Mark Van Raamsdonk

* [webpage](http://www.phas.ubc.ca/~mav/vanraamsdonk.html)

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

## Selected writings

On the [[non-abelian DBI action]] for [[intersecting branes]] with non-abelian [[gauge enhancement]] on their worldvolume:

* [[Washington Taylor]], [[Mark Van Raamsdonk]], _Multiple D$p$-branes in Weak Background Fields_, Nucl. Phys. B573:703-734, 2000 ([arXiv:hep-th/9910052](https://arxiv.org/abs/hep-th/9910052))

On the [[BMN matrix model]]:

* {#DSJVR02} [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Matrix Perturbation Theory For M-theory On a PP-Wave_, JHEP 0205:056, 2002 ([arXiv:hep-th/0205185](https://arxiv.org/abs/hep-th/0205185))

* [[Keshav Dasgupta]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Protected Multiplets of M-Theory on a Plane Wave_, JHEP 0209:021, 2002 ([arXiv:hep-th/0207050](https://arxiv.org/abs/hep-th/0207050))

On [[M5-branes]] in the [[BMN matrix model]]:

* [[Juan Maldacena]], [[Mohammad Sheikh-Jabbari]], [[Mark Van Raamsdonk]], _Transverse Fivebranes in Matrix Theory_, JHEP 0301:038, 2003 ([arXiv:hep-th/0211139](https://arxiv.org/abs/hep-th/0211139))

and a limit where aspects of [[little string theory]] become visible:

* Henry Ling, Ali Reza Mohazab, Hsien-Hang Shieh, Greg van Anders, [[Mark Van Raamsdonk]], *Little String Theory from a Double-Scaled Matrix Model*, JHEP 0610:018 (2006) $[$[arXiv:hep-th/0606014](https://arxiv.org/abs/hep-th/0606014), [doi:10.1088/1126-6708/2006/10/018](https://doi.org/10.1088/1126-6708/2006/10/018)$]$

On the [[mass-deformed ABJM model]] and the [[Myers effect]] for [[M2 branes]] ([[M2-M5 brane bound states]]):

* [[Jaume Gomis]], Diego Rodriguez-Gomez, [[Mark Van Raamsdonk]], [[Herman Verlinde]], _A Massive Study of M2-brane Proposals_, JHEP 0809:113, 2008 ([arXiv:0807.1074](https://arxiv.org/abs/0807.1074))

On a variant of the [[Witten-Sakai-Sugimoto model]] for [[holographic QCD]] with [[D4-brane|D4]] [[flavor branes]]:

* [[Mark Van Raamsdonk]], Kevin Whyte, _Baryon charge from embedding topology and a continuous meson spectrum in a new holographic gauge theory_, JHEP 1005:073, 2010 ([arxiv:0912.0752](https://arxiv.org/abs/0912.0752))

On [[holographic entanglement entropy]]:

* [[Mark Van Raamsdonk]], _Building up spacetime with quantum entanglement_, Gen. Rel. Grav. 42:2323-2329, 2010; Int. J. Mod. Phys.D19:2429-2435,2010 ([arXiv:1005.3035](https://arxiv.org/abs/1005.3035))

* [[Mark Van Raamsdonk]], _Building up spacetime with quantum entanglement II: It from BC-bit_ ([arXiv:1809.01197](https://arxiv.org/abs/1809.01197))

* [[Mark Van Raamsdonk]], _Lectures on Gravity and Entanglement_, chapter 5 in  New Frontiers in Fields and Strings, TASI 2015 Proceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics 2015 Theoretical Advanced Study Institute in Elementary Particle Physics ([arXiv:1609.00026](https://arxiv.org/abs/1609.00026))

* {#HMPSR19}  Felix M. Haehl, Eric Mintun, Jason Pollack, Antony J. Speranza, [[Mark Van Raamsdonk]], _Nonlocal multi-trace sources and bulk entanglement in holographic conformal field theories_, 	J. High Energ. Phys. (2019) 2019: 005
([arxiv:1904.01584](https://arxiv.org/abs/1904.01584), [talk recording](https://youtu.be/kRCwzyliJ1M))

and specifically for [[AdS2/CFT1]] with the [[BFSS matrix model]]:

* Tarek Anous, Joanna L. Karczmarek, Eric Mintun, [[Mark Van Raamsdonk]], Benson Way, _Areas and entropies in BFSS/gravity duality_ ([arXiv:1911.11145](https://arxiv.org/abs/1911.11145))

On [[D-branes]]:

> *[The Amazing Matrix in String Theory](http://www.physics.mcgill.ca/seminars/PSC_van-raamsdonk.html)* (2006)

> Most of the dramatic progress in string theory over the past decade has in some way involved the degrees of freedom of D-branes, solitonic objects in string theory whose low energy physics is described by quantum field theories living on the branes. Essential to many of the applications of D-brane physics is the fact that the number of degrees of freedom of a collection of branes scales not as the number of branes, but as the square of the number of branes. In particular, the fields describing the low energy physics of the branes, including even the coordinates describing the transverse fluctuations of D-branes, become matrix valued. At the classical level, this implies that D-brane configurations are described by a new kind of geometry, with such dramatic possibilities as objects that can change their dimension dynamically. Quantum mechanically, these matrix degrees of freedom can give rise to quantum field theories unlike any studied previously, and most dramatically, to new nonperturbative formulations of string theory and quantum gravity. In this talk, I will review some basic aspects of D-branes and their matrix degrees of freedom, some of the fascinating classical and quantum phenomena associated with these matrix degrees of freedom, and some aspects of the truly remarkable new nonperturbative formulations of string theory.

## Related $n$Lab entries

* [[holographic entanglement entropy]]

[[!redirects Mark van Raamsdonk]]

category: people


Mark van Raamsdonk > history

see at *[[Mark Van Raamsdonk]]*


Mark Weber

* [website](http://sites.google.com/site/markwebersmaths/)

## Selected writings

On [[Yoneda structures]] and [[2-toposes]]:

* [[Mark Weber]], *Yoneda structures from 2-toposes*, Appl Categor Struct **15** (2007) 259–323 [[doi:10.1007/s10485-007-9079-2](https://doi.org/10.1007/s10485-007-9079-2), [pdf](https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes)]

Proving the [[biequivalence]] between [[Feynman categories]] and [[colored operads]]:

* {#BataninKockWeber15} [[Michael Batanin]], [[Joachim Kock]], [[Mark Weber]], _Regular patterns, substitudes, Feynman categories and operads_, Theory Appl. Categ. 33 (2018), 148--192 ([arXiv:1510.08934](https://arxiv.org/abs/1510.08934))

category: people


marked 2-limit

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

## Motivation

In [[category theory]], there is one sensible notion of [[limit]], which is defined in terms of [[universal property|universal]] [[cones]]. However, when we move to [[enriched category theory]], such limits (which we now call **conical limits** to distinguish them from more general notions of limits) are no longer sufficient. By "sufficient", we mean here that there are notions in enriched category theory that are limit-like (such as [[powers]]) but that are not examples of conical limits. Furthermore, it is not true that the [[presheaf category|copresheaf construction]] exhibits the free completion under conical limits. This motivates the move to [[weighted limits]], which are the appropriate notion of limit for enriched categories. In particular, the copresheaf construction is the free completion under weighted limits, and powers are examples of weighted limits.

For a general [[monoidal category]] $V$, weighted limits are more general than conical limits. However, when $V = Set$, every weighted limit can be expressed as a conical limit. This is useful, as the conical limits are simpler to describe than weighted limits.

A particular [[base of enrichment]] of interest is $Cat$, so that enriched categories are precisely [[2-categories]]. The appropriate notion of limit for 2-categories is a weighted 2-limit, i.e. the usual notion of weighted limit for enriched categories (see [[2-limit]] for more details). Unlike for $V = Set$, conical 2-limits do not suffice to capture all weighted 2-limits. Intuitively, cones only capture 1-cells, whereas it is also necessary to capture 2-cells for the appropriate notion of 2-limit.

However, it turns out that, just as every weighted limit for $V = Set$ can be reduced to a conical limit, every weighted limit for $V = Set$ can be reduced to a simpler kind of limit, called a **marked limit**. This is useful, as it makes working with 2-limits easier than working with general weighted limits.

## History and terminology

While the notion of marked 2-limit goes back a long way, for one reason or another they have remained overlooked until recently.

* Marked 2-limits were introduced by [[John Gray]] in [[Adjointness for 2-Categories]] as **cartesian quasi-limits**.
* [[Ross Street]] proved the equivalence between marked 2-limits and weighted 2-limits.
* Szyld studied such 2-limits, calling them **$\sigma$-s-limits**.
* The terminology **marked** appears in [Gagna–Harpaz–Lanari](#GHL22).
* A particular class of marked 2-limits, called **lax normal conical 2-limits**, (which suffices to capture all 2-limits) was studied by Mesiti.

## Definition

Let $F : A \to B$ be a [[2-functor]] between [[2-categories]], and let $A'$ be a locally full sub-2-category of $A$ (alternatively: a class of morphisms of $A$ closed under identities and composition). A **marked-lax limit** of $(A', F)$ is an object $m_l lim (A', F)$ together with a family of isomorphisms
$$B(b, m_l lim (A', F)) \cong [A, B]_{l, A'}(\Delta(b), F)$$
natural in $b \in B$, where $[A, B]_{l, A'}(\Delta(b), F)$ is the 2-category of [[2-functors]] $A \to B$, marked-lax natural transformations ([[lax natural transformations]] $\alpha$ such that $\alpha_f = 1$ if $f \in A'$), and [[modifications]].

## Related pages

* [[2-limit]]
* [[double limit]]
* [[F-category]]
* [[marked simplicial set]]

## References

The concept is introduced in the following, where marked 2-limits are called *cartesian quasi-limits*:

* [[Adjointness for 2-Categories]]

It was generalised in section 0.2 of:

* [[Robert Blackwell]], _Some existence theorems in the theory of doctrines_, PhD thesis, UNSW Sydney, 1976.

The equivalence to weighted 2-limits is first proven in:

* {#StreetLimitsIndexed} [[Ross Street]], _Limits indexed by category-valued 2-functors_ Journal of Pure and Applied Algebra **8**, Issue 2 (1976) pp 149-181. doi:[10.1016/0022-4049(76)90013-X](https://doi.org/10.1016/0022-4049(76%2990013-X)

* M.E. Descotte, [[Eduardo J. Dubuc]], M. Szyld, _Sigma limits in 2-categories and flat pseudofunctors_, (v1: On the notion of flat 2-functors) [arXiv:1610.09429](https://arxiv.org/abs/1610.09429) Adv. Math. __333__ (2018) 266--313

* Martin Szyld. _Lifting PIE limits with strict projections_ (2018), ([arXiv:1809.04712](https://arxiv.org/abs/1809.04712))

* [[Luca Mesiti]], *The 2-Set-enriched Grothendieck construction and the lax normal conical 2-limits* (2023), [[arXiv:2302.04566](https://arxiv.org/abs/2302.04566)]

On **marked bilimits**:

* {#GHL22} Andrea Gagna, Yonatan Harpaz, and Edoardo Lanari, _Bilimits are bifinal objects_, Journal of Pure and Applied Algebra 226.12 (2022): 107137.

* [[Ivan Di Liberti]], Axel Osmond, *Bi-accessible and bipresentable 2-categories.* [[arXiv:2203.07046](https://arxiv.org/abs/2203.07046)]

A proof of the equivalence between weighted 2-limits and marked 2-limits is given in the following, as well as a generalisation of these ideas to weighted limits for [[F-categories]]:

* Joanna Ko. _Dotted Limits_ (2023), ([arXiv:2306.01625](https://arxiv.org/abs/2306.01625))

[[!redirects marked 2-limits]]
[[!redirects marked limit]]
[[!redirects marked limits]]
[[!redirects marked 2-colimit]]
[[!redirects marked 2-colimits]]
[[!redirects marked colimit]]
[[!redirects marked colimits]]


marked extensional well-founded order

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

\tableofcontents

## Idea

Marked extensional well-founded orders are important in constructing the [[sets]] of the [[cumulative hierarchy]] $V$ in a model of set theory, in the same way that [[well-orders]] or sets with [[transitive relation|transitive]] [[extensional relation|extensional]] [[well-founded relations]] are important in constructing the [[ordinals]] of $V$. These become important in [[constructive mathematics]] where one cannot in general show that the set of [[cardinals]] in $V$ and the set of ordinals in $V$ are in bijection with each other.

## Definition

A **marked extensional well-founded order** or **mewo** is a set $S$ with an [[extensional relation|extensional]] [[well-founded relation]] $a \prec b$ on a set $S$ and a [[predicate]] $m(a)$ on $S$.

## Properties

### Marked elements

An element $x \in S$ is **marked** if $m(x)$ is true. By the [[axiom schema of bounded separation]], one can construct the subset of marked elements in $S$:

$$M_S \coloneqq \{x \in S \vert m(x)\}$$

### Marked extensional well-founded order homomorphisms

A **marked extensional well-founded order homomorphism** between two marked extensional well-founded order $S$ and $T$ is a function $f:S \to T$ such that

* $f$ preserves the marking predicate: for all $x \in S$, $m(x)$ implies that $m(f(x))$

* $f$ preserves the well-founded relation: for all $x \in S$ and $y \in S$, $x \prec y$ implies that $f(x) \prec f(y)$

A **marked extensional well-founded order isomorphism** is a marked extensional well-founded order homomorphism $f:S \to T$ with a marked extensional well-founded order homomorphism $g:T \to S$ such that for all elements $s \in S$, $g(f(s)) = s$ and for all elements $t \in T$ $f(g(t)) = t$.

The category of marked extensional well-founded order homomorphisms is the [[concrete category]] $\mathrm{MEWO}$ whose [[objects]] are marked extensional well-founded orders and whose [[morphisms]] are marked extensional well-founded order homomorphisms.

### Trivialization

There is a [[forgetful functor]] $U:\mathrm{MEWO} \to \mathrm{EWO}$ which takes a marked extensional well-founded order $(S, \prec, m)$ to its underlying extensional well-founded order $(S, \prec)$. There is also a [[functor]] $F:\mathrm{EWO} \to \mathrm{MEWO}$ which takes an extensional well-founded order $(S, \prec)$ to the marked extensional well-founded order $(S, \prec, \top_S)$, where $\top_S$ is the always true predicate on $S$. The **trivialization** of a marked extensional well-founded order is defined as the composite of the two functors above $\overline{S} \coloneqq F(U(S))$.

### Simulations

A [[simulation]] between marked extensional well-founded orders $S$ and $T$ is a marked extensional well-founded order homomorphism $f:S \to T$ such that its [[image]] is downwards closed: for all $x \in S$ and $y \in T$, if $y \prec f(x)$, then there exists $z \in S$ such that $z \prec x$ and $f(z) \prec y$.

We denote the set of simulations between marked extensional well-founded orders $S$ and $T$ by $S \sqsubseteq T$.

The category $\mathrm{MEWO}_\mathrm{Sim}$ is the [[subcategory]] of $\mathrm{MEWO}$ whose objects are marked extensional well-founded orders and whose [[morphisms]] are simulations. We show that the above category is a [[poset]]:

Since the relation $\prec$ is extensional and well-founded, the following is true of simulations:

* Every simulation $f:S \to T$ is an [[injection]].

* Given marked extensional well-founded orders $S$ and $T$, the set of simulations $S \sqsubseteq T$ has at most one element.

As a result, one could define the relation $S \leq T$ on marked extensional well-founded orders by $\exists x.x \in S \sqsubseteq T$. The relation $S \leq T$ between marked extensional well-founded orders is a [[partial order]]

* for all marked extensional well-founded orders $S$, $S \leq S$
* for all marked extensional well-founded orders $S$, $T$, and $U$, $S \leq T \wedge T \leq U \implies S \leq U$
* for all marked extensional well-founded orders $S$ and $T$, $S \leq T \wedge T \leq S \implies S = T$

### Covered elements and initial segments

An element $x \in S$ is **covered** if there exists a marked element $y \in S$ such that $x \prec y$ is true. Given a marked extensional well-founded order $S$ and an element $x \in S$, the **initial segment** of $S$ is the set of all elements covered by $x \in S$

$$S \downarrow^+ x \coloneqq \{t \in S \vert t \prec x\}$$

The initial segment $S \downarrow^+ x$ is a marked extensional well-founded order, with the well-founded relation defined as $y \prec z$ for elements $y \in S \downarrow^+ x$ and $z \in S \downarrow^+ x$, and the marking relation defined as $m_x(y) \coloneqq y \prec x$ for elements $y \in S \downarrow^+ x$.

A marked extensional well-founded order is **covered** if every element $x \in S$ is covered; i.e.

$$\forall x \in S.\exists y \in S.m(y) \wedge (x \prec y)$$

It follows by well-founded induction that $S \downarrow^+ x$ is a covered marked extensional well-founded order.

\begin{theorem}
Given a marked extensional well-founded order $S$, the function $f:S \downarrow^+ x \to \overline{S}$, defined by the composite of the canonical subset injection $i:S \downarrow^+ x \to S$ and the canonical function $j:S \to \overline{S}$ given by the identity function on the carrier set, is a simulation.
\end{theorem}

\begin{theorem}
Given a marked extensional well-founded order $S$ and elements $x \in S$ and $y \in S$, $S \downarrow^+ x = S \downarrow^+ y$ implies that $x = y$.
\end{theorem}

\begin{theorem}
Given marked extensional well-founded orders $S$ and $T$ and a function $f:S \to T$ which preserves the marking predicate; i.e. $m(x)$ implies that $m(f(x))$ for all $x \in S$, then $f$ is a simulation if and only if $S \downarrow^+ x = T \downarrow^+ f(x)$.
\end{theorem}

### Bounded simulations

A bounded simulation between marked extensional well-founded orders $S$ and $T$ consists of a marked element $y \in M_T$ and a simulation $f:S \to T$ such that the [[image]] of $f$ is equal to the initial segment at $y$, $\mathrm{im}(f) = T \downarrow^+ y$. We denote the set of bounded simulations between marked extensional well-founded orders $S$ and $T$ by $S \sqsubset T$.

\begin{theorem}
Given marked extensional well-founded orders $S$ and $T$, the set of bounded simulations $S \sqsubset T$ has at most one element.
\end{theorem}

As a result, one could define the relation $S \lt T$ on marked extensional well-founded orders by $\exists x.x \in S \sqsubset T$. The relation satisfies the following properties

* for all marked extensional well-founded orders $S$ and $T$, $S \lt T$ implies that $S \leq \overline{T}$

* for all marked extensional well-founded orders $S$ and $T$ and $U$, $S \lt T$ and $T \lt U$ implies that $S \leq \overline{U}$

* for all marked extensional well-founded orders $S$ and $T$ and $U$, $S \lt T$ and $T \leq U$ implies that $S \lt U$

## Related concepts

* [[well-order]]

* [[extensional relation]]

* [[well-founded relation]]

## References

* {#deJongEtAl23} [[Tom de Jong]], [[Nicolai Kraus]], [[Fredrik Nordvall Forsberg]], [[Chuangjie Xu]], *Set-Theoretic and Type-Theoretic Ordinals Coincide* ([arXiv:2301.10696](https://arxiv.org/abs/2301.10696))

[[!redirects marked extensional well-founded order]]
[[!redirects marked extensional well-founded orders]]

[[!redirects mewo]]
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[[!redirects MEWO]]
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[[!redirects covered marked extensional well-founded order]]
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[[!redirects marked extensional well-founded relation]]
[[!redirects marked extensional well-founded relations]]

[[!redirects covered marked extensional well-founded relation]]
[[!redirects covered marked extensional well-founded relations]]

[[!redirects marked element]]
[[!redirects marked elements]]

[[!redirects subset of marked elements]]
[[!redirects subsets of marked elements]]

[[!redirects covered element]]
[[!redirects covered elements]]


marked simplicial set > history

< [[marked simplicial set]]


Marko Berghoff

* [webpage](http://www2.mathematik.hu-berlin.de/~berghoff/)

## Selected writings

* [[Marko Berghoff]], _Wonderful renormalization_, 2014 ([pdf](http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf), [doi:10.18452/17160](https://doi.org/10.18452/17160))

* [[Marko Berghoff]], _Wonderful compactifications in quantum field theory_,  Communications in Number Theory and Physics Volume 9 (2015) Number 3 ([arXiv:1411.5583](https://arxiv.org/abs/1411.5583))

## Related $n$Lab entries

* [[Feynman amplitudes on compactified configuration spaces of points]]

* [[wonderful compactification]]

* [[Chern-Simons propagator]]

category: people


Marko Stošić

* [Mathematics Genealogy Page](http://genealogy.math.ndsu.nodak.edu/id.php?id=171810)

## Selected writings

Introducing the [[knots-quivers correspondence]]:

* {#KRSS19} [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *Knots-quivers correspondence*, Adv. Theor. Math. Phys. **23** 7 (2019) 1849-1902 [[arXiv:1707.04017](https://arxiv.org/abs/1707.04017), [doi:10.4310/ATMP.2019.v23.n7.a4](https://doi.org/10.4310/ATMP.2019.v23.n7.a4)]

* [[Piotr Kucharski]], [[Markus Reineke]], [[Marko Stosic]], [[Piotr Sułkowski]], *BPS states, knots and quivers*, Phys. Rev. D **96** 121902 (2017) [[arXiv:1707.02991](https://arxiv.org/abs/1707.02991), [doi:10.1103/PhysRevD.96.121902](https://doi.org/10.1103/PhysRevD.96.121902)]

Further discussion:

* [[Marko Stošić]], *Generalized knots-quivers correspondence* [[arXiv:2402.03066](https://arxiv.org/abs/2402.03066)]

## Related entries

* [[knot invariant]]

category: people

[[!redirects Marko Stosic]]