open morphism

> This page is about the general concept. For open [[continuous functions]] see at _[[open map]]_.

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## Definition

### For maps between topological spaces

A [[function]]  $f : X \to Y$ between [[topological space]]s  is called an __[[open map]]__ if the [[image]] of every [[open set]] in $X$ is also open in $Y$.

Recall that $f$ is a __[[continuous map]]__ if the [[preimage]] of every [[open subspace|open set]] in $Y$ is open in $X$.  For defining open maps typically one restricts attention to open [[continuous map]]s, although it also makes sense to speak of open functions that are not continuous.

#### Examples

* For any two topological spaces $X$, $Y$, the projection map $\pi \colon X \times Y \to Y$ out of the [[product topological space]] is open.

* If $G$ is a [[topological group]] and $H$ is a subgroup, then the projection to the coset space $p \colon G \to G/H$, where $G/H$ is provided with the quotient topology (making $p$ a quotient map), is open. This follows easily from the observation that if $U$ is open in $G$, then so is
$$p^{-1}(p(U)) = U H = \bigcup_{h \in H} U h$$

* If $p \colon A \to B$ and $q \colon C \to D$ are open maps, then their product $p \times q \colon A \times C \to B \times D$ is also an open map.

### For morphisms between locales

A continuous map $f\colon X \to Y$ of topological spaces defines a homomorphism $f^*\colon Op(Y) \to Op(X)$ between the [[frames]] of open sets of $X$ and $Y$.  If $f$ is open, then this frame homomorphism is also a [[complete lattice|complete]] [[Heyting algebra]] homomorphism; the converse holds if $Y$ is a [[T-D space]].  Accordingly, we define a map $f\colon X \to Y$ of [[locales]] to be __open__ if it is, as a frame homomorphism $f^*\colon Op(Y) \to Op(X)$, a complete Heyting algebra homomorphism, i.e. it preserves arbitrary [[meets]] and the Heyting implication.

This is equivalent to saying that $f^*\colon Op(Y) \to Op(X)$ has a left adjoint $f_!$ (by the [[adjoint functor theorem]] for posets) which satisfies the [[Frobenius reciprocity]] condition that $f_!(U \cap f^* V) = f_!(U) \cap V$.

In the special case that $Y$ is the one-point locale, the Frobenius reciprocity condition is automatically satisfied.

### For geometric morphisms of toposes

[[categorification|Categorifying]], a [[geometric morphism]] $f\colon X \to Y$ of [[toposes]] is an [[open geometric morphism]] if its [[inverse image functor]] $f^*\colon Y \to X$ is a [[Heyting functor]].

### For morphisms in a topos

A [[class]] $R \subset Mor(\mathcal{E})$ of [[morphism]]s in a [[topos]] $\mathcal{E}$ is called a **class of open maps** if it satisfies the following axioms.

1. Every [[isomorphism]] belongs to $R$;

1. The [[pullback]] of a morphism in $R$ belongs to $R$.

1. If the pullback of a morphism $f$ along an [[epimorphism]] lands in $R$, then $f$ is also in $R$.

1. For every [[set]] $S$ the canonical morphism $(\coprod_{s \in S} *) \to *$ from the $S$-fold [[coproduct]] of the [[terminal object]] to the terminal object is in $R$.

1. For $\{X_i \stackrel{f_i}{\to} Y_i\}_{i \in I} \subset R$ then also the [[coproduct]] $\coprod_i X_i \to \coprod_i Y_i$ is in $R$.

1. If in a diagram of the form

   $$
     \array{
        Y &&\stackrel{p}{\to}&& X
        \\
        & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}}
        \\
        && B
     }
   $$

   we have that $p$ is an [[epimorphism]] and $g$ is in $R$, then $f$ is in $R$.

The class $R$ is called a class of **étale maps** if in addition to the axioms 1-5 above it satisfies

1. for $f : X \to Y$ in $R$ also the [[diagonal]] $Y \to Y \times_X Y$ is in $R$.

1. If in

   $$
     \array{
        Y &&\stackrel{p}{\to}&& X
        \\
        & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}}
        \\
        && B
     }
   $$

   we have that $p$ is an epimorphism, and $p, g \in R$, then $f\in R$.

For instance ([JoyalMoerdijk, section 1](#JoyalMoerdijk)).

## Related concepts

* [[open geometric morphism]]

* [[essential sublocale]]

* [[closed morphism]], [[proper morphism]]

* [[étale map]]

* [[separated morphism]]

## References

* [[Peter Johnstone]],  _Complemented sublocales and open maps_ , Annals of Pure and Applied Logic **137** (2006) pp.240–255.

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_, volume II, Oxford University Press 2002. (Section C1.5, pp.520-522)

* {#JoyalMoerdijk} [[André Joyal]], [[Ieke Moerdijk]], _A completeness theorem for open maps_, Annals of Pure and Applied Logic __70__ no.1 (1994) pp.51-86. [MR95j:03104](http://www.ams.org/mathscinet-getitem?mr=1303663), doi

Application to [[transition systems]] and [[bisimulations]]

* {#JoyalNielsenWisnkel94} [[André Joyal]], [[Mogens Nielsen]], [[Glynn Winskel]]: *Bisimulation from open maps*, BRICS report series RS-94-7 (1994) [[doi:10.7146/brics.v1i7.21663](https://doi.org/10.7146/brics.v1i7.21663), [pdf](http://www.brics.dk/RS/94/7/BRICS-RS-94-7.pdf)], Information and Computation **127** 2 (1996) 164-185 [[doi:10.1006/inco.1996.0057](https://doi.org/10.1006/inco.1996.0057)]

[[!redirects open morphisms]]


open point

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## Definition

Let $(X, \tau)$ be a [[topological space]]. Then a point $x \in X$ in the underlying [[set]] is called an _open point_ if the [[singleton]] [[subset]] $\{x\} \in X$ is an [[open subset]], i.e. $\{x\} \in \tau$.

## Related concepts

* [[closed point]]

* [[interior point]]

[[!redirects open points]]


open problem of confinement -- references

### The open problem of confinement
 {#OpenProblemConfinementReferences}

While [[experiment]] as well as [[lattice gauge theory]]-computer simulation clearly show that [[confinement]] takes place, a real theoretical understanding has been missing (though [[AdS-QCD]] is now on a good track). This is the _confinement problem_. The same problem from the point of view of [[mathematics]] is called the _[[mass gap problem|mass gap]] [Millennium Problem](quantization+of+Yang-Mills+theory#ReferencesNonperturbativeTheoryAndMassGap)_. A related problem is the _[flavor problem](flavor+problem#TheFlavorProblem)_.

The following is a list of quotes highlighting the open problem of confinement:

\linebreak

* {#Witten79} [[Edward Witten]], _Baryons in the $1/n$ Expansion_, Nucl. Phys. B160 (1979) 57-115 ([spire:140391](http://inspirehep.net/record/140391), doi:10.1016/0550-3213(79)90232-3)

> many of the essential  properties that the theory $[$QCD$]$ is presumed to have, including  confinement, dynamical mass generation, and chiral symmetry  breaking, are only poorly understood. And apart from the low-lying bound states of heavy quarks, which we believe can be  described by a nonrelativistic Schroedinger equation, we are unable to derive from the basic theory even the grossest  features of the particle spectrum, or of traditional strong  interaction phenomenology

\linebreak

* {#Machleidt89} [[Rupert Machleidt]], _The Meson Theory of Nuclear Forces and Nuclear Structure_, In: Negele J.W., Vogt E. (eds.) _Advances in Nuclear Physics_ Advances in Nuclear Physics, vol 19. Springer 1989 ([doi:10.1007/978-1-4613-9907-0_2](https://doi.org/10.1007/978-1-4613-9907-0_2))

> There  are theoretical attempts to connect the fundamental theory of [[QCD]] with the very successful [[meson]] picture at low energy. The [[Skyrme model]] is an  example. In other attempts, one tries to derive the NN interaction more or less directly from [[QCD]].  At present, the predictions are more of a qualitative kind. For quantitative results, the one-[[pion]] and two-[[pion]] contributions have to be added by hand, as they do not emerge naturally out of [[QCD]]-inspired models. Knowing  that $\pi$ and $2\pi$ are the most important parts of the [[nuclear force]], this defect of present [[quark]] model calculations is serious.

\linebreak

* {#Kutschke96} Robert Kutschke, section 3.1 _Heavy flavour spectroscopy_, in D. Bugg (ed.), _Hadron Spectroscopy and the Confinement Problem_, Proceedings of a NATO Advanced Study Institute, Plenum Press 1996  ([doi:10.1007/978-1-4613-0375-6](https://www.springer.com/cn/book/9780306453038))

> While it is generally believed that QCD is the correct fundamental theory of the strong interactions there are, as yet, no practical means to produce full QCD calculations of hadron masses and their decay widths.

\linebreak

* {#Holstein99} Barry R. Holstein, _A Brief Introduction to Chiral Perturbation Theory_, Czech. J. Phys. 50S4:9-23, 2000 ([arXiv:hep-ph/9911449](https://arxiv.org/abs/hep-ph/9911449))

> the holy grail sought by particle/nuclear knights has been to verify the correctness of the "ultimate" theory of strong interactions  -- quantum chromodynamics (QCD).

> The theory is, of course, deceptively simple on the surface. $[...]$ So why are we still not satisfied? While at the very largest energies, asymptotic freedom allows the use of [[perturbative quantum field theory|perturbative techniques]], for those who are interested in making contact with low energy experimental findings there exist at least three fundamental difficulties:

> i) QCD is written in terms of the "wrong" degrees of freedom -- [[quarks]] and [[gluons]] -- while low energy experiments are performed with [[hadron|hadronic]] [[bound states]];

> ii) the theory is non-linear due to [[gluon]] self-[[interactions]];

> iii) the theory is one of strong coupling so that [[perturbative quantum field theory|perturbative methods]] are not practical

\linebreak

* [[Igor Klebanov]], Oyvind Tafjord, "Can we quantitatively understand quark and gluon confinement in Quantum Chromodynamics and the existence of a mass gap?", 10th of *[10 Physics Problems for the Next Millennium](http://theory.caltech.edu/~preskill/millennium.html)* selected at [Strings 2000](https://inspirehep.net/conferences/972466)

\linebreak

* [[Yakov Shnir]], *Magnetic Monopoles*, Springer 2005 ([ISBN:978-3-540-29082-7](https://www.springer.com/gp/book/9783540252771))

  Section 9.1:

  > $[$confinement$]$ still remains one of the very few Big Unsolved Problems of the theoretical physics of the XXI-st century. Moreover, this is the "classic question that has resisted solution over the years", which was included by the Clay Mathematics Institute in the list of seven Millennium Prize Problems [$[14]$](mass+gap#JaffeWitten00). The award will amount to USD 1,000,000, thus, it is worth-while to account for more information about the matter.

\linebreak

* {#CsakiReece06} Csaba Csaki, [[Matthew Reece]], _Toward a Systematic Holographic QCD: A Braneless Approach_, JHEP 0705:062, 2007 ([arxiv:hep-ph/0608266](https://arxiv.org/abs/hep-ph/0608266))

  (in motivation of [[AdS/QCD]])

> QCD is a perennially problematic theory. Despite its decades of experimental support, the detailed low-energy physics remains beyond our calculational reach. The lattice provides a
technique for answering nonperturbative questions, but to date there remain open questions that have not been answered.

\linebreak

* {#Guidry08} Mike Guidry, _Gauge Field Theories: An Introduction with Applications_, Wiley 2008 ([ISBN:978-3-527-61736-4](https://www.wiley.com/en-us/Gauge+Field+Theories%3A+An+Introduction+with+Applications-p-9783527617364))

  Section 13.1.9:

> The holy grail of QCD is the proof that a color SU(3) gauge theory
confines in the non-perturbative regime.

> This is not difficult to show for lattices with large spacing; unfortunately, such a demonstration does not constitute a proof of QCD confinement: to do that we must also demonstrate that the same theory that confines at large lattice spacing (strong coupling) has a continuum limit (weak coupling) that is consistent with the asymptotically free short distance behavior of QCD.

\linebreak

* {#Yi09} [[Piljin Yi]], _Holographic Baryons_ ([arXiv:0902.4515](https://arxiv.org/abs/0902.4515), [doi:10.1142/9789814280709_0016](https://www.worldscientific.com/doi/abs/10.1142/9789814280709_0016)), Chapter 16 in: [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

> [[QCD]] is a challenging theory. Its most interesting aspects, namely the [[confinement]] of [[color charge|color]] and the [[chiral symmetry breaking]], have defied all analytical approaches. While there are now many data accumulated from the [[lattice gauge theory]], the methodology falls well short of giving us insights on how one may understand these phenomena analytically, nor does it give us a systematic way of obtaining a low energy theory of [[QCD]] below the [[confinement]] [[scale]].

\linebreak

* {#HashimotoIizukaYi10} [[Koji Hashimoto]], [[Norihiro Iizuka]], [[Piljin Yi]], _A Matrix Model for Baryons and Nuclear Forces_, JHEP 1010:003, 2010 ([arXiv:1003.4988](https://arxiv.org/abs/1003.4988))

> Nuclear physics is one of the oldest branches of high energy physics, yet remains one of more difficult. Despite the fact that we know the underlying fundamental theory, i.e. QCD, we are still unable to predict, reliably and analytically, behavior of nuclei or even a single proton. The problem is of course that one must understand the strong-coupling regime of QCD, which by and large remains inaccessible except by large-scale lattice simulations. Traditionally, this sets nuclear physics apart from the rest of high energy physics in many aspects. However, [[Witten-Sakai-Sugimoto model|recent developments]] in the so-called [[AdS-CFT|gauge/gravity duality]] began to solve certain strongly coupled field theories,possibly including QCD or its close relatives, allowing the two communities to merge with each other.

\linebreak

* {#Goldstein11} Jeff Greensite, _An Introduction to the Confinement Problem_, Lecture Notes in Physics, Volume 821, 2011 ([doi:10.1007/978-3-642-14382-3](https://link.springer.com/book/10.1007%2F978-3-642-14382-3))

> Because of the great importance of the standard model, and the central role it plays in our understanding of particle physics, it is unfortunate that, in one very important respect, we don’t really understand how it works. The problem lies in the sector dealing with the interactions of quarks and gluons, the sector known as Quantum Chromodynamics or QCD. We simply do not know for sure why quarks and gluons, which are the fundamental fields of the theory, don’t show up in the actual spectrum of the theory, as asymptotic particle states. There is wide agreement  about  what  must  be  happening  in  high  energy  particle  collisions:  the  formation of color electric flux tubes among quarks and antiquarks, and the eventual fragmentation of those flux tubes into mesons and baryons, rather than free quarks and gluons. But there is no general agreement about why this is happening, and that limitation exposes our general ignorance about the workings of non-abelian gauge theories in general, and QCD in particular, at large distance scales.

\linebreak

* {#MachleidtEntem11} [[Rupert Machleidt]], [[David Rodríguez Entem]], _Chiral effective field theory and nuclear forces_, Phys. Rept. 503:1-75, 2011 ([arXiv:1105.2919](https://arxiv.org/abs/1105.2919), [doi:10.1016/j.physrep.2011.02.001](https://doi.org/10.1016/j.physrep.2011.02.001))

> The problem with a derivation of [[nuclear forces]] from [[QCD]] is that this theory is [[non-perturbative QFT|non-perturbative]] in the low-energy regime characteristic of [[nuclear physics]], which makes direct solutions very difficult. Therefore, during the first round of new attempts, [[QCD]]-inspired [[quark]] models became popular. The positive aspect of these models is that they try to explain [[hadron]] structure and hadron-hadron [[interactions]] on an equal footing and, indeed, some of the gross features of the [[nucleon]]-[[nucleon]] interaction are explained successfully.

> However, on a critical note, it must be pointed out that these quark-based approaches are nothing but another set of models and, thus, do not represent fundamental progress. For the purpose of describing hadron-hadron interactions, one may equally well stay with the simpler and much more quantitative [[meson]] models.

\linebreak

* [[Koji Hashimoto]], _D-Brane -- Superstrings and New Perspective of Our World_, Springer (2012) $[$[doi:10.1007/978-3-642-23574-0](https://link.springer.com/book/10.1007%2F978-3-642-23574-0), [spire:1188897](http://inspirehep.net/record/1188897)$]$

> pp 126:  Heuristically, one may speculate that, since the coupling constant of QCD is too large, quarks bind together so that any single quark may not be able to exist alone. However, it has never been proved theoretically. After all, at present we have not understood yet QCD, the theory with a large coupling constant in which quantum theoretical effects (many loops in Feynman graphs as in Fig. 2.18) are essential. This is really a serious problem. Actually, one of big seven mathematical millennium problems proposed by Clay mathematical institute in the United States is concerning this QCD, and one million dollars will be awarded for a solution of the problem
related with this.

> D-branes give us quite an innovative idea, as an approach to this problem. It is called "holography" which is introduced in this chapter.

\linebreak

* {#Brambilla14} Brambilla et al., _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981 ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723). [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

>  The success of the technique does not remove the challenge of understanding the non-perturbative aspects of the theory. The two aspects are
deeply intertwined. The Lagrangian of QCD is written in terms of quark and gluon degrees of freedom which become apparent at large energy but remain hidden inside hadrons in the low-energy regime. This confinement property is related to the increase of $\alpha_s$ at low energy, but it has never been
demonstrated analytically.

> We have clear indications of the confinement of quarks into hadrons from both experiments and lattice QCD. Computations of the heavy quark–antiquark potential, for example, display a linear behavior in the quark–antiquark distance, which cannot be obtained in pure perturbation theory. Indeed the two main characteristics of QCD: confinement and the appearance of nearly massless pseudoscalar mesons, emergent from the spontaneous breaking of chiral symmetry, are non-perturbative phenomena whose precise understanding continues to be a target of research.

> Even in the simpler case of gluodynamics in the absence of quarks, we do not have a precise understanding of how a gap in the spectrum is formed and the glueball spectrum is generated.

\linebreak

* {#INFN15} [Istituto Nazionale di Fisica Nucleare](https://web.infn.it/csn1/index.php/en/), _What Next: White Paper of the INFN-CSN1_ _Proposal for a long term strategy for accelerator based experiments_, Frascati Phys.Ser. 60 (2015) 1-302 (2015-05-29) ([spire:1374543](http://inspirehep.net/record/1374543/), [pdf](http://www.lnf.infn.it/sis/frascatiseries/Volume60/Volume60.pdf) ) chapter 7: _Hadron Physics and non-perturbative QCD
_  ([pdf](http://www.infn.it/csn1/White_paper_documents/NPQCD.pdf))

> Experimentally, there is a large number of facts that lack a detailed qualitative and quantitative explanation. The most spectacular manifestation of our lack of theoretical understanding of QCD is the failure to observe the elementary degrees of freedom, quarks and gluons, as free asymptotic states (color con- finement) and the occurrence, instead, of families of massive mesons and baryons (hadrons) that form approximately linear Regge trajectories in the mass squared. The internal, partonic structure of hadrons, and nucleons in particular, is still largely mysterious. Since protons and neutrons form almost all the visible matter of the Universe, it is of basic importance to explore their static and dynamical properties in terms of quarks and gluons interacting according to QCD dynamics.

\linebreak

* {#CobosMartinez16} J J Cobos-Martínez, _Non-perturbative QCD and hadron physics_ 2016  J. Phys.: Conf. Ser. 761 012036 ([doi:10.1088/1742-6596/761/1/012036](http://iopscience.iop.org/article/10.1088/1742-6596/761/1/012036))

> $[\cdots]$ the QCD Lagrangian does not by itself explain the data on strongly interacting matter, and it is not clear how the observed bound states, the hadrons, and their properties arise from QCD. Neither confinement  nor dynamical chiral symmetry breaking (DCSB) is apparent in QCD’s lagrangian, yet they play a dominant role in determining the observable characteristics of QCD. The physics of strongly interacting matter is governed by emergent phenomena such as these, which can only be elucidated through the use of non-perturbative methods in QCD [4, 5, 6, 7]

\linebreak

* {#SNM16} Hideo Suganuma, Yuya Nakagawa, Kohei Matsumoto, _1+1 Large $N_c$ QCD and its Holographic Dual -Soliton Picture of Baryons in Single-Flavor World_, Proceedings of the 14th International Conference on Meson-Nucleon Physics and the Structure of the Nucleon (MENU2016) ([arxiv:1610.02074](https://arxiv.org/abs/1610.02074))

> Since 1973, quantum chromodynamics (QCD) has been established as the fundamental theory of the strong interaction. Nevertheless, it is very difficult to solve QCD directly in an analytical manner, and many effective models of QCD have been used instead of QCD, but most models cannot be derived from QCD and its connection to QCD is unclear. To analyze nonperturbative QCD, the lattice QCD Monte Carlo simulation has been also used as a first-principle calculation of the strong interaction. However, it has several weak points. For example, the state information (e.g. the wave function) is severely limited, because lattice QCD is based on the path-integral formalism. Also, it is difficult to take the chiral limit, because zero-mass pions require infinite volume lattices. There appears a notorious "sign problem" at finite density.

> On the other hand, [[holographic QCD]] has a direct connection to QCD, and can be derived from QCD in some limit. In fact, [[holographic QCD]] is equivalent to infrared QCD in large $N_c$ and strong 't Hooft coupling $\lambda$, via gauge/gravity correspondence. Remarkably, holographic QCD is successful to reproduce many hadron phenomenology such as [[vector meson dominance]], the KSRF relation, hidden local symmetry, the GSW model and the [[Skyrmion|Skyrme soliton]] picture. Unlike [[lattice QCD]] simulations, [[holographic QCD]] is usually formulated in the chiral limit, and does not have the sign problem at finite density.

\linebreak

* {#Petrov19} V. A. Petrov, _Asymptotic Regimes of Hadron Scattering in QCD_ ([arXiv:1901.02628](https://arxiv.org/abs/1901.02628))

> This is a commonplace that so far we do not have a full-fledged theory of interaction of [[hadrons]], derived from the first principles of [[QCD]] and having a regular way of calculating of hadronic [[scattering amplitude|amplitudes]], especially at high energies and small momentum transfers. The problem is related to a more general problem that QCD Lagrangian would yield [[confinement|colour confinement]] with massive
colourless physical states (hadrons).

\linebreak

* {#HashimotoMatsuoMorita19} [[Koji Hashimoto]], [[Yoshinori Matsuo]], [[Takeshi Morita]], _Nuclear states and spectra in holographic QCD_, JHEP12 (2019) 001 ([arXiv:1902.07444](https://arxiv.org/abs/1902.07444))

> One of the long-standing problems in QCD is to reproduce profound nuclear physics. The strong coupling nature of QCD prevents us from solving it analytically, and even numerical simulations have a limitation such as the volume of the atomic nucleus versus the lattice size. It is quite important to bridge the particle physics and the nuclear physics, by solving QCD to derive typical fundamental notions of the nuclear physics, such as the magic numbers, the nuclear binding energy and the nuclear shell model.

> [[holographic QCD|Holographic QCD]] is an analytic method to approach these problems in the strong coupling limit and at a large $N_c$. The [[nuclear matrix model]] is a many-body quantum effective mechanics for multiple baryons, derived by the [[AdS/CFT correspondence]] applied to QCD.

\linebreak

* {#BarnafoldiGogokhia19} Gergely Gábor Barnaföldi, Vakhtang Gogokhia, *The Mass Gap Approach to QCD* ([arXiv:1904.07748](https://arxiv.org/abs/1904.07748))

> QCD as a fundamental quantum field gauge theory still suffers from a few important conceptual problems. We focus on some of them as follows: (A) The dynamical generation of a mass squared at the fundamental quark-gluon level, since the QCD Lagrangianforbids such kind of terms apart from the current quark mass. $[\ldots]$ (D) The non-observation of the colored objects as physical states which does not follow from the QCD Lagrangian,i.e., it cannot explain confinement of gluons and quarks.

\linebreak

* {#HKL19} Yonggoo Heo, C. Kobdaj, M.F.M. Lutz, _Constraints from a large-$N_c$ analysis on meson-baryon interactions at chiral order $Q^3$_, Phys. Rev. D 100, 094035 (2019) ([arXiv:1908.11816](https://arxiv.org/abs/1908.11816))

> Still after many decades of vigorous studies the outstanding challenge of modern physics
is to establish a rigorous link of QCD to low-energy hadron physics as it is observed in
the many experimental cross section measurements.

\linebreak

* {#DHMMNVW19} Christian Drischler, Wick Haxton, Kenneth McElvain, Emanuele Mereghetti, Amy Nicholson, Pavlos Vranas, André Walker-Loud, _Towards grounding nuclear physics in QCD_ ([arxiv:1910.07961](https://arxiv.org/abs/1910.07961))

> the entirety of the rich field of [[nuclear physics]] emerges from [[QCD]]: from the [[forces]] binding [[protons]] and [[neutrons]] into the nuclear landscape, to the fusion and fission reactions between [[atomic nucleus|nuclei]], to the prospective interactions of nuclei with BSM physics, and to the unknown [[phase of matter|state of matter]] at the cores of [[neutron stars]].

> How does this emergence take place exactly? How is the clustering of [[quarks]] into [[nucleons]] and [[alpha particles]] realized? What are the mechanisms behind collective phenomena in nuclei as strongly correlated many-body systems? How does the extreme fine-tuning required to reproduce nuclear binding energies proceed? – are big open questions in [[nuclear physics]].

\linebreak

* {#RobertsSchmidt20} [[Craig Roberts]], [[Sebastian M. Schmidt]], _Reflections upon the Emergence of Hadronic Mass_, The European Physical Journal Special Topics volume 229, pages 3319–3340 (2020) ([arXiv:2006.08782](https://arxiv.org/abs/2006.08782), [doi:10.1140/epjst/e2020-000064-6](https://doi.org/10.1140/epjst/e2020-000064-6))

> More than 98% of visible mass is contained within [[atomic nuclei|nuclei]].  In first approximation, their atomic weights are simply the sum of the masses of all the [[neutrons]] and [[protons]] ([[nucleons]]) they contain.  Each nucleon has a [[mass]] $m_N \sim 1$ [[GeV]], i.e. approximately 2000-times the [[electron]] mass. The [[Higgs  boson]] produces the latter, but  what produces the masses of the neutron and proton?  This is the question posed above, which is pivotal to the development of modern physics: how can science explain the emergence of [[hadron|hadronic]] mass (EHM)? $[\cdots]$

> Modern science is thus encumbered with the fundamental problem of [[gluon]]  and [[quark]] [[confinement]]; and confinement is crucial because it ensures absolute stability of the [[proton]].  $[\cdots]$ Without confinement,our Universe cannot exist.

> As the 21st Century began, the Clay Mathematics Institute established seven Millennium Prize Problems. Each represents one of the toughest challenges in mathematics. The set contains the [[mass gap problem|problem of confinement]]; and  presenting a sound solution will win its discoverer 1,000,000 bucks. Even with  such motivation, today, almost fifty years after the discovery of [[quarks]], no rigorous solution has been found. Confinement and EHM are inextricably linked. Consequently, as  science plans for the next thirty  years, solving the problem of EHM has become a _grand challenge_. $[\cdots]$

> In trying to match [[QCD]] with Nature, one confronts the many complexities of strong, nonlinear dynamics in relativistic quantum field theory, e.g. the loss of particle number conservation, the frame and scale dependence of the  explanations and interpretations of observable processes, and the evolving character of the relevant degrees-of-freedom. [[electroweak field|Electroweak theory]] and phenomena are essentially [[perturbative quantum field theory|perturbative]]; hence, possess little of this complexity.  Science has never before encountered an [[interaction]] such as that at work in [[QCD]]. Understanding this interaction, explaining everything of which it is capable, can potentially change the way we look at the Universe.

\linebreak

* {#Chatterjee20} [[Sourav Chatterjee]], _A probabilistic mechanism for quark confinement_, Comm. Math. Phys. 2021 ([arXiv:2006.16229](https://arxiv.org/abs/2006.16229))

> The confinement of quarks is one of the enduring mysteries of modern physics. $[\ldots]$ In spite of many decades of research, physically relevant quantum gauge theories have not yet been constructed in a rigorous mathematical sense. $[$ [[non-perturbative quantum field theory|non-perturbatively]], that is $]$ $[\ldots]$ Taking this program to its completion is one of the Clay millennium problems. $[\ldots]$ Perhaps the most important example is four-dimensional [[SU(3)]]-[[lattice gauge theory]]. If one can show that this theory has a [[mass gap]] at all values of the [[coupling constant|coupling strength]], that would explain why particles known as [[glue-balls]] in the [[QCD|theory of strong interactions]] have mass. All such questions remain open.

> The second big open question is the problem of quark confinement. Quarks are the constituents of various elementary particles, such as protons and neutrons. It is an enduring mystery why quarks are never observed freely in nature. The problem of quark confinement has received enormous attention in the physics literature, but the current consensus seems to be that a satisfactory theoretical explanation does not exist.

\linebreak

* Michael Creutz, _QCD beyond diagrams_ ([arXiv:2010.04774](https://arxiv.org/abs/2010.04774))

> QCD, the theory of the strong interactions, involves quarks interacting with non-Abelian gluon fields. This theory has many features that are difficult to impossible to see in conventional diagrammatic perturbation theory. This includes quark confinement, mass generation, and chiral symmetry breaking.

\linebreak

* {#Roberts21} [[Craig Roberts]], _On Mass and Matter_, AAPPS Bulletin volume 31, Article number: 6 (2021) ([arXiv:2101.08340](https://arxiv.org/abs/2101.08340), [doi:10.1007/s43673-021-00005-4](https://doi.org/10.1007/s43673-021-00005-4))

> The origin of the proton mass, and with it the basic mass-scale for all nuclear physics, is one of the most profound puzzles in Nature.

> Although QCD is defined by a seemingly simple Lagrangian, it specifies a problem that has defied solution for more than forty years. The key challenges in modern nuclear and high-energy physics are to reveal the observable content of strong QCD and, ultimately, therefrom derive the properties of nuclei.

\linebreak

* [[Guy de Teramond]], [[Stanley Brodsky]], _Longitudinal dynamics and chiral symmetry breaking in holographic light-front QCD_ ([arXiv:2103.10950](https://arxiv.org/abs/2103.10950))

> In spite of the important progress of Euclidean lattice gauge theory, a  basic  understanding  of  the  mechanism of  color  confinement and its  relation  to  chiral  symmetry breaking in QCD has remained an unsolved problem.

> Recent [[holographic light front QCD|developments]] based on superconformal quantum mechanics in  light-front  quantization  and  its holographic  embedding on a higher  dimensional  gravity theory (gauge/gravity correspondence) have led to new analytic insights into the structure of hadrons and their  dynamics.

\linebreak

* {#Klebanov21} [[Igor Klebanov]], *Remarks on Color Confinement*, talk at [Strings 2021](https://www.ictp-saifr.org/strings2021/) ([pdf](https://www.ictp-saifr.org/wp-content/uploads/2021/06/DiscussionConfinementKlebanov.pdf), [video](https://youtu.be/B9NI2LE7d68?t=262))

> Perhaps the gauge/string duality has provided us with a "physicist's proof of confinement" in some exotic gauge theories like the one described by the warped deformed conifold. Yet, we still don't have a quantitative handle on the Asymptotically Free theories in 3+1 dimensions. $[\cdots]$ Don’t take confinement for granted, even in 1+1 dimensions where it seems obvious. Proof of Color Confinement in 2+1 and 3+1 dimensions would be very important.

\linebreak

* {#Dubovsky21} [[Sergei Dubovsky]], *Comments on (mostly long) QCD strings*, talk at [Strings 2021](https://www.ictp-saifr.org/strings2021/) ([pdf](https://www.ictp-saifr.org/wp-content/uploads/2021/06/Sergei.pdf), [[Dubovsky_QCDStrings.pdf:file]], [video](https://youtu.be/B9NI2LE7d68?t=1183))

  concludes in view of [[AdS/QCD]]:

  > With  modern  lattice  tools  and  understanding  of  QFT/string theory we are in a good position to solve this problem. It may take another 30 years to build a new collider and get new data in  particle  physics.  But  we  are  lucky  to  have  a  tractable  and fundamental  problem  to  solve  while  waiting.  Participants  of STRINGS 2051 will be puzzled if we don’t make an effort now  and leave it to them.

\linebreak

* Clara Peset, Antonio Pineda, Oleksandr Tomalak, *The proton radius (puzzle?) and its relatives* ([arXiv:2106.00695](https://arxiv.org/abs/2106.00695))

  > (in the context of the [[proton radius puzzle]])

> In the beginning God created quarks, And made them interact through the strong forces, And it was dark... And God said, "I do not understand a damn thing"

\linebreak

* Sophia K Domokos, Robert Bell, Trinh La, Patrick Mazza, *A Pedagogical Introduction to Holographic Hadrons*, published as: *Holographic hadron masses in the language of quantum mechanics*,  European Journal of Physics **42** 6 (2021) 065801 $[$[arXiv:2106.13136](https://arxiv.org/abs/2106.13136), [doi:10.1088/1361-6404/ac1abb](https://iopscience.iop.org/article/10.1088/1361-6404/ac1abb)$]$

> We have known for over fifty years that protons are made up of quarks and gluons. We have pinned down the masses and couplings of quarks to a startling degree of accuracy. Yet we still don’t know why the proton’s mass is almost exactly a factor of 100 greater than the sum of the masses of its constituent quarks. This mystery persists because quarks and gluons are strongly coupled at low energies: they interact so forcefully and often that our usual calculational tools – based almost entirely on perturbation theory – fail.

> Holographic duality, or “[[holographic QCD]]” (hQCD) when applied to hadrons, emerged in the early 2000s as a new way to tackle strongly coupled systems.

\linebreak

* Roman Pasechnik, Michal Šumbera, *Different faces of confinement*, Universe 7 (2021) 9, 330 ([arXiv:2109.07600](https://arxiv.org/abs/2109.07600), [doi:10.3390/universe7090330](https://doi.org/10.3390/universe7090330))

> in the infrared (IR) limit, QCD enters entirely different, strongly-coupled domain, rendering the Perturbation Theory inapplicable and creating substantial problems for making reliable predictions at intermediate and low momentum transfers, i.e. at large distances. While it is conventionally believed that QCD should remain the correct theory of strong interactions also at large distances, in the so-called confined regime, deriving reliable predictions remains a big theoretical challenge.

> $[\cdots]$

> The *problem of confinement* concerns the strongly-coupled sector of QCD composed of interacting colored partons (quarks and gluons). $[\cdots]$ Despite the major efforts of the research community and tremendous progress made over last few decades, it does not appear to be fully and consistently resolved yet.

\linebreak

* {#Koppenburg21} [[Patrick Koppenburg]] for the [[LHCb collaboration]]: *List of hadrons observed at the LHC* (2021) $[$[pdf](https://cds.cern.ch/record/2749030/files/FIGURE-2021-001.pdf), [webpage](https://www.nikhef.nl/~pkoppenb/particles.html), [cds:2749030](https://cds.cern.ch/record/2749030)$]$

> While the LHC is best known for the observation of the first fundamental scalar particle, the Higgs Boson, it has also yielded many observations of yet unknown hadrons.  Hadrons are composed of quarks and are thus not fundamental particles of the Standard Model. However, their properties follow from yet unsolved mysteries of the strong interaction.

> Although the QCD Lagrangian is expected to completely describe the spectrum of hadrons and all of their properties, there is no rigorous first-principle way of expressing this mathematically. The quark confinement conjecture is experimentally well tested, but mathematically still unproven. And it is still unknown which combinations of quarks may or may not form hadrons. Experimental guidance is needed to help improving theoretical models.

\linebreak

* Daniele Binosi, *Emergent Hadron Mass in Strong Dynamics*, Few-Body Syst **63** 42 (2022) $[$[arXiv:2203.00942](https://arxiv.org/abs/2203.00942), [doi:10.1007/s00601-022-01740-6](https://doi.org/10.1007/s00601-022-01740-6)$]$

> [[asymptotic freedom]] implies that the theory is innately [[nonperturbative quantum field theory|nonperturbative]] at short distances, which in turn means that it is extremely difficult to describe [[QCD]]’s dynamics at the [[fm|$fm$]] scale. Indeed, QCD’s colour-charged degrees of freedom, the massless gluons and the light up and down quarks, have never been observed in isolation: they are confined in colour-neutral bound-states, the hadrons, of which the (light) pion and the (heavy) proton are primary examples. It should be noticed that without a mass-scale confinement would not be possible: colour-singlet combinations of quarks would still be there, but the participating particles would need not be close together, since in a scale invariant theory all lengths are equivalent. Accordingly, the question "how does confinement appear in QCD" is inextricably connected to the question “how does mass emerge in strong dynamics”.

\linebreak

* [[Mikhail Shifman]], *OPE-based Methods in Nonperturbative QCD*, Eur. Phys. J. C (2022) $[$[arXiv:2208.10600](https://arxiv.org/abs/2208.10600)$]$

> QCD and its relatives are special because QCD is the theory of nature. QCD is strongly coupled in the infrared domain where it is impossible to treat it quasi-classically -- perturbation theory fails even qualitatively. It does not capture drastic rearrangement of the vacuum structure related to confinement. The Lagrangian is defined at short distances in terms of gluons and quarks, while at large distances we deal with [[hadrons]], e.g. pions, ρ mesons, protons, etc. Certainly, the latter are connected with quarks and gluons in a divine way, but this connection is highly nonlinear and non-local; even now, 50 years later, the full analytic solution of QCD
is absent.

> [[non-perturbative effect|Non-perturbative methods]] were desperately needed.

\linebreak

* Minghui Ding, [[Craig D. Roberts]], Sebastian M. Schmidt, *Emergence of Hadron Mass and Structure* $[$[arXiv:2211.07763](https://arxiv.org/abs/2211.07763)$]$

* {#Roberts22} [[Craig D. Roberts]], *Origin of the Proton Mass*, at: *8th International Symposium on Symmetries in Subatomic Physics (SSP 2022)* $[$[arXiv:2211.09905](https://arxiv.org/abs/2211.09905), [doi:10.1051/epjconf/202328201006](https://doi.org/10.1051/epjconf/202328201006)$]$

  > (see the articles for a wealth of relevant commentary)

\linebreak

* [[Guy de Téramond]], *Emergent phenomena in QCD: The holographic perspective* $[$[arXiv:2212.14028](https://arxiv.org/abs/2212.14028)$]$

  > (discussion in [[holographic light front QCD]])

> At large distances, however, the nonperturbative nature of the strong interactions becomes dominant and a basic understanding of the essential features of hadron physics from first principles QCD has remained an important unsolved problem in the standard model of particle physics. Hadronic characteristics are not explicit properties of the QCD Lagrangian but emergent phenomena, notably, the origin of the hadron mass scale, the mechanism of color confinement, the relation between chiral symmetry breaking and confinement, the massless pion in the chiral limit (the limit of zero quark masses), bound states and the pattern of hadron excitations.

\linebreak

* Seth Grable, [[Paul Romatschke]], *Elements of Confinement for QCD with Twelve Massless Quarks* $[$[arXiv:2310.12203](https://arxiv.org/abs/2310.12203)$]$

> Quantum Chromodynamics (QCD) is notorious for being a quantum field theory that is difficult to solve on large length scales $[\ldots]$ what is missing in current QCD literature is a (potentially approximate) solution to continuum QCD that captures the main qualitative features such as confinement in the IR and asymptotic freedom in the UV $[\ldots]$ if such a solution does exist, it cannot be based on a perturbative weak-coupling expansion, because QCD becomes strongly coupled towards long length scales. So the putative solution must arise from non-perturbative methods.


open problems in homotopy type theory

A list of (believed to be) open problems in [[homotopy type theory]].  To add more detail about a problem (such as why it is hard or interesting, or what ideas have been tried), make a link to a new page.

# Contents
* table of contents
{: toc}

## Semantics

* Construct a [[model of type theory in an (infinity,1)-topos]]. What remains open is the issue of [[weak Tarski universes]].

* Construct an [[(infinity,1)-topos]], or precisely an [[elementary (infinity,1)-topos]], out of (the syntactic category of) a system of type theory. Kapulkin [1507.02648](http://arxiv.org/abs/1507.02648) has shown that we obtain a locally cartesean closed quasicategory. [[Urs Schreiber]] proposed the following question:
"Given an (infinity-)functor from (any flavor of) homotopy type theories to infinity-categories as above, characterize its essential image."

* Define a notion of [[elementary (infinity,1)-topos]] and show that any such admits a model of homotopy type theory.

* The [[semantics]] for strict [[resizing rules]]. Probably refers to the rules introduced in Voevodsky's [Bergen lecture](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2011_Bergen.pdf), as opposed to the ones in the book.

* Construct (higher) [[inductive-recursive type|inductive-recursive types]] in univalent models, or show that they don't exist.

* Connect [[univalence]] and [[parametricity]] as suggested [here](http://bentnib.org/dtt-parametricity.pdf). One answer is [here](https://arxiv.org/abs/1909.05027).

* Show that when the construction of the [[cumulative hierarchy]] in chapter 10 of the book is interpreted in the [[simplicial set model]], it reproduces (up to equivalence) the usual cumulative hierarchy of the ambient ZFC.  Conclude that if HoTT+AC proves that the cumulative hierarchy satisfies some statement of set theory, then that statement follows from ZFC; see also [this comment](http://math.andrej.com/2014/01/13/univalent-foundations-subsume-classical-mathematics/comment-page-1/#comment-34342) and surrounding discussion.

* The [[model invariance problem]]: show that the interpretation of type theory is independent of the [[model category]] chosen to present an [[(infinity,1)-category]].

* Treat co-inductive types in HoTT. Part of the problem is that general co-inductive types are not fully understood in ordinary type theory.
Using extensionality, we obtain [[M-types]] from W-types.

## Metatheory

* "Writing all inductive definitions in terms of Sigma-types and W-types is theoretically possible, but extremely tedious ... it’s unknown whether any similar reduction for HITs is possible." ([source](http://homotopytypetheory.org/2014/06/08/hiru-tdd/))

* Give a [[computational interpretation]] of univalence and HITs. The cubical set model makes progress on this. [PDF](http://www.cse.chalmers.se/~coquand/mod1.pdf)

* Is there a [[cubical type theory]] that describes more exactly what happens in the [[model of type theory in cubical sets]]? Partial solutions: the [cubical language](https://github.com/simhu/cubical), [this](http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf) and [this](http://www.cs.nott.ac.uk/~txa/publ/ctt.pdf)

* Show in HoTT that the $n^{th}$ universe is a model of HoTT with $(n-1)$ universes. Part of this question is: what is an [internal model of HoTT](http://homotopytypetheory.org/2014/03/03/hott-should-eat-itself/).

* What is the [[proof theoretic strength of univalent type theory plus HITs]]? In particular, can they be predicatively justified? One could start by considering simple classes of HITs; e.g. [here](http://reports-archive.adm.cs.cmu.edu/anon/2014/CMU-CS-14-101R.pdf) before continuing to, say, pushouts, the cumulative sets, the Cauchy reals and the surreals.

* Is univalence consistent with Induction-Recusion? This would allow us to build a non-univalent universe inside a univalent one. Related to this, higher inductive types [can](http://homotopytypetheory.org/2014/06/08/hiru-tdd/) be used to define a univalent universe.

* Consider the type theory with a sequence of universes plus for each $n\gt 0$, an axiom asserting that there are [[n-type|n-types]] that are not $(n-1)$-types; see [here](http://arxiv.org/abs/1311.4002). Does adding axioms asserting that each universe is [[univalence|univalent]] increase the logical strength? Of course, univalence gives funext.

* Prove or disprove the conjecture that every fibrant type in [[HTS]] is equivalent to one definable in [[Martin-Löf type theory|MLTT]].  A potential method of disproof would be to solve the [[model invariance problem]] positively for MLTT but negatively for fibrant types in HTS.

* Define [[higher inductive types]] in [[higher observational type theory]].

## Homotopy theory and algebraic topology

* [[Homotopy theory in HoTT]]

* Similarly to the torus, consider the projective plane, Klein bottle, ... as discussed in the book (sec 6.6). Show that the Klein bottle is not orientable.  (This requires defining "orientable".)
    * This also requires defining what a surface is.

* Calculate more [[homotopy groups of spheres]].

* Show that the [[homotopy groups of spheres]] are all finitely generated, and are finite with the same exceptions as classically.

    * The classical proof requires Hurewicz, and now that spectral sequences are around should be possible.

* Define the Toda bracket.

* Prove that $n$-spheres are $\infty$-truncated.

* Prove that $S^2$ is not an $n$-type.

* Define the/a delooping of $S^3$.

* Can we verify [computational algebraic topology](http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo/) using HoTT?

* Bott periodicity

* Develop synthetic stable homotopy theory

## Higher algebra and higher category theory

* Define [[semi-simplicial type|semi-simplicial types]] in single-level non-modal dependent type theory, or show that this is not possible. Define [[Segal space]] [[complete Segal space]].

* Define a [[weak omega-category in type theory]].

* Is it possible to have limits be judgmentally the same as opposite colimits, and simultaneously have (co)limits be judgmentally the same as particular Kan extensions.  As partially detailed [on an issue on the HoTT/HoTT repo](https://github.com/HoTT/HoTT/issues/284), part of the problem is that the functor ᵒᵖ : (D → C)ᵒᵖ → (Dᵒᵖ → Cᵒᵖ) has a judgmental inverse (the composition is judgmentally the identity functor on objects and morphisms), and, assuming univalence, the precategories (D → C)ᵒᵖ and (Dᵒᵖ → Cᵒᵖ) are propositionally but not judgmentally equal.

* [[Eric Finster, Towards Higher Universal Algebra in Type Theory]]

* Construct the first 8 $\mathbb{N}$-indexed families of $\infty$-groups in the [[Whitehead tower]] of the [[orthogonal group]]s.

## Other mathematics

* What axioms of set theory are satisfied by the [[HIT model of ZFC]] constructed in chapter 10 of the book?  For instance, does it satisfy collection or REA?

* Are any of the stronger forms of the [[axiom of choice]] mentioned in the book (ex 7.8) consistent with univalence?

* Do the higher inductive-inductive [[real numbers]] from the book coincide with the Escardo-Simpson reals, [here](http://www.cs.bham.ac.uk/~mhe/papers/lics2001-revised.pdf) and [here](http://www.cs.bham.ac.uk/~mhe/papers/realadt.pdf)? [Solved](https://arxiv.org/abs/1706.05956) by Auke Booij using resizing/impredicativity. Can a predicative treatment be obtained from hSets as a predicative topos (by [Rijke/Spitters](http://arxiv.org/abs/1305.3835)) and the predicative [formalization](https://arxiv.org/abs/1610.05072) of Cauchy reals by Gilbert.

* Can multiplication be defined for the higher inductive-inductive [[surreal numbers]] from the book?

* Can Rezk complete categories remove the non-constructivity from the applications of [[Freyd's adjoint functor theorem]]?

* Is there a predicative and constructive abstract definition of the category of real Hilbert spaces, in a similar fashion as what Chris Heunen and Andre Kornell did in classical mathematics in their article ([Axioms for the category of Hilbert spaces](https://arxiv.org/abs/2109.07418))? Solér's theorem, which is used in their proof, is only valid in classical mathematics.

## In cohesive homotopy type theory
 {#InCohesiveHomotopyTypeTheory}

Problems in [[cohesive homotopy type theory]]:

* Does the [[axiom of real cohesion]] imply the [[axiom of localic real cohesion]]?

* Assuming the [[axiom of real cohesion]], is the [[shape]] of every [[subset]] of the [[Dedekind real numbers]] a [[set]]?

* Assuming the [[axiom of real cohesion]] and [[excluded middle]], prove that the [[analytic Markov's principle]] is true.

* What is the $(\infty,1)$-topos theoretic interpretation of the [[axiom of real cohesion]]?

* Prove the approximate [[intermediate value theorem]] in [[Mike Shulman]]'s model of [[real-cohesive homotopy type theory]] without resorting to crispness.

* Construct a real-cohesive homotopy type theory in [[multimodal dependent type theory]] a la [GKNB21](https://arxiv.org/abs/2011.15021) and prove the results of [Shulman16](https://arxiv.org/abs/1509.07584) in the multimodal dependent type theory

See also the commented list of problems at:

* *[[schreiber:Some thoughts on the future of modal homotopy type theory]]* (2015):

  * [solved problems](https://ncatlab.org/schreiber/show/Some+thoughts+on+the+future+of+modal+homotopy+type+theory#SolvedProblems)

  * [open problems](https://ncatlab.org/schreiber/show/Some+thoughts+on+the+future+of+modal+homotopy+type+theory#OpenProblems)

## In cubical type theory
 {#InCubicalTypeTheory}

* Does [[cubical type theory]] with [[regularity]] have [[canonicity]]?

* Does [[cubical type theory]] with [[regularity]] have an [[algorithm]] to compute the [[canonical forms]] of [[closed terms]]?

* Does [[cubical type theory]] with [[regularity]] have [[normalization]]?

* Does [[cubical type theory]] with [[regularity]] have an [[algorithm]] to compute [[normal forms]]?

## In weak type theory

* Does [[univalence]] imply [[function extensionality]] for types in the universe in [[weak type theory]]?

* Is (the [[homotopy type theory]] based upon) [[Martin-Löf type theory]] conservative over (the [[homotopy type theory]] based upon) [[weak type theory|weak]] Martin-Löf type theory?

* How much of the [[HoTT book]] could be done in [[weak type theory]]?

* Does [[weak type theory]] have [[homotopy canonicity]] and [[normalization]]?

* What is the [[categorical semantics]] of the [[homotopy type theory]] based upon [[weak type theory]], with all [[higher inductive types]] and [[weakly Tarski]] [[univalent universes]]?

* Is [[weak function extensionality]] equivalent to [[function extensionality]] in [[weak type theory]]?

* Does [[product extensionality]] hold in [[weak type theory]]? Namely, given types $A$ and $B$ and elements $a:A \times B$ and $b:A \times B$, is the canonical function $\mathrm{idtoprojectionids}:(a =_{A \times B} b) \to (\pi_1(a) \simeq \pi_1(b)) \times (\pi_2(a) \simeq \pi_2(b))$ an [[equivalence of types]]?

* Is [[function extensionality]] still provable in weak [[cubical type theory]]?

## Formalization

* Formalize the construction of models of type theory using [[contextual categories]].

* Formalize [[semi-simplicial types in homotopy type theory]].

* Formalize $\infty$-groupoids, $\infty$-categories within HoTT.

* Formalize what remains to be done from chapters 8, 10, and 11 of the book.
  In particular, develop the Higher Inductive-Inductive real numbers in some language, such as Coq (the basics of the surreal numbers have been done).

  In general, [this file](https://github.com/HoTT/HoTT/blob/master/contrib/HoTTBook.v) contains a Coq outline of the book. Instructions for how to contribute are [here](https://github.com/HoTT/HoTT/blob/master/README.markdown).

## Other lists of open problems

* Coquand's five open problems:
Coquand listed five open problems [here](https://eutypes.cs.ru.nl/eutypes_pmwiki/uploads/Meetings/Coquand.pdf)

* Open problems stated at the HoTT2019 Summer School [[HoTT2019 Summer School open problems list]]

* [Interpretation of Coq in higher toposes](https://cstheory.stackexchange.com/questions/44403/model-of-coq-pcuic-in-higher-toposes).

## Closed problems

+-- {: .query}
How about keeping a running list of solutions like this:?

Maybe only things not in the HoTT book?  Else the list of solved problems gets very long!

OK, here's the rule: if it was stated here (or on the UF-wiki) as an open problem, then it gets recorded here once it's solved.
=--

* There is a [[nlab:model structure on semi-simplicial sets]].

* Prove that the [[torus]] (with the HIT definition involving a 2-dimensional path) is equivalent to a product of two circles. See Sojakova's proof: [[torus.pdf:file]]. A shorter formalized proof is [here](http://homotopytypetheory.org/2015/01/20/ts1s1-cubically/)

* Construct the Hopf fibration. (Lumsdaine gave the construction in 2012 and Brunerie proved it was correct in 2013.)

* Construct the "super [[Hopf fibration]]" (AKA [[nlab:quaternionic Hopf fibration]]) $S^3 \to S^7 \to S^4$. (Buchholtz and Rijke solved this early 2016 through a [homotopy version of the Cayley-Dickson construction](https://arxiv.org/abs/1610.01134).)

*  Prove the [[Seifert-van Kampen theorem]]. (Shulman did it in 2013.)

* Construct Eilenberg--MacLane spaces and use them to define cohomology. (Licata and Finster did it in 2013, written up in [this paper](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf) (pdf).)

* Define the Whitehead product.
Guillaume Brunerie did this in 2017, written up in [this paper](https://arxiv.org/abs/1710.10307). He also gives a constructive proof of $\pi_4( S^3)= Z/n Z$ for some $n$ and a construction of the [reduced James product](https://ncatlab.org/nlab/show/EHP+spectral+sequence#via_the_james_model) as a HIT and a homotopy colimit. It is also proven that $\Omega \Sigma X \simeq J(X)$ for some pointed connected type $X$. All of these constructions can be found in detail in [his thesis](https://arxiv.org/abs/1606.05916).

* Does having an [[interval type]] with only typal [[beta conversion]] rules imply [[function extensionality]] in [[Martin-Löf type theory]]? If yes, does the above still hold if the [[function types]] only have a typal [[eta conversion]] rule? Answered in the negative in [this section about the relation between the interval type and function extensionality](https://ncatlab.org/nlab/show/interval+type#relation_to_function_extensionality): where judgmental beta conversion was used in the original proof becomes function extensionality in the typal case, resulting in circularity.

* Is the [[propositional truncation]] of the [[boolean domain]], both of which only have typal [[computation rules]] rules, the [[interval type]]? The case with judgmental computation rules was done [here](https://www.cs.bham.ac.uk/~mhe/truncation-and-extensionality/hsetfunext.html). Answered in the positive in [this section about the relation between the interval type and propositional truncations](https://ncatlab.org/nlab/show/interval+type#relation_to_propositional_truncations): one uses the definition of the interval type with a function $j:\mathbb{2} \to \mathbb{I}$ and use double induction on $\mathbb{2}$ to inductively define a function $f:\prod_{a:2} \prod_{b:2} j(a) =_\mathbb{I} j(b)$.

* Define the Hurewicz map and prove the Hurewicz theorem. A proof of the Hurewicz theorem in HoTT can be [found here](https://arxiv.org/abs/2007.05833).

* What is the [[loop space of a wedge of circles]] indexed by a set without decidable equality? It is a set, and can be shown to be using the [zigzag construction](https://arxiv.org/abs/2402.12339).


open quantum system

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

#Contents#
* table of contents
{:toc}

## Idea

Consider a [[pair]] of [[quantum systems]], $Q$ and $E$, where $Q$ is some system of interest and $E$ is some system that is external to $Q$ and that is in some fixed [[pure state]] $\vert e \rangle$.  Now let us suppose that the two systems [[interaction|interact]] and evolve via some [[unitary operator]] on the [[tensor product of Hilbert spaces|tensor product]] [[Hilbert space]] of each, $\mathscr{Q} \otimes \mathscr{E}$.  If one is interested in the $\mathscr{E}$-[[trace]] of [[density operators]] on this system, then $Q$ is known as an _open system_ and $E$ is the _environment_.

## Related concepts

* [[quantum measurement]]

* [[quantum decoherence]]

## References

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems I -- The Hamiltonian Approach*, Lecture Notes in Mathematics **1880**, Springer (2006) [[doi:10.1007/b128449](https://doi.org/10.1007/b128449)]

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems II -- The Markovian approach*,  Lecture Notes in Mathematics **1881**, Springer (2006) [[doi:10.1007/b128451](https://doi.org/10.1007/b128451)]

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems III -- Recent Developments*, Lecture Notes in Mathematics **1882**, Springer (2006) [[doi:10.1007/b128453](https://doi.org/10.1007/b128453)]

* [[Heinz-Peter Breuer]], [[Francesco Petruccione]], *The Theory of Open Quantum Systems*, Oxford University Press (2007) [[doi:10.1093/acprof:oso/9780199213900.001.0001](https://doi.org/10.1093/acprof:oso/9780199213900.001.0001)]

* {#SchumacherWestmoreland10} [[Benjamin Schumacher]], [[Michael Westmoreland]], *Quantum Processes, Systems, and Information*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511814006](https://doi.org/10.1017/CBO9780511814006)]

See also:

* Wikipedia, *[Open quantum systems](https://en.wikipedia.org/wiki/Open_quantum_system)*

[[!redirects open quantum systems]]


open subscheme

#Contents#
* table of contents
{:toc}

## Idea

An open subscheme is the analogue of an open subset of a [[topological space]] for schemes.

## Definition

An __open subscheme__ of a [[scheme]] $(Y,\mathcal{O}_Y)$ is a scheme $(U,\mathcal{O}_Y)$ whose underlying space is the subspace $U$ of $Y$ together with an isomorphism of the structure sheaf $\mathcal{O}_U$ with the restriction $\mathcal{O}_Y|_U$ of the structure sheaf $\mathcal{O}_Y$ to $U$. An isomorphism of a scheme $(X,\mathcal{O}_X)$ and an open subscheme $(U,\mathcal{O}_Y)$ of another scheme $(Y,\mathcal{O}_Y)$ amounts to an [[open immersion of schemes]] $(X,\mathcal{O}_X)\hookrightarrow(Y,\mathcal{O}_Y)$.

## Open subsets of the underlying topological space of a scheme

Before embarking upon the proof of Proposition \ref{PropositionOpenSubsetAsOpenSubscheme}, we shall need a few preliminaries. Let $A$ be a commutative ring. Let $Spec A$ denote the set of prime ideals of $A$. Given an element $a$ of $A$, we denote by $D_{A}(a)$ the set of [[prime ideal|prime ideals]] of $A$ to which $a$ does not belong. Either by definition, or by a little basic commutative algebra, we have that $\{ D_{A}(a) | a \in A \}$ is a [[topological base|basis]] for the [[Zariski topology]] on $Spec A$.

An [[affine scheme]] is by definition the locally ringed space $(Spec A, \mathcal{O}_{Spec A})$, where $Spec A$ is the set we have just defined equipped with the Zariski topology, and $\mathcal{O}_{Spec A}$ is a certain sheaf of rings on this space.

A basic result in commutative algebra is that, for any $a \in A$, $(D_{A}(a), \mathcal{O}_{Spec A} | D_{A}(a))$ is an affine scheme, isomorphic to $(Spec A_{a}, \mathcal{O}_{Spec A_{a}})$. Here $A_{a}$ is the [[localisation of a commutative ring at an element | localisation]] of $A$ at $a$.

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

Let $(X, \mathcal{O}_{X})$ be a scheme, and let $U$ be an open subset of $X$. Then $(U, \mathcal{O}_{X} | U)$, in the same notation as at [[scheme]], is a scheme.

=--

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

Let $x \in U$. We must prove that $x$ has an open neighbourhood $W$ in $U$ such that $(W, \mathcal{O}_{U} | W)$ is isomorphic to an [[affine scheme]].

Immediately from the definition of a scheme, there is an open neighbourhood $N$ of $x$ in $X$ such that $(N, \mathcal{O}_{X} | N)$ is isomorphic to an [[affine scheme]], that is to say, a pair $(Spec A, \mathcal{O}_{Spec A})$ as defined above, for some commutative ring $A$. This isomorphism in particular involves an isomorphism of topological spaces $N \rightarrow Spec A$, which we shall denote by $i$.

Since $\{ D_{A}(a) | a \in A \}$ is a basis for the Zariski topology on $Spec A$, there is some $a \in A$ such that $i(x) \in D_{A}(a)$ and $D_{A}(a) \subset i(N \cap U)$, noting that since $U$ is open in $X$, $N \cap U$ is open in $N$.

Now, as we have remarked, $(D_{A}(a), \mathcal{O}_{Spec A} | D_{A}(a))$ is isomorphic to an affine scheme. Hence $(i^{-1}(D_{A}(a)), i^{*}(\mathcal{O}_{Spec A} | D_{A}(a)))$ is isomorphic to an affine scheme, where $i^{*}$ is the [[inverse image]] functor from sheaves of commutative rings on $D_{A}(a)$ to sheaves of commutative rings on $i^{-1}(D_{A}(a))$.

But $i^{*}(\mathcal{O}_{Spec A} | D_{A}(a))$ is isomorphic to $\mathcal{O}_{U} | i^{-1}(D_{A}(a))$. Since $i^{-1}(D_{A}(a)) \subset i^{-1}(i(N \cap U)) = N \cap U \subset U$, we conclude that we can take $W$ to be $i^{-1}(D_{A}(a))$.

=--

## Examples

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

Let $x$ be a point of the underlying topological space $X$ of a scheme $(X, \mathcal{O}_{X})$. The set $\{ x \}$ need not be a closed subset of $X$, but if it is, then, by Proposition \ref{PropositionOpenSubsetAsOpenSubscheme}, $(X \setminus \{ x \}, \mathcal{O}_{X} | X \setminus \{ x \})$ defines an open subscheme of $(X, \mathcal{O}_{X})$.

=--

category: algebraic geometry
[[!redirects open subschemes]]


open subspace

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

#Contents#
* table of contents
{:toc}

## Idea

An **open subspace** $U$ of a [[space]] $X$ is a [[subspace]] of $X$ whose elements are "stable to small perturbations", or can be "observed to belong to $U$ by measurements with finite precision".

When ordered by inclusion the open subspaces form a poset, the [[frame of opens]].

## Definition

Depending on what sort of "space" $X$ is, this may be defined in different ways.

### For topological spaces

If $X$ is a [[topological space]] in the classical sense, then it is *defined* by specifying a collection of subsets to be called "open" (that are closed under finite intersections and arbitrary joins, i.e. are a sub-[[frame]] of the [[power set]] of $X$).

### For convergence spaces

If $X$ is a [[convergence space]] (or some variation such as a [[subsequential space]]), we define a subset $U\subseteq X$ to be **open** if any filter/net/sequence converging to a point of $U$ must be eventually in $U$.  This defines an "underlying topological space" of $X$.

### For locales

In [[locale]] theory, every open $U$ in the locale defines an "open [[sublocale]]" which is given by the __open [[nucleus]]__
$$ j_{U}\colon V \mapsto U \Rightarrow V .$$
The idea is that this subspace is the part of $X$ which involves only $U$, and we may identify $V$ with $U \Rightarrow V$ when we are looking only at $U$.  If $X$ is a ([[sober space|sober]]) topological space regarded as a locale, then any such open sublocale is spatial and coincides with the subspace determined by the subset $U$ (and this is true even in [[constructive mathematics]]).

### For toposes

See [[open subtopos]].

### In synthetic topology {#SynTop}

In [[synthetic topology]], we interpret 'space' as meaning simply 'set' (or [[type]], i.e. the basic objects of our foundational system).  There are then multiple ways to define "open subset".

One is to use a [[dominance]], which specifies the open subsets representably by a set of "open truth values".

Another possibility is the following definition due to Penon: $U\subseteq X$ is open if for any $x\in U$ and $y\in X$, either $x\neq y$ or $y\in U$.  This definition does not require a choice of dominance, but it is generally only correct for [[Hausdorff spaces]]; for instance, the open point in the [[Sierpinski space]] is not open in this sense.  However, in various [[gros toposes]] of topological, smooth, or algebraic spaces it does induce the correct notion of open subsets for Hausdorff spaces; see [Dubuc-Penon](#DubucPenon).

## Properties

A [[subspace]] $A$ of a [[space]] $X$ is __open__ if the [[inclusion function|inclusion map]] $A \hookrightarrow X$ is an [[open map]].

The __[[topological interior|interior]]__ of any subspace $A$ is the largest open subspace contained in $A$, that is the [[union]] of all open subspaces of $A$.  The interior of $A$ is variously denoted $Int(A)$, $Int_X(A)$, $A^\circ$, $\overset{\circ}A$, etc.

## Related concepts

* [[closed subset]], [[clopen subset]]

* [[open point]]

* [[open cover]]

* [[open map]]

* [[overt space]]

* [[saturated subset]]

## References

* [[Jacques Penon]], *Topologie et intuitionnisme*

* [[Eduardo Dubuc]] and [[Jacques Penon]], *Objets compactes dans les topos*
 {#DubucPenon}

[[!redirects open subspace]]
[[!redirects open subspaces]]
[[!redirects open subset]]
[[!redirects open subsets]]
[[!redirects open set]]
[[!redirects open sets]]

[[!redirects open sublocale]]
[[!redirects open sublocales]]
[[!redirects open nucleus]]
[[!redirects open nuclei]]

[[!redirects open]]


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

#Contents#
* table of contents
{:toc}

## Background

+-- {: .num_defn #LocallyCompactSpace}
###### Definition
**([[locally compact topological space]])**

A [[topological space]] $X$ is called _[[locally compact topological space|locally compact]]_
if for every point $x \in X$ and every [[open neighbourhood]] $U_x \supset \{x\}$
there exists a smaller open neighbourhood $V_x \subset U_x$ whose [[topological closure]]
is [[compact topological space|compact]] and still contained in $U$:

$$
  \{x\} \subset V_x \subset \underset{\text{compact}}{Cl(V_x)} \subset U_x
  \,.
$$

=--

## Statement and proof

+-- {: .num_prop #OpenSubspacesOfCompactHausdorffSpacesAreLocallyCompact}
###### Proposition
**(open subspaces of compact Hausdorff spaces are locally compact)**

Every [[open subset|open]] [[topological subspace]] $X \underset{\text{open}}{\subset} K$
of a [[compact topological space|compact]]
[[Hausdorff space]] $K$ is a [[locally compact topological space]].

In particular every [[compact Hausdorff space]] itself is [[locally compact topological space|locally compact]].

=--

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

Let $x \in X$ be a point and let $U_x \subset X$ an [[open neighbourhood]].  We need to produce a small open neighbourhood
whose closure is compact and still contained in $U_x$.

By the nature of the [[subspace topology]] there exists an open subset $V_x\subset K$ such that $U_x = X \cap V_x$.
Since $X$ is assumed to be open, it follows that $U$ is also open as a subset of $K$.
Since [[compact Hausdorff spaces are normal]] it follows (by [this prop.](Introduction+to+Topology+--+1#T4InTermsOfTopologicalClosures)) that
there exists a smaller open neighbourhood $W_x \subset K$ whose [[topological closure]] is still contained in $U_x$,
and since [[closed subspaces of compact spaces are compact]]:

$$
  \{x\} \subset W_x \subset \underset{\text{cpt}}{Cl(W_x)} \subset V_x \subset K
  \,.
$$

The intersection of this situation with $X$ is the required smaller compact neighbourhood $Cl(W_x) \cap X$:

$$
  \{x\} \subset W_x \cap X \subset \underset{\text{cpt}}{Cl(W_x)} \cap X \subset U_x \subset X
  \,.
$$

=--

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

Conversely, every locally compact Haudorff space $X$ arises as in prop. \ref{OpenSubspacesOfCompactHausdorffSpacesAreLocallyCompact}, since it may be considered an [[open subspace]] in its [[one-point compactification]] $X \sqcup \{\infty\}$. See there _[this example](one-point+compactification#LocallyCompatcHausdorffSpaceIsOpenSubspaceOfCompactHausdorffSpace)_.

=--

## Related statements

* [[closed subspaces of compact spaces are compact]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[compact Hausdorff spaces are normal]]

* [[a CW-complex is a Hausdorff space]]

* [[Hausdorff implies sober]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

* [[continuous images of compact spaces are compact]]

[[!redirects locally compact Hausdorff spaces are equivalently open subspaces of compact Hausdorff spaces]]


open subtopos

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

# Contents
* table of contents
{:toc}

## Idea

The concept of an **open subtopos** generalizes the concept of an open subspace from topology to toposes.

## Definition

Let $U$ be a [[subterminal object]] of a [[topos]].  Then $o_U(V)\coloneqq (U\Rightarrow V)$ defines a [[Lawvere-Tierney topology]] on $\mathcal{E}$, whose corresponding [[subtopos]] is called the _open subtopos_ associated to $U$.

The reflector into the topos of sheaves can be constructed explicitly as $O_U(X) = X^U$.

A _topology_ that is of this form for some subterminal object $U$ is called _open_.

## Example

In case $\mathcal{E}=Sh(X)$ is the topos of sheaves on a topological space $X$, a subterminal object is just an open subset $U$ of $X$ and the open subtopos corresponding to it is equivalent to $Sh(U)$.

As one would expect from the topological situation, for any topos $\mathcal{E}$, the empty subtopos (given by $o(V) \coloneqq (\bot \Rightarrow V) = \top$) and $\mathcal{E}$ itself (given by $o(V) \coloneqq (\top \Rightarrow V) = V$) are open subtoposes of $\mathcal{E}$.

### Remark

The [[subterminal object]] $U$ in $\mathcal{E}$ is associated with a [[closed subtopos]] $Sh_{c(U)}(\mathcal{E})$ as well e.g. in the case of $\mathcal{E}=Sh(X)$ on a space $X$ this yields $Sh(X\setminus U)$.

Moreover, given a [[Lawvere-Tierney topology]] $j$ on a topos $\mathcal{E}$ with corresponding subtopos $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$, we a get a canonical [[subterminal object]] $ext(j)$  associated to $j$ by taking the $j$-closure of $O\rightarrowtail 1$. The corresponding closed and open subtoposes associated to $ext(j)$ provide a '_closure_' $Sh_{c(ext(j))}(\mathcal{E})$, respectively, an '_exterior_' $Sh_{o(ext(j))}(\mathcal{E})$ for $Sh_j(\mathcal{E})$ (cf. [SGA4](#SGA4), p.461). More on this [below](#ext_int).

## Properties

+-- {: .num_prop}
###### Proposition
Let $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ be a subtopos with corresponding topology $j$. The following are equivalent:

* $j$ is open.
* The [[associated sheaf functor]] $L:\mathcal{E}\to Sh_j(\mathcal{E})$ is [[logical functor|logical]].
* $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is an [[open geometric morphism]].

=--

See Johnstone ([1980](#Johnstone80), pp.219-220; [2002](#Johnstone02), pp.609-610).

+-- {: .num_prop}
###### Proposition
Let $U$ a [[subterminal object]] and $Sh_{c(U)}(\mathcal{E})$ and $Sh_{o(U)}(\mathcal{E})$ the corresponding closed, resp. open subtoposes. Then $Sh_{c(U)}(\mathcal{E})$ and $Sh_{o(U)}(\mathcal{E})$ are [[complement|complements]] for each other in the [[lattice of subtoposes]].
=--

See Johnstone [(2002, pp.212,215)](#Johnstone).

### Remark

Whereas, general [[open geometric morphisms|open morphisms]] are only bound to preserve [[first order logic]], open _inclusions_ preserve also [[higher order logic]] since their [[inverse image|inverse images]] are logical.

That the inverse image is logical is a special case of the general fact that the pullback functor $\mathcal{E}\to\mathcal{E}/X$ along $X\to 1$ is logical for arbitrary objects $X$. In particular, $Sh_{o(U)}(\mathcal{E})\cong\mathcal{E}/U$.

## Open localizations

Subtoposes of a topos $\mathcal{E}$ correspond to [[localization|localizations]] of $\mathcal{E}$ i.e. replete, reflective subcategories whose reflector preserves finite limits. Just as this notion makes sense more generally for categories with finite limits, the notion of open localization makes sense more generally for [[locally presentable category|locally presentable categories]]:

Given a locally $\alpha$-presentable category $\mathcal{C}$ with subcategory of $\alpha$-presentable objects $\mathcal{P}$, the subobject $\Omega_\mathcal{C}$ of the subobject classifier of $Set^{\mathcal{P}^{op}}$ given by $\Omega_\mathcal{C}(P):=$ set of $\alpha$-exact subpresheaves of $\mathcal{P}(\text{_},P)$, classifies subobjects of $\mathcal{C}$. Furthermore, localizations of $\mathcal{C}$ correspond to topologies $j:\Omega_\mathcal{C}\to\Omega_\mathcal{C}$.

At this level of generality, an open subtopos of a Grothendieck topos corresponds to the notion of an **open localization** of a [[locally presentable category]] that is studied in [Borceux-Korotenski (1991)](#BK91). The main result in their paper is the following

+-- {: .num_prop}
###### Proposition
Let $l\dashv i: \mathcal{D}\hookrightarrow\mathcal{C}$ be a localization of a [[locally presentable category]] $\mathcal{C}$. The localization is called _open_ if the following equivalent conditions are satisfied:

* The corresponding [[closure operator]] admits a universal dense interior operator.

* The associated [[Lawvere-Tierney topology|topology]] $j_\mathcal{D}:\Omega_\mathcal{C}\to\Omega_\mathcal{C}$ has a left adjoint.

* The localization is [[essential geometric morphism|essential]] with $k\dashv l$ and, additionally, the first of the following diagrams[^adj] being a pullback implies the second being a pullback, too:

$$\array{
D&\overset{d}{\to}&D'\\
f\downarrow&p.b.&\downarrow g\\
lC&\underset{lc}{\to}&lC'
}

\qquad\Rightarrow
\qquad

\array{
kD&\overset{kd}{\to}&kD'\\
\bar{f}\downarrow&p.b.&\downarrow \bar{g}\\
C&\underset{c}{\to}&C'
}\quad .$$
=--

[^adj]: Here $f$ corresponds to $\bar{f}$, resp. $g$ to $\bar{g}$, under the adjunction $k\dashv l$.

In case $\mathcal{C}$ is a topos, $i:\mathcal{D}\hookrightarrow\mathcal{C}$ is an open subtopos.

Open localizations are special cases of **essential localizations**, and are in general better behaved than the latter. For example, the meet of two essential localizations in the lattice of essential localizations does not coincide with their meet in the lattice of localizations. Compare this with the following

+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ a [[locally presentable category]]. The meet of two open localizations in the lattice of localizations is an open localization.
=--

cf. [Borceux-Korotenski (1991, p.235)](#BK91).

+-- {: .num_prop #open_supremum}
###### Proposition
Let $\mathcal{C}$ a [[locally presentable category]] where unions are [[universal colimit|universal]]. The supremum of a family of open localizations in the lattice of all localizations is again an open localization.
=--

cf. [Borceux-Korotenski (1991, p.236)](#BK91).

This applies e.g. to Grothendieck toposes since they are locally presentable and colimits are universal.

+-- {: .num_prop #open_supremum}
###### Proposition
Let $\mathcal{C}$ a [[locally presentable category]] where unions are [[universal colimit|universal]]. The open localizations in $\mathcal{C}$ constitute a [[locale]].
=--

cf. [Borceux-Korotenski (1991, p.237)](#BK91).

The following proposition closes the circle and recovers the primordial example of sheaf subtoposes $Sh(U)$ on open subsets $U$ as a special case:

+-- {: .num_prop}
###### Proposition
Let $\mathcal{C}$ a [[locally presentable category]] in which colimits are [[universal colimit|universal]]. Then the [[locale]] of open localizations of $\mathcal{C}$ is isomorphic to the locale of [[subterminal object|subterminal objects]] of $\mathcal{C}$.
=--

cf. [Borceux-Korotenski (1991, p.238)](#BK91).

## Open subtoposes associated to a subtopos {#ext_int}

Let $Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ be a subtopos of a (Grothendieck) topos with corresponding [[Lawvere-Tierney topology|topology]] $j$. From [prop. \ref{open_supremum}](#open_supremum) it follows that the supremum of the family of all open subtoposes contained in $Sh_j(\mathcal{E})$ is open again and, since it coincides with the supremum in the lattice of all localizations, is contained in $Sh_j(\mathcal{E})$. Clearly, it is the biggest open subtopos contained in $Sh_j(\mathcal{E})$ and therefore called the **interior** of $Sh_j(\mathcal{E})$, denoted by $Sh_{o(int(j))}(\mathcal{E})$ and the corresponding [[subterminal object]] by $int(j)$.

Whereas the other open subtopos $Sh_{o(ext(j))}(\mathcal{E})$ connected with $Sh_j(\mathcal{E})$ corresponding to $ext(j)$ is the biggest open subtopos disjoint from $Sh_j(\mathcal{E})$ i.e. its exterior. Then the sum $Sh_{o(ext(j))}(\mathcal{E}) \vee Sh_{o(int(j))}(\mathcal{E})$ is open again and corresponds to the subterminal object $ext(j)\vee int(j)$. Its closed complement $Sh_{c(ext(j)\vee int(j))}(\mathcal{E})$ is called the boundary of $Sh_j(\mathcal{E})$ in ([SGA 4](#SGA4), p. 461).

For some further details see at [[dense subtopos]].

## Related pages

* [[open geometric morphism]]
* [[locally closed subtopos]]
* [[closed subtopos]]
* [[dense subtopos]]
* [[localization]]
* [[level]]
* [[Artin gluing]]
* [[co-Heyting boundary]]

## References

* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (Exposé IV 9.2, 9.3.4-9.4., pp.451ff)

* {#BK91}[[Francis Borceux|F. Borceux]], M. Korostenski, _Open localizations_ , JPAA **74** (1991) 229-238 doi

* C. Getz, M. Korostenski, _Open localizations and factorization systems_ , Quest. Math. **17** no.2 (1994) 225-230 [doi](https:/doi.org/10.1080/16073606.1994.9631761)
* {#Johnstone77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint 2014, 93-95)

* {#Johnstone80}[[Peter Johnstone]], _Open maps of toposes_ , Manuscripta Math. **31** (1980) 217-247. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002222744))

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]] vols. I,II_, Oxford UP 2002. (A4.5., pp.204-220; C3.1.5-7, pp.609f)

[[!redirects open subtoposes]]
[[!redirects open topology]]
[[!redirects open localization]]
[[!redirects open localisation]]


open-closed homotopy algebra

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

#Contents#
* table of contents
{:toc}

## Definition
Let $\mathcal{H}=\mathcal{H}_o\oplus\mathcal{H}_c$ be a $\mathbb{Z}$-[[graded vector space]], and $\mathfrak{k}$ a collection of $n$-ary maps that make $(\mathcal{H}_c,\mathfrak{l})$ an [[L-infinity-algebra|$L_\infty$-algebra]]. If there is a collection of maps

$$
\mathfrak{n}
  \;=\;
\big\{
n_{k,l} \colon (\mathcal{H}_c ^{\otimes k})\otimes (\mathcal{H}_o )^{\otimes l} \to \mathcal{H}_o
\big\}_{k,l\geq 0}
$$

each graded-symmetric for $(\mathcal{H}_c ^{\otimes k})$ satisfying the identities

$$
\begin{array}{l}
\sum_{p+r=n} \sum_{\sigma\in S_n} \frac{(-1)^{\epsilon(\sigma)}}{p! r!} n_{1+r,m}(l_p (c_{\sigma (1)} , \cdots , c_{\sigma (p)}),c_{\sigma (p+1)}, \cdots, c_{\sigma (n)}; o_1, \cdots, o_m)
\\
\;+\;
\sum_{p+r=n} \sum_{i+s+j=m} \sum_{\sigma\in S_n} \frac{(-1)^{\mu_{p,i}(\sigma)}}{p! r!} n_{p,i+1+j} (c_{\sigma (1)}, \cdots , c_{\sigma (p); o_1, \cdots, o_i, n_{r,s}(c_{\sigma (p+1), \cdots, c_{\sigma(n)};o_{i+1}, \cdots, o_{i+s}}}), o_{i+s+1}, \cdots, o_m)
\\
\;=\;
0
\,,
\end{array}
$$

where the sign exponent $\mu_{p,i}(\sigma)$ is defined as

$$
\mu_{p,i}(\sigma) = \epsilon(\sigma)+ (c_{\sigma (1)}+ \cdots + c_{\sigma(p)}) + (o_1+ \cdots + o_i) + (o_1+ \cdots + o_i)(c_{\sigma (p+1}+ \cdots + c_{\sigma (n)})
$$

then $(\mathcal{H},\mathfrak{l},\mathfrak{n})$ is called a **weak open-closed homotopy algebra** (weak OCHA).

If furthermore one has that $l_0=n_{0,0} =0$ then this is an **open-closed homotopy algebra** (OCHA).

## Properties

By definition, $(\mathcal{H}_c,\mathfrak{l})$ is a Lie-infinity-algebra. Furthermore, given a (weak) OCHA one can show that $(\mathcal{H}_o,\{n_{0,k}\})$ is an (weak) [[A-infinity algebra]]. These are interpreted as the closed, and open sectors, respectively (see [[string field theory]] for more on this).

## References

General:

* [[Hiroshige Kajiura]], [[Jim Stasheff]]. *Open-closed homotopy algebra in mathematical physics*. J. Math. Phys. 47, 023506 (2006). ([doi](https://doi.org/10.1063/1.2171524)).


open/closed string duality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]] a given [[worldsheet]] [[manifold with boundary|with boundaries]] ([[Riemann surface]] with punctures) may typically be read in at least two different ways as describing either a [[closed string]] or an [[open string]] [[string scattering|scattering]]-process.

> graphics grabbed from [Schomerus 07, Figure 4](#Schomerus07)

But since the corresponding [[string scattering amplitude]] is the same no matter which perspective is taken, this exhibits an equivalence ("[[duality in string theory|duality]]") between these open- and closed string amplitudes.

For example, a [[cylinder]] [[worldsheet]] stretching between two [[D-brane]] [[worldvolumes]] manifestly represents equivalently the following two rather different-seeming [[string scattering]]-processes:

1. a [[closed string]] emitted from one of the branes and absorbed by the other;

1. an open string with ends on both branes going around a [[1-loop]] [[vacuum diagram]].

> graphics grabbed from [Schomerus 07, Figure 4](#Schomerus07), see also e.g.
[Peschanski 09, Figure 1](#Peschanski09)

Since

1. closed string excitations include [[gravitons]] inducing [[gravity]]

   (in the above example: the gravitational attraction between the two branes)

1. open string excitations include [[gluons]] inducing the [[force]] described by [[Yang-Mills fields|Yang-Mills]] [[gauge fields]] --e.g. the [[strong nuclear force]] in [[quantum chromodynamics]]

   (in the above example: the [[worldvolume]] gauge field on [[coincident branes]])

open/closed string duality implies _gauge/gravity duality_ between these different  [[effective field theories]].
This is exhibited by phenomena such as

* the [[KLT relations]]

* the [[AdS/CFT correspondence]]

* the [[AdS/QCD correspondence]]

* the [[gauge-string correspondence]]

## Applications

### In large $N$ limit and holography

Open/closed string duality plays a subtle role when interpreting the [['t Hooft double line notation]] of [[gauge theory]] [[Feynman diagrams]] in the [[large N limit]] alternatively as [[open string]] or as [[closed string]] [[worldsheets]] ([Gopakumar-Vafa 98](#GopakumarVafa98), [Gaiotto-Rastelli 03](#GaiottoRastelli03), [Gopakumar 04](#Gopakumar04)).

## Related concepts

* [[duality in physics]]

* [[duality in string theory]]

* [[classical double copy]]

* [['t Hooft double line construction]]

* [[large N limit]], [[small N limit]]

* [[KLT relations]]

* [[AdS/CFT correspondence]]

* [[AdS/QCD correspondence]]

* [[string theory results applied elsewhere]]

* [[effective QFT]] incarnations of [[open/closed string duality]],

  relating ([[supergravity|super]]-)[[gravity]] to ([[super Yang-Mills theory|super]]-)[[Yang-Mills theory]]:

  * [[KLT relations]]

  * [[BCJ relations]]

  * [[classical double copy]]

## References

[[!include Polyakov gauge-string duality -- references]]

### Open/closed string duality

Review:

* [[Leonardo Rastelli]], _Lectures on open/closed string duality_, PiTP 2004 ([pdf](http://www.sns.ias.edu/pitp2/2004/IASrastelli.pdf), [[RastelliStringDuality.pdf:file]])

* {#Schomerus07} [[Volker Schomerus]], _Strings for quantum chromodynamics_, International journal of modern physics A 22:30 (2007) ([doi:10.1142/S0217751X07038827](https://doi.org/10.1142/S0217751X07038827))

* {#Peschanski09} R. Peschanski, _Gauge-gravity duality and high energy collisions_, Nuclear Physics B - Proceedings Supplements, Volume 191, June 2009, Pages 320-330 ([doi:10.1016/j.nuclphysbps.2009.03.138](https://doi.org/10.1016/j.nuclphysbps.2009.03.138))

Original articles:

* {#GopakumarVafa98} [[Rajesh Gopakumar]], [[Cumrun Vafa]], _On the Gauge Theory/Geometry Correspondence_, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 ([arXiv:hep-th/9811131](https://arxiv.org/abs/hep-th/9811131))

* [[Justin Khoury]], [[Herman Verlinde]], _On Open/Closed String Duality_ ([arXiv:hep-th/0001056](http://arxiv.org/abs/hep-th/0001056))

* [[Andreas Karch]], [[Lisa Randall]], _Open and Closed String Interpretation of SUSY CFT's on Branes with Boundaries_, JHEP 0106:063, 2001 ([arXiv:hep-th/0105132](https://arxiv.org/abs/hep-th/0105132))

* {#GaiottoRastelli03} [[Davide Gaiotto]], [[Leonardo Rastelli]], _A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model_, JHEP 0507:053,2005 ([hep-th/0312196](https://arxiv.org/abs/hep-th/0312196))

* {#Gopakumar04} [[Rajesh Gopakumar]], _Free Field Theory as a String Theory?_, Comptes Rendus Physique 5 (2004) 1111-1119 ([hep-th/0409233](https://arxiv.org/abs/hep-th/0409233))

[[!redirects gauge/gravity duality]]
[[!redirects gauge-gravity duality]]
[[!redirects gauge/gravity dualities]]
[[!redirects gauge-gravity dualities]]
[[!redirects gauge/gravity correspondence]]
[[!redirects gauge-gravity correspondence]]


Operad

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

## Idea

The [[category]] whose [[objects]] are [[operads]], and whose morphisms are operad [[homomorphisms]].

Details depend on the particular flavor of operad being used (the choice of enrichment, the choice of symmetry or not, etc).

## Structures

* [[canonical model structure on Operad]]

## Related concepts

There are natural functors from and to $Operad$ to [[Cat]], [[dSet]], etc.

* [[Cat]], **Operad**

* [[2Cat]]

* [[(∞,1)Cat]], [[(∞,1)Operad]]

* [[(∞,n)Cat]]

category: category