William Arveson

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

## Selected writings

On [[completely positive maps]] on [[C-star algebras|$C^\ast$-algebras]]:

* {#Arveson69} [[William B. Arveson]], *On subalgebras of $C^\ast$-algebras*, Bull. Amer. Math. Soc. **75** (1969) 790-794 [[doi:1969-75-04/S0002-9904-1969-12293-7](https://www.ams.org/journals/bull/1969-75-04/S0002-9904-1969-12293-7), [pdf](https://www.ams.org/journals/bull/1969-75-04/S0002-9904-1969-12293-7/S0002-9904-1969-12293-7.pdf)]

On [[operator algebra]] and [[spectral theory]]:

* [[William Arveson]], *A Short Course on Spectral Theory*, Graduate Texts in Mathematics **209**, Springer (2002) [[doi:10.1007/b97227](https://link.springer.com/book/10.1007/b97227)]

category: people

[[!redirects William B. Arveson]]


William Barlow

* William Barlow (geologist)

## Selected writings

Early discussion of the internal structure of [[crystals]]:

* [[William Barlow]], *Probable Nature of the Internal Symmetry of Crystals*, Nature **29** (1883) 186–188 $[$[doi:10.1038/029186a0](https://doi.org/10.1038/029186a0)$]$

category: people


William Browder

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/William_Browder_(mathematician))

## Selected writings

On [[homology of H-spaces]] and their [[Pontryagin product]]:

* [[William Browder]], *Homology and Homotopy of H-Spaces*, Proceedings of the National Academy of Sciences of the United States of America **46** 4 (1960) 543-545 [[jstor:70867](https://www.jstor.org/stable/70867)]

* [[William Browder]], p. 36 of: *Torsion in H-Spaces*, Annals of Mathematics, Second Series **74** 1 (1961) 24-51  [[jstor:1970305](https://www.jstor.org/stable/1970305)]

* [[William Browder]], *Homology Rings of Groups*, American Journal of Mathematics **90** 1 (1968) [[jstor:2373440](https://www.jstor.org/stable/2373440)]

category: people


William Burnside

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

* [St. Andrews History page](http://www-groups.dcs.st-and.ac.uk/history/Biographies/Burnside.html)

## Selected writings

On [[group theory]] of [[finite groups]]:

* {#Burnside1897} [[William Burnside]], _Theory of Groups of Finite Order_, 1897 ([pdf](http://www.gutenberg.org/files/40395/40395-pdf.pdf)), reprinted by Cambridge University Press 2012 ([doi:10.1017/CBO9781139237253]( https://doi.org/10.1017/CBO9781139237253))

See also:

* {#NMT04} Peter M. Neumann, A.J.S. Mann, and Julia Tompson (eds.) _The Collected Papers of William Burnside_, Oxford University Press 2004 ([ISBN:9780198505853](https://global.oup.com/academic/product/the-collected-papers-of-william-burnside-9780198505853))

## Related $n$Lab entries

* [[Burnside ring]]

* [[Burnside ring is equivariant stable cohomotopy of the point]]

category: people


William Clifford

Discoverer of _[[Clifford algebra]]_, following [[Hermann Grassmann]]'s [[Grassmann algebra]].

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

## Selected writings

Early speculation on what much later came to be called *[[geometrodynamics]]*:

* [[William Clifford]], *[[On the Space-Theory of Matter]]*, Proceedings of the Cambridge Philosophical Society **2** (1864-1876 - Printed 1876) 157-158 reprinted in: *The Concepts of Space and Time* Boston Studies in the Philosophy of Science **22** Springer (1976) [[doi:10.1007/978-94-010-1727-5_50](https://doi.org/10.1007/978-94-010-1727-5_50)]

English translation of [[Bernhard Riemann]]'s, *[[Über die Hypothesen, welche der Geometrie zu Grunde liegen]]*, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]:

* [[William Clifford]]: *[[On the hypotheses which underlie geometry]]*, Nature **VIII** (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]

Introducing the notion of [[Clifford algebra]]:

* [[William Clifford]], *Applications of Grassmann's extensive algebra*, American Journal of Mathematics **1** 4 (1878) 350-358 [[doi:10.2307/2369379](https://doi.org/10.2307/2369379), [jstor:2369379](https://www.jstor.org/stable/2369379)]

  > (referring to [[Hermann Grassmann]]'s *[[Ausdehnungslehre]]*)

## Related entries

* _[[On the Space-Theory of Matter]]_

category: people

[[!redirects William Kingdon Clifford]]


William Crawley-Boevey

__William Crawley-Boevey__ is an English mathematician, mainly algebraist. Topics he contributed to include representations of quivers, quiver varieties, [[Deligne-Simpson problem]], Kleinian singularities, [[double derivation]]s.

* [Webpage](https://www.math.uni-bielefeld.de/~wcrawley/)

* wikipedia:[William_Crawley-Boevey](https://en.wikipedia.org/wiki/William_Crawley-Boevey); Google Scholar [page](https://scholar.google.com/citations?user=Ozh9p84AAAAJ)

He introduced _deformed_ [[preprojective algebra]]s of quivers in connection with study of Kleinian singularities in

* W. Crawley-Boevey, Martin P. Holland, _Noncommutative deformations of Kleinian singularities_, Duke Math. J. 92 (3): 605–635 (1998) [doi](https://doi.org/10.1215/S0012-7094-98-09218-3) [MR1620538](https://mathscinet.ams.org/mathscinet-getitem?mr=1620538)

A construction generalizing deformed preprojective algebra of quivers and assigning to a $K$-algebra $A$ with an element $\lambda\in K\otimes_{\mathbf{Z}}K_0(A)$ a new algebra $\pi^\lambda(A)$ and a canonical homomorphism $A\to \pi^\lambda(A)$ is described in

* _Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities_, Comment. Math. Helv. 74 (1999) 548-574 [doi](https://doi.org/10.1007/S000140050105)

Flavour of noncommutative differential geometry based on [[double derivation]]s:

* [[William Crawley-Boevey]], [[Pavel Etingof]], [[Victor Ginzburg]], _Noncommutative geometry and quiver algebras_,  Adv. Math. __209__:1 (2007) 274-336 [doi](https://doi.org/10.1016/j.aim.2006.05.004)

Some papers

* _Poisson structures on moduli spaces of representations_, J. Algebra 325, 205–215 (2011) [MR2745537](https://mathscinet.ams.org/mathscinet-getitem?mr=2745537) [doi](https://doi.org/10.1016/j.jalgebra.2010.09.033)
* _Geometry of the moment map for representations of quivers_, Comp. Math. __126__ (3): 257–293 [doi](https:/doi.org/10.1023/A:1017558904030) [MR1834739](https://mathscinet.ams.org/mathscinet-getitem?mr=1834739)

category: people

[[!redirects W. Crawley-Boevey]]
[[!redirects Bill Crawley-Boevey]]


William Detmold

* [personal page](https://wdetmold.mit.edu/welcome)

* [institute page](https://physics.mit.edu/faculty/william-detmold/)

* [inspire page](https://inspirehep.net/authors/1019639)

## Selected writings

On the [[pressure]] of the proton, via [[lattice QCD]]:

* [[Phiala Shanahan]], [[William Detmold]], *Pressure Distribution and Shear Forces inside the Proton*, Physical Review Letters **122** 7 (2019) [[doi:10.1103/PhysRevLett.122.072003](https://doi.org/10.1103/PhysRevLett.122.072003), [arXiv:1810.07589](https://arxiv.org/abs/1810.07589)]

category: people


William Dunbar

* [Institute page](https://simons-rock.edu/academics/faculty-bios/science-mathematics-and-computing-faculty/william-dunbar.php)

## Selected writings

On the classification of 3-dimensional compact [[Riemannian orbifolds]]:

* {#Dunbar88} [[William Dunbar]], _Geometric orbifolds_, Rev. Mat. Univ. Complutense Madr. 1, No.1-3, 67-99 (1988)

category: people


William Dwyer

William G. Dwyer is a mathematician at the University of Notre Dame.

* [website](http://www.nd.edu/~wgd/)

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

## Selected articles

On the $G$-[[Borel model structure]] and its [[Quillen equivalence]] with the [[slice model structure]] over the [[simplicial classifying space]] $\overline W G$:

* {#DDK80} [[Emmanuel Dror]], [[William Dwyer]], [[Daniel Kan]], _Equivariant maps which are self homotopy equivalences_, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 ([jstor:2043448](http://www.jstor.org/stable/2043448))

On [[simplicial localization]]:

* {#DwyerKanLocalizations} [[William Dwyer]], [[Daniel Kan]], _Simplicial localizations of categories_, J. Pure Appl. Algebra **17** 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3, [pdf](http://www.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf)]

* {#DwyerKanCalculating} [[William Dwyer]], [[Daniel Kan]], _Calculating simplicial localizations_, J. Pure Appl. Algebra 18 (1980), 17-35 [doi:10.1016/0022-4049(80)90113-9, [pdf](http://www.nd.edu/~wgd/Dvi/CalculatingSimplicialLocalizations.pdf)]

* {#DwyerKan80FunctionComplexes} [[William Dwyer]], [[Daniel Kan]], _Function complexes in homotopical algebra_, Topology 19 (1980), 427-440 [doi:10.1016/0040-9383(80)90025-7, [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dwyer-kan-3.pdf)]

* {#DwyerKanEquivalences} [[William Dwyer]], [[Daniel Kan]], *Equivalences between homotopy theories of diagrams*, in: *Algebraic topology and algebraic K-theory*, Ann. of Math. Stud. **113**, Princeton University Press (1988) [[doi:10.1515/9781400882113-009](https://doi.org/10.1515/9781400882113-009)]

On [[derived hom-spaces]] ([[function complexes]]) in [[projective model structures on simplicial presheaves]]:

* [[William Dwyer]], [[Daniel Kan]], *Function complexes for diagrams of simplicial sets*, Indagationes Mathematicae (Proceedings) **86** 2 (1983) 139-147 [doi:10.1016/1385-7258(83)90051-3, [pdf](https://core.ac.uk/download/pdf/82652265.pdf)]

On (enhancement and generalization of) [[Elmendorf's theorem]] in [[equivariant homotopy theory]]:

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], _Singular functors and realization functors_, Indagationes Mathematicae (Proceedings) Volume 87, Issue 2, 1984, Pages 147-153 (doi:10.1016/1385-7258(84)90016-7)

Introducing the [[model structure on simplicial groupoids]]:

* [[William Dwyer]], [[Daniel Kan]],  *Homotopy theory and simplicial groupoids*, Indagationes Mathematicae (Proceedings) **87** 4 (1984) 379-385 [doi:10.1016/1385-7258(84)90038-6]

On [[homotopy commutative diagrams]]:

* [[William Dwyer]], [[Dan Kan]], [[Jeff Smith]], _Homotopy commutative diagrams and their realizations_, Journal of Pure and Applied Algebra 57 (1989) 5-24 (doi:10.1016/0022-4049(89)90023-6)

On [[p-compact groups]]:

* {#DwyerWilkerson94} [[William Dwyer]], [[Clarence Wilkerson]], _Homotopy Fixed-point Methods for Lie Groups and Finite Loop Spaces_,  Annals of Mathematics Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 395-442 ([jstor:2946585](https://www.jstor.org/stable/2946585))

Introducing the [[Dwyer-Wilkerson H-space]]:

*  {#DwyerWilkerson93} [[William Dwyer]], [[Clarence Wilkerson]], _A new finite loop space at the prime two_, J. Amer. Math. Soc. 6 (1993), 37-64  ([doi:10.1090/S0894-0347-1993-1161306-9](https://doi.org/10.1090/S0894-0347-1993-1161306-9))

On [[homotopy theory]] and [[model categories]]:

* {#DwyerSpalinski95} [[William Dwyer]], [[Jan Spalinski]], *[[Homotopy theories and model categories]]* ([[DwyerSpalinski_HomotopyTheories.pdf:file]])

  in: [[Ioan Mackenzie James|I. M. James]], *[[Handbook of Algebraic Topology]]*, North Holland 1995 ([ISBN:9780080532981](https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2), [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

On [[homotopy theory|homotopy theoretic]] methods in [[group cohomology]]:

* [[William G. Dwyer]], [[Hans-Werner Henn]], *Homotopy Theoretic Methods in Group Cohomology*, Advanced Courses in Mathematics - CRM Barcelona, Springer (2001) [[doi:10.1007/978-3-0348-8356-6](https://doi.org/10.1007/978-3-0348-8356-6)]

On [[derived functors]] such as [[homotopy limit]]-functors on [[model categories]] and more general [[homotopical categories]]:

* [[William Dwyer]], [[Philip Hirschhorn]], [[Daniel Kan]], [[Jeff Smith]], **[[Homotopy Limit Functors on Model Categories and Homotopical Categories]]**, Mathematical Surveys and Monographs **113**, AMS 2004 ([ISBN: 978-1-4704-1340-8](https://bookstore.ams.org/surv-113-s), [pdf](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf))

On [[localization]] in [[homotopy theory]]:

* [[William Dwyer]], _Localizations_, in: _Axiomatic, enriched and motivic homotopy theory_, NATO Sci. Ser. II **131**, Kluwer Acad. Publ. (2004) 3-28 [[doi:10.1007/978-94-007-0948-5](https://doi.org/10.1007/978-94-007-0948-5)]

On [[homotopy theory]] and [[classifying spaces]]:

* {#Dwyer2008} [[William Dwyer]], _Homotopy theory of classifying spaces_, Lecture notes, Copenhagen 2008, ([pdf](http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf), [[Dwyer_HomotopyTheoryOfClassifyingSpaces.pdf:file]])

## Related entries

* [[Dwyer-Kan localization]]

* [[Dwyer-Kan loop groupoid]]

[[!redirects William Gerard Dwyer]]

[[!redirects W. G. Dwyer]]

[[!redirects William G. Dwyer]]

[[!redirects Bill Dwyer]]

[[!redirects Dwyer]]

category: people


William F. Sawin

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

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

## Selected writings

On [[commutativity of limits with colimits]]:

* [[Marie Bjerrum]], [[Peter Johnstone]], [[Tom Leinster]], [[William F. Sawin]], *Notes on commutation of limits and colimits*, Theory and Applications of Categories **30** (2015), 527-532 [[arXiv:1409.7860](http://arxiv.org/abs/1409.7860), [tac:3015](http://www.tac.mta.ca/tac/volumes/30/15/30-15abs.html)]

category: people

[[!redirects William Sawin]]

[[!redirects Will Sawin]]
[[!redirects Will F. Sawin]]


William Fulton

* [webpage](http://www.math.lsa.umich.edu/~wfulton/)

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

## Selected writings

On [[representation theory]]:

* {#FultonHarris91} [[William Fulton]], [[Joe Harris]], _Representation Theory: a First Course_, Springer, Berlin, 1991 ([doi:10.1007/978-1-4612-0979-9](https://link.springer.com/book/10.1007/978-1-4612-0979-9))

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

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

On [[algebraic topology]]:

* [[William Fulton]], *Algebraic Topology -- A First Course*, Graduate Texts in Mathematics **153**, Springer (1995) \[doi:10.1007/978-1-4612-4180-5\]

On [[Young tableaux]] in [[representation theory]] and geometry:

* [[William Fulton]], _Young Tableaux, with Applications to Representation Theory and Geometry_, Cambridge U. Press, 1997 ([doi:10.1017/CBO9780511626241](https://doi.org/10.1017/CBO9780511626241))

## Related $n$Lab entries

* [[Fulton-MacPherson compactification]]

* [[configuration spaces of points]]

* [[Hilbert scheme]]

category: people


William G. Dwyer > history

> See at *[[William Gerard Dwyer]]*


William Hodge

* [Wikipedia entry](http://en.wikipedia.org/wiki/W._V._D._Hodge)

## Related entries

* [[Hodge theory]]

  [[Hodge theorem]]

category: people

[[!redirects W. V. D. Hodge]]
[[!redirects W.V.D. Hodge]]
[[!redirects Hodge]]


William Howard

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

## Related entries

* [[propositions as types]]

category: people

[[!redirects Bill Howard]]
[[!redirects William Alvin Howard]]


William J. Munro

* [personal page](https://www.brl.ntt.co.jp/people/bilmun/)

* [institute page](https://www.oist.jp/research/william-john-munro)

## Selected writings

On "continuous variable" or *analog* [[quantum computation]]:

* [[Vivien M. Kendon]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Analogue Computing*, Phil. Trans. R. Soc. A **368** 1924 (2010) 3621-3632 [[arXiv:1001.2215](https://arxiv.org/abs/1001.2215), [doi:10.1098/rsta.2010.0017](https://doi.org/10.1098/rsta.2010.0017)]

On [[quantum error correction]]:

* {#DevittNemotoMunro09} [[Simon J. Devitt]], [[Kae Nemoto]], [[William J. Munro]], *Quantum Error Correction for Beginners*, Rep. Prog. Phys. **76** (2013) 076001 [[arXiv:0905.2794](https://arxiv.org/abs/0905.2794), [doi:10.1088/0034-4885/76/7/076001](https://iopscience.iop.org/article/10.1088/0034-4885/76/7/076001)]

category: people


William K. Wootters

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

## Selected writings

Introducing the [[no-cloning theorem]] in [[quantum information theory]]:

* [[William Wooters]], [[Wojciech Zurek]], *A single quantum cannot be cloned*, Nature **299** (1982) 802-803 [[doi:10.1038/299802a0](https://doi.org/10.1038/299802a0)]

Introducing the notion of [[quantum teleportation]]:

* {#BBCJPW93} [[Charles H. Bennett]], [[Gilles Brassard]], [[Claude Crépeau]], [[Richard Jozsa]], [[Asher Peres]], [[William K. Wootters]]: *Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels*, Phys. Rev. Lett. **70** 1895 (1993) [[doi:10.1103/PhysRevLett.70.1895](https://doi.org/10.1103/PhysRevLett.70.1895)]

category: people


William Keigher

* [Mathematics Genealogy page](http://genealogy.math.ndsu.nodak.edu/id.php?id=4264)

category: people


William Kinney

* [webpage](http://arts-sciences.buffalo.edu/physics/faculty/william-h-kinney.html)

## selected writings

* {#BarenboimKinneyPark16} Gabriela Barenboim, [[William Kinney]], Wan-Il Park, _Eternal Hilltop Inflation_,  Journal of Cosmology and Astroparticle Physics, Volume 2016, May 2016 ([arXiv:1601.08140](https://arxiv.org/abs/1601.08140))

  ([[eternal inflation]] in [[Starobinsky model of cosmic inflation|Starobinsky-type models]] of [[cosmic inflation]])

## related $n$Lab entries

* [[Starobinsky model of cosmic inflation]]

* [[eternal inflation]]

category: people


William Kirwin

* [webpage](http://www.mi.uni-koeln.de/~wkirwin/index.php?nav=3)

## related $n$Lab entries

* [[coherent state in geometric quantization]]

category: people


William Kronholm

* [webpage](http://www.whittier.edu/academics/math/Kronholm)

## Selected writings

On the [[Serre spectral sequence]] in [[equivariant cohomology]] ([[RO(G)-grading|RO(G)-graded]] [cohomology with coefficients in a Mackey functor](Mackey+functor#Cohomology)):

* {#Kronholm10} [[William C. Kronholm]], *The $RO(G)$-graded Serre spectral sequence*, Homology Homotopy Appl. **12** 1 (2010) 75-92. [[arXiv:0908.3827](https://arxiv.org/abs/0908.3827), [doi:10.4310/HHA.2010.v12.n1.a7](https://doi.org/10.4310/HHA.2010.v12.n1.a7),  [euclid:hha/1296223823](https://projecteuclid.org/euclid.hha/1296223823)]

Introducing [[Kronholm's freeness theorem]] for [[cyclic group of order 2|$\mathbb{Z}/2$]]-[[equivariant cohomology|equivariant]] [[Bredon cohomology]]:

* [[William C. Kronholm]], *A Freeness Theorem for $RO(\mathbb{Z}/2)$-graded Cohomology*, Topology and its Applications **157** 5 (2010) 902-915 [[arXiv:0908.3825](https://arxiv.org/abs/0908.3825), [doi:10.1016/j.topol.2009.12.006](https://doi.org/10.1016/j.topol.2009.12.006)]

## Related entries

* [[Serre spectral sequence]]

[[!redirects William C. Kronholm]]

category: people


William Lawvere

[[!redirects Bill Lawvere]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabiJndex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Overview

> Talking with Bill, I often feel like a fly buzzing around a cow. (It seems to me I can liken Bill to a cow, if I'm just a fly myself.) On any easy question, I'll probably see the answer first. But his thoughts seem to move on a level where I don't function, I can barely see down there. (From an [interview](http://at.yorku.ca/t/o/p/c/06.htm) with [[John Isbell]].)

F. William Lawvere (February 9, 1937 -- January 23, 2023)

* ([web site](http://www.acsu.buffalo.edu/~wlawvere/), [Wikipedia entry](http://en.wikipedia.org/wiki/William_Lawvere))

was an influential [[category theory|category theorist]].

See

* {#Interview07} _An interview with F. William Lawvere_, [Bulletin of the International Center for Mathematics](http://www.cim.pt/?q=publications) ([part 1, December 2007](http://www.cim.pt/files/publications/b23www.pdf), [part 2, June 2008](http://www.cim.pt/files/publications/b24www.pdf),  [full pdf](http://www.mat.uc.pt/~picado/lawvere/interview.pdf), [[LawvereInterview.pdf:file]])

for a survey of his academic path and work. See also

* Interview by Felice Cardone, March 2013 ([link](https://dl.dropboxusercontent.com/u/92056191/Archive/temporary_new_material/lawvere-cardone-2000.html))

for an interview on his contributions to [[categorical logic]].

A recording of the commemorative special session in honour of Bill Lawvere from CT2023 can be found here: ([video](https://www.youtube.com/watch?v=x23KS3VOpRg&list=PLu4STGsfbix9l6rPxGsjG6Gl822k1hj2X&index=1&pp=iAQB))

For a (somewhat random) list of further links see also at "[conceptual mathematics](http://conceptualmathematics.wordpress.com/2013/02/09/happy-birthday-professor-f-william-lawvere/)".

## Topics

### Main contributions

Lawvere invented [[categorical logic]] and introduced the eponymous [[Lawvere theories]] as a [[category theory|category-theoretic]] way to describe finitary [[algebraic theories]].  He generalised [[Grothendieck toposes]] to [[elementary toposes]], revolutionising the [[foundations]] of [[mathematics]]; in this vein, he developed the [[foundation]] in [[structural set theory]] called [[ETCS]].  He also introduced and worked on [[synthetic differential geometry]] as a foundation for [[differential geometry]] and [[equations of motion]] in [[continuum physics]]. Later he introduced the notion of _[[cohesive topos]]_ as a more general foundation of [[geometry]].

### Mathematics relating to Physics
 {#MotivationFromFoundationsOfPhysics}

A central motivation for Lawvere's work is the search for a good  mathematical [[foundations]] of [[physics]], specifically of ([[classical physics|classical]]) [[continuum mechanics]] (or at least some [[kinematics|kinematical]] aspects thereof, Lawvere does not seem to mention [[Hamiltonians]], [[Lagrangians]] or [[action functionals]]).

In ([interview, p. 8](http://www.mat.uc.pt/~picado/lawvere/interview.pdf)) he recalls:

> I had been a student at Indiana University from 1955 to January 1960. I liked [[experiment|experimental]] [[physics]] but did not appreciate the imprecise reasoning in some theoretical courses. So I decided to study [[mathematics]] first. [[Clifford Truesdell|Truesdell]] was at the Mathematics Department but he had a great knowledge in Engineering Physics. He took charge of my education there. $[...]$ in 1955 (and subsequently) had advised me on pursuing the study of [[continuum mechanics]] and [[kinetic theory]].

> In Summer 1958 I studied [[topological dynamics|Topological Dynamics]] with George Whaples, with the agenda of understanding as much as possible in categorical terms.  $[...]$ Categories would clearly be important for simplifying the foundations of [[continuum physics]]. I concluded that I would make category theory a central line of my study.

Then in ([interview, p. 11](http://www.mat.uc.pt/~picado/lawvere/interview.pdf)) about the early 1960s:

> I felt a strong need to learn more [[set theory]] and [[logic]] from experts in that field, still of course with the aim of clarifying the foundations of [[category theory]] and of [[physics]].

The title of the early text _[[Toposes of laws of motion]]_, which is often cited as the text introducing [[synthetic differential geometry]], clearly witnesses the origin and motivation of these ideas in [[classical mechanics]].

On this, in ([interview, p. 15](http://www.mat.uc.pt/~picado/lawvere/interview.pdf)):

> Q: As an assistant professor in Chicago, in 1967, you taught with Mac Lane a course on [[mechanics|Mechanics]], where you "started to think about the justification of older intuitive methods in geometry" You called it "[[synthetic differential geometry]]". How did you arrive at the program of [[Categorical dynamics|Categorical Dynamics]] and [[synthetic differential geometry|Synthetic Differential Geometry]]?

> A: From January 1967 to August 1967 I was Assistant Professor at the University of Chicago. Mac Lane and I soon organized to teach a joint course based on Mackey's book "Mathematical Foundations of [[quantum mechanics|Quantum Mechanics]]".

> Q: So, Mackey, a functional analyst from Harvard mainly concerned with the relationship between quantum mechanics and representation theory, had some relation to category theory.

Then ([interview, p. 16](http://www.mat.uc.pt/~picado/lawvere/interview.pdf)):

> Q: Back to the origins of Synthetic Differential Geometry, where did the idea of organizing such a joint course on Mechanics originate ? Apparently, Chandra had suggested that Saunders give some courses relevant to physics, and our joint course was the first of a sequence. Eventually Mac Lane gave a talk about the [[Hamilton-Jacobi equation]] at the Naval Academy in summer 1970 that was published in the American Mathematical Monthly

> A: $[...]$ I began to apply the Grothendieck topos theory that I had learned from Gabriel to the problem of simplied foundations of [[continuum mechanics]] as it had been inspired by [[Clifford Truesdell|Truesdell]]'s teachings, [[Walter Noll|Noll]]'s [[axiom|axiomatizations]], and by my 1958 efforts to render categorical the subject of [[topological dynamics]].

A review of this with more comments on more relations to physics is in the introduction to the book collection _[[Categories in Continuum Physics]]_, which is the proceedings of a meeting organized by Lawvere in 1982.

In the talk _[[Toposes of laws of motion]]_ in 1997, Lawvere starts with the following remark

> I read somewhere recently that the basic program of infinitesimal [[calculus]], [[continuum mechanics]], and [[differential geometry]] is that all the world can be reconstructed from the [[infinitesimal object|infinitely small]]. One may think this is not possible, but nonetheless it's certainly a program that has been very fruitful over the last 300 years. I think we are now finally in a position to actually make more explicit what that program amounts to.

> $[...]$ I think that on the basis of these developments we can focus on this question of making very explicit how [[continuum physics]] etc. can be built up mathematically from very simple ingredients.

In the same talk, a few lines later after discussion of [[infinitesimally thickened points]] $T$, it says:

> The basic [[spaces]] which are needed for [[functional analysis]] and theories of [[physical fields]] are thus in some sense available in any [[topos]] with a suitable [[object]] $T$.

In 2000 in the article _[[Comments on the development of topos theory]]_ Lawvere writes in the closing section 7 titled "From and to continuum physics":

> What was the impetus which demanded the simplification and generalization of [[Grothendieck]]'s concept of [[topos]], if indeed the application to [[logic]] and [[set theory]] were not decisive. $[...]$ My own motivation came from my earlier study of [[physics]]. The [[foundation]] of the [[continuum physics]] of general materials, in the spirit of [[Clifford Truesdell|Truesdell]], [[Walter Noll|Noll]] and others, involves powerful and clear physical ideas, which unfortunately have been submerged under a mathematical apparatus $[...]$. But, as Fichera $[25]$ lamented, all this apparatus gives often a very uncertain fit to the phenomena. The apparatus may well be helpful in the solution of certain problems, but can the problems themselves and the needed axioms be stated in a direct and clear manner? And might this not lead to a simpler, equally rigorous account? These were the questions to which I began to apply the [[topos]] method in my 1967 Chicago lectures $[$ _[[Categorical dynamics]]_ $]$. It was clear that work on the notion of topos itself would be needed to achieve the goal. I had spent 1961-62 with the Berkeley logicians, believing that listening to experts on [[foundations]] might be the road to clarifying foundational questions.

> $[...]$ Several books treating the simplified [[topos theory]] ([[Sheaves in Geometry and Logic|MacLane-Moerdijk]] being the most recent and readable text), together with the three excellent books on [[synthetic differential geometry]] $[...]$ provide a solid basis on which further treatment of [[functional analysis]] and the needed development of [[continuum physics]] can be based.

The [Wikipedia entry](http://en.wikipedia.org/wiki/William_Lawvere) concludes:

> Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications.

See also at _[[higher category theory and physics]]_ for more on this.

### Mathematics relating to Philosophy
 {#RelationToPhilosophy}

Lawvere has proposed formalizations in [[category theory]], [[categorical logic]] and [[topos theory]] of concepts which are motivated from [[philosophy]], notably in [[Georg Hegel]]'s _[[Science of Logic]]_ (see there for more). This includes for instance definitions of concepts found there such as:

* _[[objective and subjective logic]]_, _[[abstract general]]_, _[[concrete general]]_, _[[concrete particular]]_, _[[unity of opposites]]_, _[[Aufhebung]]_, _[[being]]_, _[[becoming]]_, _[[space and quantity]]_, _[[cohesion]]_, _[[intensive and extensive quantity]]_

(see the references in these entries for pointers).

In ([Lawvere 92](#Lawvere92)) it says:

> It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.

A precursor to this undertaking is [[Hermann Grassmann]] with his _[[Ausdehnungslehre]]_ ([Lawvere 95](#Lawvere95)), see there for more.

## Talks, Lecture notes and Publications
 {#Talks}

The following is a list of texts by Lawvere, equipped with brief comments and hyperlinks to further material on the $n$Lab. See also the

* [list of publications on Lawvere's website](http://www.acsu.buffalo.edu/~wlawvere/list.html)

and also this

* [repository of Lawvere's publications](https://github.com/mattearnshaw/lawvere), made available by [[Matt Earnshaw]].

(Some of Lawvere's writings don't exist as published articles, but circulate in some form or other. Notably the "[[Archive for Mathematical Sciences & Philosophy]]" run by [[Michael Wright]] has a lot of recordings or lectures by Lawvere, though presently few or none of the files in the archive are available online.)

* _[[Functorial Semantics of Algebraic Theories]]_ Originally published as: Ph.D. thesis, Columbia University, 1963 and
in Reports of the Midwest Category Seminar II, 1968, 41-61, Republished in:
Reprints in Theory and Applications of Categories, No. 5 (2004) pp 1-121 ([tac](http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html))

  (on [[algebraic theories]], now also called [[Lawvere theories]])

* _An elementary theory of the category of sets_, 1964, Proceedings of the National Academy of Science of the U.S.A 52, 1506-1511. Expanded version with commentary by [[Colin McLarty]] and the author in:
Reprints in Theory and Applications of Categories, No. 11 (2005) pp. 1-35 ([tac](http://www.tac.mta.ca/tac/reprints/articles/11/tr11abs.html))

  > (axiomatizing the category [[Set]] of sets; see at *[[ETCS]]*)

*  _The Category of Categories as a Foundation for Mathematics_, pp. 1-20 in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

   > (axiomatizing the [[2-category]] [[Cat]] of categories; see at *[[ETCC]]*)

* _[[Categorical dynamics]]_, 1967 Chicago lectures ([pdf](http://www.mat.uc.pt/~ct2011/abstracts/lawvere_w.pdf))

  _Categorical Dynamics Revisited_, talk at _Sets Within Geometry_, Nancy, France 26-29 July 2011 ([recording](https://www.youtube.com/watch?v=sXATg9_6iWE))

  (on [[synthetic differential geometry]] as a context for axiomatizing [[ordinary differential equations]] as they appear in [[classical physics]])

* {#AdjointnessInFoundations} _[[Adjointness in Foundations]]_, ([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296

  (on the interpretation of [[quantifiers]] in [[categorical logic]] as [[base change]] [[adjoints]])

* [[H. Applegate]], [[M. Barr]], [[J. Beck]], [[F. W. Lawvere]], [[F. E. J. Linton]], [[E. Manes]], [[M. Tierney]], [[F. Ulmer]]: _[[Seminar on Triples and Categorical Homology Theory]]_, ETH 1966/67, edited by [[Beno Eckmann]] and [[Myles Tierney]], **[[LNM 80]]**, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories **18** (2008) 1-303 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

  (on [[monads]] in [[universal algebra]] and ([[cohomology|co-]])[[homology]]-theory:
)

* (with [[Max Kelly]]) *On the complete lattice of essential localizations*, Bull. Soc. Math. Belgique **41** (1989) 289–319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

  > (on [[essential localizations]])

* _Equality in hyperdoctrines and comprehension schema as an adjoint functor_.  In A. Heller, ed., _Proc. New York Symp. on Applications of Categorical Algebra_, pp. 1--14.  AMS, 1970. ([[LawvereComprehension.pdf:file]])

    (on [[comprehension]] in [[hyperdoctrines]])

* *Quantifiers and sheaves*, Actes Congrès intern. Math. **1** (1970) 329-334 [[pdf](https://raw.githubusercontent.com/mattearnshaw/lawvere/master/pdfs/1970-quantifiers-and-sheaves.pdf), [[Lawvere-QuantifiersAndSheaves.pdf:file]]]

    on [[geometric modalities]]/[[Lawvere-Tierney topology]] and [[universal quantifier|universal]]/[[existential quantifier|existential]] [[quantifiers]] related to [[dependent product]]/[[dependent sum]] further developed in:

*  _Equality in hyperdoctrines and
comprehension schema as an adjoint functor_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

* _Teoria delle categorie sopra un topos di base_, lecture notes from Perugia (1972--73) ([pdf](https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1972-perugia-lecture-notes.pdf), [[LawverePerugia1972.pdf:file]])

    (on basic category theory, the [[category of sets]] and [[elementary toposes]], and an approach to [[indexed categories]])

* _Metric spaces, generalized logic and closed categories_ Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 ([tac](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html))

     (A classic text from 1973 on [[Cauchy complete categories]] and [[enriched category theory]])

* _Variable sets etendu and variable structure in topoi_ ,
Lecture notes University of Chicago 1975.

    (a review of topos theory: on structural sets, role of [[forcing]] and non-standard analysis and [[étendue]] toposes)

* *Variable quantities and variable structures in topoi*, in [[Alex Heller]], [[Myles Tierney]] (eds.), *Algebra, Topology and Category Theory -- A Collection of Papers in Honor of [[Samuel Eilenberg]]*, Academic Press New York (1976) 101-131 [[doi:10.1016/C2013-0-10841-0](https://doi.org/10.1016/C2013-0-10841-0)]

    (reviews toposes in [[algebraic geometry]], first attention to [[gros topos]])

* _Topos-theoretic methods in geometry_, Aarhus 1997

* _Extensive and intensive quantities_ Workshop on Categorical Methods in Geometry, Aarhus 1983

  (a proposal for a general axiomatization of [[homotopy]]/[[homology]]-like "[[extensive]] quantities" and [[cohomology]]-like "[[intensive]] quantities")

* _State Categories, Closed Categories, and the Existence Semi-Continuous Entropy Functions_ , IMA reprint 86, 1984 ([pdf](https://conservancy.umn.edu/bitstream/handle/11299/4672/86s.pdf?sequence=1&isAllowed=y))

* _Functional Remarks on the General Concept of Chaos_ , IMA reprint 87, 1984 ([pdf](http://www.ima.umn.edu/sites/default/files/87s.pdf))

  (on a formalization of the concept of [[chaos]])

* _Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body_, Cah.top.géo.diff.cat, 21 nr.4, 1980, pp.377-392 ([Numdam](http://www.numdam.org/item?id=CTGDC_1980__21_4_377_0))

   (contains a section with some remarks on Lagrangian mechanics)

* with [[Stephen Schanuel]], _[[Categories in Continuum Physics]]_, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 1986

  (on [[continuum mechanics]], [[variational calculus]] and [[laws of motion]] in [[synthetic differential geometry]] and in [[diffeological spaces]] and [[Frölicher spaces]], and on [[intensive and extensive]] properties in terms of [[ring objects]] and [[modules]] in a [[topos]].)

* {#TakingCategoriesSeriously} _Taking categories seriously_, Revista Colombiana de Matemáticas **XX** (1986) 147-178, Reprints in Theory and Applications of Categories, **8** (2005) 1-24. [[tac:tr8](http://www.tac.mta.ca/tac/reprints/articles/8/tr8abs.html), [pdf](http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf)]

  (on the relevance of fundamental concepts in [[category theory]], such as [[Isbell duality]])

* *Categories of spaces may not be generalized spaces, as exemplified by directed graphs*, Revista Colombiana de Matemáticas **XX** (1986) 179-186, reprinted as: Reprints in Theory and Applications of Categories, **9** (2005) 1-7 [[tac:tr9](http://www.tac.mta.ca/tac/reprints/articles/9/tr9abs.html)]

  (on the notion of [[gros topos|gros toposes]], details of his claims are worked out at [[sufficiently cohesive topos]])

* *Qualitative Distinctions between some Toposes of Generalized Graphs*, in *Categories in Computer Science and Logic*, Cont. Math. **92** (1989) 261-299 [[doi:10.1090/conm/092](http://dx.doi.org/10.1090/conm/092), [pdf](https://github.com/mattearnshaw/lawvere/raw/master/pdfs/1989-qualitative-distinctions-between-some-toposes-of-generalized-graphs.pdf)]

     (Uses toposes of graphs to introduce his ideas on petit/gros toposes)

* _Display of graphics and their applications, as exemplified by 2-categories and the Hegelian "taco"_, Proceedings of the first international Conference on algebraic methodology and software technology University of Iowa, May 22-24 1989, Iowa City pp. 51-74

  (on [[Aufhebung]], [[graphic category|graphic categories]], the [[Hegelian taco]]; the most explicit text with Lawvere's ideas on [[Hegel]])

* _More on graphic toposes_, Cah. Top. Géom. Diff. Cat. **XXXII** no. 1 (1991) pp.5-10. ([Numdam](http://www.numdam.org/item?id=CTGDC_1991__32_1_5_0))

  (on [[graphic category|graphic categories]] and their presheaf categories)

* _[[Some Thoughts on the Future of Category Theory]]_ in A. Carboni, M. Pedicchio, G. Rosolini (eds.), _Category Theory_  , [[Como|Proceedings of the International Conference held in Como]], LNM **1488** Springer Heidelberg 1991.

  (on -- implicitly -- [[cohesive topos|cohesive toposes]]);

* with [[Stephen Schanuel]], _Conceptual Mathematics - A first  introduction to categories_ , Cambridge UP 1997.

  (first steps into category theory; the Spanish translation is available [here](http://www.acsu.buffalo.edu/~wlawvere/concep-3.pdf))

* {#Lawvere92} _Categories of space and quantity_ in: J. Echeverria et al (eds.), _The Space of mathematics_ , de Gruyter, Berlin, New York (1992)

  (on [[space and quantity]] and on formalizing concepts from [[philosophy]] as in the _[[Science of Logic]]_ in terms of [[category theory]] and [[categorical logic]])

* {#IntroductionToLinearCategories} [[William Lawvere]], *Introduction to Linear Categories and Applications*, course lecture notes (1992) [[pdf](https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf), [[Lawvere-LinearCategories.pdf:file]]]

  (on [[additive categories]])

* _[[Cohesive Toposes and Cantor's "lauter Einsen"]]_ Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. ([[LawvereCohesiveToposes.pdf:file]])

  (an early version of the notion of [[cohesive topos|cohesive toposes]])

* {#Lawvere94b} _[[Tools for the advancement of objective logic]]: closed categories and toposes_, in J. Macnamara and [[Gonzalo Reyes]] (Eds.), _The Logical Foundations of Cognition_, Oxford University Press 1993 (Proceedings of the Febr. 1991 Vancouver Conference "Logic and Cognition"),
pages 43-56, 1994.

  (on formalizing [[philosophy]] as in the _[[Science of Logic]]_ in terms of [[category theory]] and [[categorical logic]].)

* {#Lawvere95} _A new branch of mathematics, "The Ausdehnungslehre of 1844," and other works. Open Court (1995), Translated by Lloyd C. Kannenberg, with foreword by Albert C. Lewis_, _Historia Mathematica Volume 32, Issue 1, February 2005, Pages 99–106_ ([publisher](http://www.sciencedirect.com/science/article/pii/S031508600400059X))

  (on [[Hermann Grassmann]]'s _[[Ausdehnungslehre]]_ and hence also on _[[extensive and intensive quantity]]_)

* _Grassmann's Dialectics and Category Theory_, in _Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar_, Boston Studies in the Philosophy of Science Volume 187, 1996, pp 255-264 ([publisher](http://link.springer.com/chapter/10.1007%2F978-94-015-8753-2_21))

* _Volterra's functionals and covariant cohesion of space_ Perugia Studies in Mathematics (Proceedings of the May 1997 Meeting in Perugia) ([pdf](http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf))

  (with first remarks on [[cohesive topos|cohesion]])

* _[[Toposes of laws of motion]]_ , transcript of a talk in Montreal, Sept. 1997 ([pdf](http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf))

  (on the description of [[differential equations]] in terms of [[synthetic differential geometry]] and the notion of _[[toposes of laws of motion]]_)

* _Outline of synthetic differential geometry_ , seminar notes (1998) ([pdf](http://www.acsu.buffalo.edu/~wlawvere/SDG_Outline.pdf))

  (the origin of the concept of [[synthetic differential geometry]])

* _Kinship and Mathematical Categories_ , pp.411-425 in Jackendoff, Bloom, Wynn (eds.), _Language, Logic, and Concepts_ , MIT Press Cambridge 1999.

   (Lawvere's venture into anthropology; a summary is at [[kinship]])

* _[[Comments on the development of topos theory]]_ , pp.715-734 in Jean-Paul Pier (ed.), _Development of mathematics 1950-2000_, Birkhäuser Basel 2000.

     (review of the history of topos theory)

* _Linearization of graphic toposes via Coxeter groups_,  JPAA **168** (2002) pp. 425-436. ([pdf](http://www.sciencedirect.com/science/article/pii/S0022404901001074/pdfft?md5=a4ca9bc67df6ae63ddf53c559bd71315&pid=1-s2.0-S0022404901001074-main.pdf))

* _Categorical algebra for continuum micro physics_,  JPAA **175** (2002) pp. 267-287.  ([pdf](http://www.sciencedirect.com/science/article/pii/S002240490200138X/pdfft?md5=ebd435105778d597f25141bf065978e3&pid=1-s2.0-S002240490200138X-main.pdf))

*  _Foundations and Applications: Axiomatization and Education_, Bulletin of Symbolic Logic **9** no.2 (2003) pp.213-224. ([ps-preprint](https://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps))

   (a review of (not only set-theoretic) foundations in mathematics)

* {#LawvereRosebrugh} with [[Robert Rosebrugh]], _[[Sets for Mathematics]]_ , Cambridge UP  2003. ([web](http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics))

  (on [[set theory]] from a practical [[category theory|theoretic]] point of view (informal [[ETCS]]))

* _Axiomatic cohesion_, Theory and Applications of Categories **19** 3 (2007) 41-49 [[tac:19-03](http://www.tac.mta.ca/tac/volumes/19/3/19-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf)]

  (a more formal definition of [[cohesive topos|cohesive toposes]], [[sufficiently cohesive topos]] works out details of his claims concerning 'sufficient cohesion')

* _[[Cohesive Toposes -- Combinatorial and Infinitesimal Cases]]_ , lecture in Como (2008)

  (on [[cohesive topos|cohesive toposes]])

* _Open problems in topos theory_, April 2009 ([pdf](https://web.archive.org/web/20170421103453/http://cheng.staff.shef.ac.uk/pssl88/lawvere.pdf))

  (problem 7 presented there "The algebra of time", concerns the characterization of _[[Toposes of laws of motion]]_)

* _The Dialectic of the Continuous and the Discrete in the history of the struggle for a usable guide to mathematical thought_, talk at _Sets Within Geometry_, Nancy, France 26-29 July 2011 ([recording](https://www.youtube.com/watch?v=OE-R3565jrs))

* _What are Foundations of Geometry and Algebra_, Keynote lecture at the _[Fifty Years of Functorial Semantics](http://www.math.union.edu/~niefiels/13conference/Web/)_ conference, Union College, October 2013 ([recording](https://www.youtube.com/watch?v=ZYGyEPXu8as))

*  _Alexander Grothendieck and the Concept of Space_, address CT15 Aveiro 2016. ([link](http://www.acsu.buffalo.edu/~wlawvere/GrothendiekAveiro15.htm))

   (a talk given at CT15 reviewing 50 years of developments since the 1965 La Jolla conference)

* {#Lawvere21} _Toposes generated by codiscrete objects in combinatorial topology and functional analysis_, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, [pdf](http://tac.mta.ca/tac/reprints/articles/27/tr27abs.html).

category: people

[[!redirects William Lawvere]]
[[!redirects Lawvere]]
[[!redirects F. W. Lawvere]]
[[!redirects F.W. Lawvere]]
[[!redirects W. Lawvere]]
[[!redirects F. William Lawvere]]


William Linch III

* [webpage](http://people.physics.tamu.edu/wdlinch3)

## Selected writings

On [[D=5 supergravity]]:

* [[William Linch III]], Markus A. Luty, J. Phillips, _Five dimensional supergravity in $N = 1$ superspace_, Phys.Rev.D68:025008,2003 ([arXiv:hep-th/0209060](https://arxiv.org/abs/hep-th/0209060))

* [[Katrin Becker]], [[Melanie Becker]], Daniel Butter, [[William Linch III]], Stephen Randall, _Five-dimensional Supergravity in N = 1/2 Superspace_ ([arXiv:1909.09208](https://arxiv.org/abs/1909.09208))

category: people

[[!redirects William Linch]]


William Massey

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

## Selected writings

On [[algebraic topology]]:

* [[William S. Massey]], *Algebraic Topology: An Introduction*, Harcourt Brace & World 1967, reprinted in: Graduate Texts in Mathematics, Springer 1977 ([ISBN:978-0-387-90271-5](https://link.springer.com/book/9780387902715))

The [[Arnold-Kuiper-Massey theorem]]:

* [[William Massey]], _The quotient space of the complex projective space under conjugation is a 4-sphere_, Geometriae Didactica 1973 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/massey5.pdf))

## Related $n$Lab entries

* [[Massey product]]

* [[Blakers-Massey theorem]]

* [[covering space]], [[universal cover]]

category: people

[[!redirects William S. Massey]]


William Messing

+ [website](http://www.math.umn.edu/~messing/)

#related $n$Lab entries#

* [[gerbe]]

* [[infinitesimal singular simplicial complex]]

category: people


William Mycroft

## Selected writings

On [[cohomology operations]]:

* [[William Mycroft]], _Unstable Cohomology Operations: Computational Aspects of Plethories_, 2017 ([pdf](https://etheses.whiterose.ac.uk/20524/1/wm-thesis-final.pdf), [[MycroftUnstableCohomologyOperations.pdf:file]])

specifically on [[topological K-theory]]:

* [[William Mycroft]], [[Sarah Whitehouse]], _The plethory of operations in complex topological K-theory_ ([arXiv:2001.01608](https://arxiv.org/abs/2001.01608))

category: people


William Rowan Hamilton

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

## related $n$Lab entries

* [[Hamiltonian mechanics]]

  * [[Hamiltonian]], [[Hamiltonian vector field]]

* [[quaternions]]

* [[nabla]]

category: people

[[!redirects William Hamilton]]


William S. Massey

[[!redirects W. S. Massey]]
William S. Massey was an American algebraic topologist (born 1920).  He is best known for the development of [[Massey products]] and for contributions to the theory of spectral sequences, via his work on [[exact couples]].  He wrote several influential textbooks.

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

category:people


William Stein

William Stein is lead developer of *[[SageMath]]*

* [personal page](https://wstein.org/)

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

* [MathematicsGenealogy page](https://mathgenealogy.org/id.php?id=36697)

category: people


William T. Tutte

[[!redirects W. T. Tutte]]
Bill Tutte (b. May 14, 1917; d. May 2, 2002) was a British and Canadian mathematician.

Tutte made seminal contributions to [[graph theory]] and the theory of [[matroids]]. He [published](#Tutte1958I) the [[Tutte homotopy theorem]] which can be regarded as a generalization of the notion of [[paths]] to matroids, and about which [[Gian-Carlo Rota]] wrote

> The (...) motivation, (...) in Tutte's vertiginously profound papers, was graph-theoretic. Matroids are objects that play the role of the dual graph of a graph when the graph is not [[planar]]. Almost every fact about graphs that can be formulated without using the term "vertex" has a matroidal analogue. The deepest insight obtained from this yoga is Tutte's homotopy theorem, a unique combinatorial achievement whose far reaching implications are a long way from being fully understood. -- Gian-Carlo Rota: Indiscrete Thoughts. Modern Birkhäuser Classics. Chapter "Book Reviews"

His writing style is sometimes described to be extraordinarily compressed, in cautionary remarks like

> If you read Tutte, remember: it's all there, but take a semester off -- and plan the first ten pages.
(Personal recollections from a seminar)

## References

* William T. Tutte: _A homotopy theorem for matroids. I_, Transactions of the American Mathematical Society, 88: 144–160 (1958)
{#Tutte1958I}

* William T. Tutte: _A homotopy theorem for matroids. II_, Transactions of the American Mathematical Society, 88: 161–174, (1958)
{#Tutte1958II}

* W. T. Tutte (1984), _Graph Theory_, Addison-Wesley.

* W. T. Tutte (1998), _Graph Theory As I Have Known It_, Oxford.

* [Wikipedia page](https://en.wikipedia.org/wiki/W._T._Tutte)

* [St. Andrews History page](http://www-history.mcs.st-and.ac.uk/Biographies/Tutte.html)
category:people


William Tait

* [webpage](http://home.uchicago.edu/~wwtx/)

## Related entries

* [[propositions as types]]

category: people

[[!redirects Bill Tait]]


William Thurston

Bill Thurston was an American mathematician (Fields Medalist, 1982) who made important contributions to the theory of [[orbifolds]], the theory of [[foliation|foliations]], [[low-dimensional topology]] (especially in three dimensions), and to [[geometric group theory]].  He is perhaps best known for his [[geometrization conjecture]] (which was proved by Thurston in special cases, and by [[Richard Hamilton]] and [[Grigori Perelman]] in complete generality, and which in turn implies the [[Poincaré conjecture]]).

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

* [AMS obituary](http://www.ams.org/news?news_id=1602)

* [NYT obituary](http://www.nytimes.com/2012/08/23/us/william-p-thurston-theoretical-mathematician-dies-at-65.html?_r=1)

* [St Andrew's History entry](http://www-history.mcs.st-and.ac.uk/Biographies/Thurston.html)

* [[C. T. C. Wall]], _On the work of W Thurston_, Proceedings of the International Congress of Mathematicians, Warsaw 1983 1 (Warsaw, 1984), 11-14. ([pdf](www.mathunion.org/ICM/ICM1983.1/ICM1983.1.ocr.pdf))

## Selected publications

* The theory of foliations of codimension greater than one, Comm. Math. Helv. 49 (1974), 214-231.

* Existence of codimension-one foliations, Annals of Math. (2), 104 (1976), 249-268.

* Three-Dimensional Geometry and Topology, Vol. I (ed. Silvio Levy), Princeton University Press, 1997.

* On proof and progress in mathematics, Bull. AMS 30 (1994), 161-177. ([web](http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/home.html))

On [[3-manifolds]]:

* Geometry and topology of three-manifolds (1980), electronic version 1.1 (2002) available from MSRI ([web](http://library.msri.org/books/gt3m/))

On [[hyperbolic 3-manifolds]]:

* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, I: Deformation of acylindrical manifolds_, Annals of Math, 124 (1986), 203--246 ([jstor:1971277](https://www.jstor.org/stable/1971277), [arXiv:math/9801019](https://arxiv.org/abs/math/9801019))

* [[William Thurston]], _Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle_ ([arXiv:math/9801045](https://arxiv.org/abs/math/9801045))

* [[William Thurston]], _Three dimensional manifolds, Kleinian groups and hyperbolic geometry_, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 3 (1982), 357-381 ([euclid.bams/1183548782](https://projecteuclid.org/euclid.bams/1183548782))

category: people

[[!redirects William P. Thurston]]
[[!redirects William P Thurston]]
[[!redirects William Thurston]]
[[!redirects Bill Thurston]]
[[!redirects Thurston]]


William Wooters

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

## related $n$Lab entries

* [[no-cloning theorem]]

category: people


Willis Lamb

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

## related $n$Lab entries

* [[Lamb shift]]

category: people


Willy Fischler

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

## Selected writings

Introducing the [[BFSS matrix model]] ([[M-theory]] as [[D0-brane]] [[quantum mechanics]] in [[non-perturbative effect|non-perturbative]] [[type IIA string theory]]):

* {#BanksFischlerShenkerSusskind96} [[Tom Banks]], [[Willy Fischler]], [[Stephen Shenker]], [[Leonard Susskind]], _M Theory As A Matrix Model: A Conjecture_  Phys. Rev. D55 (1997). ([arXiv:hep-th/9610043](http://arxiv.org/abs/hep-th/9610043))

On [[black holes in string theory]] via the [[BFSS matrix model]]:

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]],  _Schwarzschild Black Holes from Matrix Theory_, Phys.Rev.Lett.80:226-229,1998 ([arXiv:hep-th/9709091](https://arxiv.org/abs/hep-th/9709091))

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzchild Black Holes in Matrix Theory II_, JHEP 9801:008,1998 ([arXiv:hep-th/9711005](https://arxiv.org/abs/hep-th/9711005))

## Related $n$Lab entries

* [[axion]]

* [[BFSS matrix model]]

category: people


Wilson line > history

see _[[Wilson loop]]_.


Wilson loop

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _Wilson loop_ or _Wilson line_ is an [[observable]] in (both classical and quantum) [[gauge theory]] obtained from the [[holonomy]] of the [[gauge field|gauge]] [[connection on a bundle|connection]].

Hence if the gauge connection is given by a globally defined 1-form $A$, then the __Wilson loop__ along a closed [[loop]] $C$ is the trace of the [[Dyson formula|path-ordered exponential]]

$$
  W_C = Tr(\mathcal{P}exp(i\oint_C A_\mu d x^\mu))
$$

where $\mathcal{P}$ is the "path-ordering operator" and $A_\mu$ are the components of the connection.

## Properties

### Relation to 1d Chern-Simons theory

For $G$ a suitable [[Lie group]] (compact, semi-simple and simply connected) the Wilson loops of $G$-[[principal connections]] are equivalently the [[partition functions]] of a [[1-dimensional Chern-Simons theory]].

This appears famously in the formulation of [[Chern-Simons theory]] [with Wilson lines](Chern-Simons+theory#WithWilsonLineObservables). More detailes are at _[[orbit method]]_.

### As the universal Vassiliev invariant

The un-traced [[Wilson loop observable]] of [[perturbative quantization of 3d Chern-Simons theory|perturbative Chern-Simons theory]] is the [[universal Vassiliev invariant]] (see there for more):

> from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

### Relation to defects

Wilson loop insertions may be thought of or at least related to _defects_ in the sense of  _[[QFT with defects]]_.

### Duality with 't Hooft operators under S-duality and geometric Langlands

[[S-duality]] of 4d [[super Yang-Mills theory]] may exchange Wilson loop operators with [['t Hooft operators]], in an incarnation of the [[geometric Langlands correspondence]] ([Kapustin-Witten 06](#KapustinWitten06))

[[!include geometric Langlands QFT -- table]]

## Examples

### In Chern-Simons theory

In $SU(2)$-[[Chern-Simons theory]] the Wilson line observables compute the [[Jones polynomial]] of the given curve. See there for more details.

### In Rozansky-Witten theory

+-- {: .num_prop #RozanskyWittenWilsonLoopOfUnknotIsSquareRootOfAHat}
###### Proposition
**([[Rozansky-Witten Wilson loop of unknot is A-hat genus|Rozansky-Witten Wilson loop of unknot is square root of A-hat genus]])**

For $\mathcal{M}^{4n}$ a [[hyperkähler manifold]] (or just a [[holomorphic symplectic manifold]]) the [[Rozansky-Witten invariant]] [[Wilson loop observable]] associated with the [[unknot]] in the [[3-sphere]] is the [[square root]] $\sqrt{{\widehat A}(\mathcal{M}^{4n})}$ of the [[A-hat genus]] of $\mathcal{M}^{4n}$.

=--

This is [Roberts-Willerton 10, Lemma 8.6](Rozansky-Witten+Wilson+loop+of+unknot+is+A-hat+genus#RobertsWillerton10), using the [[Wheels theorem]] and the [[Hitchin-Sawon theorem]].

## Related entries

* [[flux tube]]

* [[Polyakov gauge-string duality]]

* [['t Hooft operator]]

* [[Wilson surface]]

## References

### General

* [[Kenneth Wilson]], _Confinement of quarks_, Physical Review __D 10__ (8): 2445. [doi](http://dx.doi.org/10.1103/PhysRevD.10.2445) (1974)

* [[Yuri Makeenko]], *Methods of contemporary gauge theory*, Cambridge Monographs on Math. Physics, Cambridge University Press (2002) [[doi:10.1017/CBO9780511535147]( https://doi.org/10.1017/CBO9780511535147), [gBooks](http://books.google.com/books?id=9W-W2w75ulAC)]

* Wikipedia [Wilson loop](http://en.wikipedia.org/wiki/Wilson_loop)

* R. Giles, _Reconstruction of gauge potentials from Wilson loops_, Physical Review __D 24__ (8): 2160, [doi](http://dx.doi.org/10.1103/PhysRevD.24.2160)

* A. Andrasi, J. C. Taylor, _Renormalization of Wilson operators in Minkowski space_, Nucl. Phys. B516 (1998) 417, [hep-th/9601122](http://arxiv.org/abs/hep-th/9601122)

* Amit Sever, Pedro Vieira, Luis F. Alday, Juan Maldacena, [[Davide Gaiotto]], _An Operator product expansion for polygonal null Wilson loops_, [arxiv.org/abs/1006.2788](http://arxiv.org/abs/1006.2788)

On super-[[Wilson lines]] via [[integration over supermanifolds]]:

* C.A. Cremonini, [[Pietro Grassi]], S. Penati, _Supersymmetric Wilson Loops via Integral Forms_ ([arXiv:2003.01729](https://arxiv.org/abs/2003.01729))

### In Chern-Simons theory

See at [[perturbative quantization of 3d Chern-Simons theory]]/[[Vassiliev knot invariants]].

Relation between [[Dehn surgery]] and [[Wilson loop observables]] in [[Chern-Simons theory]]:

* E. Guadagnini, _Surgery rules in quantum Chern-Simons field theory_, Nuclear Physics B
Volume 375, Issue 2, 18 May 1992, Pages 381-398 (doi:10.1016/0550-3213(92)90037-C)

* Boguslaw Broda, _Chern-Simons theory on an arbitrary manifold via surgery_ ([arXiv:hep-th/9305051](https://arxiv.org/abs/hep-th/9305051))

The [[Poisson bracket]] of Wilson line observables in [[3d Chern-Simons theory]] was obtained in

* W. Goldman, _Invariant functions on Lie groups and Hamiltonian flow of surface
group representations_, Inventiones Math., 85 (1986), 263–302.

For more see

* {#GelcaUribe10b} [[Razvan Gelca]], [[Alejandro Uribe]], section 4 of _Quantum mechanics and non-abelian theta functions for the gauge group $SU(2)$_ ([arXiv:1007.2010](http://arxiv.org/abs/1007.2010))

In [[Chern-Simons theory as a topological string theory]]:

* [[Hirosi Ooguri]], [[Cumrun Vafa]], _Knot Invariants and Topological Strings_, Nucl. Phys. B 577:419-438, 2000 ([arXiv:hep-th/9912123](https://arxiv.org/abs/hep-th/9912123))

[[!include Jones polynomial as Wilson loop observable -- references]]

[[!include Chern-Simons Wilson lines in AdS3-CFT2 -- references]]

### In QFT with defects

Relation to [[QFT with defects]] is discussed in

slide 17 of

* [[Constantin Bachas]], _Conformal defects in/and String theory_ ([pdf](http://www.ggi.fi.infn.it/talks/talk106.pdf))

slide 5 of

* [[Constantin Bachas]], _Conformal defects in gauged WZW models_ ([pdf](http://hep.physics.uoc.gr/conf09/TALKS/Bachas.pdf))

### As partition functions of 1d Chern-Simons theory

Expression of Wilson loops as [[partition functions]] of [[1-dimensional Chern-Simons theories]]  by the [[orbit method]] (as used notably in [[Chern-Simons theory]]) is in section 4 of

* [[Chris Beasley]], _Localization for Wilson Loops in Chern-Simons Theory_, in J. Andersen, H. Boden, A. Hahn, and B. Himpel (eds.) _Chern-Simons Gauge Theory: 20 Years After_, AMS/IP Studies in Adv. Math., Vol. 50, AMS, Providence, RI, 2011. ([arXiv:0911.2687](http://arxiv.org/abs/0911.2687))

referring to

* S. Elitzur, [[Greg Moore]], A. Schwimmer, [[Nathan Seiberg]], _Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory_, Nucl. Phys. B 326 (1989) 108–134.

in the context of [[Chern-Simons theory]] and in more general gauge theory to

* A. P. Balachandran, S. Borchardt, and A. Stern, _Lagrangian And Hamiltonian Descriptions of Yang-Mills Particles_, Phys. Rev. D 17 (1978) 3247–3256

### In S-duality and relation to t Hoofts operators

* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_, Communications in Number Theory and Physics Volume 1 (2007) Number 1 ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))

[[!redirects Wilson loop operator]]
[[!redirects Wilson loop operators]]

[[!redirects Wilson loop]]
[[!redirects Wilson loops]]
[[!redirects Wilson-loop]]
[[!redirects Wilson-loops]]

[[!redirects Wilson line]]
[[!redirects Wilson-line]]
[[!redirects Wilson lines]]
[[!redirects Wilson-lines]]

[[!redirects Wilson operator]]
[[!redirects Wilson operators]]

[[!redirects Wilson loop observable]]
[[!redirects Wilson loop observables]]

[[!redirects Wilson line observable]]
[[!redirects Wilson line observables]]


Wilson surface

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] the term _[[Wilson loop]]_ refers to an [[observable]] in [[gauge field theory]] given by the [[holonomy]] of the given [[gauge field]] [[connection on a principal bundle|connection]] along a fixed curve.

In [[higher gauge theory]] one correspondingly has [[observables]] given by [[higher holonomy]] around higher dimensional manifolds, such as [[surfaces]] etc. These are hence sometimes called "Wilson surfaces" etc.

## References

e.g.

* [[John Baez]], _[TWF 235](http://math.ucr.edu/home/baez/week235.html)_

* [[Anton Alekseev]], Olga Chekeres, [[Pavel Mnev]], _Wilson surface observables from equivariant cohomology_, JHEP11(2015)093 ([arXiv:1507.06343](https://arxiv.org/abs/1507.06343))

[[!redirects Wilson surfaces]]


Wim Hordijk

* [personal page](https://worldwidewanderings.net/)

## Selected writings

On [[Cayley graph spectra]]:

* {#RockmoreKostelecHordijkStadler02} [[Dan Rockmore]], [[Peter Kostelec]], [[Wim Hordijk]], [[Peter F. Stadler]], *Fast Fourier Transform for Fitness Landscapes*, Applied and Computational Harmonic Analysis Volume 12, Issue 1, January 2002, Pages 57-76 ([doi:10.1006/acha.2001.0346](https://doi.org/10.1006/acha.2001.0346))

category: people


Wim Ruitenberg

* [personal page](https://www.mscsnet.mu.edu/~wim/)

## Selected writings

On [[roots]] of [[polynomials]] in [[constructive mathematics]]:

* {#Ruitenberg91} [[Wim Ruitenberg]], *Constructing Roots of Polynomials over the Complex Numbers*, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract, **84** Centre for Mathematics and Computer Science, Amsterdam (1991) 107–128 [[pdf](https://www.mscsnet.mu.edu/~wim/publica/roots_new.pdf), [[Ruitenberg-Roots.pdf:file]]]

category: people


Wim van Dam

* [institute page](https://engineering.ucsb.edu/people/wim-van-dam)

## Selected writings

On [[adiabatic quantum computation]]:

* [[Dorit Aharonov]], [[Wim van Dam]], [[Julia Kempe]], [[Zeph Landau]], [[Seth Lloyd]], [[Oded Regev]], *Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation*, SIAM Journal of Computing **37** 1 (2007) 166-194 [[arXiv:quant-ph/0405098](https://arxiv.org/abs/quant-ph/0405098), [jstor:20454175](https://www.jstor.org/stable/20454175), [doi:10.1109/FOCS.2004.8](https://doi.org/10.1109/FOCS.2004.8), [doi:10.1137/080734479](https://doi.org/10.1137/080734479)]

category: people


winding number

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

#Contents#
* table of contents
{:toc}

## Idea

A [[continuous function]] between two [[topological spaces]] that are each [[weak homotopy equivalence|weakly homotopy equivalent]] to the [[circle]] is, up to homotopy, defined by how much it "winds" one circle around the other. This is its _winding number_, an [[integer]]. Equivalently this is the element in the [[fundamental group]] of the codomain which is represented by the map.

A special case of the _[[degree of a continuous function]]_.

## Related construction

* [[clutching construction]]

* [[fundamental group of the circle is the integers]]

[[!redirects winding numbers]]


Winter 2016-2017 seminar on higher structures

## Seminar on Higher Structures at Goettingen: 2016 Fall - 2017 Winter  ##
***

**Place: Hörsaal 6.


Winter 2017-2018 seminar on higher structures (Higher Lie Groupoids)

First all, we have our [studip link](
https://www.studip.uni-goettingen.de/dispatch.php/course/overview?cid=cd3c8d093de4d8fd9c8245f25e75f851). You need to register in Flexnow and present at least once, if you want to obtain your credit for this seminar.

Our seminar follows mostly Goerss-Jardine (GJ)
 [Simplicial Homotopy Theory](http://dodo.pdmi.ras.ru/~topology/books/goerss-jardine.pdf)

1. Simplicial set (GJ I.1)  by Daniel (02.11.2017)

2. Simplicial set (GJ I.1) by Daniel and realization by Laura   (09.11.2017, 16.11.2017)

2.5 Kan Complex I  by Laura(GJ I.3, first three pages) (23.11.2017)

3. Kan Complex II towards n-groupoids (GJ I.3, the rest of it) by Richard (23.11.2017)

4. [n-groupoid objects in a category with topology] (https://arxiv.org/pdf/0801.2057.pdf) section 1, 2.1, 2.2

5. simplicial homotopy group (GJ I.7) Laura 2 (but maybe we need more people, it's a long chapter)

6. Category of fibrant objects (GJ I.9 + factorization Lemma) Dennis

7. [Category of fibrant objects for n-groupoids with topology](https://arxiv.org/abs/1609.01394) Chenchang

8. Dold-Kan (GJ III.2) Madleine

9. Lie n-algebroids Malte


Wirthmüller context

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Wirthmüller context_ is a pair of two [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ which are connected by an [[adjoint triple]] of [[functors]] such that the middle one is a [[closed monoidal functor]].

This is the variant/special case of the [[yoga of six operations]] consisting of two [[adjoint pairs]] $(f_! \dashv f^!)$ and $(f^\ast \dashv f_\ast)$ and the [[tensor product]]/[[internal hom]] [[adjunctions]] $((-)\otimes B \dashv [B,-])$,  specialized to the case that  $f^! \simeq f^\ast$:

\begin{tikzcd}
  \mathcal{X}
    \ar[
      rr,
      shift left=20pt,
      "f_!"{above}
    ]
    \ar[
      rr,
      phantom,
      shift left=10.5pt,
      "\scalebox{.62}{$\bot$}"{description}
    ]
    \ar[
      rr,
      phantom,
      shift right=10.5pt,
      "\scalebox{.62}{$\bot$}"{description}
    ]
    \ar[
      rr,
      shift right=20pt,
      "f_\ast"{below}
    ]
    &&
    \mathcal{Y}
      \ar[
        ll,
        "f^! = f^\ast"{description}
      ]
\end{tikzcd}

(The other specialization of the [[six operations]] to $f_\ast \simeq f_!$ is called the _[[Grothendieck context]]_).

Often one is interested in the case that there is an [[object]] $C \in \mathcal{X}$ and an [[equivalence]]

$$
  f_\ast 1_{X} \simeq f_! C
  \,.
$$

If this induces a [[natural equivalence]]

$$
  f_\ast A  \simeq f_!(A \otimes_X C)
$$

for $A \in \mathcal{X}$, then one says this is a _Wirthmüller isomorphism_, following ([Wirthmueller 74](#Wirthmueller74)).
In particular there is a canonical [[natural transformation]]

$$
  f_\ast A  \longrightarrow f_!(A \otimes_X C)
$$

and one can ask this to be an equivalence, hence a Wirthmüller isomorphism ([May 05](#May05)).

## Definition

+-- {: .num_defn #WirthmullerContext}
###### Defininition

Let $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ be two [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] and let

\begin{tikzcd}
  \mathcal{X}
    \ar[
      rr,
      shift left=20pt,
      "f_!"{above}
    ]
    \ar[
      rr,
      phantom,
      shift left=10.5pt,
      "\scalebox{.62}{$\bot$}"{description}
    ]
    \ar[
      rr,
      phantom,
      shift right=10.5pt,
      "\scalebox{.62}{$\bot$}"{description}
    ]
    \ar[
      rr,
      shift right=20pt,
      "f_\ast"{below}
    ]
    &&
    \mathcal{Y}
      \ar[
        ll,
        "f^! = f^\ast"{description}
      ]
\end{tikzcd}

be an [[adjoint triple]] of [[functors]] between them. We call this setup

* a _pre-Wirthmüller context_ if $f^\ast$ is a [[strong monoidal functor]]:

* a _Wirthmüller context_ if $f^\ast$ is in addition a [[strong closed functor]], hence a strongly [[closed monoidal functor]].

=--

([May 05, def. 2.12](#May05))

+-- {: .num_defn }
###### Definition (Notation)

We write $[-,-]$ for the [[internal hom]] functors. For $A \in \mathcal{X}$ we write

$$
  \mathbb{D}A \coloneqq [A,1_X]
$$

for the [[internal hom]] from $A$ into the [[unit object]], hence for dual of $A$ with respect to the [[closed category]] structure, its [[dual object in a closed category]].

We say "$A$ is dualizable" to mean that it is a [[dualizable object]] with respect, instead, to the (symmetric) [[monoidal category]] structure $\otimes_X$. If $A$ is dualizable we write $A^\vee$ for its monoidal [[dual object]].

Similarly for $B \in \mathcal{Y}$.

=--

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

If all objects in $\mathcal{X}$ and $\mathcal{Y}$ are [[dualizable object|dualizable]], hence if they are [[compact closed categories]], then they are in particular also [[star-autonomous categories]] with [[dualizing object]] the [[tensor unit]]. As such their [[internal logic]] is _[[linear logic]]_ and their [[type theory]] is _[[linear type theory]]_. In terms of this a Wirthmüller morphism, def. \ref{WirthmullerContext}, is the linear analog of a [[context extension]] morphism in a [[hyperdoctrine]]: it defines a _[[dependent linear type theory]]_. See there for more.

=--

## Properties
 {#Properties}

### The comparison maps
 {#TheComparisonMaps}

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

In a pre-Wirthmüller context, def. \ref{WirthmullerContext},
there is a canonical [[natural transformation]]

$$
  f_!(A \otimes_X B)
    \longrightarrow
  (f_! A) \otimes_Y (f_! B)
  \,,
$$

(not necessarily an [[equivalence]]) being the [[adjunct]] of the composite

$$
  A \otimes_X B
  \overset
    { \eta_A \otimes \eta_B }
    {\longrightarrow}
  (f^\ast f_! A) \otimes_X (f^\ast f_! B)
  \overset
    {\simeq}
    {\longrightarrow}
  f^\ast ( (f_! A) \otimes_Y (f_! B) )
  \,,
$$

where the first morphism is the [[tensor product]] of two copies of the
[[adjunction unit]] and where the second
is the [[equivalence]] that exhibits $f^\ast$ as a [[strong monoidal functor]].

=--

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

In a pre-Wirthmüller context, def. \ref{WirthmullerContext},
write $\overline {\pi}$ for the [[natural transformation]]

$$
  \overline{\pi}
  \;\colon\;
  f_! ((f^\ast B) \otimes A)
  \longrightarrow
  B \otimes f_! A
$$

given as the composite

$$
  \overline{\pi}
  \;\colon\;
  f_!
  \big(
    (f^\ast B) \otimes A
  \big)
   \overset
     { \widetilde{ \eta_{f^\ast B} \otimes \eta_A }  }
     {\longrightarrow}
   (f_! f^\ast B) \otimes_Y (f_! A)
   \overset
     { \epsilon_B \otimes_Y id }
     {\longrightarrow}
  B \otimes f_! A
  \,,
$$

where

* the first morphism is that of remark \ref{QuasiMonoidalnessOfLeftAdjoint}

* the second is the $(f_! \dashv f^\ast)$ [[counit of an adjunction|counit]] (tensored with an [[identity morphism]]).

Also write

$$
  \overline{\gamma}
    \;\colon\;
  [f_! A ,\, B]
   \longrightarrow
  f_\ast [A ,\, f^\ast B]
$$

for the $(f^\ast \dashv f_\ast)$ [[adjunct]] of the [[natural transformation]] given as the composite

\[
  \label{MayProp211AdjunctMorphism}
  f^\ast [f_! A ,\, B]
   \stackrel{\simeq}{\longrightarrow}
  [f^\ast f_! A ,\, f^\ast B]
   \longrightarrow
  [A ,\, f^\ast B]
  \,,
\]

where the first map exhibits $f^\ast$ as a [[closed monoidal functor]] and where the second is the $(f_! \dashv f^\ast)$-[[unit of an adjunction|unit]] (under the [[internal hom]]).

=--

see ([May 05, prop. 2.11](#May05))

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

In a pre-Wirthmüller context, def. \ref{WirthmullerContext}, the comparison maps of def. \ref{ComparisonMaps} are [[equivalences]]
when the argument $B \in (\mathcal{Y}, \otimes_Y, 1_Y)$ is a [[dualizable object]].

If either of the two happens to be a [[natural equivalence]] (hence an equivalence for all arguments), then so is the other.

=--

([May 05, prop. 2.8 and prop. 2.11](#May05))

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

Precisely when the pre-Wirthmüller context is a Wirthmüller context, def. \ref{WirthmullerContext}, both comparison maps of def. \ref{ComparisonMaps} are natural equivalences.

=--

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

For all $A \in \mathcal{X}$ and $B,C \in \mathcal{Y}$ we have
by the $(f_! \dashv f^\ast)$-[[adjunction]] and the
tensor$\dashv$hom-adjunction a [[commuting diagram]] of the form

$$
  \array{
    \mathcal{Y}(B \otimes f_! A, \, C  )
     &
      \stackrel{
        \mathcal{Y}(\overline{\pi}(A,B), C)
      }{
        \longrightarrow
     }
     &
    \mathcal{Y}(f_! ((f^\ast B) \otimes A),\, C)
    \\
    \downarrow^{\mathrlap{\simeq}}
     &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathcal{X}(A, f^\ast [B,C])
    &\stackrel{}{\longrightarrow}&
    \mathcal{X}(A, [(f^\ast B), (f^\ast C)])
  }
  \,.
$$

By naturality in $A$ and by the [[Yoneda lemma]] this shows
that $\overline{\pi}$ is an equivalence precisey if
$f^\ast$ is strong closed.

For $\overline{\gamma}$ the same statement follows from this with
prop. \ref{ComparisonIsEquivalenceOnDualizables}.

=--

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

The first [[natural equivalence]] of prop. \ref{InWirthmuellerContextProjectionIsEquivalence}

$$
  f_!(f^\ast A \otimes B) \simeq A \otimes f_!(B)
$$

is often called the *[[projection formula]]*. In [[representation theory]] this is also sometimes called *[[Frobenius reciprocity]]*, though mostly that term is used for (just) the existence of the $(f_! \dashv f^\ast)$-[[adjunction]], where in representation theory the [[left adjoint]] $f_!$ forms [[induced representations]].

=--

### Consequences
 {#Consequences}

Assuming a Wirthmüller context, the [[projection formula]] has the following implications.

\begin{proposition}\label{ProjectionComposedWithCounitInRemainingVariable}
  In the special case that both of its variables are in the image of $f^\ast$, then
  composing the [[projection formula]]-map from Def. \ref{ComparisonMaps} with the [[adjunction counit]] in the first variable equals the adjunction unit of the full term, in that the following [[commuting diagram|diagram commutes]] (where we are leaving the structure isomorphism of the [[strong monoidal functor]] $f^\ast$ notationally implicit):
  $$
    \array{
      f_!
      \big(
        (f^\ast A) \otimes (f^\ast B)
      \big)
      &\overset{ \overline{\pi} }{\longrightarrow}&
      (f_! f^\ast A) \otimes B
      \\
      \mathllap{{}^{\epsilon_{A \otimes B}}}\Big\downarrow
      &&
      \Big\downarrow
      \mathrlap{ {}^{ \epsilon \otimes id } }
      \\
      A \otimes B
      &=&
      A \otimes B
    }
  $$
\end{proposition}
\begin{proof}
The [[adjunct]] of the composite of the top and the right morphism is the composite of

* the [[adjunct]] of the top morphism, which by Def. \ref{ComparisonMaps}, is

  $f^\ast A \otimes f^\ast B \overset{\eta_{f^\ast A} \otimes \eta_{f^\ast B}}{\longrightarrow} f^\ast f_! f^\ast A \otimes f^\ast f_! f^\ast B \overset{id \otimes f^\ast \epsilon_B}{\longrightarrow} f^\ast f_! f^\ast A \otimes f^\ast  B$

* the image under $f^\ast$ of the right morphism, which is $f^\ast f_! f^ \ast A \otimes f^\ast  B \overset{f^\ast \epsilon_A \otimes id}{\longrightarrow} f^\ast A \otimes f^\ast B$.

That composite is the identity by the [[triangle identity]] in each of the two variables.

In conclusion, the left vertical morphism is the adjunct of the identity, and hence the counit as claimed.
\end{proof}

\begin{proposition}
\label{AdjunctionCounitByTensoringWithCounitOnTensorUnit}
  The $(f_! \dashv f^\ast)$-[[adjunction counit]] on any object is, up to [[isomorphism]] given by the [[tensor product]] with the adjunction counit on the [[tensor unit]]:
$$
  \epsilon_{\mathscr{V}}
  \,\colon\,
  f_! f^\ast \mathscr{V}
  \,\simeq\,
  \mathscr{V} \otimes f_! f^\ast(\mathscr{V})
  \overset{
    id_{\mathscr{V}}
    \,\otimes\,
    \epsilon_{\mathbb{1}}
  }{\longrightarrow}
  \mathscr{V} \,\otimes\, \mathbb{1}
  \,\simeq\,
  \mathscr{V}
$$
\end{proposition}
\begin{proof}
The identification is given by the following [[commuting diagram]]:

\begin{tikzcd}
    f_! f^\ast \mathcal{V}
    \ar[
      rr,
      "{ \epsilon_{\mathcal{V}} }"
    ]
    \ar[
      d,
      "{ f_! f^\ast (r_{\mathcal{V}}) }",
      "{\sim}"{sloped, swap}
    ]
    &&
    \mathcal{V}
    \ar[
      d,
      "{ r_{\mathcal{V}} }",
      "{ \sim }"{sloped, swap}
    ]
    \\
    f_! f^\ast
    \big(
      \mathcal{V}_{\mathcal{Y}}
      \otimes
      1
    \big)
    \ar[
      rr,
      "{ \epsilon_{\mathcal{V} \otimes 1} }"
    ]
    \ar[
      d,
      "{ \sim }"{sloped, swap}
    ]
    &&
    \mathcal{V} \otimes 1
    \ar[d, equals]
    \\
    f_!
    \big(
      f^\ast \mathcal{V}
      \otimes
      f^\ast 1
    \big)
    \ar[
      d,
      "{ \overline{\pi} }",
      "{ \sim }"{sloped}
    ]
    \ar[rr]
    &&
    \mathcal{V} \otimes 1
    \ar[d, equals]
    \\
    \mathcal{V} \otimes f_! f^\ast 1
    \ar[
      rr,
      "{
        \mathcal{V}
          \,\otimes\,
        \epsilon_{1}
      }"
    ]
    &&
    \mathcal{V} \otimes 1
\end{tikzcd}

Here

1. the top square is the [[naturality square]] of the [[adjunction counit]] on the right [[unitor]],

1. the bottom square is that from Prop. \ref{ProjectionComposedWithCounitInRemainingVariable}.

\end{proof}

### Comparison of push-forwards and Wirthmüller isomorphism
 {#ComparisonOfPushForwardsAndWirthmuelleriso}

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

In a pre-Wirthmüller context, def. \ref{WirthmullerContext}, the functors $f_!$ and $f_\ast$ are intertwined by dualization, in that there is a [[natural equivalence]]

$$
  \mathbb{D}(f_! A) \simeq f_\ast(\mathbb{D} A)
  \,.
$$

=--

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

This is the special case of the property of $\overline{\gamma}$ in  prop. \ref{ComparisonIsEquivalenceOnDualizables}  for $B = 1_Y$:

$$
  \mathbb{D}(f_! A)
  =
  [f_! A, 1_Y]
   \underoverset{\simeq}{\overline{\gamma}}{\longrightarrow}
  f_\ast [A, f^\ast 1_Y]
  \simeq
  f_\ast [A, 1_Y]
  =
  f_\ast \mathbb{D} A
  \,.
$$

=--

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

With a Wirthmüller context regarded as [[categorical semantics]] for [[dependent linear type theory]] (see there), then the statement of cor. \ref{PushforwardsIntertwinedByDuality} is an instance of [[de Morgan duality]] where linear [[dual object|dualization]] intertwines linear [[dependent sum]] $\sum$ and [[dependent product]]  $\prod$

$$
  \prod_f \mathbb{D} \simeq \mathbb{D} \sum_f
  \,.
$$

For more on this see also ([[schreiber:Quantization via Linear homotopy types|Schreiber 14, section 3.3]]).

=--

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

In a pre-Wirthmüller context

$$
  f_\ast 1_X \simeq \mathbb{D}(f_! 1_X)
  \,.
$$

=--

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

By cor. \ref{PushforwardsIntertwinedByDuality},
since $\mathbb{D} 1_X \simeq 1_X$.

=--

+-- {: .num_prop #AbstractWirthmuellerIso}
###### Proposition
**(Wirthmüller isomorphism)**

In a Wirthmüller context, def. \ref{WirthmullerContext} if $f_! 1_X$ is a [[dualizable object]] with dual $f_! C$, then there is a [[natural equivalence]]

$$
  \omega
   \;\colon\;
  f_\ast f^\ast A \stackrel{\simeq}{\longrightarrow} f_!((f^\ast A) \otimes C)
  \,.
$$

=--

([May 05, prop. 4.13](#May05)).

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

In particular if there is $D \in \mathcal{Y}$ with

$$
  \mathbb{D}(f_! f^\ast 1_Y) \simeq f_! f^\ast D
$$

(hence if $C \simeq f^\ast D$ in the notation of prop. \ref{AbstractWirthmuellerIso}) and using that by cor. \ref{PushforwardsIntertwinedByDuality},
$
  f_\ast f^\ast  \mathbb{D}B
  \simeq
  f_\ast \mathbb{D} f^\ast B
  \simeq
  \mathbb{D}(f_! f^\ast B)$
then prop. \ref{AbstractWirthmuellerIso} gives a natural equivalence of the form

$$
  \mathbb{D}(f_! f^\ast B)
    \stackrel{\simeq}{\longrightarrow}
   f_! f^\ast ((\mathbb{D}B) \otimes D)
  \,,
$$

saying that the [[comonad]] $f_! f^\ast$ commutes with dualization up to a "twist" given by tensoring with $D$.

=--

## Examples

### Cartesian Wirthmüller contexts in toposes
 {#CartesianWirthmuellerContexts}

For $\mathbf{H}$ a [[topos]] and $f \colon X \longrightarrow Y$
any [[morphism]], then in the induced [[base change]] [[etale geometric morphism]]

$$
  (\sum_f \dashv f^\ast \dashv \prod_f)
  \;\colon\;
  \mathbf{H}_{/X}
  \longrightarrow
  \mathbf{H}_{/Y}
$$

the [[inverse image]]/[[context extension]] is a [[cartesian closed functor]] (see there for the proof). Therefore any base change of [[toposes]] constitutes a cartesian Wirthmüller context.

Conversely, this means that systems of Wirthmüller contexts are generalizations of [[categorical logic]] ([[hyperdoctrines]]) to non-cartesian contexts (see at _[[dependent linear type theory]]_).

Notice that in a cartesian Wirthmüller context duality is trivial, in that $\mathbb{D}X \simeq 1$ for all objects $X$. Therefore to the extent that the [[six operations]] yoga involves duality, it is interesting only the more non-cartesian (non-classical) the ambient Wirthmüller context is.

For instance the [[projection formula]] $\overline{\gamma}$ in def. \ref{ComparisonMaps} for base change along a pointed connected type $\mathbf{B}G \to \ast$ equivalently says that genuine ([[Bredon cohomology|Bredon]]) [[equivariant cohomology]] reduces to [[Borel equivariant cohomology]] when the action on the coefficients is trivial. See at _[[equivariant cohomology]]_ for more on this.

### Pointed objects with smash product
 {#PointedObjects}

A first step away from the Cartesian example [above](#CartesianWirthmuellerContexts) is the following.

Let $\mathbf{H}$ be a [[topos]]. For $X \in \mathbf{H}$ any [[object]], write

$$
  \mathcal{C}_X \coloneqq \mathbf{H}_{/X}^{X/}
$$

for the [[category of pointed objects]] in the [[slice topos]] $\mathbf{H}_{/X}$. Equipped with the [[smash product]] $\wedge_X$ this is a [[closed monoidal category|closed]] [[symmetric monoidal category]] $(\mathcal{C}_X, \wedge_X, X \coprod X)$.

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

For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$, the [[base change]] [[inverse image]] $f^\ast$ restricts to a functor $f^\ast \colon \mathcal{C}_Y \longrightarrow \mathcal{C}_X$ which is a Wirthmüller context.

=--

This appears as ([Shulman 07, examples 12.13 and 13.7](#Shulman07)) and ([Shulman 12, example 2.33](#Shulman12)).

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

For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$ then the [[base change]] [[inverse image]] $f^\ast \colon \mathbf{H}_{/Y} \longrightarrow \mathbf{H}_{/X}$ preserves pointedness, and the [[pushout]] functor $f_! \colon \mathbf{H}^{X/} \longrightarrow \mathbf{H}^{/Y}$ preserves co-pointedness. These two functors hence form an [[adjoint pair]]
$(f_1 \dashv f^\ast) \colon \mathcal{C}_X \longrightarrow \mathcal{C}_Y$.
Moreover, since [[colimits]] in the under-over category $\mathbf{H}_{/X}^{X/}$ are computed as colimits in $\mathbf{H}$ of [[diagrams]] with an [[initial object]] adjoined, and since by the [[Giraud axioms]] in the [[topos]] $\mathbf{H}$ [[pullback]] preserves these colimits, it follows that $f^\ast \colon \mathcal{C}_Y \to \mathcal{C}_X$ preserves colimits.
Finally by the discussion at _[[category of pointed objects]]_ we have that $\mathcal{C}_X$ and $\mathcal{C}_Y$ are [[locally presentable categories]], so that by the [[adjoint functor theorem]] it follows that $f^\ast$ has also a [[right adjoint]] $f_\ast \colon \mathcal{C}_X \to \mathcal{C}_Y$.

To see that $f^\ast$ is a [[strong monoidal functor]] observe that the [[smash product]] is, by the discussion there, given by a [[pushout]] over [[coproducts]] and [[products]] in the [[slice topos]]. As above these are all preserved by [[pullback]]. Finally to see that $f^\ast$ is also a [[strong closed functor]] observe that the [[internal hom]] on [[pointed objects]] is, by the discussion there, a [[fiber product]] of cartesian internal homs. These are preserved by [the above case](#CartesianWirthmuellerContexts), and the fiber product is preserved since $f^\ast$ preserves all limits. Hence $f^\ast$ preserves also the internal homs of pointed objects.

=--

### Bundles of modules

For $R$ a [[ring]], $R Mod$ its [[category of modules]], there is a [[functor]]

$$
  [-, R Mod] \;\colon\; Set^{op} \longrightarrow ClMonCat
$$

which sends a [[set]] $X$ to the [[closed monoidal category]] of $R$-[[modules]] parameterized over $X$.

This takes values in Wirthmüller morphisms. ([Shulman 12, example 2.2, 2.17](#Shulman12)).

### Bundles of $\infty$-modules

for [[(infinity,1)-module bundles]]: ([[schreiber:master thesis Nuiten|Nuiten 13]], [Hopkins-Lurie](#HopkinsLurie), [[schreiber:Homotopy-type semantics for quantization|Schreiber 14]])

### Becker-Gottlieb transfer

A _[[transfer context]]_ is a Wirthmüller context in which also $f_\ast$ satisfies its [[projection formula]]. In this context there is an abstract concept of _[[Becker-Gottlieb transfer]]_. See there for more.

### In equivariant stable  homotopy theory

In [[equivariant stable homotopy theory]], see ([May 05b](#May05b)).

### For quasicoherent sheaves (in $E_\infty$-geometry)
 {#ForQuasicoherentSheaves}

Pull-push of [[quasicoherent sheaves]] is usually discussed as a [[Grothendieck context]] of [[six operations]], but under some conditions it also becomes a Wirthmüller context.

Using results of Lurie this follows in the full generality of [[E-∞ geometry]] ([[spectral geometry]]).

Consider quasi-compact and quasi-separated [[E-∞ algebraic spaces]] ([[spectral algebraic spaces]]). (This includes precisely those [[spectral Deligne-Mumford stacks]] which have a [[scallop decomposition]], see [here](derived+Deligne-Mumford+stack#RelationToDerivedAlgebraicSpaces).)

If $f \;\colon\; X \longrightarrow Y$ is a map between these which is

1. locally almost of finite presentation;

1. strongly proper;

1. has finite [[Tor-amplitude]]

then the left adjoint to pullback of [[quasicoherent sheaves]] exists

$$
  (f_! \dashv f^\ast)
    \;\colon\;
  QCoh(X)
    \stackrel{\overset{f_!}{\longrightarrow}}{\underset{f^\ast}{\longleftarrow}}
  QCoh(Y)
  \,.
$$

([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 3.3.23]])

If $f$ is

* quasi-affine

then the right adjoint exists

$$
  (f^\ast \dashv f_\ast)
    \;\colon\;
  QCoh(X)
    \stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}
  QCoh(Y)
  \,.
$$

([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|LurieQC, prop. 2.5.12]], [[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 2.5.12]])

The [[projection formula]] in the dual form

$$
  f_\ast A \otimes B \longrightarrow f_\ast (A\otimes f^\ast B)
$$

for $f$ quasi-compact and quasi-separated appears as ([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, remark 1.3.14]]).

Now if all the conditions on $f$ hold, so that $(f_! \dashv f^\ast \dashv f_\ast) \;\colon\; QCoh(X) \longrightarrow QCoh(Y)$, then passing to opposite categories $QCoh(X)^{op} \longrightarrow QCoh(Y)^{op}$ exchanges the roles of $f_!$ and $f_\ast$, makes the projection formula be as in the above discussion and hence yields a Wirthmüller context.

The existence of [[dualizing modules]] $K$

$$
  D X = [X,K]
$$

is discussed in ([[Representability theorems|Lurie, Representability theorems, section 4.2]].)

## Related concepts

* [[Verdier-Grothendieck context]]

* [[Grothendieck context]]

* [[Beck-Chevalley condition]]

* [[ambidextrous adjunction]]

## References

The Wirthmüller isomorphism in [[equivariant stable homotopy theory]] goes back to and is named after

* {#Wirthmueller74} [[Klaus Wirthmüller]], _Equivariant homology and duality_. Manuscripta Math. 11(1974), 373-390

see Theorem II.6.2 in

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], and Mark Steinberger (with contributions by J.E. McClure), _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics Vol.1213. 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))

see also

* {#Nardin12} [[Denis Nardin]], section 2.6 and A.3 of _Stability and distributivity over orbital ∞-categories_, 2012 ([hdl.handle.net/1721.1/112895]( http://hdl.handle.net/1721.1/112895), [pdf](https://www.math.univ-paris13.fr/~nardin/thesis.pdf))

A clear discussion of axioms of [[six operations]] and their consequences, with emphasis on the Wirthmüller isomorphisms, is in

* {#May05} [[Halvard Fausk]], [[Po Hu]], [[Peter May]],  *Isomorphisms between left and right adjoints*, Theory and Applications of Categories, **11** 4 (2003) 107-131 [[tac:11-04](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf)]

* [[Paul Balmer]], [[Ivo Dell'Ambrogio]], [[Beren Sanders]], _Grothendieck-Neeman duality and the Wirthmüller isomorphism_, Compositio Mathematica 152.8 (2016): 1740-1776 ([arXiv:1501.01999](http://arxiv.org/abs/1501.01999))

Discussion of the original Wirthmüller isomorphism in [[equivariant stable homotopy theory]], based on this, is in

* {#May05b} [[Peter May]], _The Wirthmüller isomorphism revisited_, Theory and Applications of Categories 11.5 (2003): 132-142 ([TAC:11/5/11-05](http://www.tac.mta.ca/tac/volumes/11/5/11-05abs.html), [K-theory:0574](http://www.math.uiuc.edu/K-theory/0574/WirthRev.pdf))

More elaboration of the Wirthmüller context is in

* Baptiste Calmès, Jens Hornbostel, section 4 of _Tensor-triangulated categories and dualities_, Theory and Applications of Categories, Vol. 22, 2009, No. 6, pp 136-198 ([TAC](http://www.tac.mta.ca/tac/volumes/22/6/22-06abs.html), [arXiv:0806.0569](http://arxiv.org/abs/0806.0569))

Discussion in the context of [[pure motives]] includes

* {#Deglise} [[Frédéric Déglise]], around prop. 1.34 of _Finite correspondences and transfers over a regular base_, [pdf](http://www.math.uiuc.edu/K-theory/0765/regular_base.pdf).

Wirthmüller morphisms between pointed objects and between bundles of modules are discussed in

* {#Shulman07} [[Mike Shulman]], _Framed bicategories and monoidal fibrations_, in  Theory and Applications of Categories,  Vol. 20, 2008, No. 18, pp 650-738.  ([arXiv:0706.1286](http://arxiv.org/abs/0706.1286), [TAC](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html))

* {#Shulman12} [[Mike Shulman]], _Enriched indexed categories_ ([arXiv:1212.3914](http://arxiv.org/abs/1212.3914))

A twisted version of the Wirthmüller isomorphism is discussed in

* Bastiaan Cnossen, _Twisted ambidexterity in equivariant homotopy theory_ ([arXiv:2303.00736](https://arxiv.org/abs/2303.00736))

Discussion in [[E-∞ geometry]] is in

* {#Lurie} [[Jacob Lurie]], section 3.3. of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

* [[Jacob Lurie]], section 4.2 of _[[Representability theorems]]_

* {#HopkinsLurie} [[Michael Hopkins]], [[Jacob Lurie]], _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_

[[!redirects Wirthmüller contexts]]

[[!redirects Wirthmueller context]]
[[!redirects Wirthmueller contexts]]

[[!redirects Wirthmuller context]]
[[!redirects Wirthmuller contexts]]

[[!redirects Wirthmüller isomorphism]]
[[!redirects Wirthmüller isomorphisms]]

[[!redirects Wirthmueller isomorphism]]
[[!redirects Wirthmueller isomorphisms]]

[[!redirects Wirthmuller isomorphism]]
[[!redirects Wirthmuller isomorphisms]]