Hisham Sati

Hisham Sati ([faculty page](https://nyuad.nyu.edu/en/academics/divisions/science/faculty/hisham-ahmad-sati.html)) is working on [[non-perturbative effects|non-perturbative]] phenomena  in [[string theory]]/[[M-theory]] using tools of [[cohomology]], [[homotopy theory]], [[algebraic topology]] and [[higher category theory]]. His thesis advisor was [[Michael Duff]].

Hisham Sati is an associate  professor of [mathematics at NYU Abu Dhabi](https://nyuad.nyu.edu/en/academics/divisions/science/academic-programs/math.html) and lead-PI of the *[[Center for Quantum and Topological Systems]]*.

* [webpage](https://nyuad.nyu.edu/en/academics/divisions/science/faculty/hisham-sati.html)

From his [cached Pitt faculty page](https://web.archive.org/web/20170811155442/http://www.mathematics.pitt.edu/person/hisham-sati):

> My research is interdisciplinary and lies in the intersection of [[differential geometry]], [[algebraic topology]], and [[mathematical physics|mathematical/theoretical physics]]. I am mainly interested in geometric and topological structures arising from [[quantum field theory|quantum (topological) field theory]], [[string theory]], and [[M-theory]].  This includes [[orientation in generalized cohomology|orientations]] with respect to [[generalized cohomology theories]], and corresponding description via [[higher geometry|higher geometric]], [[topology|topological]], and [[category theory|categorical]] notions of [[bundles]].

## Selected talks

* _Geometric perspectives on topological action functionals_, talk at _[The CUNY Workshop on differential cohomology](http://qcpages.qc.cuny.edu/~swilson/cunyworkshop14.html)_, New York, August 2014 ([video recording](http://videostreaming.gc.cuny.edu/videos/video/1802/))

* _M-theory and cohomotopy_, talk at _[[M-Theory and Mathematics]]_, NYU AD 2020 ([pdf slides](https://ncatlab.org/nlab/files/Sati_MTheoryCohomotopy_2020.pdf))

  > (on [[Hypothesis H]]_)

## Selected publications

* Hisham Sati, _[[Geometric and topological structures related to M-branes]]_,

  part I, in: *Superstrings, Geometry, Topology, and $C^{\ast}$-algebras*, Proc. Symp. Pure Math. **81** (2010) 181-236 [[arXiv:1001.5020](http://arXiv.org/abs/1001.5020), [ams:pspum/081](http://www.ams.org/books/pspum/081)],

  part _II: Twisted $String$ and $String^c$ structures_, J. Australian Math. Soc. 90 (2011), 93-108 ([arXiv:1007.5419](http://arxiv.org/abs/1007.5419));

  part _III: Twisted higher structures_, Int. J. Geom. Meth. Mod. Phys. 8 (2011), 1097-1116 ([arXiv:1008.1755](http://arxiv.org/abs/1008.1755))

on [[cohomology]] and [[twisted cohomology]] structures in [[string theory]]/[[M-theory]]. See also _[[twisted smooth cohomology in string theory]]_.

On [[modular equivariant elliptic cohomology]] in [[type II string theory]]/[[F-theory]]:

* {#SatiKriz04} [[Igor Kriz]], [[Hisham Sati]], _M-theory, type IIA superstrings, and elliptic cohomology_, Adv. Theor. Math. Phys. 8 (2004), no. 2, 345–394 ([euclid:atmp/1091543172](http://projecteuclid.org/euclid.atmp/1091543172), [arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))

* {#KrizSati04b} [[Igor Kriz]], [[Hisham Sati]], _Type IIB String Theory, S-Duality, and Generalized Cohomology_, Nucl.Phys. B715 (2005) 639-664 ([arXiv:hep-th/0410293](http://arxiv.org/abs/hep-th/0410293))

* {#KrizSati05} [[Igor Kriz]], [[Hisham Sati]], _Type II string theory and modularity_, JHEP 0508 (2005) 038 ([arXiv:hep-th/0501060](http://arxiv.org/abs/hep-th/0501060))

On the [[Diaconescu-Moore-Witten anomaly]] interpreted in [[integral Morava K-theory]]:

* [[Hisham Sati]], [[Craig Westerland]], _Twisted Morava K-theory and E-theory_ ([arXiv:1109.3867](http://arxiv.org/abs/1109.3867))

  (this is followed up on in remark 5.4.12 in [[Michael Hopkins|Hopkins]], [[Jacob Lurie|Lurie]], _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_ )

On [[F4]] and [[Cayley plane]]-[[fiber bundles]] in [[M-theory]], relating to [[bosonic M-theory]]:

* [[Hisham Sati]], _$\mathbb{O}P^2$-bundles in M-theory_, Commun. Num. Theor. Phys 3:495-530,2009 ([arXiv:0807.4899](http://arxiv.org/abs/0807.4899))

* [[Hisham Sati]], _On the geometry of the supermultiplet in M-theory_, Int. J. Geom. Meth. Mod. Phys. 8 (2011) 1-33 ([arXiv:0909.4737](https://arxiv.org/abs/0909.4737))

On mathematical foundations of [[quantum field theory]] and [[perturbation theory|perturbative]] [[string theory]]:

* [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, [volume 83](http://www.ams.org/bookstore?fn=20&arg1=pspumseries&ikey=PSPUM-83) AMS (2011)

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:L-∞ algebra connections]]_ in _Quantum Field Theory_, Birkhäuser (2009), 303-424, DOI: 10.1007/978-3-7643-8736-5_17  ([publisher link](http://www.springerlink.com/content/p421153213548t31/), [arXiv:0801.3480](http://arxiv.org/abs/0801.3480))

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Twisted Differential String and Fivebrane Structures]]_ Communications in Mathematical Physics  October 2012, Volume 315, Issue 1, pp 169-213
([arXiv:0910.4001](http://arxiv.org/abs/0910.4001))

* H. S. , [[Urs Schreiber]], [[Jim Stasheff]] _[[schreiber:Fivebrane structures]]_ Rev. Math. Phys.21:1197-1240 (2009) ([arXiv:0805.0564](http://arxiv.org/abs/0805.0564))

* [[Domenico Fiorenza]], H. S., [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane]]_ ([arXiv:1201.5277](http://arxiv.org/abs/1201.5277))

* [[Domenico Fiorenza]], [[Hisham Sati]] , [[Urs Schreiber]], _[[schreiber:The moduli 3-stack of the C-field]]_ ([arXiv:1202.2455](http://arxiv.org/abs/1202.2455))

* [[Domenico Fiorenza]], [[Hisham Sati]] , [[Urs Schreiber]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_ Journal of Geometry and Physics, Volume 74, 2013, Pages 130–163 ([arXiv:1207.5449](http://arxiv.org/abs/1207.5449))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]],  _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_ ([arXiv:1301.2580](http://arxiv.org/abs/1301.2580)) in  Damien Calaque et al. (eds.) _Mathematical Aspects of Quantum Field Theories_  Springer 2014

* {#Sati13} [[Hisham Sati]], _Framed M-branes, corners, and topological invariants_, J. Math. Phys. 59 (2018), 062304 ([arXiv:1310.1060](http://arxiv.org/abs/1310.1060))

  (on the [[Hopf-Wess-Zumino term]] of the [[M5-brane]], [[rational Cohomotopy]] in the [[11d supergravity]] [[equations of motion]] and ...)

* [[Domenico Fiorenza]], H. S., [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_ ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

* {#MathaiSati14} [[Varghese Mathai]], [[Hisham Sati]], _Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality_, J. Geom. Phys. 92:240-251, 2015 ([arXiv:1404.2257](https://arxiv.org/abs/1404.2257))

  (on a [[non-commutative geometry]]-deformation of the [[self-dual higher gauge theory]] as it appears on the [[M5-brane]] [[worldvolume]], and its [[S-duality]])

On [[ninebrane structures]]:

* {#Sati14} H. S. , _Ninebrane structures_ ([arXiv:1405.7686](http://arxiv.org/abs/1405.7686))

On [[spherical T-duality]] in [[iterated algebraic K-theory]]:

* {#LindSatiWesterland16} [[John Lind]], [[Hisham Sati]], [[Craig Westerland]], _A higher categorical analogue of topological T-duality for sphere bundles_, Annals of K-Theory, Vol. 5 (2020), No. 1, 1–42 ([arXiv:1601.06285](http://arxiv.org/abs/1601.06285), [doi:doi:10.2140/akt.2020.5.1](https://msp.org/akt/2020/5-1/p01.xhtml))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]_, [ATMP Volume 22 (2018) Number 5](http://www.intlpress.com/site/pub/pages/journals/items/atmp/content/vols/0022/0005/) ([arXiv:1611.06536](https://arxiv.org/abs/1611.06536))

Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type II string theory]] (see also [there](D-brane#ReferencesKTheoryDescription)):

* {#GradySati19a} [[Daniel Grady]], [[Hisham Sati]], _Ramond-Ramond fields and twisted differential K-theory_ ([arXiv:1903.08843](https://arxiv.org/abs/1903.08843))

Discussion of [[twisted K-theory|twisted]] [[differential K-theory|differential]] [[KO-theory|orthogonal]] [[topological K-theory|K-theory]] and its relation to [[D-brane charge]] in [[type I string theory]] (on [[orientifolds]]):

* {#GradySati19b} [[Daniel Grady]], [[Hisham Sati]], _Twisted differential KO-theory_ ([arXiv:1905.09085](https://arxiv.org/abs/1905.09085))

On ([[cohomotopy|co-]])[[homotopy theory|homotopical]] foundations of [[M-theory]] ([[Hypothesis H]]):

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]] _[[schreiber:The rational higher structure of M-theory]]_, _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/):_ _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html), August 2018_. Fortschritte der Physik, 2019 ([doi:10.1002/prop.201910017](https://doi.org/10.1002/prop.201910017), [arXiv:1903.02834](https://arxiv.org/abs/1903.02834))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds]]_ ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

In relation to the [[Hopf WZ term]] and [[Page charge]]:

* {#FSS19WZ} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]_ ([arXiv:1906.07417](https://arxiv.org/abs/1906.07417))

On [[differential cohomotopy|differential]] [[Cohomotopy]] and [[Hypothesis H]]:

* [[Daniel Grady]], [[Hisham Sati]], *Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers*, Journal of Geometry and Physics **165** (2021) 104203 [[arXiv:2001.07640](https://arxiv.org/abs/2001.07640), [doi:10.1016/j.geomphys.2021.104203](https://doi.org/10.1016/j.geomphys.2021.104203)]

On [[U-duality]] (and possibly [[mysterious duality]]) via [[Hypothesis H]] as [[automorphisms]] of [[iterated loop space|iterated]] ([[Sullivan model of loop space|rational]]) [[cyclic loop spaces]] of the ([[rational n-sphere|rational]]) [[4-sphere]]:

* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious Triality and Rational Homotopy Theory* [[arXiv:2111.14810](https://arxiv.org/abs/2111.14810)]

* [[Hisham Sati]], [[Alexander Voronov]], *Mysterious Triality and M-Theory* [[arXiv:2212.13968](https://arxiv.org/abs/2212.13968)

On [[smooth sets]] as a [[convenient category of spaces|convenient category]] for [[variational calculus]] of [[Lagrangian quantum field theory|Lagrangian]] [[classical field theory]]:

* [[Grigorios Giotopoulos]], [[Hisham Sati]], *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

category: people


historical note on characteristic classes

Excerpt from the Preface to _Characteristic classes_ by [[John Milnor]] and [[Jim Stasheff]].

> The theory of characteristic classes began in the year 1935 with almost simultaneous work by HASSLER WHITNEY in the United States and EDUARD STIEFEL in Switzerland. Stiefel's thesis, written under the direction of Heinz Hopf, introduce and studied certain "characteristic" homology classes determined by the tangent bundle of a smooth manifold. Whitney, then at Harvard University, treated the case of an arbitrary sphere bundle. Somewhat later he invented the language of cohomology theory, hence the concept of a characteristic cohomology class, and proved the basic product theorem.

>In 1942 LEV PONTRJAGIN of Moscow University began to study the homology of Grassmann manifolds, using a cell subdivision due to Charles Ehresmann. This enabled him to construct important new characteristic classes. (Pontrjagin's many contributions to mathematics are the more remarkable in that he is totally blind, having lost his eyesight in an accident at the age of fourteen.)

> In 1946 SHING-SHEN CHERN, recently arrived at the Institute for Advanced Study from Kunming in southwestern China, defined  classes for complex vector bundles. In fact he showed that the complex Grassmann manifolds have a cohomology structure which is much easier to understand than that of the real Grassmann manifolds.

According to [this site](http://www-history.mcs.st-and.ac.uk/Biographies/Stiefel.html), E. Stiefel

> was awarded his doctorate for his thesis Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten from ETH Zürich in 1935. He published the results of this thesis in a paper in 1936.

the 1936., is of course Stiefel's paper on Stiefel-Whitney characteristic classes.

That sites quotes

* V Szebehely, D Saari, J Waldvogal, U Kirchgraber, _Eduard L Stiefel (1909-1978)_, in Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics, Math. Forschungsinst., Oberwolfach, 1978, Celestial Mech. __21__ (1) (1980), 3-4.

who say in the paper

> ... perhaps Stiefel's most famous contribution to pure mathematics, was dedicated to a fundamental study of the theory of vector fields on manifolds. Generalising the classical notion of the Eulerian characteristic of a manifold, he introduced the idea of the characteristic classes.

In that paper,

* Eduard Stiefel, _Richtungsfelder und Fernparallelismus in $n$-dimensionalen Mannigfaltigkeiten_, Comm. Math. Helv. __8__, 305-353 (1935-1936) [perm URL](http://resolver.sub.uni-goettingen.de/purl?GDZPPN002052075)

page 324, "Characteristik" (number) seems to appear for the first time:

> Die Zahl a heisst die Charakteristik des $m$-Feldes $\mathfrak{F}$ auf der Sphäre $S^r = S^{n-m}$.

* E. Stiefel at German [wikipedia](http://de.wikipedia.org/wiki/Eduard_Stiefel)
* Eduard Stiefel, _Über Richtungsfelder in den projektiven Räumen und einen Satz aus der reellen Algebra_, Comm. Math. Helv. _13_, 201-218 (1940-1941) [perm URL](http://resolver.sub.uni-goettingen.de/purl?GDZPPN002053217)

Pontrjagin's 1942 paper is

* Lev Pontrjagin, _Characteristic cycles on manifolds_, C. R. (Doklady) Acad. Sci. URSS (N.S.) 35, (1942). 34–37.

[[!redirects historical notes on characteristic classes]]


historical notes on quasigroups

This entry is about terminological and historical issues (and matters of opinion) on binary operations/algebraic structures, in particular nonassociative ones: magmas, quasigroups, racks and quandles.

Historically, one most often considered everywhere-defined binary algebraic operations; however partial operations were also sometimes considered. [[Oystein Ore]] termed everywhere defined operations groupoidal, hence a set with one binary operation that is everywhere defined has been called a groupoid. If the operation is not everywhere defined, then according to the textbook of [Bruck](#Bruck), people talked about halfgroupoids. In the $n$Lab, a [[groupoid]] is another structure (a small category where all morphisms are isomorphisms). This however comes from Brandt's groupoids, which were considered as sets with partially defined operations, equivalent to what we call connected groupoids. As Brandt groupoids assumed primacy, what had been the usual groupoids were now renamed magmas (after [[Nicolas Bourbaki]]). In categorical universal algebra, "magma" usually means unital magma. Bourbaki coined some other names of common structures, including [[semigroup]] and [[distributive lattice]]. In choosing the word "magma", Bourbaki means to suggest amorphous structure, as a _weakening_ of the common and strong notion of a group. There is actually a tradition in such terminology of weakenings for groups: to use terms suggesting loose matter. For example, [[heap]] (synonymous with the earlier and similar *flock*) for a group with forgotten identity, and *wrack* (or what is more usual nowadays, [[rack]]) for a left distributive left quasigroup. Here *wrack* was introduced by [[John Conway]] (cf. the verb "wreck" and the noun "wreckage"), evidently to conjure up the phrase *wrack and ruin*, but Conway was also punning on the similar spelling and sound in the name of his coauthor [[Gavin Wraith|Wraith]]. (Meanwhile, the term "[[shelf]]", introduced by [[Alissa Crans]], would appear to be based on the homophone "rack" which has a different meaning, thus departing from the tradition, or starting a new one.)

Books on binary structures are quite dominated by the study of nonassociative ones, where quasigroups are the most prominent class (including one sided versions, like left quasigroups and their subclasses, e.g., left [[rack]]s).

## Quasigroups

Some consider the concept of quasigroup to be an example of [[centipede mathematics]], and uninteresting due to their lack of deep applications (the latter opinion on quasigroups and loops is however becoming obsolete in view of modern examples and applications).  For example, one mathematician has written:

>The meeting was dominated by algebraic loop theory.  It occurred to me that as a way to use your intellectual resources this was very akin in significance to doing a difficult sudoku, a thought that was made very ironic when one speaker started making loops out of what were essentially sudoku squares.

Nonetheless it can be instructive to ponder these concepts, and there are some nontrivial examples.

## Literature

* wikipedia [magma](http://en.wikipedia.org/wiki/Magma_%28algebra%29), [Øystein Ore](http://en.wikipedia.org/wiki/%C3%98ystein_Ore), [Nicolas Bourbaki](http://en.wikipedia.org/wiki/Nicolas_Bourbaki)
* R.H. Bruck, _A survey of binary systems_, Springer-Verlag 1958
 {#Bruck}
* Hala Orlik Pflugfelder, _Historical notes on loop theory_, Comment. Math. Univ. Carolin. __41__,2 (2000) 359–370 [pdf](http://www.emis.de/journals/CMUC/pdf/cmuc0002/pflug.pdf)

category: algebra


history of cohomology with local coefficients

The subject of [[cohomology]] with local coefficients ([[twisted cohomology]]) begins with [[Reidemeister]] in his study of lens spaces in 1938 in the article _Topologie der Polyeder und kombinatorischen Topologie der Komplexe_. He considered the cohomology of a polyhedron (there called a manifold -- such terminology hadn't yet settled down and so is probably best considered as a combinatorial manifold) $M$ in terms of cochains of the universal covering space $U$ together with the action of the fundamental group of $M$ as deck transformations. His terminology was _Überdeckung_ (covering space). Interestingly, this approach underlies the [[crossed complex|crossed complexes]] studied extensively by Ronnie Brown, Phil Higgins and others in the past four decades.

+--{.query}
[[Zoran Škoda]] J. Nielsen, _Über die Minimalzahl der Fixpunkte bei den Abbildungstypen der Ringflächen_, Math. Ann. __82__(1-2):83–93, 1920. is probably the reference responsible for so called Nielsen invariant. Nielsen invariant and Reidemeister torsion are rather related (there are many article now on this relation, for example by A. Felshtyn); there are also some precursors like dilogarithm functions from the end of 19th century, closely related to the subject. Finally the first article having homological algebra by Cayley from 1840-s is claimed by Gelfand-Kapranov-Zelevinsky to use at one point an invariant which is in the calculations with Koszul complexes the analogue of the Whitehead torsion, but I was not able to do the comparison. There is a book

* J.L. Dupont, <a href="http://home.imf.au.dk/dupont/scissors.ps">Scissors congruences, group homology and characteristic classes (.ps) </a>, Lecture notes from Nankai Institute of Mathematics, 1998, Manuscript, University of Aarhus, Department of Math., Aarhus, 1999, 122 pp., and in: Nankai Tracts in Mathematics 1, World Scientific, Singapore, 2001.

 which explains the relations of dilogarithms, [[Reidemeister torsion]], Dehn invariant (which is from 1900) and its role in understanding the 19th century scissors congruence...The subject is very active now. After physicist Anatole Kirillov found the [[quantum dilogarithm]] and L. D. Fadeeev and Kashaev published a paper about it (and discovered the pentagon relation), this became an important topic in quantization (Fadeev's modular double, quantization of Teichmuller spaces, work of A. Fock, L. Takhdajan, Aldrovandi etc.) of hyperbolic spaces and generally the hyperbolic geometry and with relations, together with classical [[dilogarithm]] to many other subjects (e.g. to the work of Reznikov and theory or regulators, specially Borel regulator in higher algebraic K-theory, important for study of motives (Goncharov et al.) and, as W. Nahm shown, also relevant for understanding some phenomena in CFT).
=--

[[Norman Steenrod|Steenrod]] in 1942 independently generalized the concept of ordinary cohomology to *cohomology with coefficients in a local system of groups*, later known as *cohomology with local coefficients*. In 1943, he acknowledged Reidemeister's precedence  showing that (after a suitable interpretation) the theory includes Reidemeister's concept of Überdeckung.

In 1947, [[Eilenberg]] showed that the cohomology theory with a local system of groups is naturally equivalent to a theory of equivariants with respect to the [[deck transformation]]s of cohomology with ordinary coefficients on the corresponding universal [[covering space]].

In 1950, Olum extended Eilenberg's [[obstruction]] theory to the non-simple case, i.e. when the [[fundamental group]] acts non-trivially on the higher [[homotopy group]]s involved.

* _Obstructions to extensions and homotopies_, Annals 52 (1950) 1--52

In the 1960s and early 1970s, there were several papers addressing operations on cohomology with local coefficients:

* 1963, [[Sam Gitler]] (Steenrod mod $p$), _Cohomology operations with twisted coefficients_, AJM 85 (1963)156--188

* 1966, McClendon, thesis -- summarized in

* 1967, Emery Thomas, tc ops

* 1967, Larmore, tc ops

* 1969, McClendon, _Higher order twisted cohomology operations_, Invent. Math. 7 (1969) 183--214

* 1969, Larmore, tc

* 1970, Peterson, tc ops

* 1971, McClendon, tc ops

In 1972, the phrase _twisted cohomology_ was  used by Larmore to describe
$H(-; \mathcal{E})$, cohomology with coefficients in a special kind of [[spectrum]] $\mathcal{E}$  related to a [[fibration]] $p:E\to B.$ The result is what May and Sigurdsson call a parameterized spectrum,
the _parameters_ being the points of $B$, which might also be called, in the older topological terminology, an _ex-spectrum_.

In recent years, especially after  the invention of [[twisted K-theory]], 'twisted cohomology' has become the generic term, including not only twisted [[Whitehead-generalized cohomology]] but even what is known as twisted _[[nonabelian cohomology|non-abelian]]_ cohomology. Among other things, non-abelian refers to the fact
that no spectrum need be involved, but only a single target space, preferably at least a [[loop space]].

Also in 1972 Robinson constructed Moore--[[Postnikov system]]s for non-simple fibrations. In particular, he
provided twisted $K(\pi,n)$s corresponding to cohomology with local coefficients.

##Comments##

It is useful to show that  the work of Reidemeister was extended in another way by [[J.H.C. Whitehead]] in his paper [CHII] referenced below. [[Graham Ellis]] writes in his paper (E) as follows:

.... J.H.C. Whitehead showed in
(CHII)  that, for connected $CW$-complexes $X, Y$ with
dim $X \leq n$ and $\pi_i Y = 0$ for $2\leq i \leq  n - 1$, the
homotopy classification of maps $X \to  Y$ can be reduced to a
purely algebraic problem of classifying, up to an appropriate
notion of homotopy, the $\pi_1$-equivariant chain homomorphisms $C_*
\widetilde{X} \to  C_* \widetilde{Y}$ between the cellular chain complexes of the
universal  covers. The classification of homotopy equivalences $Y
\simeq Y$ can similarly be reduced to a purely algebraic problem.
Moreover, the algebra of the cellular chains of the universal
covers closely reflects the topology, and provides pleasant and
interesting exercises.

These results ought to be a standard piece of elementary algebraic
topology. Yet, perhaps because of the somewhat esoteric exposition given in (CHII), and perhaps because of a lack of
worked examples, they have remained largely ignored. The purpose
of the present paper is to rectify this situation.  We shall show
the utility of Whitehead's results by using them to give new and
clearer treatments of various known classifications.

In fact this work includes that of Olum referenced above.

(End of quote)

Whitehead introduced what he called _homotopy systems_ which we now call free, reduced [[crossed complex]]es, and these were a major tool in his work. He related these to chain complexes with a group of operators, and, for CW-complexes, to the chains of the universal cover.

However Whitehead remarks that the homotopy systems have better realisation properties than the chain complexes with operators, and this we can now see is reflected in the classifying space $BC$ of a crossed complex $C$ defined in (BH). By choosing the crossed complex $C$ to be a group $G$ in dimension 1 and a $G$-module $A$ in dimension $n$ we easily get the classifying space for local coefficients defined by various authors. This is discussed in more generality in Section 7 of (BH).

##References##

* (E) [[G.J.Ellis]],  Homotopy classification the J.H.C. Whitehead way.
_Exposition. Math._ 6 (1988) 97--110.

* (BH) Brown, R. and Higgins, P.J.
The classifying space of a crossed complex.
_Math. Proc. Cambridge Philos. Soc._ 110 (1991)
  95--120.

* (CHII) Whitehead, J. H.C.
Combinatorial homotopy. II.
_Bull. Amer. Math. Soc._ 55 (1949) 453--496.

[[!redirects History of cohomology with local coefficients]]


history of inductive types -- references

### History of inductive types
 {#ReferencesHistoryOfInductiveTypes}

Historical references on the definition of [[inductive types]].

#### Precursors

A first [[type theory|type theoretic]] formulation of general [[inductive definitions]]:

* [[Per Martin-Löf]], *Hauptsatz for the Intuitionistic Theory of Iterated Inductive Definitions*, Studies in Logic and the Foundations of Mathematics **63** (1971) 179-216 $[$doi:10.1016/S0049-237X(08)70847-4$]$

The [[induction principle]] for [[identity types]] (also known as "path induction" or the "[[J-rule]]") is first stated in:

* {#MartinLof75} [[Per Martin-Löf]], §1.7 and p. 94 of: _An intuitionistic theory of types: predicative part_, in: H. E. Rose, J. C. Shepherdson (eds.), *Logic Colloquium '73, Proceedings of the Logic Colloquium*, Studies in Logic and the Foundations of Mathematics **80** (1975) 73-118 (doi:10.1016/S0049-237X(08)71945-1, [CiteSeer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926))

and in the modern form of [[inference rules]] in:

* {#NordströmPeterssonSmith90} Bengt Nordström, Kent Petersson, Jan M. Smith, §8.1 of: *Programming in Martin-Löf's Type Theory*, Oxford University Press (1990) $[$[webpage](https://www.cse.chalmers.se/research/group/logic/book/), [pdf](https://www.cse.chalmers.se/research/group/logic/book/book.pdf), [[NordstromPeterssonSmith-TypeTheory.pdf:file]]$]$

The special case of inductive types now known as *[[W-types|$\mathcal{W}$-types]]* is first formulated in:

* [[Per Martin-Löf]], pp. 171 of: *Constructive Mathematics and Computer Programming*, in: *Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science (1979)*, Studies in Logic and the Foundations of Mathematics **104** (1982) 153-175 $[$doi:10.1016/S0049-237X(09)70189-2, [ISBN:978-0-444-85423-0](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/104/suppl/C)$]$

* [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]), _Intuitionistic type theory_, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) $[$[pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]]$]$

Early proposals for a general formal definition of inductive types:

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

* [[Paul Francis Mendler]], *Inductive Definition in Type Theory*, Cornell (1987) $[$[hdl:1813/6710](https://hdl.handle.net/1813/6710)$]$

* [[Roland Backhouse]], *On the Meaning and Construction of the Rules of Martin-Löf's Theory of Types*, in: [Workshop on General Logic](https://www.lfcs.inf.ed.ac.uk/reports/88/ECS-LFCS-88-52/), ECS-LFCS-88-52 (1988) $[$[pdf](https://www.bcs.org/media/5076/bcs-facs-martin-lof-theory-of-types-a-computing-scientists-perspective-roland-backhouse-sep1987-file-4.pdf), [pdf](http://www.lfcs.inf.ed.ac.uk/reports/88/ECS-LFCS-88-52/ECS-LFCS-88-52.pdf)$]$

#### Modern definition

The modern notion of [[inductive types]] and [[inductive families]] in [[intensional type theory]] is independently due to

* [[Peter Dybjer]], *Inductive sets and families in Martin-Löf's type theory and their set-theoretic semantics*, Logical frameworks (1991) 280-306 $[$[doi:10.1017/CBO9780511569807.012](https://doi.org/10.1017/CBO9780511569807.012), [pdf](http://www.cse.chalmers.se/~peterd/papers/Setsem_Inductive.pdf)$]$

* [[Peter Dybjer]], *Inductive families*, Formal Aspects of Computing **6** (1994) 440–465 $[$[doi:10.1007/BF01211308](https://doi.org/10.1007/BF01211308), [doi:10.1007/BF01211308](https://doi.org/10.1007/BF01211308), [pdf](http://www.cse.chalmers.se/~peterd/papers/Inductive_Families.pdf)$]$

and due to

* [[Thierry Coquand]], [[Christine Paulin]], *Inductively defined types*, COLOG-88 Lecture Notes in Computer Science **417**, Springer (1990) 50-66 $[$[doi:10.1007/3-540-52335-9_47](https://doi.org/10.1007/3-540-52335-9_47)$]$

which became the basis of the [[calculus of inductive constructions]] used in the [[Coq]]-[[proof assistant]]:

* [[Frank Pfenning]], [[Christine Paulin-Mohring]], *Inductively defined types in the Calculus of Constructions*, in: *Mathematical Foundations of Programming Semantics MFPS 1989*,  Lecture Notes in Computer Science **442**, Springer (1990) $[$[doi:10.1007/BFb0040259](https://doi.org/10.1007/BFb0040259)$]$

* [[Christine Paulin-Mohring]], *Inductive definitions in the system Coq -- Rules and Properties*, in: *Typed Lambda Calculi and Applications* TLCA 1993, Lecture Notes in Computer Science **664** Springer (1993) $[$[doi:10.1007/BFb0037116](https://doi.org/10.1007/BFb0037116)$]$

reviewed in

* [[Zhaohui Luo]], §9.2.2 in: *Computation and Reasoning -- A Type Theory for Computer Science*, Clarendon Press (1994) $[$[ISBN:9780198538356](https://global.oup.com/academic/product/computation-and-reasoning-9780198538356), [pdf](https://www.researchgate.net/profile/Zhaohui-Luo-4/publication/243770570_Computation_and_Reasoning_A_Type_Theory_for_Computer_Science/links/54d0dc6b0cf29ca81103f26a/Computation-and-Reasoning-A-Type-Theory-for-Computer-Science.pdf)$]$

* [[Christine Paulin-Mohring]], §2.2. in: *Introduction to the Calculus of Inductive Constructions*, contribution to: *Vienna Summer of Logic* (2014) $[$[hal:01094195](https://hal.inria.fr/hal-01094195), [pdf](https://hal.inria.fr/hal-01094195/document), [pdf slides](https://easychair.org/smart-program/VSL2014/APPA-invited-slides-6.pdf)$]$

with streamlined exposition in:

* [[Univalent Foundations Project]], §5.6 of: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) $[$[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)$]$

The generalization to [[inductive-recursive types]] is due to

* [[Peter Dybjer]], *A general formulation of simultaneous inductive-recursive definitions in type theory*, The Journal of Symbolic Logic **65** 2 (2000) 525-549 $[$[doi:10.2307/2586554](https://doi.org/10.2307/2586554), [pdf](http://www.cse.chalmers.se/~peterd/papers/Inductive_Recursive.pdf)$]$

* [[Peter Dybjer]], [[Anton Setzer]], *Indexed induction-recursion*, in *Proof Theory in Computer Science* PTCS 2001. Lecture Notes in Computer Science**2183** Springer (2001) $[$[doi:10.1007/3-540-45504-3_7](https://doi.org/10.1007/3-540-45504-3_7), [pdf](http://www.cse.chalmers.se/~peterd/papers/InductionRecursionInitialAlgebras.pdf)$]$

See also:

* [[Peter Dybjer]], *[Inductive and inductive-recursive definitions in type theory](http://www.cse.chalmers.se/~peterd/papers/inductive.html)* (link list)


history of mathematics

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## Overview

For the moment, we list here only a few works in the history of mathematics:
 some with a focus on fields particularly emphasised in the $n$Lab, and some general. [Wikipedia](https://en.wikipedia.org/wiki/History_of_mathematics) has a nice list of general works.

## Spaces, categories, topology, sheaves, topoi, cohomology, ...

* Mathieu Bélanger. _Grothendieck et les topos : rupture et continuité dans les modes
d'analyse du concept d'espace topologique_, thesis, Université de Montréal 2010 [pdf](https://papyrus.bib.umontreal.ca/jspui/bitstream/1866/4759/2/Belanger_Mathieu_2010_these.pdf)
* [[Pierre Cartier]], _A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry_, Bull. Amer. Math. Soc. __38__ (2001), 389-408, [pdf](http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf)
* Ralf Krömer, _Tool and object: A history and philosophy of category theory_, Birkhäuser 2007
* [[John Bell]], _The development of categorical logic_ ([pdf](http://publish.uwo.ca/~jbell/catlogprime.pdf))
* Jean-Pierre Marquis, Gonzalo E. Reyes, _The history of categorical logic 1963–1977_, 111 pp. [pdf](https://www.webdepot.umontreal.ca/Usagers/marquisj/MonDepotPublic/HistofCatLog.pdf)

## General history of mathematics

* C. B. Boyer, _A history of mathematics_, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989
* [[Felix Klein]], _The development of mathematics in 19th century_
* Julian Lowell Coolidge, _A history of geometrical methods_, 1940, Dover Phoenix Editions, Oxford University Press 1940.
* J. Gray, _The real and the complex: a history of analysis in the 19th
century_, Springer Undergraduate Mathematics Series
* Dirk Jan Struik, _A concise history of mathematics_
* [[Umberto Bottazzini]], _Il flauto di Hilbert: Storia della matematica_, UTET Università, 473pp.; _The higher calculus: A history of real and complex analysis from Euler to Weierstrass_, Springer 1986
* B. L. van  der  Waerden, _Geometry and algebra in ancient civilizations_, Springer 1983 [gBooks](https://books.google.hr/books?id=-pTzCAAAQBAJ)

## Wiki, collection and encyclopaedia articles

* The MacTutor History of Mathematics archive: [html](http://www.gap-system.org/~history)
* Stanford Encyclopaedia of Philosophy [online](http://plato.stanford.edu), [contents](http://plato.stanford.edu/contents.html)
* wikipedia [history of mathematics](http://en.wikipedia.org/wiki/History_of_mathematics)

## Researchers and institutes for history of mathematics
* [hsm](http://hsm.institut.math.jussieu.fr) at Jussieu

## $n$Lab entries with historical information

* [[history of cohomology with local coefficients]]
* [[historical notes on characteristic classes]]
* [[geometry of 19th century]], [[Erlangen program]]
* [[Čech methods]]
* [[philosophy]]
* lots of entries on individuals, including for some of the most influential mathematicians in history, e.g. [[Alexander Grothendieck]], [[Jacob Lurie]], [[Maxim Kontsevich]], [[Saunders MacLane]], [[Bourbaki]], [[Alfred Tarski]] etc.
* [[list of theses in category theory]], [[list of lost manuscripts in category theory]]

## Other literature

* G.A. Miller, _Felix Klein and the history of modern mathematics_, PNAS 13, 1927, 611-613

The relation of the techniques of the pioneers of infinitesimal calculus and the modern [[nonstandard analysis]] is discussed in

* Piotr Blaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, David Sherry, _Toward a history of mathematics focused on procedures_, [arxiv/1609.04531](http://arxiv.org/abs/1609.04531)

A collection of articles on XX-century mathematics is in

* Michele Audin (editor), _[[Matériaux pour l’histoire des mathématiques au XXe siècle]]_ - Actes du colloque à la mémoire de Jean Dieudonné (Nice 1996) Séminaires & Congrès. Numèro 3. Paris (Socièté Mathématique de France). 1998. 283 pp.

* Glenn Shafer, Vladimir Vovk, _The sources of Kolmogorov’s Grundbegriffe_, Statistical Science 21(1) [doi](https://dx.doi.org/10.1214/088342305000000467) [arXiv:math/0606533](http://arxiv.org/abs/math/0606533)

[[!redirects history]]


Hitchin connection

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* table of contents
{:toc}

## Idea

For $G$ a [[compact Lie group]] and $\Sigma$ a [[surface]], the _Hitchin connection_ ([Hitchin 90](#Hitchin90)) is a [[projectively flat connection]] over the [[moduli space of Riemann surfaces]] $\mathcal{M}_\Sigma$ whose [[fiber]] over a [[complex structure]] on $\Sigma$ is a [[space of quantum states]] which [[quantization of Chern-Simons theory]] for [[gauge group]] $G$ assigns to the induced [[Kähler polarization]] on the [[moduli space of flat connection]] $Loc_G(\Sigma)$ (the [[phase space]] of [[Chern-Simons theory]]), hence whose fiber is the space of [[holomorphic sections]] of the [[prequantum line bundle]] of the CS theory with respect to the induced Kähler structure on $Loc_G(\Sigma)$ (induced, for abelian $G$, via the [Weil complex structure](intermediate+Jacobian#WeilIntermediateJacobian) and in the general nonabelian case via the [[Narasimhan-Seshadri theorem]]/[[Donaldson-Uhlenbeck-Yau theorem]]).

Therefore the existence and (projective) flatness of the Hitchin connection exhibits the relative independence of the [[geometric quantization]] of [[Chern-Simons theory]] from the choice of [[polarization]].

The Hitchin connection is akin to the _[[Knizhnik-Zamolodchikov connection]]_, which is built by thinking [[AdS3-CFT2 and CS-WZW correspondence|holographic dually]] of the vector bundle of [[conformal blocks]] of the corresponding $G$-[[Wess-Zumino-Witten model]] [[2d CFT]] on $\Sigma$.

A more axiomatic characterization of such projectively flat connections in terms of [[modular functors]] is due to ([Segal 88, prop. 5.4](#Segal88)). See ([Segal 88, around (5.9)](#Segal88)) for discussion of how to turn these projectively flat connections into genuine [[flat connections]].

Reviews of the Hitchin connection include ([Lauridsen 10, section 2](#Lauridsen10)).

## Related concepts

* [[quantization of Chern-Simons theory]]

* [[AdS3-CFT2 and CS-WZW correspondence]]

* [[modular functor]]

## References

The original article is

* {#Hitchin90} [[Nigel Hitchin]], _Flat connections and geometric quantization_, : Comm. Math. Phys. Volume 131, Number 2 (1990), 347-380. ([Euclid](http://projecteuclid.org/euclid.cmp/1104200841))

(What is now called the Hitchin connection appears in theorem 3.6 there, its expression in local coordinates is around (3.15). That for abelian gauge group $U(1)$ the classical [[Riemann theta functions]] constitute the covariantly constant sections of the Hitchin connection in these coordinates is remark 4.12.)

More axiomatic/abstract discussion of these projectively flat connections in terms of [[modular functors]] is in

* {#Segal88} [[Graeme Segal]], section 5 of _The definition of conformal field theory_ , preprint, 1988; also in [[Ulrike Tillmann]] (ed.) _Topology, geometry and quantum field theory_ , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. ([pdf](https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf))

More explicit expressions are discussed in

* Bert van Geemen, [[Aise Johan de Jong]], _On Hitchin's Connection_ ([arXiv:alg--geom/9701007](http://arxiv.org/abs/alg--geom/9701007))

A nice review and new concise account is in

* {#Looijenga10} [[Eduard Looijenga]], _From WZW models to Modular Functors_ ([arXiv:1009.2245](http://arxiv.org/abs/1009.2245))

* {#Andersen11} Johan Martens [[Jørgen Andersen]], notes by Søren Jørgensen, section 1.2.1 and 4 of _Topological quantum field theories and moduli spaces_, 2011 ([pdf](http://maths.fuglede.dk/noter/tqftms.pdf))

Discussion for [[deformation quantization]] instead of [[geometric quantization]] is in

* [[Jørgen Andersen]], _Hitchin's connection, Toeplitz operators and symmetry invariant deformation quantization_ ([arXiv:math/0611126](http://arxiv.org/abs/math/0611126))

This also reproduces the original construciton in the context of [[Chern-Simons theory]] in

* {#AxelrodPietraWitten91} [[Scott Axelrod]], S. Della Pietra, [[Edward Witten]], _Geometric quantization of Chern-Simons gauge theory_, Jour. Diff. Geom. 33 (1991), 787-902.  ([EUCLID](http://projecteuclid.org/euclid.jdg/1214446565))

More details (on [[metaplectic correction]]) and generalization to connections over more general manifolds are in

* {#ScheinostSchottenloher96} Peter Scheinost, [[Martin Schottenloher]], _Metaplectic quantization of the moduli spaces of flat and parabolic bundles_, J. reine angew. Mathematik, 466 (1996) ([web](https://eudml.org/doc/153753))

Similar generalization away from [[moduli spaces of flat connections]] to general [[symplectic manifolds]] with [[Kähler structure]] on them is also in

* {#Lauridsen10} Lauridsen, _Aspects of quantum mathematics  -- Hitchin connections and the AJ conjecture_, PhD thesis Aarhus 2010 ([pdf](http://pure.au.dk/portal/files/41741925/imf_phd_2010_mrl.pdf))

Relation between the Hitchin connection and bundles of [[conformal blocks]] is discussed in

* {#Laszlo98} [[Yves Laszlo]] _Hitchin's and WZW connection are the same_, J. Differential Geom. 49 (1998), no. 3, 547–576 ([pdf](http://www.emis.de/journals/NYJM/JDG/archive/vol.49/3_5.pdf))

For more on this see at _[[quantization of 3d Chern-Simons theory]]_.

[[!redirects Hitchin connections]]


Hitchin fibration

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## Overview

Liouville [[integrable systems]] may always be phrased in the form of a [[Lax pair]] depending on a spectral parameter belonging to what is called a _[[spectral curve]]_. This way the [[analytic geometry|complex analytic]] theory of [[Riemann surfaces]] comes into play.

The landmark [Hitchin 87](#Hitchin87) introduced more refined viewpoint to a rather general class of [[integrable systems]]. The spectral curve is a [[hyperelliptic curve]] (genus 2 or more) for Sp(m,C)-[[Higgs bundles]]. This system is defined on the [[cotangent bundle]] of [[moduli space]] of [[stable bundles]] studied by Ramanan, Narasimhan, Hitchin, Nitsure, [[Gerd Faltings|Faltings]], [[Carlos Simpson|Simpson]] (introduction of [[Higgs bundles]]!) and others.

Some of the features of that system are more readily seen from consideration of a canonical map from the cotangent bundle of the moduli space to the so-called [[Hitchin base]], and the map is called the
_Hitchin fibration_.

The [[elliptic Calogero-Moser system]] is an example of a Hitchin system.

For other purposes than integrable systems, e.g. in [[representation theory]], one can consider Hitchin fibration over other ground fields. [[Alexander Beilinson]] and [[Vladimir Drinfeld]] have shown the importance of Hitchin fibration in the
[[geometric Langlands program]]. Ngô Bảo Châu (with Laumon) has studied a Hitchin fibration to prove an important step in Langlands program, so called Fundamental Lemma, which was conjectural from 1970s and is one of the main mathematical discoveries in last decade. For this achievement, Ngô  was awarded the Fields medal at ICM 2010 in Hyderabad ([link](http://www.icm2010.org.in/imu-prizes/prize-winners-2010/fields-medal-ngo-bao-chau)).

## Related entries

* [[moduli space of bundles]]

* [[topological recursion]]

## References

* {#Hitchin87} [[Nigel Hitchin|N.J. Hitchin]], _Stable bundles and integrable systems_,  Duke Math. J. , 54  (1987)  pp. 91–114 [MR89a:32021](http://www.ams.org/mathscinet-getitem?mr=89a:32021) [doi](http://dx.doi.org/10.1215/S0012-7094-87-05408-1) [euclid](http://projecteuclid.org/euclid.dmj/1077305506)

* N.J. Hitchin, _Flat connections and geometric quantization_, Comm. Math. Phys. __131__, n 2 (1990), 347-380, [euclid](http://projecteuclid.org/euclid.cmp/1104200841)

* Emma Previato, _Hitchin system_, Springer Online [Enc. of Math.](http://eom.springer.de/H/h120090.htm)

* A. Beilinson, V. Drinfeld, _Quantization of Hitchin's integrable system and Hecke eigensheaves_, [pdf](http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf)

* Ngô Bảo Châu, _Le lemme fondamental pour les algebres de Lie_, [arxiv/0806.4566](https://arxiv.org/abs/0806.4566)

* [[Carlos Simpson|Carlos T. Simpson]], _Higgs bundles and local systems_, Publ. Mathématiques de l'IHÉS, 75 (1992), p. 5-95, [numdam](http://www.numdam.org/item?id=PMIHES_1992__75__5_0)

* R. Donagi, _Spectral covers_, MSRI series 28, [pdf](http://www.msri.org/publications/books/Book28/files/donagi.pdf)

* Tamas Hausel, _Global topology of the Hitchin system_, [arxiv/1102.1717](https://arxiv.org/abs/1102.1717)

* David Nadler, _Springer theory via the Hitchin fibration_, Compositio Math. __147__ (2011) 1635--1670
[doi](https://doi.org/10.1112/S0010437X1100546X) [arxiv/0806.4566](https://arxiv.org/abs/0806.4566) ($n$POV emphasis to an example in representation theory)

* James Arthur, _The Work of Ngô Bảo Châu_, Proc. ICM Hyderabad 2010, [pdf](http://www.icm2010.org.in/wp-content/icmfiles/laudaions/fields2.pdf)

* Ken-ichi Sugiyama, _A quantization of the Hitchin hamiltonian system and the Beilinson-Drinfeld isomorphism_, [arxiv/0708.2957](https://arxiv.org/abs/0708.2957)

Treatment of [[wall crossing]], relation of Hitchin system to [[Donaldson-Thomas invariant]]s and theories of [[class S]]

* [[Davide Gaiotto]], [[Greg Moore]], [[Andrew Neitzke]], _Wall-crossing, Hitchin systems, and the WKB approximation, Advances in Mathematics __234__ (2013) 239--403 [arXiv:0907.3987](https://arxiv.org/abs/0907.3987) [doi](https://doi.org/10.1016/j.aim.2012.09.027)

Discussion in the context of the [[topological recursion]] includes

* {#BaragliaHuang17} [[David Baraglia]], Zhenxi Huang, _Special Kähler geometry of the Hitchin system and topological recursion_, Adv. Theor. Math. Phys. __23__ No. 8 (2019) 1981--2024 ([arXiv:1707.04975](https://arxiv.org/abs/1707.04975) [doi](https://doi.org/10.4310/ATMP.2019.v23.n8.a2)

reviewed in

* {#Baraglia18} [[David Baraglia]], _Topological recursion and Bergman tau functions for the Hitchin system_, talk at _[Tau Functions of Integrable Systems and Their Applications](http://www.birs.ca/events/2018/5-day-workshops/18w5025)_, Banff 2018 ([recording](http://www.birs.ca/events/2018/5-day-workshops/18w5025/videos/watch/201809060845-Baraglia.html))

See also

* Wikipedia, _[Hitchin system](https://en.wikipedia.org/wiki/Hitchin_system)_

[[!redirects Hitchin system]]
[[!redirects Hitchin systems]]
[[!redirects Hitchin integrable system]]
[[!redirects Hitchin integrable systems]]
[[!redirects Hitchin fibrations]]


Hitchin functional

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## Idea

What are now called _Hitchin functionals_ are [[functions]]/[[functionals]] on spaces of [[differential n-forms]] on [[manifolds]] of [[dimension]] 6 or 7, for $n = 3$ and/or $n =4$. They are such that their [[critical points]] characterize holomorphic 3-forms of [[complex manifolds]] in dimension 6, as [[associative 3-forms]] of [[G2-manifolds]] in dimension 7.

## Definition

### In 6 dimensions

Let $X$ be a [[closed manifold|closed]] [[orientation|oriented]] [[smooth manifold]] of [[dimension]] 6.

For $\Omega \in \wedge^3 \Gamma(T^* X)$ a [[differential n-form|differential 3-form]] on $X$, write

* ${\vert \lambda \Omega \vert} \in (\wedge^6 \Gamma(T^* X))^{\otimes 2}$

* $\sqrt{\vert \lambda \Omega \vert} \in \wedge^6 \Gamma(T^* X)$.

The _Hitchin function_ on 3-forms is the function

$$
  \wedge^3 \Gamma(T^* X) \to \mathbb{R}
$$

which sends 3-form to the [[integration of differential forms]] of this 6-form over $X$

$$
  \Omega \mapsto \int_X \sqrt{\vert \lambda (\Omega) \vert}
  \,.
$$

### In 7 dimensions

Let $X$ be 7-dimensional.

Let $\Omega \in \wedge^3 \Gamma(T^* X)$ be a [[stable differential form]]. This determines a [[Riemannian metric]] $\g_\Omega$ on $X$. Write $\star_\Omega$ for the corresponding [[Hodge star operator]]. The _Hitchin functional_ now is the function that takes stable forms to

$$
  \int_X \Omega \wedge \star_\Omega \Omega
  \,.
$$

## Properties

### In 6 dimensions

A differential 3-form $\Omega$ such that $\lambda(\Omega) \lt 0$ is a [[critical point]] of the above functional precisely if there is the structure of a [[complex manifold]] on $X$ such that $\Omega$ is the real part of a non-vanishing holomorphic 3-form.

This is ([Hitchin, theorem 13](#Hitchin00)).

### In 7 dimensions

(...)

## Related concepts

* [[Calabi-Yau manifold]], [[G2-manifold]]

* [[topological M-theory]]

## References

The original articles are

* [[Nigel Hitchin]], _The geometry of three-forms in six and seven dimensions_, ([arXiv:math/0010054](http://arxiv.org/abs/math/0010054))
 {#Hitchin00}

* [[Nigel Hitchin]], _Stable forms and special metrics_ ([arXiv:math/0107101](http://arxiv.org/abs/math/0107101))
 {#Hitchin01}

Reviews with a perspective on the role of the functionals in [[physics]]/[[gravity]]/[[string theory]]/[[topological M-theory]] include

* [[Robbert Dijkgraaf]], [[Sergei Gukov]], [[Andrew Neitzke]], [[Cumrun Vafa]], _Topological M-theory as Unification of Form Theories of Gravity_ ([arXiv:hep-th/0411073v2](http://arxiv.org/abs/hep-th/0411073v2))

[[!redirects Hitchin functionals]]

[[!redirects Hitchin's functional]]
[[!redirects Hitchin's functionals]]


Hitoshi Murakami

* [webpage](http://www.math.is.tohoku.ac.jp/~hitoshi/)

## Selected writings

On the [[volume conjecture]]:

* {#MurakamiMurakami01} [[Hitoshi Murakami]], [[Jun Murakami]], _The Colored Jones Polynomial And The Simplicial Volume Of A Knot_, Acta Math. 186 (2001) 85-104 ([euclid.acta/1485891370](https://projecteuclid.org/euclid.acta/1485891370))

* {#MMOTY02} [[Hitoshi Murakami]], [[Jun Murakami]], M. Okamoto, T. Takata, and Y. Yokota, _Kashaev's Conjecture And The Chern-Simons Invariants Of Knots And Links_, Experiment. Math. 11 (2002) 427-435 ([arXiv:math/0203119](https://arxiv.org/abs/math/0203119))

* {#Murakami04} [[Hitoshi Murakami]], _Asymptotic Behaviors Of The Colored Jones Polynomials Of A Torus Knot_, Internat. J. Math. 15 (2004) 547-555.

* {#Murakami11} [[Hitoshi Murakami]], _An Introduction to the Volume Conjecture_ ([arXiv:1002.0126](https://arxiv.org/abs/1002.0126)) In: _Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory_, Contemporary Mathematics Volume 541, AMS 2011  ([doi:10.1090/conm/541](http://dx.doi.org/10.1090/conm/541))

## Related $n$Lab entries

* [[volume conjecture]]

category: people


Hitoshi Murayama

* [webpage](http://hitoshi.berkeley.edu/)

## related $n$Lab entries

* [[quantum electrodynamics]]

* [[Lamb shift]]

category: people


Hitoshi Nishino

* [webpage](https://web.csulb.edu/depts/physics/people/nishino_h.shtml)

## Selected writings

On [[D=12 supergravity]]:

* {#Nishino97} [[Hitoshi Nishino]], _Supergravity in 10 + 2 Dimensions as Consistent Background for Superstring_, ([arXiv:hep-th/9703214](https://arxiv.org/abs/hep-th/9703214))

* {#Nishino97b} [[Hitoshi Nishino]], _N=2 Chiral Supergravity in (10 + 2)-Dimensions As Consistent Background for Super (2 + 2)-Brane_, Phys. Lett. B437 (1998) 303-314 ([arXiv:hep-th/9706148](https://arxiv.org/abs/hep-th/9706148))

* {#Nishino98} [[Hitoshi Nishino]], _Supergravity Theories in $D \geq 12$ Coupled to Super p-Branes_, Nucl.Phys. B542 (1999) 217-261 ([arXiv:hep-th/9807199](https://arxiv.org/abs/hep-th/9807199))

On [[D=14 supersymmetry]]:

* [[Hitoshi Nishino]], _Supersymmetric Yang-Mills Theories in $D \geq 12$_, Nucl.Phys. B523 (1998) 450-464 ([arXiv:hep-th/9708064](https://arxiv.org/abs/hep-th/9708064))

category: people


HNN-extension

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## Idea

In [[combinatorial group theory]], by the *HNN construction* --- named after [Higman, Neumann and Neumann (1949)](#HigmanNeumannNeumann49) --- one means a [[universal construction]] which from a [[group]] and two *abstractly* [[isomorphism|isomorphic]] [[subgroups]] produces a group extension in which these two become [[conjugate subgroups]].

## Definition

\begin{definition}\label{HNNConstruction}
Given a [[group]], $G$, and a [[subgroup]], $H \subset G$, equipped with a(nother) [[monomorphism]] $\theta \colon  H \hookrightarrow G$, then the **HNN-extension** $G \ast_H$ is obtained by adjoining an [[element]] $t$ to $G$ subject to the condition:

$$
  \underset{h \in H}{\forall}
  \;\;\;\;
  t^{-1} \cdot h \cdot t \;=\; \theta(h)
  \,.
$$

\end{definition}

Notice that there is a canonical [[subgroup]]-inclusion

$$
  G \hookrightarrow G \ast_H
$$

under which the two copies of $H$ in $G$ (given by $H$ itself and by the [[image]] $\theta(H)$) become [[conjugate subgroups]] in $G \ast_H$.

## Properties

### Link with graphs of groups

Examine the [[fundamental group of a graph of groups|fundamental group]] of the [[graph of groups]], $\mathcal{G}$, with [[underlying]] [[graph]] the graph with one [[vertex]], $v$ and one [[edge]], $e$ and nothing else.

Take

* the vertex group, $G_v$, to be $G$,

* the edge group, $G_e$, to be $H$,

* the two morphisms from $G_e$ to $G_v$ to be the canonical inclusion of $H$ into $G$ and the given monomorphism, $\theta$,

then $\Pi_1(\mathcal{G})(v) \cong G \ast_H$.

### As a 2-colimit
 {#AsATwoColimit}

Consider groups $G$ seen in terms of their [[delooping groupoid]] $\mathbf{B}G$ as objects of the [[(2,1)-category]] [[Grpd]] of [[groupoids]].

> Beware that this construction $\mathbf{B}(-) \;\colon\; Grps \to Grpds$ is not a [[fully faithful (infinity,1)-functor|fully faithful]] [[(2,1)-functor]]: The fully-faithful embedding is obtained only by regarding the delooping groupoids as *[[pointed objects]]* in [[Grpd]], see at *[[looping and delooping]]*. The [[2-morphisms]] between ([[1-morphisms]] between) delooping groupoids which do not respect the base point are precisely those [[natural transformations]] referred to in the following.

Explicitly, this means that for a [[parallel pair]] of [[group homomorphisms]] $(\varphi,\theta) \colon H \rightrightarrows G$, a [[2-morphism]] $\mathbf{B}\varphi \Rightarrow \mathbf{B}\theta$ between their [[deloopings]] is equivalently (a [[natural isomorphism]] whose unique component is) an element $g \in G$ such that $g\varphi g^{-1}=\theta$.

Now within this [[(2,1)-category]], the ([[delooping]] of the) HNN-extension $G\ast_H$ (Def. \ref{HNNConstruction}) is a [[2-colimit]]:

Namely, if $\iota \colon H \hookrightarrow G$ denotes the given [[subgroup]] inclusion and $\theta \colon H \hookrightarrow G$ the (other) monomorphism, then $\mathbf{B}(G\ast_H)$ is the *[[coinserter]]* of $(\mathbf{B}\theta,\mathbf{B}\iota) \colon \mathbf{B}H \rightrightarrows \mathbf{B}G$.

## References

The original article:

* {#HigmanNeumannNeumann49} Graham Higman, B. H. Neumann, Hanna Neuman, *Embedding Theorems for Groups*, J. of the London Mathematical Society **s1-24** 4 (1949)  247-254 [[doi:10.1112/jlms/s1-24.4.247](https://doi.org/10.1112/jlms/s1-24.4.247)]

See also:

* Wikipedia, *[HNN extension](https://en.wikipedia.org/wiki/HNN_extension)*

* [[Jean-Pierre Serre]], _Arbres, amalgames, $SL_2$_, volume 46 of _Astérisque , Société mathématique
de France (1977)

* [[Jean-Pierre Serre]], , _Trees_ , Springer Monographs in Mathematics, Springer-Verlag Berlin.
150, 157, 161 (2003)


Ho(Cat)

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## Definition of Ho(Cat)

**Ho(Cat)** is a name for the [[homotopy category]] of [[Cat]].  That is, $Ho(Cat)$ is the [[category]]

* whose objects are ([[small category|small]]) categories, and
* whose morphisms are [[natural isomorphism]] classes of [[functors]].

This is an instance of a general construction which, given a [[2-category]], or more generally an [[n-category]], produces a [[1-category]] with the same objects and whose morphisms are [[equivalence]] classes of 1-morphisms in the original $n$-category.  Sometimes this is called the 1-[[truncation]] and denoted $\tau_1$.

It can also be viewed as an instance of the homotopy category of a [[model category]] (or more generally a [[category with weak equivalences]]).  The category $Ho(Cat)$ as defined above is equivalent to the category obtained from $Cat$ by forcing all [[equivalences of categories]] to be isomorphisms (by [[localization|localizing]]).  This is for the same reason that the category $hTop$ of [[topological spaces]] and [[homotopy]] classes of [[continuous maps]] is equivalent to the category obtained from $Top$ by inverting the homotopy equivalences (namely, the existence of [[cylinder objects]] and/or [[path objects]]).  Indeed, a [[cylinder object]] for a category $C$ is the [[product category]] $C \times I$ where $I$ is the category with two objects 0 and 1 and an isomorphism $0 \to 1$.  It is not difficult to see that an isomorphism of functors is the same as a [[homotopy]] of functors with the respect to the [[canonical model structure]] on $Cat$.

## Subcategories of Ho(Cat)

Some notable full subcategories of $Ho(Cat)$ include

* $Ho(Gpd)$, the homotopy category of the category [[Gpd]] of [[groupoids]].  Note that this is equivalent to the homotopy category of (unbased) [[homotopy 1-types]].
* The category whose objects are [[groups]] and whose morphisms are [[conjugacy class]]es of [[group homomorphism]]s.  This can be identified with the full subcategory of $Ho(Gpd)$ whose objects are the connected groupoids.  This category sometimes arises in the study of [[gerbes]].

## Ho(Cat)-categories

Like the homotopy category of any model category, $Ho(Cat)$ has [[products]] and [[coproducts]], and is in particular a [[cartesian monoidal category]].  Therefore, we can talk about categories [[enriched category|enriched over]] $Ho(Cat)$.  Such a "$Ho(Cat)$-category" consists of

* a collection of objects $x,y,z$
* for each pair of objects, a category $C(x,y)$
* for each object $x$, an objects $id_x\in C(x,x)$
* for each triple of objects, a functor $C(y,z)\times C(x,y)\to C(x,z)$

such that the usual associativity and unit diagrams for an enriched category commute up to isomorphism.  The difference between a $Ho(Cat)$-category and a [[bicategory]] is that in a $Ho(Cat)$-category, *no coherence axioms* are required of the [[associator]] and [[unitor]] isomorphisms; they are merely required to *exist*.  Thus a $Ho(Cat)$-category can be thought of as an "incoherent bicategory."  In particular, any bicategory has an underlying $Ho(Cat)$-category.

Although $Ho(Cat)$-categories are not very useful, there are some interesting things that can be said about them.  For instance:

* Any $Ho(Cat)$-category which is equivalent, as a $Ho(Cat)$-category, to a bicategory, is itself in fact a bicategory.
* Any [[2-functor]] between bicategories which induces an equivalence of underlying $Ho(Cat)$-categories is in fact itself an equivalence of bicategories (or "biequivalence").

An example of a $Ho(Cat)$-category that does not come from any bicategory is sketched in [this MathOverflow answer](https://mathoverflow.net/a/346613).

## Other limits and colimits {#OtherLimits}

Although $Ho(Cat)$ has products and coproducts, like most homotopy categories it is not well-endowed with other [[limits]].  The following concrete example shows that it (and also $Ho(Gpd)$) fails to have [[pullbacks]].

Consider the [[cospan]]
$$\array{&& \mathbb{Z}/3\\
  && \downarrow^j\\
  \mathbb{Z}/2 & \underset{i}{\to} & S_3}$$
where the two arrows are inclusions of subgroups.  That is, we choose a 2-cycle and a 3-cycle in $S_3$, say $a=(1,2)$ and $b=(1,2,3)$, and identify $\mathbb{Z}/2$ and $\mathbb{Z}/3$ with the subgroups generated by $a$ and $b$ respectively.  Regard these groups as connected groupoids and thus as objects of $Ho(Cat)$, and suppose that this cospan had a pullback
$$\array{P & \overset{f}{\to} & \mathbb{Z}/3\\
  ^g \downarrow && \downarrow^j\\
  \mathbb{Z}/2 & \underset{i}{\to} & S_3}$$
in $Ho(Cat)$ or $Ho(Gpd)$.

Note that for any category $C$, the set $Ho(Cat)(1,C)$ is the set of isomorphism classes of objects in $C$ (where $1$ is the [[terminal category]]).  Therefore, any pullbacks that exist in $Ho(Cat)$ must induce pullbacks of sets of isomorphism classes of objects, and so $P$ must also have only one isomorphism class of objects; i.e. it must be a [[monoid]], regarded as a one-object category.  We choose monoid homomorphisms $P\to \mathbb{Z}/2$ and $P\to \mathbb{Z}/3$ representing $f$ and $g$, respectively.  We also choose a natural isomorphism $\sigma\colon j f \cong i g$, which consists of an element $\sigma\in S_3$ such that $j(f(c)) = \sigma \cdot i(g(c)) \cdot \sigma^{-1}$ for all $c\in P$.

Now let $Q$ be the [[2-pullback]]
$$\array{Q & \overset{h}{\to} & \mathbb{Z}/3\\
  ^k \downarrow & \cong & \downarrow^j\\
  \mathbb{Z}/2 & \underset{i}{\to} & S_3.}$$
Then the objects of $Q$ are the elements of $S_3$, and the morphisms from $x$ to $y$ consist of pairs $(u,v)\in \mathbb{Z}/2 \times \mathbb{Z}/3$ such that $i(u)\cdot x = y \cdot j(v)$.  Since the square defining $Q$ commutes in $Ho(Cat)$, there must be a functor $\ell\colon Q\to P$ such that $f\ell\cong h$  and $g\ell\cong k$.

Now every element of $\mathbb{Z}/2$ or $\mathbb{Z}/3$ is the image of some morphism of $Q$ under $k$ or $h$, respectively.  For instance, $a\in \mathbb{Z}/2$ is the image of $(a,1)\colon 1 \to a$ and $b\in \mathbb{Z}/3$ is the image of $(1,b)\colon b\to 1$.  Therefore, since $h$ and $k$ factor through $f$ and $g$ up to isomorphism, $f$ and $g$ must be surjective as monoid homomorphisms.

Let $c_1$ be such that $f(c_1)=b$.  If $g(c_1)$ is not the identity, let $c=c_1$.  Otherwise $g(c_1)=1$ and there is some $c_2$ with $g(c_2)=a$.  If $f(c_2)$ is not the identity, then let $c=c_2$.  Otherwise $f(c_2)=1$ and let $c = c_1\cdot c_2$.  In either case, neither $f(c)$ nor $g(c)$ is the identity.  Therefore, neither $j(f(c))$ nor $i(g(c))$ is the identity, and moreover $j(f(c))$ is a 3-cycle and $i(g(c))$ is a 2-cycle in $S_3$.  But the element $\sigma$ conjugates $i(g(c))$ to $j(f(c))$, a contradiction.

Since all the categories involved were groupoids (except possibly $P$), the same argument shows that $Ho(Gpd)$ doesn't have pullbacks.  Moreover, basically the same argument, regarding groupoids as connected 1-types, shows that the homotopy category of topological spaces doesn't have pullbacks either (in this case the final contradiction is derived from $\pi_1(P)$ instead of $P$ itself).

## Related concepts

* [[canonical model structure]]

* [[Ho(CombModCat)]]

[[!include categories of categories - contents]]

category: category


Ho(sSet)

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## Definition

The [[homotopy category]]/[[localization]] of [[sSet]] with respect to the [[weak homotopy equivalences]].

The corresponding [[simplicial localization]] is the [[(∞,1)-category]] of [[∞-groupoids]]/[[homotopy types]].

## Related concepts

* [[Ho(Top)]]

category: category


Ho(Top)

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## Idea

The "classical homotopy category" $Ho(Top)$ typically refers to the [[category]] of [[topological spaces]] with morphisms between them the [[homotopy classes]] of [[continuous functions]], or (slightly less classically but more commonly these days) to its [[full subcategory]] on those topological spaces [[homeomorphism|homeomorphic]] to a [[CW-complex]]. The latter is technically the [[homotopy category]] obtained by [[localization|localizing]] the category of topological spaces at those [[continuous functions]] that are  [[weak homotopy equivalences]], hence it is also the [[homotopy category of a model category]] of the [[classical model structure on topological spaces]].

## Definition

By $Ho(Top)$ one denotes the [[category]] which is the [[homotopy category]] of [[Top]] with respect to [[category with weak equivalences|weak equivalences]] given

* either by [[homotopy equivalence]]s -- $Ho(Top)_{he}$.

* or by [[weak homotopy equivalence]]s -- $Ho(Top)_{whe}$.

Depending on context here [[Top]] contains all topological spaces or is some subcategory of [[nice topological space]]s.

The study of $Ho(Top)$ was the motivating example of [[homotopy theory]]. Often $Ho(Top)$ is called _the_ homotopy category.

The [[simplicial localization]] of [[Top]] at the [[weak homotopy equivalences]] yields the [[(∞,1)-category]] of [[∞-groupoids]]/[[homotopy types]].

## Compactly generated spaces

Let now $Top$ denote concretely the category of [[compactly generated space|compactly generated]] [[weakly Hausdorff space]]s. And Let $CW$ be the subcategory on [[CW-complex]]es. We have $Ho(CW)_{whe} = Ho(CW)_{he} = Ho(CW)$.

There is a functor

$$
  Top \to Ho(CW)
$$

that sends each topological space to a weakly homotopy equivalent CW-complex.

By the [[homotopy hypothesis]]-theorem $Ho(CW)$ is equivalent for instance to the [[homotopy category of a model category]] $Ho(sSet_{Quillen})$ of the [[classical model structure on simplicial sets]] as well as $Ho(Top_{Quillen})$of the [[classical model structure on topological spaces]].

## Shape theory

The category $Ho(Top)_{he}$ can be studied by testing its objects with objects from $Ho(CW)$. This is the topic of [[shape theory]].

## Related concepts
 {#RelatedConcepts}

* [[hypothèse inspiratrice]]

* [[Top]], [[Ho(Top)]]

* [[sSet]], [[Ho(sSet)]]

* [[∞Grpd]], [[Ho(∞Grpd)]]

* [[stable homotopy category]]

## References

The [[exact completion]] of the category $Ho(Top)$ is shown to be a [[pretopos]] in:

* [[Marino Gran]], [[Enrico Vitale]], *On the exact completion of the homotopy category*, Cahiers de Topologie et Géométrie Différentielle Catégoriques **39** 4 (1998)  287-297 [[numdam:CTGDC_1998__39_4_287_0](http://www.numdam.org/item/CTGDC_1998__39_4_287_0)]

category: category

[[!redirects HoTop]]
[[!redirects homotopy category of spaces]]
[[!redirects homotopy category of topological spaces]]

[[!redirects classical homotopy category]]
[[!redirects classical homotopy categories]]

[[!redirects classical homotopy category of topological spaces]]
[[!redirects classical homotopy categories of topological spaces]]

[[!redirects classical pointed homotopy category]]
[[!redirects classical pointed homotopy categories]]


Ho(∞Grpd)

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## Idea

The [[homotopy category of an (∞,1)-category]] of [[∞Grpd]]. [[equivalence of categories|equivalent]] to [[Ho(Top)]]. Obtained as the [[localization]] of [[sSet]] at the [[weak homotopy equivalences]]. This is the classical [[homotopy category]].

## Properties

### H-Monoids

A [[monoid object]] in $Ho(\infty Grpd)$ is an _[[H-monoid]]_.

A [[group object]] in $Ho(\infty Grpd)$ is an _[[H-group]]_.

## Related concepts

* [[Ho(sSet)]]

category: category


Hoang Kim Nguyen

* [personal page](https://hk-nguyen-math.github.io/)

## Selected writings

On the [[(infinity,1)-Grothendieck construction]] via [[straightening and unstraightening]] entirely within the context of [[quasi-categories]]:

* [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], *The universal coCartesian fibration* [[arXiv:2210.08945](https://arxiv.org/abs/2210.08945)]

On the resulting interpretation of the [[universal coCartesian fibration]] as [[categorical semantics]] for a [[univalence axiom|univalent]] [[type universe]] in [[directed homotopy type theory]]:

* {#CisinskiEtAl23} [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], Tashi Walde: *Univalent Directed Type Theory*, lecture series in the *[CMU Homotopy Type Theory Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)* (13, 20, 27 Mar 2023) [[web](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html#230313), video 1:[YT](https://www.youtube.com/watch?v=5YOltuTcBK8), 2:[YT](https://www.youtube.com/watch?v=xWmELBvHMPo), 3:[YT](https://www.youtube.com/watch?v=P0Cfb4eUJo4); slides 0:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-intro_talk1.pdf), 1:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk1.pdf), 2:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk2.pdf), 3:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf)]

On the [[directed univalence axiom]] in this context:

* [[Hoang Kim Nguyen]], *Directed univalence in simplicial sets*, talk in *[Homotopy Type Theory Electronic Seminar Talks](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html)* (March 2023) [[video](https://www.youtube.com/watch?v=GgcJqzGvq80), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Nguyen-2023-03-09-HoTTEST.pdf)]

category: people


Hoare logic

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## Idea

*Hoare logic* [[Hoare (1969)](#Hoare69)] is a form of [[formal program verification]].

> [[from Streicher (1993, p.1-2):](#Streicher93)] In theoretical computer science the issue of *correctness of programs* has become more and more important over the years. Early approaches to establishing correctness  were *program verification logics* such as Hoare calculi for verifying [[imperative programs]] and the method of *algebraic specification* for specifying modules in functional languages.

> Program logics as e.g. the Hoare calculus are oriented towards state-oriented programming languages and provide calculi for deriving correct "Hoare triples", i.e. expressions of the form $\{P\}S\{Q\}$ with the intuitive meaning that whenever the initial state of a program satisfies [[proposition]] $P$ then the execution of the program $S$ [[termination|terminates]] and results in a state satisfying proposition $Q$.

> Although this method works  quite well for "pidgin" programs (i.e. using only `assignment`, `if_then_else` and `while` over a *single* [[data type]]) there are big problems with dealing with procedures, modules and general user defined data types. In general a main drawback is that the world of programs and the world of specifications are conceptually quite different. Of course, this problem is typical for all state-oriented imperative programming languages.

> This conceptual gap between programs and their specification has been overcome by the appearance and development of [[functional programming languages]]. [...] for languages supporting the construction and verification of large modular systems the concept of [[dependent type]] is inevitable.

## Related concepts

* [[separation logic]]

## References

The original article:

* {#Hoare69} [[C. A. R. Hoare]], *An axiomatic basis for computer programming*, Communications of the ACM **12** 10 (1969) 576–580 [[doi:10.1145/363235.363259](https://doi.org/10.1145/363235.363259), [pdf](https://sites.cs.ucsb.edu/~kemm/courses/cs266/acmhoare69.pdf)]

  also:

  in: *Programming Methodology. Texts and Monographs in Computer Science*, Springer (1978) 89-100 [[doi:10.1007/978-1-4612-6315-9_9](https://doi.org/10.1007/978-1-4612-6315-9_9)]

See also:

* Wikipedia, *[Hoare logic](https://en.wikipedia.org/wiki/Hoare_logic)*

The above quote is from:

* {#Streicher93} [[Thomas Streicher]], p. 1-2 of: *Investigations into Intensional Type Theory*, Habilitation Thesis, Darmstadt (1993) [[pdf](https://www2.mathematik.tu-darmstadt.de/~streicher/HabilStreicher.pdf), [[Streicher-IntensionalTT.pdf:file]]]

which otherwise is about [[intensional type theory|intensional]] [[dependent type theory]].

[[!redirects Hoare logics]]

[[!redirects Hoare calculus]]
[[!redirects Hoare calculi]]