1016/B978-044481779-2/50018-3), [ZB](http://mirrors.library.cornell.edu/ZMATH/zmath/en/zmath/search/?an=0865.55009))

based on (including discussion of globally [[Kan fibration|Kan fibrant]] [[simplicial topological spaces|topological]] [[simplicial groups]]):

* {#BrownSzczarba89} [[Edgar H. Brown]], [[Robert H. Szczarba]], *Continuous cohomology and real homotopy type*, Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 ([doi:10.1090/S0002-9947-1989-0929667-6](https://doi.org/10.1090/S0002-9947-1989-0929667-6), [jstor:2001017](https://www.jstor.org/stable/2001017))

* {#BrownSzczarba90} [[Edgar Brown]], [[Robert H. Szczarba]], _Continuous cohomology and Real homotopy type II_ Asterisque 191, Societe Mathematique De France (1990) ([numdam:AST_1990__191__45_0](http://www.numdam.org/book-part/AST_1990__191__45_0))

* [[Edgar Brown]], [[Robert H. Szczarba]], *Real and rational homotopy theory for spaces with arbitrary fundamental group*, Duke Mathematical Journal 71. (1993): 229-316 ([doi:10.1215/S0012-7094-93-07111-6](https://projecteuclid.org/journals/duke-mathematical-journal/volume-71/issue-1/Rational-and-real-homotopy-theory-with-arbitrary-fundamental-groups/10.1215/S0012-7094-93-07111-6.short))

category: people

[[!redirects Roberts Henry Szczarba]]


Robert Harper

* [website](http://www.cs.cmu.edu/~rwh/),

* [blog](http://existentialtype.wordpress.com/)

## Selected writings

On [[logical frameworks]]:

* [[Robert Harper]], [[Furio Honsell]], [[Gordon Plotkin]], _A framework for defining logics_, J. ACM __40__:1 (1993) 143--184 [[doi:10.1145/138027.138060](https://doi.org/10.1145/138027.138060)]

An introduction to [[program|programming]] languages in terms of [[type theory]]:

* [[Robert Harper]], _[[Practical Foundations for Programming Languages]]_, Cambridge University Press (2016) [[ISBN:9781107150300](http://www.cambridge.org/us/academic/subjects/computer-science/programming-languages-and-applied-logic/practical-foundations-programming-languages-2nd-edition?format=HB), [pdf](https://www.cs.cmu.edu/~rwh/pfpl/2nded.pdf)]

An abridged preview, author comments and supplementary material are available on the [PFPL Home Page](http://www.cs.cmu.edu/~rwh/pfpl.html).

An introduction to programming in Standard ML

*  _Programming in Standard ML_, [draft pdf](http://www.cs.cmu.edu/~rwh/smlbook/book.pdf)

On [[guarded recursion]]:

* [[Jonathan Sterling]], [[Robert Harper]], *Guarded Computational Type Theory*, LICS '18 (2018) 879–888 [[arxiv:1804.09098](https://arxiv.org/abs/1804.09098), [doi:10.1145/3209108.3209153](https://doi.org/10.1145/3209108.3209153)]

On ([[homotopy types]] of) [[covering spaces]] via [[homotopy type theory]]:

* [[Kuen-Bang Hou]], [[Robert Harper]], *Covering Spaces in Homotopy Type Theory*, Leibniz International Proceedings in Informatics (LIPIcs) **97** (2018) $[$[doi:10.4230/LIPIcs.TYPES.2016.11](https://doi.org/10.4230/LIPIcs.TYPES.2016.11), [pdf](https://drops.dagstuhl.de/opus/volltexte/2018/9851/pdf/LIPIcs-TYPES-2016-11.pdf)$]$

## Related entries

* [[computational trinitarianism]]

category: people

[[!redirects Robert Harper]]

[[!redirects Bob Harper]]


Robert J. Aumann

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_Aumann)

## Selected writings

On [[epistemic logic|epistemic]] ([[S5 modal logic|S5]]) [[modal logic]] in [[game theory]]:

* [[Robert J. Aumann]], *Interactive epistemology I: Knowledge*, International Journal of Game Theory **28** (1999) 263–300 $[$[doi:10.1007/s001820050111](https://doi.org/10.1007/s001820050111)$]$

category: people

[[!redirects Robert Aumann]]


Robert J. Baston

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=129025)

## Selected writings

On [[twistors]]:

* {#BastonEastwood89} [[Robert J. Baston]], [[Michael G. Eastwood]], _The Penrose transform its interaction with representation theory_ Oxford Science Publications, Clarendon Press, 1989 (ISBN:9780198535652, [spire:290342](https://inspirehep.net/literature/290342), [pdf](http://www.math.utah.edu/~ptrapa/math-library/baston-eastwood/baston-eastwood.pdf))

category: people

[[!redirects Robert Baston]]


Robert J. Harris

* [Institute page](https://equs.org/users/rob-harris)

* [GoogleScholar page](https://scholar.google.com/citations?user=OsvwY2MAAAAJ&hl=en)

## Selected writings

On [[quantum error correcting codes]]:

* [[Robert J. Harris]], [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], *Calderbank-Steane-Shor Holographic Quantum Error Correcting Codes*, Phys. Rev. A 98, 052301 (2018) ([arXiv:1806.06472](https://arxiv.org/abs/1806.06472))

* [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], [[Robert J. Harris]], Elliot Coupe, *Decoding Holographic Codes with an Integer Optimisation Decoder* ([arXiv:2008.10206](https://arxiv.org/abs/2008.10206))

* [[Terry Farrelly]], [[Robert J. Harris]], [[Nathan A. McMahon]], [[Thomas M. Stace]], *Tensor-network codes* ([arXiv:2009.10329](https://arxiv.org/abs/2009.10329))

category: people

[[!redirects Robert Harris]]


Robert Langlands

* [Wikipedia entry](http://en.wikipedia.org/wiki/Robert_Langlands)

* [[James Arthur]], _The work of Robert Langlands_ ([arXiv:2307.02571](https://arxiv.org/abs/2307.02571))

## Selected writings

* [[Robert Langlands]], _Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele_, in Emil Artin and beyond --- class field theory and L-functions, European Math. Soc., 2015, pp. 175--209. Translated by [[James Milne]] as _Functoriality in the theory of automorphic forms: its discovery and aims_, ([pdf](https://www.jmilne.org/math/Documents/LanglandsF.pdf))

## Quotes
 {#Quotes}

From *[[Problems in the theory of automorphic forms -- 45 years later]]*, in the file "Video for Fesenko" ([[Langlands-VideoForFosenko.pdf:file]]):

> [p. 5](https://ncatlab.org/nlab/files/Langlands-VideoForFosenko.pdf#page=5): "Since Fields medals are taken quite seriously by a large number of mathematicians and by whatever part of the general public takes an interest in mathematics, the lack of perspective is regrettable. […] These specialists are all excellent mathematicians, but one wonders again whether the influence of these medals, and other prizes with wide recognition, on the course of mathematics and on our understanding of it is entirely beneficial."

> [p. 8](https://ncatlab.org/nlab/files/Langlands-VideoForFosenko.pdf#page=5): "I find, however, that [these specialists] are excessively focussed. We can, I am sure, not do without them; they however appear to prefer problems with circumscribed, more modest goals! I fear that those with a larger view of the subject will have to learn from them; on the other hand, they themselves, so far as I have observed, have no desire to learn anything *was ihnen nicht in den Kram paßt* [D: that does not suit them]."

## Related entries

* [[Langlands program]], [[geometric Langlands correspondence]]

  [[automorphic representation]]

  [[Langlands dual group]]

* _[[Problems in the theory of automorphic forms -- 45 years later]]_

category: people


Robert Leigh

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_Leigh)

## Selected writings

Introducing [[orientifolds]]:

* {#DaiLinPolchinski89} Jin Dai, [[Robert Leigh]], [[Joseph Polchinski]], p. 12 of _New Connections Between String Theories_, Mod. Phys. Lett. A4 (1989) 2073-2083 ([spire:25758](http://inspirehep.net/record/25758), [pdf scan](https://lib-extopc.kek.jp/preprints/PDF/1989/8905/8905564.pdf))

On [[chiral fermions]] in [[intersecting D-brane models]]:

* {#BerkoozDouglasLeigh96} [[Micha Berkooz]], [[Michael Douglas]], [[Robert Leigh]], _Branes Intersecting at Angles_, Nucl. Phys. B480:265-278, 1996 ([arXiv:hep-th/9606139](https://arxiv.org/abs/hep-th/9606139))

## Related $n$Lab entries

* [[intersecting D-brane models]]

category: people


Robert Loren Jaffe

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_Jaffe)

## Selected writings

On [[physical units]]:

* [[Robert Loren Jaffe]], *Natural Units and the Scales of Fundamental Physics*, Course Notes 2007 ([pdf](https://stuff.mit.edu/afs/athena/course/8/8.06/spring08/handouts/units.pdf), [[Jaffe_NaturalUnits.pdf:file]])

category: people

[[!redirects R. L. Jaffe]]
[[!redirects Robert L. Jaffe]]


Robert Lubarsky

* [personal page](http://www.math.fau.edu/people/faculty/lubarsky.php)

## Selected writings

On the [[Cauchy real numbers]] in [[constructive analysis]]:

* {#Lubarsky} [[Robert Lubarsky]], _On the Cauchy Completeness of the Constructive Cauchy Reals_, Electronic Notes in Theoretical Computer Science **167** (2007) 225-254 [[doi:10.1016/j.entcs.2006.09.012](https://doi.org/10.1016/j.entcs.2006.09.012)]

category: people


Robert MacPherson

* [Wikipedia entry](http://en.wikipedia.org/wiki/Robert_MacPherson_%28mathematician%29)

## Selected writings

The [[Fulton-MacPherson compactification]] of [[configuration spaces of points]]:

* {#FultonMacPherson94} [[William Fulton]], [[Robert MacPherson]], _A compactification of configuration spaces_, Ann. of Math. (2), 139(1):183–225, 1994 ([jstor:2946631](https://www.jstor.org/stable/2946631))

On [[persistent homology]]:

* [[Robert MacPherson]], Benjamin Schweinhart, _Measuring shape with topology_, J. Math. Phys. __53__, 073516 (2012); [doi](http://dx.doi.org/10.1063/1.4737391)

* [[Robert MacPherson]], Amit Patel, _Persistent Local Systems_ ([arXiv:1805.02539](https://arxiv.org/abs/1805.02539))

## Related $n$Lab entries

* [[Fulton-MacPherson compactification]]

* [[configuration spaces of points]]

* [[Hilbert scheme]]

category: people


Robert McDowell Thrall

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=4047)

## Selected writings

Introducing the [[hook length formula]] in the [[representation theory of the symmetric group]]:

* {#FrameRobinsonThrall54} [[James Sutherland Frame]], [[Gilbert de Beauregard Robinson]], [[Robert McDowell Thrall]], *The Hook Graphs of the Symmetric Group*, Canadian Journal of Mathematics, Volume 6, 1954, pp. 316 - 324 ([doi:10.4153/CJM-1954-030-1](https://doi.org/10.4153/CJM-1954-030-1))

category: people


Robert Mills

* [wikipedia entry](http://en.wikipedia.org/wiki/Robert_Mills_%28physicist%29)

## Selected writings

Introducing [[Yang-Mills theory]]:

* [[Chen Ning Yang]], [[Robert Mills]], _Conservation of Isotopic Spin and Isotopic Gauge Invariance_. Physical Review 96 (1): 191– 195. (1954) ([web](http://prola.aps.org/abstract/PR/v96/i1/p191_1))

## Related entries

* [[Yang-Mills theory]]

category: people


Robert Mosher

* [Mathematics Genealogy Page](http://www.genealogy.ams.org/id.php?id=12004)

## Selected writings

On [[stable homotopy groups]] of [[complex projective space]]:

* [[Robert E. Mosher]],  *Some stable homotopy of complex projective space*, Topology **7** (1968) 179-193 [doi:10.1016/0040-9383(68)90026-8, [pdf](https://core.ac.uk/download/pdf/82547916.pdf)]

On [[cohomology operations]]:

* [[Robert Mosher]], [[Martin Tangora]], _Cohomology Operations and Application in Homotopy Theory_, Harper and Row (1968), Dover (2008) ([ISBN10:0486466647](https://store.doverpublications.com/0486466647.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf))

## Related $n$Lab entries

* [[cohomology operation]]

* [[Steenrod algebra]]

* [[Hopf construction]]

* [[Hopf invariant]]

category: people

[[!redirects Robert E. Mosher]]


Robert Myers

* [webpage](https://www.perimeterinstitute.ca/people/robert-myers)

* Wikipedia entry

## Selected writings

On [[black holes]] in [[D=5 supergravity]]:

* {#GauntlettMyersTownsend98} [[Jerome Gauntlett]], [[Robert C. Myers]], [[Paul Townsend]], *Black Holes of $D=5$ Supergravity*, Class.Quant.Grav. **16** (1999) 1-21 [[arXiv:hep-th/9810204](http://arxiv.org/abs/hep-th/9810204), [doi:10.1088/0264-9381/16/1/001](https://doi.org/10.1088/0264-9381/16/1/001)]

On the [[non-abelian DBI action]] and introducing the [[Myers effect]]:

* {#Myers99} [[Robert Myers]], _Dielectric-Branes_, JHEP 9912 (1999) 022 ([arXiv:hep-th/9910053](https://arxiv.org/abs/hep-th/9910053))

On [[D1-D3-brane intersections]] as [[fuzzy funnels]]/[[BIons]] on [[fuzzy 2-spheres]]

* [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _The Noncommutative Bion Core_, Phys. Rev. D61 (2000) 106009 ([arXiv:hep-th/9911136](https://arxiv.org/abs/hep-th/9911136))

On [[D1-D5-brane intersections]] as [[fuzzy funnels]] on [[fuzzy 4-spheres]]:

* [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _Non-abelian Brane Intersections_, JHEP 0106:023, 2001 ([arXiv:hep-th/0102080](https://arxiv.org/abs/hep-th/0102080))

Generally on [[noncommutative geometry]] in [[D-brane]]-physics:

* [[Robert Myers]], _Nonabelian D-branes and Noncommutative Geometry_, J. Math. Phys. 42: 2781-2797, 2001 ([arXiv:hep-th/0106178](https://arxiv.org/abs/hep-th/0106178))

On [[S-branes]]:

* {#KruczenskiMyersPeet02} [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], _Supergravity S-Branes_, JHEP 0205 (2002) 039 ([arXiv:hep-th/0204144](https://arxiv.org/abs/hep-th/0204144))

On [[D-brane polarization]] into [[supertubes]]:

* [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], David J. Winters, _Aspects of supertubes_, JHEP 0205:017, 2002 ([arXiv:hep-th/0204103](https://arxiv.org/abs/hep-th/0204103))

## Related entries

* [[Myers effect]]

* [[ALE space]]

category: people

[[!redirects Robert C. Myers]]


Robert Oppenheimer

* [Wikipedia entry](http://en.wikipedia.org/wiki/J._Robert_Oppenheimer)

category: people


Robert Paré

* [website](http://www.mscs.dal.ca/~pare/)

## Selected writings

On [[indexed categories]]:

* [[P. T. Johnstone]], [[R. Paré]], [[R. D. Rosebrugh]], [[D. Schumacher]], [[R. J. Wood]], [[G. C. Wraith]], _Indexed Categories and Their Applications_, Lecture Notes in Mathematics **661** (1978) [[doi:10.1007/bfb0061362](https://doi.org/10.1007/bfb0061362)]

On the [[indexed adjoint functor theorem]]:

* [[Robert Paré]], Dietmar Schumacher, *Abstract families and the adjoint functor theorems*, in _Indexed categories and their applications_, Lecture Notes in Math. **661** Springer (1978) [[doi:10.1007/BFb0061361](https://doi.org/10.1007/BFb0061361)]

On [[category]]-valued [[stacks]] ([[2-sheaves]]) as [[internal categories]] in a [[sheaf topos]] and the notion of [[weak equivalences of internal categories]]:

* {#BungePare} [[Marta Bunge]], [[Robert Paré]], _Stacks and equivalence of indexed categories_, [[Cahiers|Cahiers de Top. et Géom. Diff. Catég]] **20** 4 (1979) 373-399 [[numdam:CTGDC_1979__20_4_373_0](http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0)]

On [[accessible categories]] and [[categorical semantics|categorical]] [[model theory]] via [[sketches]]:

* {#MakkaiPare} [[Michael Makkai]], [[Robert Paré]],  _Accessible categories: The foundations of categorical model theory_,  Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989 ([ISBN:978-0-8218-7692-3](https://bookstore.ams.org/conm-104))

On [[reflective factorization systems]]:

* {#CJKP} [[Aurelio Carboni]], [[George Janelidze]], [[Max Kelly]], [[Robert Paré]], _On localization and stabilization for factorization systems_, Appl. Categ. Structures 5 (1997), 1--58 ([doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444))

On [[2-colimits]] of [[accessible categories]] (i.e. in [[AccCat]]):

* [[Robert Paré]], [[Jiří Rosický]], *Colimits of accessible categories*, Math. Proc. Cambr. Phil. Soc. **155** (2013) 47-50 [[doi:10.1017/S0305004113000030](https://doi.org/10.1017/S0305004113000030), [arXiv:1110.0767](http://arxiv.org/abs/1110.0767)]

## Related entries

* [[model theory]] in [[categorical logic]]

* [[accessible category]]

## Collected writings

* [[Robert Paré]], _Embedding the Affine Group in the Projective Plane_, Master's thesis, McGill University, 1967. [[pdf](https://www.mscs.dal.ca/~pare/MScThesis.pdf)]

* [[Robert Paré]], _A Pappus Type Theorem in the Affine Group_, Canadian Mathematical Bulletin, Vol. 11, No. 4, 1968, pp. 547-554. [[doi:10.4153/CMB-1968-065-5](https://doi.org/10.4153/CMB-1968-065-5)]

* [[Robert Paré]], _Absoluteness Properties in Category Theory_, PhD thesis, McGill University, 1969. [[pdf](https://www.mscs.dal.ca/~pare/PhDThesis.pdf)]

* [[Robert Paré]], _Absolute coequalizers_, Lecture Notes in Mathematics, Vol. 86, 1969, pp. 132-145. [[doi:10.1007/BFb0079387](https://doi.org/10.1007/BFb0079387)]

* [[Robert Paré]], _On absolute colimits_, Journal of Algebra, Vol. 19, No. 1, 1971, pp. 80-95. [doi:10.1016/0021-8693(71)90116-5]

* [[Robert Paré]], _Connected components and colimits_, Journal of Pure and Applied Algebra, Vol. 3, No. 1, 1973, pp. 21-42. [doi:10.1016/0022-4049(73)90003-0]

* [[Robert Paré]], _Colimits in Topoi_, Bulletin of the American Mathematical Society, Vol. 80, No. 3, 1974, pp. 556-561. [[doi:10.1090/S0002-9904-1974-13497-X](https://doi.org/10.1090/S0002-9904-1974-13497-X)]

* [[Robert Paré]] and [[Peter T. Johnstone]] (editors), _Indexed Categories and Their Applications_, Lecture Notes in Mathematics, Vol. 661, 1978. [[doi:10.1007/BFb0061360](https://doi.org/10.1007/BFb0061360)]

* [[Robert Paré]] and [[Dietmar Schumacher]], _Abstract families and the adjoint functor theorems_, Lecture Notes in Mathematics, Vol. 661, 1978, pp. 1-125. [[doi:10.1007/BFb0061361](https://doi.org/10.1007/BFb0061361)]

* [[Robert Paré]] and [[William Schelter]], _Finite extensions are integral_, Journal of Algebra, Vol. 53, No. 2, 1978, pp. 477-479. [doi:10.1016/0021-8693(78)90293-4]

* [[F. E. J. Linton]] and [[Robert Paré]], _Injectives in topoi, I: Representing coalgebras as algebras_, Lecture Notes in Mathematics, Vol. 719, 1979, pp. 196–206. [[doi:10.1007/BFb0065272](https://doi.org/10.1007/BFb0065272)]

* [[P. T. Johnstone]], [[F. E. J. Linton]] and [[Robert Paré]], _Injectives in Topoi, II: Connections with the axiom of choice_, Lecture Notes in Mathematics, Vol. 719, 1979, pp. 207–216. [[doi:10.1007/BFb0065273](https://doi.org/10.1007/BFb0065273)]

* [[Marta Bunge]] and [[Robert Paré]], _Stacks and equivalence of indexed categories_, Cahiers de topologie et géométrie différentielle, Vol. 20, No. 4, 1979, pp. 373-399. [[cahiers](http://archive.numdam.org/item/CTGDC_1979__20_4_373_0/)]

* [[Michael Barr]] and [[Robert Paré]], _Molecular toposes_, Journal of Pure and Applied Algebra, Vol. 17, No. 2, 1980, pp. 127-152. [doi:10.1016/0022-4049(80)90080-8]

* [[Robert Paré]], _Indexed categories and generated topologies_, Journal of Pure and Applied Algebra, Vol. 19, 1980, pp. 385-400. [doi:10.1016/0022-4049(80)90107-3]

* [[L. Grunenfelder]], [[Robert Paré]] and [[H. Radjavi]], _On a commutator theorem of Robert C. Thompson_, Linear and Multilinear Algebra, Vol. 16, No. 1-4, 1984, pp. 129-131. [[doi:10.1080/03081088408817615](https://doi.org/10.1080/03081088408817615)]

* [[Robert Paré]], _A topos with no geometric morphism to any Boolean one_, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 97, No. 3, 1985, pp. 397-397. [[doi:10.1017/S0305004100062952](https://doi.org/10.1017/S0305004100062952)]

* [[Saunders Mac Lane]] and [[Robert Paré]], _Coherence for bicategories and indexed categories_, Journal of Pure and Applied Algebra, Vol. 37, 1985, pp. 59-80. [doi:10.1016/0022-4049(85)90087-8]

* [[Luzius Grunenfelder]] and [[Robert Paré]], _Families parametrized by coalgebras_, Journal of Algebra, Vol. 107, No. 2, 1987, pp. 316-375. [doi:10.1016/0021-8693(87)90093-7]

* [[G.M. Kelly]] and [[Robert Paré]], _A note on the Albert–Kelly paper “The closure of a class of colimits”_, Journal of Pure and Applied Algebra, Vol. 51, No. 1-2, 1988, pp. 19-25. [doi:10.1016/0022-4049(88)90074-6]

* [[Michael Makkai]] and [[Robert Paré]], _Accessible Categories: The Foundations of Categorical Model Theory_, Contemporary Mathematics, Vol. 104, 1989. [[doi:10.1090/conm/104](https://doi.org/10.1090/conm/104)]

* [[Robert Paré]], _Some Applications of Categorical Model Theory_, Categories in Computer Science and Logic, Contemporary Mathematics, Vol. 92, 1989, pp. . [[doi:10.1090/conm/092](https://doi.org/10.1090/conm/092)]

* [[Robert Paré]], [[R. Rosebrugh]] and [[R. J. Wood]], _Idempotents in bicategories_, Bulletin of the Australian Mathematical Society, Vol. 39, No. 3, 1989, pp. 421-434. [[doi:10.1017/S0004972700003336](https://doi.org/10.1017/S0004972700003336)]

* [[Robert Paré]] and [[Leopoldo Román]], _Monoidal categories with natural numbers object_, Studia Logica, Vol. 48, 1989, pp. 361-376. [[doi:10.1007/BF00370829](https://doi.org/10.1007/BF00370829)]

* [[Robert Paré]], _Simply Connected Limits_, Canadian Journal of Mathematics, Vol. 42, No. 4, 1990, pp. 731-746. [[doi:10.4153/CJM-1990-038-6](https://doi.org/10.4153/CJM-1990-038-6)]

* [[R. Dawson]] and [[Robert Paré]], _Characterizing tileorders_, Order, Vol. 10, 1993, pp. 111–128. [[doi:10.1007/BF01111295](https://doi.org/10.1007/BF01111295)]

* [[Robert Dawson]] and [[Robert Paré]], _General associativity and general composition for double categories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 34, No. 1, 1993, pp. 57-79. [[cahiers](http://archive.numdam.org/item/CTGDC_1993__34_1_57_0/)]

* [[Robert Paré]], _Universal Covering Categories_, Rendiconti dell'Istituto di Matematica dell'Università di Trieste, Vol. 25, 1993, pp. 391-411. [[pdf](https://rendiconti.dmi.units.it/volumi/25/25.pdf)]

* [[A. Carboni]], [[G. Janelidze]], [[G. M. Kelly]] and [[Robert Paré]], _On Localization and Stabilization for Factorization Systems_, Applied Categorical Structures, Vol. 5, 1997, pp. 1–58. [[doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444)]

* [[Robert Paré]], _A Visual Proof of Eddy and Fritsch's Minimal Area Property_, The College Mathematics Journal, Vol. 26, No. 1, 1995, pp. 43-44. [[doi:10.1080/07468342.1995.11973667](https://doi.org/10.1080/07468342.1995.11973667)]

* [[Robert Paré]] and [[Leopoldo Román]], _Dinatural numbers_, Journal of Pure and Applied Algebra, Vol. 128, No. 1, 1998, pp. 33-92. [doi:10.1016/S0022-4049(97)00036-4]

* [[Marco Grandis]] and [[Robert Paré]], _Limits in double categories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 40, No. 3, 1999, pp. 162-220. [[cahiers](http://archive.numdam.org/item/CTGDC_1999__40_3_162_0/)]

* [[Robert Paré]], _Contravariant Functors on Finite Sets and Stirling Numbers_, Theory and Applications of Categories, Vol. 6, No. 5, 1999, pp. 65-76. [[TAC](http://www.tac.mta.ca/tac/volumes/6/n5/6-05abs.html)]

* [[Robert Dawson]] and [[Robert Paré]], _What is a free double category like?_, Journal of Pure and Applied Algebra, Vol. 168, No. 1, 2002, pp. 19-34. [doi:10.1016/S0022-4049(01)00049-4]

* [[Robert Dawson]], [[Robert Paré]] and [[Dorette Pronk]], _Adjoining adjoints_, Advances in Mathematics, Vol. 178, No. 1, 2003, pp. 99-140. [doi:10.1016/S0001-8708(02)00068-3]

* [[Robert Dawson]], [[Robert Paré]] and [[Dorette Pronk]], _Undecidability of the Free Adjoint Construction_, Applied Categorical Structures, Vol. 11, 2003, pp. 403–419. [[doi:10.1023/A:1025712521140](https://doi.org/10.1023/A:1025712521140)]

* [[Robert Dawson]], [[Robert Paré]] and [[Dorette Pronk]], _Free extensions of double categories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 45, No. 1, 2004, pp. 35-80. [[cahiers](http://www.numdam.org/item/CTGDC_2004__45_1_35_0/)]

* [[Marco Grandis]] and [[Robert Paré]], _Adjoint for double categories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 45, No. 3, 2004, pp. 193-240. [[cahiers](http://archive.numdam.org/item/CTGDC_2004__45_3_193_0/)]

* [[Robert Dawson]], [[Robert Paré]] and [[Dorette Pronk]], _Universal properties of Span_, Theory and Applications of Categories, Vol. 13, No. 4, 2004, pp. 61-85. [[TAC](http://www.tac.mta.ca/tac/volumes/13/4/13-04abs.html)]

* [[Robert Dawson]], [[Robert Paré]] and [[Dorette Pronk]], _Paths in double categories_, Theory and Applications of Categories, Vol. 16, No. 18, 2006, pp. 460-521. [[TAC](http://www.tac.mta.ca/tac/volumes/16/18/16-18abs.html)]

* [[Marco Grandis]] and [[Robert Paré]], _Lax Kan extensions for double categories (on weak double categories, part IV)_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 48, No. 3, 2007, pp. 163-199. [[cahiers](http://archive.numdam.org/item/CTGDC_2007__48_3_163_0/)]

* [[Marco Grandis]] and [[Robert Paré]], _Kan extensions in double categories (On weak double categories, Part III)_, Theory and Applications of Categories, Vol. 20, No. 8, 2008, pp. 152-185. [[TAC](http://www.tac.mta.ca/tac/volumes/20/8/20-08abs.html)]

* [[Robert Dawson]], [[Robert Paré]] and [[Dorette Pronk]], _The span construction_, Theory and Applications of Categories, Vol. 24, No. 13, 2010, pp. 302-377. [[TAC](http://www.tac.mta.ca/tac/volumes/24/13/24-13abs.html)]

* [[Robert Paré]], _Mealy Morphisms of Enriched Categories_, Applied Categorical Structures, Vol. 20, 2012, pp. 251–273. [[doi:10.1007/s10485-010-9238-8](https://doi.org/10.1007/s10485-010-9238-8)]

* [[Toby Kenny]] and [[Robert Paré]], _Categories as monoids in Span, Rel and Sup_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 52, No. 3, 2011, pp. 209-240. [[cahiers](http://www.numdam.org/item/CTGDC_2011__52_3_209_0/)]

* [[Robert Paré]], _Yoneda theory for double categories_, Theory and Applications of Categories, Vol. 25, No. 17, 2011, pp. 436-489. [[TAC](http://www.tac.mta.ca/tac/volumes/25/17/25-17abs.html)]

* [[Marco Grandis]] and [[Robert Paré]], _From cubical to globular higher categories_, Diagrammes, Vol. S67-68, 2012, pp. 117-148. [[journal](http://www.numdam.org/item/DIA_2012__S67-68__117_0/)]

* [[Robert Paré]] and [[J. Rosický]], _Colimits of accessible categories_, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 155, No. 1, 2013, pp. 47-50. [[doi:10.1017/S0305004113000030](https://doi.org/10.1017/S0305004113000030)]

* [[Robert Paré]], _Composition of modules for lax functors_, Theory and Applications of Categories, Vol. 27, No. 16, 2013, pp. 393-444. [[TAC](http://www.tac.mta.ca/tac/volumes/27/16/27-16abs.html)]

* [[Robert Paré]], _Morphisms of Colimits: from Paths to Profunctors_, Applied Categorical Structures, Vol. 23, 2015, pp. 1–28. [[doi:10.1007/s10485-013-9357-0](https://doi.org/10.1007/s10485-013-9357-0)]

* [[Marco Grandis]] and [[Robert Paré]], _Intercategories_, Theory and Applications of Categories, Vol. 30, No. 38, 2015, pp. 1215-1255. [[TAC](http://www.tac.mta.ca/tac/volumes/30/38/30-38abs.html)]

* [[Marco Grandis]] and [[Robert Paré]], _Intercategories: A framework for three-dimensional category theory_, Journal of Pure and Applied Algebra, Vol. 221, No. 5, 2017, pp. 999-1054. [[doi:10.1016/j.jpaa.2016.08.002](https://doi.org/10.1016/j.jpaa.2016.08.002)]

* [[Marco Grandis]] and [[Robert Paré]], _An introduction to multiple categories (On weak and lax multiple categories, I)_, Cahiers de topologie et géométrie différentielle catégoriques, Vol. 57, No. 2, 2016, pp. 103-159. [[cahiers](http://cahierstgdc.com/index.php/volumes/volume-lvii-2016/)]

* [[Robert Paré]], _Wobbly double functors_, Tbilisi Mathematical Journal, Vol. 8, No. 1, 2015, pp. 63-84. [[doi:10.1515/tmj-2015-0005](https://doi.org/10.1515/tmj-2015-0005)]

* [[Marco Grandis]] and [[Robert Paré]], _Limits in multiple categories (On weak and lax multiple categories, II)_, Cahiers de topologie et géométrie différentielle catégoriques, Vol. 57, No. 3, 2016, pp. 163-202. [[cahiers](http://cahierstgdc.com/index.php/volumes/volume-lvii-2016/)]

* [[Marco Grandis]] and [[Robert Paré]], _Adjoints for multiple categories (On weak and lax multiple categories, III)_, Cahiers de topologie et géométrie différentielle catégoriques, Vol. 58, No. 1, 2017, pp. 3-48. [[cahiers](http://cahierstgdc.com/index.php/volume-lviii-2017/)]

* [[Marco Grandis]] and [[Robert Paré]], _Span and cospan representations of weak double categories_, Categories and General Algebraic Structures with Applications, Vol. 6, 2017, pp. 85-105. [[journal](https://cgasa.sbu.ac.ir/article_39606.html)]

* [[Marco Grandis]] and [[Robert Paré]], _A multiple category of multiple lax categories_, Cahiers de topologie et géométrie différentielle catégoriques, Vol. 58, No. 3-4, 2017, pp. 195-212. [[cahiers](http://cahierstgdc.com/index.php/volume-lviii-2017/)]

* [[Marco Grandis]] and [[Robert Paré]], _Persistent double limits_, Cahiers de topologie et géométrie différentielle catégoriques, Vol. 60, No. 3, 2019, pp. 255-297. [[cahiers](http://cahierstgdc.com/index.php/volume-lx/)]

* [[Robert Paré]], _Morphisms of Rings_, Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics, Outstanding Contributions to Logic, Vol. 20, 2021, pp. 271–298. [[doi:10.1007/978-3-030-66545-6_8](https://doi.org/10.1007/978-3-030-66545-6_8)]

* [[Robert Paré]], _Three Easy Pieces: Imaginary seminar talks in honour of Bob Rosebrugh_, Theory and Applications of Categories, Vol. 36, No. 6, 2021, pp. 171-200. [[TAC](http://www.tac.mta.ca/tac/volumes/36/6/36-06abs.html)]

* [[Robert Paré]], _Retrocells_, 2023. [[arXiv:2306.06436](https://arxiv.org/abs/2306.06436)]

category: people

[[!redirects Robert Pare]]
[[!redirects Robert Pare]]
[[!redirects Bob Pare]]
[[!redirects Bob Paré]]
[[!redirects Paré]]
[[!redirects R. Paré]]


Robert Penner

* [webpage as USC](http://physics.usc.edu/Faculty/Penner/)

* [webpage at IHES](https://www.ihes.fr/en/professeur/robert-c-penner-2/)

* [Mathematics Genealogy Page](http://www.genealogy.ams.org/id.php?id=11754)

## related $n$Lab entries

* [[super Riemann surface]]

category: people

[[!redirects Robert C. Penner]]


Robert Piacenza

* [Mathematics Genealogy Page](http://www.genealogy.ams.org/id.php?id=122354)

## Selected writings

On the [[projective model structure on enriched functors]] for the case of enrichment in the [[classical model structure on topological spaces]]:

* {#Piacenza91} Robert Piacenza, _Homotopy theory of diagrams and CW-complexes over a category_, Can. J. Math. Vol 43 (4), 1991 ([[Piazenza91.pdf:file]])

  also chapter VI of [[Peter May]] et a,, _Equivariant homotopy and cohomology theory_, 1996 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf))

## Related $n$Lab entries

* [[model structure on functors]]

* [[Elmendorf's theorem]]

* [[Model categories of diagram spectra]]

category: people


Robert Pisarski

* [webpage](https://www.bnl.gov/physics/NTG/people/pisarski.php)

## Selected writings

On the [[infinite-temperature thermal field theory]] of [[Yang-Mills theory]]:

* {#AppelquistPisarski81} [[Thomas Appelquist]], [[Robert Pisarski]], _High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics_, Phys. Rev. D **23** 2305 (1981) [[doi:10.1103/PhysRevD.23.2305](https://doi.org/10.1103/PhysRevD.23.2305)]

On the [[rho-meson]]  and its [[chiral partner]] [[a1-meson]] in the [[quark-gluon plasma]]:

* [[Robert Pisarski]], _Where does the Rho Go? Chirally Symmetric Vector Mesons in the Quark-Gluon Plasma_, Phys. Rev. D52 (1995) 3773-3776 ([arXiv:hep-ph/9503328](https://arxiv.org/abs/hep-ph/9503328))

category: people


Robert Rand

* [personal page](http://rand.cs.uchicago.edu/)

## Selected writings

Introducing *[[QWIRE]]*, a [[quantum programming language]] based on [[linear type theory]] connected with [[intuitionistic type theory]] via the [[exponential modality]]:

* {#PaykinRandZdancewic17} [[Jennifer Paykin]], [[Robert Rand]], [[Steve Zdancewic]], *QWIRE: a core language for quantum circuits*, POPL 2017: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming LanguagesJanuary 2017 Pages 846–858 ([doi:10.1145/3009837.3009894](https://doi.org/10.1145/3009837.3009894))

Application to [[verified programming]] after implementation in [[Coq]]:

* [[Robert Rand]], [[Jennifer Paykin]], [[Steve Zdancewic]], *QWIRE Practice: Formal Verification of Quantum Circuits in Coq*, EPTCS 266, 2018, pp. 119-132 ([arXiv:1803.00699](https://arxiv.org/abs/1803.00699))

* [[Robert Rand]], *Formally Verified Quantum Programming*, UPenn (2018) [[ediss:3175](https://repository.upenn.edu/edissertations/3175)]

  > [p. iv:] "We argue that [[quantum programs]] demand [[proof assistant|machine-checkable proofs]] of correctness. We justify this on the basis of the complexity of programs manipulating [[quantum states]], the expense of running [[quantum programs]], and the inapplicability of traditional debugging techniques to programs whose states cannot be examined."

  > [p. 3:] "Quantum programs are tremendously difficult to understand and implement, almost guaranteeing that they will have bugs. And traditional approaches to debugging will not help us: We cannot set breakpoints and look at our qubits without [[quantum state collapse|collapsing the quantum state]]. Even techniques like unit tests and random testing will be impossible to run on classical machines and too expensive to run on quantum computers – and failed tests are unlikely to be informative"

  > [p. 4:] "Thesis Statement: *[[quantum programming|Quantum programming]] is not only amenable to [[software verification|formal verification]]: it demands it.*"

  > "The overarching goal of this thesis is to write and verify quantum programs together. Towards that end, we introduce a quantum programming language called [[Qwire]] and embed it inside the [[Coq]] [[proof assistant]]. We give it a [[linear type system]] to ensure that it obeys the laws of [[quantum mechanics]] and a [[denotational semantics]] to prove that programs behave as desired."

* [[Robert Rand]], [[Jennifer Paykin]], [[Dong-Ho Lee]], [[Steve Zdancewic]], *ReQWIRE: Reasoning about Reversible Quantum Circuits*, EPTCS **287** (2019) 299-312 [[arXiv:1901.10118](https://arxiv.org/abs/1901.10118), [doi:10.4204/EPTCS.287.17](https://doi.org/10.4204/EPTCS.287.17)]

Further on [[quantum circuit]] [[certified programming|certification]] with the [[quantum programming language]] *[[SQIR]]*:

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Xiaodi Wu, [[Michael Hicks]], *A verified optimizer for Quantum circuits*, Proceedings of the ACM on Programming Languages **5** Issue POPL 37 (2021) 1–29 [[doi:10.1145/3434318](https://doi.org/10.1145/3434318)]

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Liyi Li, [[Michael Hicks]], *Proving Quantum Programs Correct*, in *12th International Conference on Interactive Theorem Proving (ITP 2021)*, Leibniz International Proceedings in Informatics (LIPIcs) **193** (2021) [[arXiv:2010.01240](https://arxiv.org/abs/2010.01240)]

category: people


Robert Raphael

* [institute page](https://www.concordia.ca/faculty/robert-raphael.html)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=30589)

## Selected writings

On [[product topological space|product spaces]] with [[Lindelöf topological spaces]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On productively  Lindelöf spaces*, Scient. Math. Japon. **65** (2007) 23–36 [[pdf](https://www.math.mcgill.ca/barr/papers/alster.pdf), [[BKR-ProductivelyLindeloef.pdf:file]]]

In [[Isbell duality]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Isbell Duality* Theory and Applications of Categories **20** 15 (2008) 504-542 [[tac:20-15](http://www.tac.mta.ca/tac/volumes/20/15/20-15abs.html)]

and specifically for [[modules]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Isbell duality for for modules*, Theory and Applications of Categories **22** 17 (2009) 401-419 [[tac:22-17](http://www.tac.mta.ca/tac/volumes/22/17/22-17abs.html), [pdf](https://www.math.mcgill.ca/barr/papers/rmod.pdf)]

On discrete [[dynamical systems]] (here called "flows") on [[compact Hausdorff spaces]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Flows: cocyclic and almost cocyclic*, Theory Appl. Categories **25** 18 (2010) 490–507 [[tac:25-18](http://www.tac.mta.ca/tac/volumes/25/18/25-18abs.html)]

On [[coherent spaces]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Countable meets in coherent spaces with applications to the cyclic spectrum*, Theory Appl. Categories **25** 19 (2011), 508–232 [[tac:25-19](http://www.tac.mta.ca/tac/volumes/25/19/25-19abs.html)]

On [[injective hulls]] of [[partial order|partially ordered]] [[monoids]]:

* [[Joachim Lambek]], [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Injective hulls of partially ordered monoids*, Theory Appl. Categories **26** 13 (2012) 338–348 [[tac:16-13](http://www.tac.mta.ca/tac/volumes/26/13/26-13abs.html)]

On [[star-autonomous categories]] of [[suplattice|sup]]-[[semilattices]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *The $\ast$-autonomous category of uniform sup semi-lattices*,  Theory & Applications of Categories **27** 11 (2012) 222–241 [[tac:27-11](http://www.tac.mta.ca/tac/volumes/27/11/27-11abs.html)]

On [[limits]] of [[integral domains]] among [[commutative rings]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Limit closures of classes of commutative rings*, Theory Appl. Categories **30** (2015) 229–304 [[tac:30-08](http://www.tac.mta.ca/tac/volumes/30/8/30-08abs.html), [pdf](https://www.math.mcgill.ca/barr/papers/domrefl.pdf)]

On [[reflective subcategory|reflective]] and [[coreflective subcategories]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On reflective and coreflective hulls*, [[Cahiers|Cahiers Topologie Géométrie Différentielle Catégorique]] **56** (2015) 162–208 [[pdf](https://www.math.mcgill.ca/barr/papers/biref.pdf), [[BarrKennisonRaphael-Reflective.pdf:file]]]

On contractibility of [[simplicial objects]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Contractible simplicial objects*, Comm. Math. Univ. Carol. **60** 4 (2019) 473–495 [[pdf](https://math.mcgill.ca/barr/papers/cso.pdf), [eudml:295068](https://eudml.org/doc/295068)]

On [[coequalizers]] and [[free monads]]:

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Coequalizers and free triples, II*, Theory Appl. Categories **34** (2019) 662–683 [[tac:34-23](http://www.tac.mta.ca/tac/volumes/34/23/34-23abs.html)]

category: people


Robert Raussendorf

* [institute page](https://phas.ubc.ca/~raussen/)

* [GoogleScholar page](https://scholar.google.com/citations?user=2mxCrYAAAAAJ&hl=en)

## Selected writings

Introducing [[measurement-based quantum computation]]:

* {#RaussendorfBriegel01} [[Robert Raussendorf]], [[Hans J. Briegel]], *A One-Way Quantum Computer*, Phys. Rev. Lett. **86** (2001) 5188 [[doi:10.1103/PhysRevLett.86.5188](https://doi.org/10.1103/PhysRevLett.86.5188)]

Review:

* [[Hans J. Briegel]], [[Dan E. Browne]], [[Wolfgang Dür]], [[Robert Raussendorf]], [[Maarten Van den Nest]], *Measurement-based quantum computation*, Nature Physics **5** 1 (2009) 19-26 [[arXiv:0910.1116](https://arxiv.org/abs/0910.1116), [doi:10.1038/nphys1157](https://doi.org/10.1038/nphys1157)]

Discussion of [[quantum contextuality]] via [[singular cohomology]]:

* [[Cihan Okay]], [[Sam Roberts]], [[Stephen D. Bartlett]], [[Robert Raussendorf]], *Topological proofs of contextuality in quantum mechanics*, Quantum Information and Computation **17** (2017) 1135-1166 [[arXiv:1701.01888](https://arxiv.org/abs/1701.01888), [doi:10.26421/QIC17.13-14-5](https://doi.org/10.26421/QIC17.13-14-5)]

category: people

[[!redirects Robert Raußendorf]]


Robert Rosebrugh

Bob Rosebrugh is a [[category theory|category theorist]], based in New Brunswick, Canada. He is the founding editor of [Theory and Applications of Categories](http://www.tac.mta.ca/tac/) and served as its managing editor for 25 years.

[webpage](http://www.mta.ca/~rrosebru/)

A short biography and list of publications of Bob Rosebrugh appeared in the foreword to [The Rosebrugh Festschrift](http://www.tac.mta.ca/tac/index.html#vol36):

* [[Marino Gran]], [[Michael Johnson]], [[Walter Tholen]], [[R. J. Wood]], _Foreword_, Theory and Applications of Categories, Vol. 36, 2021, pp 1-8. ([pdf](http://www.tac.mta.ca/tac/volumes/36/fwd/36-fwd.pdf))

## Selected writings:

On [[indexed categories]]:

* [[P. T. Johnstone]], [[R. Paré]], [[R. D. Rosebrugh]], [[D. Schumacher]], [[R. J. Wood]], [[G. C. Wraith]], _Indexed Categories and Their Applications_, Lecture Notes in Mathematics **661** (1978) [[doi:10.1007/bfb0061362](https://doi.org/10.1007/bfb0061362)]

Characterizing the [[category of sets]] as that whose [[Yoneda embedding]] extends to the [[left adjoint|left]] to an [[adjoint quintuple]] of [[adjoint functors]]:

* [[Robert Rosebrugh]], [[Richard J. Wood]], *An adjoint characterization of the category of sets*. PAMS **122** 2 (1994) 409-413 [[jstor:2161031](https://www.jstor.org/stable/2161031)]

On [[adjoint quadruples]] with a [[fully faithful functor|fully faithful]] [[right adjoint]]:

* [[Bob Rosebrugh]], [[Richard J. Wood]], *Distributive Adjoint Strings*, Theory and Applications of Categories, **1** 6 (1995) 119-145 [[tac:1-06](http://www.tac.mta.ca/tac/volumes/1995/n6/1-06abs.html)]

On [[distributive laws]] and [[factorization systems]]:

* {#RosebrughWood02} [[Robert Rosebrugh]], [[Richard J. Wood]], *Distributive Laws and Factorization*, Journal of Pure and Applied Algebra **175** 1–3 (2002) 327-353 [doi:10.1016/S0022-4049(02)00140-8[pdf](http://www.mta.ca/~rrosebru/articles/fac.pdf)]

Introduction to [[set theory]] and the [[foundations of mathematics]] from a practical and [[categorical logic|category theoretic]] point of view known as _[[structural set theory]]_ ("informal [[ETCS]]"):

* [[William Lawvere]], [[Robert Rosebrugh]], *[[Sets for Mathematics]]*, Cambridge UP (2003) [[doi:10.1017/CBO9780511755460](https://doi.org/10.1017/CBO9780511755460), [book homepage](http://www.mta.ca/~rrosebru/setsformath/), [GoogleBooks](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics), [pdf](http://patryshev.com/books/Sets%20for%20Mathematics.pdf)]

On [[lenses (in computer science)]]:

* {#JohnsonRosebrughWood10} [[Michael Johnson]], [[Robert Rosebrugh]], [[Richard Wood]], _Algebras and Update Strategies_, Journal of Universal Computer Science **16** (2010) [[doi:10.3217/jucs-016-05-0729](http://dx.doi.org/10.3217/jucs-016-05-0729)]

* [[Michael Johnson]], [[Robert Rosebrugh]], [[Richard J. Wood]], *Lenses, fibrations and universal translations* Math. Structures Comput. Sci. **22** 1 (2012) 25–42 [[doi:10.1017/S0960129511000442](https://doi.org/10.1017/S0960129511000442)]

* {#JohnsonRosebrugh16} [[Michael Johnson]], [[Robert Rosebrugh]], *Unifying Set-Based, Delta-Based and Edit-Based Lenses*, CEUR Workshop Proceedings **1571** (2016) [[pdf](http://ceur-ws.org/Vol-1571/paper_13.pdf)]

* [[Michael Johnson]], [[Robert Rosebrugh]], *The more legs the merrier: A new composition for symmetric (multi-)lenses*, EPTCS **333** (2021) 92-107 [[arXiv:2101.10482](https://arxiv.org/abs/2101.10482), [doi:10.4204/EPTCS.333.7](https://doi.org/10.4204/EPTCS.333.7)]

[[!redirects Bob Rosebrugh]]
[[!redirects Rosebrugh]]
[[!redirects R. Rosebrugh]]

[[!redirects R. D. Rosebrugh]]


Robert Schrader

* [obituary](http://www.math.uni-bonn.de/people/mueller/people/Obituary.pdf)

## Selected writings

On [[anyon]]/plekton-[[wavefunctions]] as [[multivalued functions]] on a [[configuration space of points]]:

* {#MundSchrader93} J. Mund, [[Robert Schrader]], *Hilbert Spaces for Nonrelativistic and Relativistic "Free" Plektons (Particles with Braid Group Statistics)*, in *Advances in dynamical systems and quantum physics* (Capri, 1993), World Sci. (1995) 235–259 $[$[arXiv:hep-th/9310054v2](https://arxiv.org/abs/hep-th/9310054)$]$

## Related entries

* [[Osterwalder-Schrader theorem]]

category: people


Robert Seely

* [webpage](http://www.math.mcgill.ca/rags/)

## Selected writings

On [[hyperdoctrines]], and [[natural deduction]] in [[categorical logic]], and the [[Beck-Chevalley condition]]:

* [[Robert A. G. Seely]], *Hyperdoctrines, Natural Deduction and the Beck Condition*, Zeitschr. f. math. Logik und Grundlagen d. Math. **29** (1983) 505-542 [[doi:10.1002/malq.19830291005]( https://doi.org/10.1002/malq.19830291005), [pdf](https://www.math.mcgill.ca/seely/ZML/ZML.PDF)]

On the [[relation between type theory and category theory]]...

...the [[equivalence of categories]] between [[first order logic|first order theories]] and [[hyperdoctrines]]:

* {#SeelyA} [[R. A. G. Seely]], _Hyperdoctrines, natural deduction, and the Beck condition_, Zeitschrift für Math. Logik und Grundlagen der Math. (1984) ([pdf](http://www.math.mcgill.ca/rags/ZML/ZML.PDF))

...the [[equivalence of categories]] between [[locally cartesian closed categories]] and [[dependent type theories]] was originally claimed in

* [[R. A. G. Seely]], _Locally cartesian closed categories and type theory_, Math. Proc. Camb. Phil. Soc. **95** (1984) 33-48  [[doi:10.1017/S0305004100061284](https://doi.org/10.1017/S0305004100061284), [pdf](http://www.math.mcgill.ca/rags/LCCC/LCCC.pdf)]

... the [[categorical semantics]] of [[linear logic]] in [[star-autonomous categories]]:

* {#Seely89} [[R. A. G. Seely]],  *Linear logic, $\ast$-autonomous categories and cofree coalgebras*, in *Categories in Computer Science and Logic*, Contemporary Mathematics **92** (1989)  [[[SeelyLinearLogic.pdf:file]], [ps.gz](http://www.math.mcgill.ca/rags/nets/llsac.ps.gz), [ISBN:978-0-8218-5100-5](https://bookstore.ams.org/conm-92)]

On [[categorical semantics]] of the [[exponential modality]] in [[linear type theory]]:

* {#BluteCockettSeely96} [[R. F. Blute]] , [[J. R. B. Cockett]], [[R. A. G. Seely]], *! and ? -- Storage as tensorial strength*, Mathematical Structures in Computer Science **6** 4 (1996) 313-351 [[doi:10.1017/S0960129500001055](https://doi.org/10.1017/S0960129500001055)]

On [[2-type theory]]:

* [[R.A.G. Seely]], *Modeling computations: a 2-categorical framework*, LICS 1987 ([pdf](http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf))

Introducing [[differential categories]]:

*  {#BluteCocketSeely06} [[R. F. Blute]], [[J. R. B. Cockett]], and [[R. A. G. Seely]]: _Differential categories_. Math. Struct. Comput. Sci. 16(06), 1049–1083 (2006) ([doi:10.1017/S0960129506005676](https://doi.org/10.1017/S0960129506005676))

* [[Richard Blute]], [[Robin Cockett]], [[Jean-Simon Lemay]] and [[Robert Seely]], _Differential categories revisited,_ Appl. Categ. Struct. 28, 171-235 (2020). ([arXiv:1806.04804](https://arxiv.org/abs/1806.04804), [doi:10.1007/s10485-019-09572-y](https://doi.org/10.1007/s10485-019-09572-y))

## Related $n$Lab entries

* [[relation between category theory and type theory]]

* [[categorical model of dependent types]]

* [[internal logic]]

category: people

[[!redirects Robert A. G. Seely]]
[[!redirects R. A. G. Seely]]
[[!redirects R.A.G. Seely]]
[[!redirects R.A.G.Seely]]


Robert Seiringer

* [webpage](http://pub.ist.ac.at/~rseiring/Robert_Seiringer/Robert_Seiringers_Homepage.html)

##Related concepts

* [[stability of matter]]

category: people


Robert Simmons

* [webpage](http://www.cs.cmu.edu/~rjsimmon/)

category: people


Robert Solovay

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_M._Solovay)

* [institute page](https://math.berkeley.edu/people/faculty/robert-m-solovay)

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=6522)

## Selected writings

On the [[Solovay-Kitaev theorem]]:

* [[Robert Solovay]], *Lie Groups and Quantum Circuits*, talk at MSRI (2000) [[webpage](https://www.msri.org/workshops/189/schedules/12826)]

category: people

[[!redirects Robert M.Solovay]]


Robert Spekkens

* [webpage](https://perimeterinstitute.ca/people/robert-spekkens)

## Related pages

* [[philosophy of physics]]

* [[interpretation of quantum mechanics]]

category: people

[[!redirects Rob Spekkens]]
[[!redirects Bob Spekkens]]


Robert Stong

* [Wikipedia entry](http://de.wikipedia.org/wiki/Robert_Stong)

## Selected writings

On the [[symplectic cobordism ring]]:

* [[Robert Stong]], _Some Remarks on Symplectic Cobordism_, Annals of Mathematics Second Series, Vol. 86, No. 3 (Nov., 1967), pp. 425-433 ([doi:10.2307/1970608](https://doi.org/10.2307/1970608))

On [[cobordism theory]], [[cobordism rings]] and [[Thom spectra]]:

* {#Stong68} [[Robert Stong]], _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html))

On [[equivariant bordism homology theory]]:

* {#Stong70} [[Robert Stong]], _Unoriented Bordism and Actions of Finite Groups_, Memoirs AMS 103, American Mathematical Society, 1970 ([ams:memo-1-103/4](https://bookstore.ams.org/memo-1-103/4))

On [[elliptic genera]]:

* {#LandweberRavenelStong93} [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], _Periodic cohomology theories defined by elliptic curves_, in [[Haynes Miller]] et al (eds.), _The Cech centennial: A conference on homotopy theory_, June 1993, AMS (1995) ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf))

category: people


Robert Switzer

* [mathematics genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=22494)

## Selected writing

On [[algebraic topology]] and [[stable homotopy theory]]:

* {#Switzer75} [[Robert Switzer]], _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 ([doi:10.1007/978-3-642-61923-6](https://link.springer.com/book/10.1007/978-3-642-61923-6))

## Related pages

* [[J-homomorphism]]

* [[algebraic topology]]

* [[stable homotopy theory]]

[[!redirects R. M. Switzer]]

category: people


Robert Thomason

Robert (Bob) Wayne Thomason (5 Nov 1952-Nov 1995) was an American mathematician. Towards the end of his short life, he worked in [[Max Karoubi]]'s lab. at Paris VII.  His main contributions were in [[homotopy theory]], [[algebraic geometry]] and [[algebraic K-theory]], where he carefully used advanced [[category theory|category theoretic]] methods.

* [biography](http://www-groups.dcs.st-and.ac.uk/history/Biographies/Thomason.html)

* [Digitization of Robert Wayne Thomason's notebooks](http://www.math-info-paris.cnrs.fr/bibli/digitization-of-robert-wayne-thomasons-notebooks/)

* [CatLab entry ](http://ncatlab.org/joyalscatlab/published/Thomason%2C+Robert)

## Selected writings

On [[homotopy fixed points]] ([[homotopy limits]] over [[group actions]]):

* [[Robert Thomason]], *The homotopy limit problem*, pages 407-419 in: [[Haynes Miller]], [[Stewart Priddy]] (eds.) *Proceedings of the Northwestern Homotopy Theory Conference*, Contemporary Mathematics **19**, AMS 1983 ([doi:10.1090/conm/019](http://dx.doi.org/10.1090/conm/019), [[ThomasonHomotopyLimit.pdf:file]])

On [[algebraic K-theory]]:

* R. W. Thomason, Thomas Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_, _The Grothendieck Festschrift_, 1990, 247-435.

Introducing the [[Thomason model structure]] on [[Cat]]:

*  {#Thomason1980} [[Robert W. Thomason]], *Cat as a closed model category*, Cahiers Topologie Géom. Différentielle **21** 3 (1980) 305-324  [[numdam:CTGDC_1980__21_3_305_0](http://www.numdam.org/item/?id=CTGDC_1980__21_3_305_0), MR0591388 (82b:18005)]

## Related $n$Lab entries

* [[Thomason model structure]]

* [[geometric realization of categories]]

* [[algebraic K-theory]]

* [[triangulated category]]

* [[thick subcategory theorem]]

category:people

[[!redirects Thomason]]
[[!redirects R. Thomason]]
[[!redirects R. W. Thomason]]
[[!redirects Bob Thomason]]
[[!redirects Robert W. Thomason]]
[[!redirects Robert Wayne Thomason]]


Robert Wald

* [Wikipedia entry](http://en.wikipedia.org/wiki/Robert_Wald)

## Selected writings

On [[general relativity]]:

* [[Robert Wald]], *General Relativity*, University of Chicago Press (1984) [[doi:10.7208/chicago/9780226870373.001.0001](https://doi.org/10.7208/chicago/9780226870373.001.0001), [pdf](http://fma.if.usp.br/~mlima/teaching/PGF5292_2021/Wald_GR.pdf)]

On [[AQFT on curved spacetime|QFT on curved spacetimes]] via [[operator product expansion]]:

* [[Stefan Hollands]], [[Robert Wald]], _Axiomatic quantum field theory in curved spacetime_, Commun. Math. Phys. 293:85-125, 2010 ([arXiv:0803.2003](https://arxiv.org/abs/0803.2003))

* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026), [doi:10.1016/j.physrep.2015.02.001](https://doi.org/10.1016/j.physrep.2015.02.001))

* [[Stefan Hollands]], [[Robert M. Wald]], *The Operator Product Expansion in Quantum Field Theory* [[arXiv:2312.01096](https://arxiv.org/abs/2312.01096)]

On [[electromagnetism]]:

* [[Robert Wald]], *Advanced Classical Electromagnetism*, Princeton University Press (2022) [[ISBN:9780691220390](https://press.princeton.edu/books/hardcover/9780691220390/advanced-classical-electromagnetism)]

## Related entries

* [[gravity]]

* [[quantum field theory on curved spacetime]]

* [[Bekenstein-Hawking entropy]]

* [[Hadamard distribution]]

* [[inhomogeneous cosmology]]

category: people

[[!redirects Robert M. Wald]]


Robert Wilson > history

see *[[Robert A. Wilson]]*


Robert Wisbauer

__Robert Wisbauer__ is an [[algebraist]] at the University of Duesseldorf.

* [webpage](http://www.math.uni-duesseldorf.de/~wisbauer)

## Selected writings

On [[distributive laws]] between [[monads]] and [[comonads]];

* Bachuki Mesablishvili, [[Robert Wisbauer]], *Bimonads and Hopf monads on categories*, Journal of K-Theory **7** 2 (2011) 349-388  [[arXiv:0710.1163](https://arxiv.org/abs/0710.1163), [doi:10.1017/is010001014jkt105](https://doi.org/10.1017/is010001014jkt105)]

## Related pages

* [[coring]],

* [[Sweedler coring]], [[Amitsur complex]]

category: people

[[!redirects R. Wisbauer]]


Robert Wolak

* [webpage](https://nauka-polska.pl/#/profile/scientist?id=28142&_k=er9mwz)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Robert_Wolak)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=60443)

## Selected writings

On [[G-structures]] on [[orbifolds]]:

* [[Robert Wolak]], _Orbifolds, geometric structures and foliations. Applications to harmonic maps_, Rendiconti del seminario matematico - Universita politecnico di Torino vol. 73/1 , 3-4 (2016), 173-187 ([arXiv:1605.04190](https://arxiv.org/abs/1605.04190))

category: people