Ross Street

Ross Street is a major member of the Australian school of [[higher category theory]], a student and collaborator of the late [[Max Kelly|Gregory Maxwell Kelly]].

* [homepage](http://www.maths.mq.edu.au/~street)

His most important works include works in [[sheaf]]-, [[fibred category]]-, and [[descent]]-theory, including the introduction of [[orientals]], a definition of [[n-category|weak higher category]], [[braided categories]] and [[quantum group]]s, [[Tannaka duality|Tannaka reconstruction]], categorical [[algebraic theory|algebra]] (e.g. formal theory of [[monad]]s), [[bicategories]], [[coherence theorems]] for [[tricategories]], etc.

## Selected writings

* [list of publications](http://science.mq.edu.au/~street/Publications.html)

On [[lax functors]]:

* R. Street, _Two constructions on lax functors_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 13 (1972) no. 3 , p. 217-264 [numdam](http://www.numdam.org/item?id=CTGDC_1972__13_3_217_0 )

On the [[2-category theory]] of [[monads]] and their [[Eilenberg-Moore categories]]:

* {#Street72} [[Ross Street]], *The formal theory of monads*, Journal of Pure and Applied Algebra **2** 2 (1972) 149-168 [doi:10.1016/0022-4049(72)90019-9]

On [[enriched category theory]] and [[cosmoi]]:

* {#Street74} [[Ross Street]],  *Elementary cosmoi I.* in *Category Seminar*, Lecture Notes in Mathematics **420**, Springer (1974) [[doi:10.1007/BFb0063103](https://doi.org/10.1007/BFb0063103)]

* {#Street80} [[Ross Street]], *Cosmoi of internal categories*, Transactions of the American Mathematical Society **258** 2 (1980) 271-318 [[doi:10.2307/1998059](https://doi.org/10.2307/1998059)]

On [[topos theory]]:

* [[Ross Street]], _A survey of topos theory_, lecture notes (1978) [[pdf](http://www.math.mq.edu.au/~street/ToposSurvey.pdf), [[Street-SurveyToposTheory.pdf:file]]]

Introducing the notion of [[traced monoidal categories]]:

* {#JoyalStreetVerity96} [[André Joyal]], [[Ross Street]], [[Dominic Verity]], _Traced monoidal categories_, Math. Proc. Camb. Phil. Soc. **119** (1996) 447-468 [[pdf](http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf), [doi:10.1017/S0305004100074338](https://doi.org/10.1017/S0305004100074338)]

* [[Stephen Lack]], [[Ross Street]], *The formal theory of monads II*, Journal of Pure and Applied Algebra
**175** 1–3 (2002) 243-265 [doi:10.1016/S0022-4049(02)00137-8]

* {#Street81} [[Ross Street]], *Cauchy characterization of enriched categories*, Rend. Sem. Mat. Fis. Milano 51 (1981): 217-233, Reprints in Theory and Applications of Categories, **4** (2004) 1-16 [[tac:tr4](http://www.tac.mta.ca/tac/reprints/articles/4/tr4abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/4/tr4.pdf)]

Introducing the notion of *[[orientals]]*:

* {#Street87} [[Ross Street]], _The algebra of oriented simplexes_, J. Pure Appl. Algebra **49** (1987) 283-335;  [[pdf](http://www.math.mq.edu.au/~street/aos.pdf), doi:10.1016/0022-4049(87)90137-X, MR89a:18019]

On basic notions of [[2-category theory]]:

* {#Street96}  [[Ross Street]], *Categorical Structures*, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam (1996) [doi:10.1016/S1570-7954(96)80019-2, [pdf](http://maths.mq.edu.au/~street/45.pdf), [[Street-CategoricalStructures.pdf:file]], [ISBN:978-0-444-82212-3](https://shop.elsevier.com/books/handbook-of-algebra/hazewinkel/978-0-444-82212-3)]

On [[Frobenius monads]]:

* [[Ross Street]], *Frobenius monads and pseudomonoids*, J. Math. Phys. **45** 3930 (2004) [[doi:10.1063/1.1788852](https://doi.org/10.1063/1.1788852)]

An attempt to fomulate [[descent]] theory using [[strict omega-categories]]:

* Ross Street, _Categorical and combinatorial aspects of descent theory_, Applied Categorical Structures 12, 537--576 (2004) [doi](https://doi.org/10.1023/B:APCS.0000049317.24861.36)

Further:

* [[André Joyal|A. Joyal]], R. Street, _Pullbacks equivalent to pseudopullbacks_, Cahiers topologie et géométrie différentielle catégoriques 34 (1993) 153-156; [numdam](http://www.numdam.org/item?id=CTGDC_1993__34_2_153_0) MR94a:18004.

* A. Joyal, R. Street, _An introduction to Tannaka duality and quantum groups_,  Category theory (Como, 1990),  413--492, Lecture Notes in Math. __1488__, Springer 1991 ([pdf](http://www.math.mq.edu.au/~street/CT90Como.pdf)).

* A. Joyal, R. Street, _The geometry of tensor calculus I_,  Adv. Math.  __88__(1991),  no. 1, 55--112, [doi](http://dx.doi.org/10.1016/0001-8708%2891%2990003-P); _Tortile Yang-Baxter operators in tensor categories_,  J. Pure Appl. Algebra  __71__  (1991),  no. 1, 43--51, [doi](http://dx.doi.org/10.1016/0022-4049%2891%2990039-5); _Braided tensor categories_, Adv. Math. __102__ (1993),  no. 1, 20--78, [doi](http://dx.doi.org/10.1006/aima.1993.1055).

* R.H. Street, _Fibrations in bicategories_, Cahiers de topologie et géométrie différentielle XXI (1980) 111–160

* R.H. Street, _Two dimensional sheaf theory_, J. Pure and Appl. Algebra __24__ (1982) [MR617138](https://mathscinet.ams.org/mathscinet-getitem?mr=617138)

* R. Street, _Characterization of bicategories of stacks_, In: Kamps, K.H., Pumplün, D., Tholen, W. (eds) Category Theory. Lect. Notes Math. __962__, Springer 1982 [doi](https://doi.org/10.1007/BFb0066909)

On [[monoidal 2-categories]], [[braided monoidal 2-categories]], [[sylleptic monoidal 2-categories]] and [[symmetric monoidal 2-categories]] (and [[pseudomonoids]]):

* [[Brian Day]], [[Ross Street]], _Monoidal bicategories and Hopf algebroids_, Adv. Math. __129__:1 (1997) 99-157 ([doi:10.1006/aima.1997.1649](https://doi.org/10.1006/aima.1997.1649))

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

* [[Robert Gordon]], [[A. John Power]], [[Ross Street]], _Coherence for tricategories_, Mem. Amer. Math Soc. __117__ 558 (1995) [[ISBN:978-1-4704-0137-5](https://bookstore.ams.org/memo-117-558)]

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

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

On [[comprehensive factorization systems]] and [[torsors]]:

* [[Ross Street]], [[Dominic Verity]], *The comprehensive factorization and torsors*, [[TAC]] **23** (2010)  42-75 [[tac:23-03](http://www.tac.mta.ca/tac/volumes/23/3/23-03abs.html)]

On the history of Australian [[higher category theory]] (such as the role of [Roberts (1979)](John+Elias+Roberts#Roberts79)):

* [[Ross Street]], *An Australian conspectus of higher category theory*, talk at Institute for Mathematics and its Applications Summer Program: *$n$-Categories: Foundations and Applications* at the University of Minnesota (Minneapolis, 7–18 June 2004), in: *[[Towards Higher Categories]]*, The IMA Volumes in Mathematics and its Applications **152**, Springer (2010) 237-264 [[pdf](http://www.math.uchicago.edu/~may/IMA/Street.pdf), [[Street-Conspectus.pdf:file]], [doi:10.1007/978-1-4419-1524-5](https://link.springer.com/book/10.1007/978-1-4419-1524-5)]

On lax [[braided monoidal categories]] with possibly non-invertible [[braiding]]:

* [[Brian Day]], [[Elango Panchadcharam]], [[Ross Street]], _Lax braidings and the lax centre_, Contemporary Mathematics **441** 1 (2007) [[pdf](http://science.mq.edu.au/~street/laxcentre.pdf), [[DayPanchadcharamStreet-LaxBraidings.pdf:file]]]

On [[monoidal categories]]:

* [[Ross Street]], *Monoidal categories in, and linking, geometry and algebra*, Bull. Belg. Math. Soc. Simon Stevin **19** 5 (2012) 769-820 [[arXiv:1201.2991](https://arxiv.org/abs/1201.2991), [doi:10.36045/bbms/1354031551](https://projecteuclid.org/journals/bulletin-of-the-belgian-mathematical-society-simon-stevin/volume-19/issue-5/Monoidal-categories-in-and-linking-geometry-and-algebra/10.36045/bbms/1354031551.full)]

## Collected writings

* [[Ross Street]], _Homotopy Classification of Filtered Complexes_, PhD thesis, University of Sydney, 1968. [[pdf](http://science.mq.edu.au/~street/Thesis.pdf)]

* [[Eduardo Dubuc]] and [[Ross Street]], _Dinatural transformations_, Lecture Notes in Mathematics, Vol. 137, 1970, pp. 126–137. [[doi:10.1007/BFb0060443](https://doi.org/10.1007/BFb0060443)]

* [[Ross Street]], _Projective diagrams of interlocking sequences_, Illinois Journal of Mathematics, Vol. 15, No. 3, 1971, pp. 429-441. [[doi:10.1215/ijm/1256052612](https://doi.org/10.1215/ijm/1256052612)]

* [[Ross Street]], _The formal theory of monads_, Journal of Pure and Applied Algebra, Vol. 2, No. 2, 1972, pp. 149-168. [doi:10.1016/0022-4049(72)90019-9]

* [[Ross Street]], _Two constructions on Lax functors_, Cahiers de topologie et géométrie différentielle, Vol. 13, No. 3, 1972, pp. 217-264. [[numdam](http://www.numdam.org/item/CTGDC_1972__13_3_217_0/)]

* [[Ross Street]], _Homotopy classification of filtered complexes_, Journal of the Australian Mathematical Society, Vol. 15, No. 3, 1973, pp. 298-318. [[doi:10.1017/S1446788700013227](https://doi.org/10.1017/S1446788700013227)]

* [[Ross Street]] and [[R. F. C. Walters]], _The comprehensive factorization of a functor_, Bulletin of the American Mathematical Society, Vol. 79, No. 5, 1973, pp. 936-941. [[doi:10.1090/S0002-9904-1973-13268-9](https://doi.org/10.1090/S0002-9904-1973-13268-9)]

* [[G. M. Kelly]] and [[Ross Street]], _Review of the elements of 2-categories_, Lecture Notes in Mathematics, Vol. 420, 1974, pp. 75–103. [[doi:10.1007/BFb0063101](https://doi.org/10.1007/BFb0063101)]

* [[Ross Street]], _Fibrations and Yoneda's lemma in a 2-category_, Lecture Notes in Mathematics, Vol. 420, 1974, pp. 104–133. [[doi:10.1007/BFb0063102](https://doi.org/10.1007/BFb0063102)]

* [[Ross Street]], _Elementary cosmoi I_, ecture Notes in Mathematics, Vol. 420, 1974, pp. 134–180. [[doi:10.1007/BFb0063103](https://doi.org/10.1007/BFb0063103)]

* [[Ross Street]], _Limits indexed by category-valued 2-functors_, Journal of Pure and Applied Algebra, Vol. 8, No. 2, 1976, pp. 149-181. [doi:10.1016/0022-4049(76)90013-X]

* [[Ross Street]], _Internally complete 2-categories_, Notices of the American Mathematical Society, Vol. 23, No. 5, August 1976, pp. A-521, abstract 737-18-7. [[pdf](https://www.ams.org/journals/notices/197608/197608FullIssue.pdf)]

* [[Ross Street]] and [[Robert Walters]], _Yoneda structures on 2-categories_, Journal of Algebra, Vol. 50, No. 2, 1978, pp. 350-379. [doi:10.1016/0021-8693(78)90160-6]

* [[Ross Street]], _Cosmoi of Internal Categories_, Transactions of the American Mathematical Society, Vol. 258, No. 2, 1980, pp. 271-318. [[doi:10.2307/1998059](https://doi.org/10.2307/1998059)]

* [[Ross Street]], _Fibrations in bicategories_, Cahiers de topologie et géométrie différentielle, Vol. 21, No. 2, 1980, pp. 111-160. [[numdam](http://www.numdam.org/item/CTGDC_1980__21_2_111_0/)]

* [[Ross Street]], [[Walter Tholen]], [[Manfred Wischnewsky]], and [[Harvey Wolff]], _Semi-topological functors III: Lifting of monads and adjoint functors_, Journal of Pure and Applied Algebra, Vol. 16, No. 3, 1980, pp. 299-314. [doi:10.1016/0022-4049(80)90035-3]

* [[Ross Street]], _Notions of topos_, Bulletin of the Australian Mathematical Society, Vol. 23, No. 2, 1981, pp. 199-208. [[doi:10.1017/S000497270000705X](https://doi.org/10.1017/S000497270000705X)]

* [[Ross Street]], _Conspectus of variable categories_, Journal of Pure and Applied Algebra, Vol. 21, No. 3, 1981, pp. 307-338. [doi:10.1016/0022-4049(81)90021-9]

* [[Ross Street]], _Cauchy characterization of enriched categories_, Rendiconti del Seminario Matematico e Fisico di Milano, Vol. 51, 1981, pp. 217–233. [[doi:10.1007/BF02924823](https://doi.org/10.1007/BF02924823)]

* [[Ross Street]], _Two-dimensional sheaf theory_, Journal of Pure and Applied Algebra, Vol. 23, No. 3, 1982, pp. 251-270. [doi:10.1016/0022-4049(82)90101-3]

* [[Ross Street]], _Characterization of bicategories of stacks_, Lecture Notes in Mathematics, Vol. 962, 1982, pp. 282–291. [[doi:10.1007/BFb0066909](https://doi.org/10.1007/BFb0066909)]

* [[Ross Street]], _Enriched categories and cohomology_, Quaestiones Mathematicae, Vol. 6, No. 1-3, 1983, pp. 265-283. [[doi:10.1080/16073606.1983.9632304](https://doi.org/10.1080/16073606.1983.9632304)]

* [[Renato Betti]], [[Aurelio Carboni]], [[Ross Street]] and [[Robert Walters]], _Variation through enrichment_, Journal of Pure and Applied Algebra, Vol. 29, No. 2, 1983, pp. 109-127. [doi:10.1016/0022-4049(83)90100-7]

* [[Ross Street]], _Absolute colimits in enriched categories_, Cahiers de topologie et géométrie différentielle, Vol. 24, No. 4, 1983, pp. 377-379. [[numdam](http://www.numdam.org/item/CTGDC_1983__24_4_377_0/)]

* [[Ross Street]], _The family approach to total cocompleteness and toposes_, Transactions of the American Mathematical Society, Vol. 284, No. 1, 1984, pp. 355-369. [[doi:10.1090/S0002-9947-1984-0742429-3](https://doi.org/10.1090/S0002-9947-1984-0742429-3)]

* [[Ross Street]], _Homotopy classification by diagrams of interlocking sequences_, Mathematics Colloquium University of Cape Town, Vol. 13, 1984, pp. 83-120. [[pdf](http://science.mq.edu.au/~street/HCDISb.pdf)]

* [[Aurelio Carboni]], [[Stefano Kasangian]] and [[Ross Street]], _Bicategories of spans and relations_, Journal of Pure and Applied Algebra, Vol. 33, No. 3, 1984, pp. 259-267. [doi:10.1016/0022-4049(84)90061-6]

* [[Aurelio Carboni]] and [[Ross Street]], _Order ideals in categories_, Pacific Journal of Mathematics, Vol. 124, No. 2, 1986, pp. 275–288. [[doi:10.2140/pjm.1986.124.275](http://dx.doi.org/10.2140/pjm.1986.124.275)]

* [[Stephen Schanuel]] and [[Ross Street]], _The free adjunction_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 27, No. 1, 1986, pp. 81-83. [[numdam](http://www.numdam.org/item/CTGDC_1986__27_1_81_0/)]

* [[Brian Day]] and [[Ross Street]], _Categories in which all strong generators are dense_, Journal of Pure and Applied Algebra, Vol. 43, No. 3, 1986, pp. 235-242. [doi:10.1016/0022-4049(86)90065-4]

* [[Ross Street]], _The algebra of oriented simplexes_, Journal of Pure and Applied Algebra, Vol. 49, No. 3, 1987, pp. 283-335. [doi:10.1016/0022-4049(87)90137-X]

* [[Ross Street]], _Correction to “Fibrations in bicategories”_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 28, No. 1, 1987, pp. 53-56. [[numdam](http://www.numdam.org/item/CTGDC_1987__28_1_53_0/)]

* [[Dietmar Schumacher]] and [[Ross Street]], _Some parametrized categorical concepts_, Communications in Algebra, Vol. 16, No. 11, 1988, pp. 2313-2347. [[doi:10.1080/00927878808823693](https://doi.org/10.1080/00927878808823693)]

* [[Ross Street]], _Fillers for nerves_, Lecture Notes in Mathematics, Vol. 1348, 1988, pp. 337–341. [[doi:10.1007/BFb0081370](https://doi.org/10.1007/BFb0081370)]

* [[Brian Day]] and [[Ross Street]], _Localisation of locally presentable categories_, Journal of Pure and Applied Algebra, Vol. 58, No. 3, 1989, pp. 227-233. [doi:10.1016/0022-4049(89)90038-8]

* [[G. J. Bird]], [[G. M. Kelly]], [[A. J. Power]], and [[Ross Street]], _Flexible limits for 2-categories_, Journal of Pure and Applied Algebra, Vol. 61, No. 1, 1989, pp. 1-27. [doi:10.1016/0022-4049(89)90065-0]

* [[Brian Day]] and [[Ross Street]], _Localisations of locally presentable categories II_, Journal of Pure and Applied Algebra, Vol. 63, No. 3, 1990, pp. 225-229. [doi:10.1016/0022-4049(90)90068-S]

* [[André Joyal]] and [[Ross Street]], _The geometry of tensor calculus, I_, Advances in Mathematics, Vol. 88, No. 1, 1991, pp. 55-112. [doi:10.1016/0001-8708(91)90003-P]

* [[André Joyal]] and [[Ross Street]], _Tortile Yang-Baxter operators in tensor categories_, Journal of Pure and Applied Algebra, Vol. 71, No. 1, year, pp. 43-51. [doi:10.1016/0022-4049(91)90039-5]

* [[Ross Street]], _Quantum Groups: an entree to modern algebra_, unpublished, 1990. [[pdf](http://maths.mq.edu.au/~street/Quantum.pdf)]

* [[André Joyal]] and [[Ross Street]], _An introduction to Tannaka duality and quantum groups_, Lecture Notes in Mathematics, Vol. 1488, 1991, pp. 413–492. [[doi:10.1007/BFb0084235](https://doi.org/10.1007/BFb0084235)]

* [[Ross Street]], _Parity complexes_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 32, No. 4, 1991, pp. 315-343. [[numdam](http://www.numdam.org/item/CTGDC_1991__32_4_315_0/)]

* [[André Joyal]] and [[Ross Street]], _Pullbacks equivalent to pseudopullbacks_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 34, No. 2, 1993, pp. 153-156. [[numdam](http://www.numdam.org/item/CTGDC_1993__34_2_153_0/)]

* [[George Janelidze]], [[Dietmar Schumacher]] and [[Ross Street]], _Galois theory in variable categories_, Applied Categorical Structures, Vol. 1, 1993, pp. 103–110. [[doi:10.1007/BF00872989](https://doi.org/10.1007/BF00872989)]

* [[André Joyal]] and [[Ross Street]], _Braided Tensor Categories_, Advances in Mathematics, Vol. 102, No. 1, 1993, pp. 20-78. [[doi:10.1006/aima.1993.1055](https://doi.org/10.1006/aima.1993.1055)]

* [[Ross Street]], _Categorical Structures_, Handbook of Algebra, Vol. 1, 1996, pp. 531-577. [doi:10.1016/S1570-7954(96)80019-2, [[Street-CategoricalStructures.pdf:file]]]

* [[Aurelio Carboni]], [[Scott Johnson]], [[Ross Street]], [[Dominic Verity]], _Modulated bicategories_, Journal of Pure and Applied Algebra, Vol. 94, No. 3, 1994, pp. 229-282. [doi:10.1016/0022-4049(94)90009-4]

* [[Robert Gordon]], [[A. J. Power]], and [[Ross Street]], _Coherence for Tricategories_, Memoirs of the American Mathematical Society, Vol. 117, No. 558, 1995. [[doi:10.1090/memo/0558](https://doi.org/10.1090/memo/0558)]

* [[Ross Street]], _Higher categories, strings, cubes and simplex equations_, Applied Categorical Structures, Vol. 3, 1995, pp. 29–77. [[doi:10.1007/BF00872948](https://doi.org/10.1007/BF00872948)]

* [[André Joyal]] and [[Ross Street]], _The Category of Representations of the General Linear Groups over a Finite Field_, Journal of Algebra, Vol. 176, No. 3, 1995, pp. 908-946. [[doi:10.1006/jabr.1995.1278](https://doi.org/10.1006/jabr.1995.1278)]

* [[Ross Street]], _Ideals, radicals, and structure of additive categories_, Applied Categorical Structures, Vol. 3, 1995, pp. 139–149. [[doi:10.1007/BF00877633](https://doi.org/10.1007/BF00877633)]

* [[André Joyal]], [[Ross Street]] and [[Dominic Verity]], _Traced monoidal categories_, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 119, No. 3, 1996, pp. 447-468. [[doi:10.1017/S0305004100074338](https://doi.org/10.1017/S0305004100074338)]

* [[Ross Street]], _Parity complexes: corrigenda_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 35, No. 4, 1994, pp. 359-361. [[numdam](http://www.numdam.org/item/CTGDC_1994__35_4_359_0/)]

* [[André Joyal]] and [[Ross Street]], _The geometry of tensor calculus II _, unpublished. [[pdf](http://science.mq.edu.au/~street/GTCII.pdf)]

* [[Brian Day]] and [[Ross Street]], _Kan extensions along promonoidal functors_, Theory and Applications of Categories, Vol. 1, No. 4, 1995, pp. 72-78. [[TAC](http://www.tac.mta.ca/tac/volumes/1995/n4/1-04abs.html)]

* [[Peter Freyd]] and [[Ross Street]], _On the Size of Categories_, Theory and Applications of Categories, Vol. 1, No. 9, 1995, pp. 174-181. [[TAC](http://www.tac.mta.ca/tac/volumes/1995/n9/1-09abs.html)]

* [[Ross Street]], _Low-dimensional topology and higher-order categories_, [Proceedings of CT95](http://www.mta.ca/~cat-dist/ct95.html), 1995. [[pdf](http://science.mq.edu.au/~street/LowDTop.pdf)]

* [[Ross Street]], _Fusion Operators and Cocycloids in Monoidal Categories_, Applied Categorical Structures, Vol. 6, 1998, pp. 177–191. [[doi:10.1023/A:1008655911796](https://doi.org/10.1023/A:1008655911796)]

* [[Ross Street]], _Descent theory_, unpublished, 1995. [[pdf](http://science.mq.edu.au/~street/Descent.pdf)]

* [[Brian Day]] and [[Ross Street]], _Monoidal Bicategories and Hopf Algebroids_, Advances in Mathematics, Vol. 129, No. 1, 1997, pp. 99-157. [[doi:10.1006/aima.1997.1649](https://doi.org/10.1006/aima.1997.1649)]

* [[Renato Betti]], [[Dietmar Schumacher]] and [[Ross Street]], _Factorizations in bicategories_, unpublished, 1999. [[pdf](http://science.mq.edu.au/~street/61.pdf)]

* [[P. J. Freyd]], [[P. W. O’Hearn]], [[A. J. Power]], [[M. Takeyama]], [[Ross Street]], and [[R. D. Tennent]], _Bireflectivity_, Theoretical Computer Science, Vol. 228, No. 1-2, 1999, pp. 49-76. [doi:10.1016/S0304-3975(98)00354-5]

* [[Ross Street]], _The quantum double and related constructions_, Journal of Pure and Applied Algebra, Vol. 132, No. 2, 1998, pp. 195-206. [doi:10.1016/S0022-4049(97)00111-4]

* [[Ross Street]], _The role of Michael Batanin’s monoidal globular categories_, Contemporary Mathematics, Vol. 230, 1998, pp. 99-116. [[doi:10.1090/conm/230](https://doi.org/10.1090/conm/230)]

* [[Ross Street]], _Parenthetic remarks_, unpublished, 1997. [[pdf](http://science.mq.edu.au/~street/Parenth.pdf)]

* [[Michael Batanin]] and [[Ross Street]], _The universal property of the multitude of trees_, Journal of Pure and Applied Algebra, Vol. 154, No. 1-3, 2000, pp. 3-13. [doi:10.1016/S0022-4049(99)00184-X]

* [[George Janelidze]] and [[Ross Street]], _Galois Theory in Symmetric Monoidal Categories_, Journal of Algebra, Vol. 220, No. 1, 1999, pp. 174-187. [[doi:10.1006/jabr.1999.7905](https://doi.org/10.1006/jabr.1999.7905)]

* [[Ross Street]], _Braids among the groups_, Seminarberichte aus dem Fachbereich Mathematik, Band 63, Teil 5, 1998, pp. 699-703. [[doi:10.18445/20180823-142243-0](https://doi.org/10.18445/20180823-142243-0)]

* [[Ross Street]], _The petit topos of globular sets_, Journal of Pure and Applied Algebra, Vol. 154, No. 1-3, 2000, pp. 299-315. [doi:10.1016/S0022-4049(99)00183-8]

* [[Max Kelly]], [[Anna Labella]], [[Vincent Schmitt]], and [[Ross Street]], _Categories enriched on two sides_, Journal of Pure and Applied Algebra, Vol. 168, No. 1, 2002, pp. 53-98. [doi:10.1016/S0022-4049(01)00048-2]

* [[Stephen Lack]] and [[Ross Street]], _The formal theory of monads II_, Journal of Pure and Applied Algebra, Vol. 175, No. 1--3, 2002, pp. 243-265. [doi:10.1016/S0022-4049(02)00137-8]

* [[Brian Day]] and [[Ross Street]], _Lax monoids, pseudo-operads, and convolution_, Diagrammatic Morphisms and Applications, Contemporary Mathematics, Vol. 318, 2003, pp. 75-96. [[doi:10.1090/conm/318](https://doi.org/10.1090/conm/318)]

* [[Julien Bichon]] and [[Ross Street]], _Militaru's D-Equation in Monoidal Categories_, Applied Categorical Structures, Vol. 11, 2003, pp. 337–357. [[doi:10.1023/A:1024458931031](https://doi.org/10.1023/A:1024458931031)]

* [[Ross Street]], _Functorial Calculus in Monoidal Bicategories_, Applied Categorical Structures, Vol. 11, 2003, pp. 219–227. [[doi:10.1023/A:1024247613677](https://doi.org/10.1023/A:1024247613677)]

* [[Brian Day]], [[Paddy McCrudden]] and [[Ross Street]], _Dualizations and Antipodes_, Applied Categorical Structures, Vol. 11, 2003, pp. 229–260. [[doi:10.1023/A:1024236601870](https://doi.org/10.1023/A:1024236601870)]

* [[Ross Street]], _Weak omega-categories_, Diagrammatic Morphisms and Applications, Contemporary Mathematics, Vol. 318, 2003, pp. 207-213. [[doi:10.1090/conm/318](https://doi.org/10.1090/conm/318)]

* [[Brian Day]] and [[Ross Street]], _Abstract substitution in enriched categories_, Journal of Pure and Applied Algebra, Vol. 179, No. 1-2, 2003, pp. 49-63. [doi:10.1016/S0022-4049(02)00291-8]

* [[A Davydov]] and [[Ross Street]], _Roots of Unity as A Lie Algebra_, Georgian Mathematical Journal, Vol. 9, No. 4, 2002, pp. 683-690. [[doi:10.1515/GMJ.2002.683](https://doi.org/10.1515/GMJ.2002.683)]

* [[Brian Day]] and [[Ross Street]], _Quantum categories, star autonomy, and quantum groupoids_, Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute Communications, Vol. 43, 2004, pp. 187-226. [[doi:10.1090/fic/043](https://doi.org/10.1090/fic/043), [arXiv:math/0301209](https://arxiv.org/abs/math/0301209)]

* [[Ross Street]], _Categorical and Combinatorial Aspects of Descent Theory_, Applied Categorical Structures, Vol. 12, 2004, pp. 537–576. [[doi:10.1023/B:APCS.0000049317.24861.36](https://doi.org/10.1023/B:APCS.0000049317.24861.36)]

* [[Ross Street]], _The monoidal centre as a limit_, Theory and Applications of Categories, Vol. 13, No. 13, 2004, pp. 184-190. [[TAC](http://www.tac.mta.ca/tac/volumes/13/13/13-13abs.html)]

* [[Ross Street]], _Frobenius monads and pseudomonoids_, Journal of Mathematical Physics, Vol. 45, No. 10, 2004, pp. 3930–3948. [[doi:10.1063/1.1788852](https://doi.org/10.1063/1.1788852)]

* [[Ross Street]], _Cauchy characterization of enriched categories_, Reprints in Theory and Applications of Categories, No. 4, 2004, pp. 1-16. [[TAC](http://www.tac.mta.ca/tac/reprints/articles/4/tr4abs.html)]

* [[Ross Street]], _An Australian Conspectus of Higher Categories_, Towards Higher Categories, The IMA Volumes in Mathematics and its Applications, Vol. 152, 2010, pp. 237–264. [[doi:10.1007/978-1-4419-1524-5_6](https://doi.org/10.1007/978-1-4419-1524-5_6)]

* [[Brian Day]], [[Elango Panchadcharam]] and [[Ross Street]], _Lax braidings and the lax centre_, Hopf Algebras and Generalizations, Contemporary Mathematics, Vol. 441, 2007, pp. 1-18. [[doi:10.1090/conm/441](https://doi.org/10.1090/conm/441)]

* [[Ross Street]], _Enriched categories and cohomology with author commentary_, Reprints in Theory and Applications of Categories, No. 14, 2005, pp. 1-18. [[TAC](http://www.tac.mta.ca/tac/reprints/articles/14/tr14abs.html)]

* [[Brian Day]] and [[Ross Street]], _Centres of monoidal categories of functors_, Categories in Algebra, Geometry and Mathematical Physics, Contemporary Mathematics, Vol. 431, 2007, pp. 187-202. [[doi:10.1090/conm/431](https://doi.org/10.1090/conm/431)]

* [[Ross Street]], _Quantum Groups: A Path to Current Algebra_, Australian Mathematical Society Lecture Series, Vol. 19, Cambridge University Press, 2007. [[doi:10.1017/CBO9780511618505](https://doi.org/10.1017/CBO9780511618505)]

* [[Eduardo Dubuc]] and [[Ross Street]], _A construction of 2-filtered bicolimits of categories_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 47, No. 2, 2006, pp. 83-106. [[numdam](http://www.numdam.org/item/CTGDC_2006__47_2_83_0/)]

* [[Elango Panchadcharam]] and [[Ross Street]], _Mackey functors on compact closed categories_, Journal of Homotopy and Related Structures, Vol. 2, No. 2, 2007, pp.  261-293. [[eudml](https://eudml.org/doc/223302)]

* [[Elango Panchadcharam]] and [[Ross Street]], _Pullback and finite coproduct preserving functors between categories of permutation representations_, Theory and Applications of Categories, Vol. 16, No. 28, 2006, pp. 771-784. [[TAC](http://www.tac.mta.ca/tac/volumes/16/28/16-28abs.html)]

* [[Elango Panchadcharam]] and [[Ross Street]], _Pullback and finite coproduct preserving functors between categories of permutation representations: Corrigendum and Addendum_, Theory and Applications of Categories, Vol. 18, No. 5, 2007, pp. 151-156. [[TAC](http://www.tac.mta.ca/tac/volumes/18/5/18-05abs.html)]

* [[Craig Pastro]] and [[Ross Street]], _Weak Hopf monoids in braided monoidal categories_, Algebra Number Theory, Vol. 3, No. 2, 2009, pp. 149-207. [[doi:10.2140/ant.2009.3.149](https://doi.org/10.2140/ant.2009.3.149)]

* [[Craig Pastro]] and [[Ross Street]], _Closed categories, star-autonomy, and monoidal comonads_, Journal of Algebra, Vol. 321, No. 11, 2009, pp. 3494-3520. [[doi:10.1016/j.jalgebra.2008.05.004](https://doi.org/10.1016/j.jalgebra.2008.05.004)]

* [[Craig Pastro]] and [[Ross Street]], _Doubles for monoidal categories_, Theory and Applications of Categories, Vol. 21, No. 4, 2008, pp. 61-75. [[TAC](http://www.tac.mta.ca/tac/volumes/21/4/21-04abs.html)]

* [[Ignacio López Franco]], [[Ross Street]] and [[Richard J. Wood]], _Duals Invert_, Applied Categorical Structures, Vol. 19, 2011, pp. 321–361. [[doi:10.1007/s10485-009-9210-7](https://doi.org/10.1007/s10485-009-9210-7)]

* [[Ronald Brown]] and [[Ross Street]], _Covering morphisms of crossed complexes and of cubical omega-groupoids are closed under tensor product_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 52, No. 3, 2011. [[numdam](http://www.numdam.org/item/CTGDC_2011__52_3_188_0/)]

* [[Ross Street]], _Weak distributive laws_, Theory and Applications of Categories, Vol. 22, No. 12, 2009, pp. 313-320. [[TAC](http://www.tac.mta.ca/tac/volumes/22/12/22-12abs.html)]

* [[Micah McCurdy]] and [[Ross Street]], _What separable Fröbenius monoïdal functors preserve?_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. 51, No. 1, 2010. [[numdam](http://www.numdam.org/item/CTGDC_2010__51_1_29_0/)]

* [[Dimitri Chikhladze]], [[Stephen Lack]], and [[Ross Street]], _Hopf monoidal comonads_, Theory and Applications of Categories, Vol. 24, No. 19, 2010, pp. 554-563. [[TAC](http://www.tac.mta.ca/tac/volumes/24/19/24-19abs.html)]

* [[Ross Street]] and [[Dominic Verity]], _The comprehensive factorization and torsors_, Theory and Applications of Categories, Vol. 23, No. 3, 2010, pp. 42-75. [[TAC](http://www.tac.mta.ca/tac/volumes/23/3/23-03abs.html)]

* [[Gabriella Böhm]], [[Stephen Lack]] and [[Ross Street]], _Weak bimonads and weak Hopf monads_, Journal of Algebra, Vol. 328, No. 1, 2011, pp. 1-30. [[doi:10.1016/j.jalgebra.2010.07.032](https://doi.org/10.1016/j.jalgebra.2010.07.032)]

* [[Thomas Booker]] and [[Ross Street]], _Torsors, herds and flocks_, Journal of Algebra, Vol. 330, No. 1, 2011, pp. 346-374. [[doi:10.1016/j.jalgebra.2010.12.009](https://doi.org/10.1016/j.jalgebra.2010.12.009)]

* [[Gabriella Böhm]], [[Stephen Lack]] and [[Ross Street]], _On the 2-Categories of Weak Distributive Laws_, Communications in Algebra, Vol. 39, No. 12, 2011, pp. 4567-4583. [[doi:10.1080/00927872.2011.616436](https://doi.org/10.1080/00927872.2011.616436)]

* [[Gabriella Böhm]], [[Stephen Lack]] and [[Ross Street]], _Idempotent splittings, colimit completion, and weak aspects of the theory of monads_, Journal of Pure and Applied Algebra, Vol. 216, No. 2, 2012, pp. 385-403. [[doi:10.1016/j.jpaa.2011.07.003](https://doi.org/10.1016/j.jpaa.2011.07.003)]

* [[Ross Street]], _Parenthetic Remarks_, In "Associahedra, Tamari Lattices and Related Structures", Progress in Mathematics, Vol. 299, 2012, pp. 251–268. [[doi:10.1007/978-3-0348-0405-9_12](https://doi.org/10.1007/978-3-0348-0405-9_12)]

* [[Thomas Booker]] and [[Ross Street]], _Tannaka duality and convolution for duoidal categories_, Theory and Applications of Categories, Vol. 28, No. 6, 2013, pp. 166-205. [[TAC](http://www.tac.mta.ca/tac/volumes/28/6/28-06abs.html)]

* [[Ross Street]], _The core of adjoint functors_, Theory and Applications of Categories, Vol. 27, No. 4, 2012, pp. 47-64. [[TAC](http://www.tac.mta.ca/tac/volumes/27/4/27-04abs.html)]

* [[Ross Street]], _Monoidal categories in, and linking, geometry and algebra_, Bulletin of the Belgian Mathematical Society Simon Stevin, Vol. 19, No. 5, 2012, pp. 769-820. [[doi:10.36045/bbms/1354031551](https://doi.org/10.36045/bbms/1354031551)]

* [[Ross Street]], _Categories in categories, and size matters_, Higher Structures, Vol. 1, No. 1, 2017, pp. 225-270. [[pdf](https://higher-structures.math.cas.cz/api/files/issues/Vol1Iss1/Street)]

* [[Stephen Lack]] and [[Ross Street]], _Skew monoidales, skew warpings and quantum categories_, Theory and Applications of Categories, Vol. 26, No. 15, 2012, pp. 385-402. [[TAC](http://www.tac.mta.ca/tac/volumes/26/15/26-15abs.html)]

* [[Ross Street]], _Skew-closed categories_, Journal of Pure and Applied Algebra, Vol. 217, No. 6, 2013, pp. 973-988. [[doi:10.1016/j.jpaa.2012.09.020](https://doi.org/10.1016/j.jpaa.2012.09.020)]

* [[Stephen Lack]] and [[Ross Street]], _A Skew-Duoidal Eckmann-Hilton Argument and Quantum Categories_, Applied Categorical Structures, Vol. 22, 2014, pp. 789–803. [[doi:10.1007/s10485-013-9356-1](https://doi.org/10.1007/s10485-013-9356-1)]

* [[Stephen Lack]] and [[Ross Street]], _Triangulations, orientals, and skew monoidal categories_, Advances in Mathematics, Vol. 258, 2014, pp. 351-396. [[doi:10.1016/j.aim.2014.03.003](https://doi.org/10.1016/j.aim.2014.03.003)]

* [[Mitchell Buckley]], [[Richard Garner]], [[Stephen Lack]] and [[Ross Street]], _The Catalan simplicial set_, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 158, No. 2, 2015, pp. 211-222. [[doi:10.1017/S0305004114000498](https://doi.org/10.1017/S0305004114000498)]

* [[Stephen Lack]] and [[Ross Street]], _Combinatorial categorical equivalences of Dold–Kan type_, Journal of Pure and Applied Algebra, Vol. 219, No. 10, 2015, pp. 4343-4367. [[doi:10.1016/j.jpaa.2015.02.020](https://doi.org/10.1016/j.jpaa.2015.02.020)]

* [[Stephen Lack]] and [[Ross Street]], _On monads and warpings_, Cahiers de topologie et géométrie différentielle, Vol. 55, No. 4, 2014, pp. 244-266. [[pdf](http://cahierstgdc.com/wp-content/uploads/2017/05/LackStreet_55-4.pdf)]

* [[Stephen Lack]] and [[Ross Street]], _Skew-monoidal reflection and lifting theorems_, Theory and Applications of Categories, Vol. 30, No. 28, 2015, pp. 985-1000. [[TAC](http://www.tac.mta.ca/tac/volumes/30/28/30-28abs.html)]

* [[Ross Street]], _Kan extensions and cartesian monoidal categories_, Seminarberichte Nr. 87, 2015, pp. 89-96. [[doi:10.18445/20180829-120304-0](https://doi.org/10.18445/20180829-120304-0)]

* [[Ross Street]], _Pointwise extensions and sketches in bicategories_, 2014. [[arXiv:1409.6427](https://arxiv.org/abs/1409.6427)]

* [[Ross Street]], _Weighted Tensor Products of Joyal Species, Graphs, and Charades_, SIGMA, Vol. 12, No. 005, 2016. [[doi:10.3842/SIGMA.2016.005](https://doi.org/10.3842/SIGMA.2016.005)]

* [[Hans-E. Porst]] and [[Ross Street]], _Generalizations of the Sweedler Dual_, Applied Categorical Structures, Vol. 24, 2016, pp. 619–647. [[doi:10.1007/s10485-016-9450-2](https://doi.org/10.1007/s10485-016-9450-2)]

* [[Richard Garner]] and [[Ross Street]], _Coalgebras governing both weighted Hurwitz products and their pointwise transforms_, Bulletin of the Belgian Mathematical Society Simon Stevin, Vol. 23, No. 5, 2016, pp. 643-666. [[doi:10.36045/bbms/1483671619](https://doi.org/10.36045/bbms/1483671619)]

* [[Ross Street]], _Wreaths, mixed wreaths and twisted coactions_, Tbilisi Mathematical Journal, Vol. 10, No. 3, 2017, pp. 1-22. [[doi:10.1515/tmj-2017-0100](https://doi.org/10.1515/tmj-2017-0100)]

* [[George Janelidze]] and [[Ross Street]], _Real sets_, Tbilisi Mathematical Journal, Vol. 10, No. 3, 2017, pp. 23-49. [[doi:10.1515/tmj-2017-0101](https://doi.org/10.1515/tmj-2017-0101)]

* [[Branko Nikolić]] and [[Ross Street]], _Comonadic base change for enriched categories_, Journal of Pure and Applied Algebra, Vol. 227, No. 8, 2023. [[doi:10.1016/j.jpaa.2023.107357](https://doi.org/10.1016/j.jpaa.2023.107357)]

* [[George Janelidze]] and [[Ross Street]], _Infinitary Addition, Real Numbers, and Taut Monads_, Applied Categorical Structures, Vol. 26, 2018, pp. 1047–1064. [[doi:10.1007/s10485-018-9524-4](https://doi.org/10.1007/s10485-018-9524-4)]

* [[Branko Nikolić]] and [[Ross Street]], _Hopf rings for grading and differentials_, 2018. [[arXiv:1806.09318](https://arxiv.org/abs/1806.09318)]

* [[Ross Street]], _Vector product and composition algebras in braided monoidal additive categories_, Commentationes Mathematicae Universitatis Carolinae, Vol. 60, No. 4, 2019, pp. 581-604. [[EuDML](http://eudml.org/doc/295076)]

* [[Ross Street]], _Polynomials as spans_, Cahiers de topologie et géométrie différentielle, Vol. 61, No. 2, 2020, pp. 113-153. [[pdf](http://cahierstgdc.com/wp-content/uploads/2020/04/STREET-LXI-2.pdf)]

* [[Ross Street]], _Span composition using fake pullbacks_, Theory and Applications of Categories, Vol. 36, No. 4, 2021, pp. 102-117. [[TAC](http://www.tac.mta.ca/tac/volumes/36/4/36-04abs.html)]

* [[Stephen Lack]] and [[Ross Street]], _Corrigendum to “Combinatorial categorical equivalences of Dold–Kan type”_, Journal of Pure and Applied Algebra, Vol. 224, No. 3, 2020, pp. 1364-1366. [[doi:10.1016/j.jpaa.2019.07.002](https://doi.org/10.1016/j.jpaa.2019.07.002)]

* [[Kevin Coulembier]], [[Ross Street]] and [[Michel van den Bergh]], _Freely adjoining monoidal duals_, Mathematical Structures in Computer Science, Vol. 31, No. 7, 2021, pp. 748-768. [[doi:10.1017/S0960129520000274](https://doi.org/10.1017/S0960129520000274)]

* [[Branko Nikolić]] and [[Ross Street]], _Monoidal centres and groupoid-graded categories_, 2022. [[arXiv:2010.10656](https://arxiv.org/abs/2010.10656)]

* [[Branko Nikolić]], [[Ross Street]] and [[Giacomo Tendas]], _Cauchy completeness for DG-categories_, Theory and Applications of Categories, Vol. 37, No. 28, 2021, pp. 940-963. [[TAC](http://www.tac.mta.ca/tac/volumes/37/28/37-28abs.html)]

* [[Ross Street]], _Variation on a comprehensive theme_, Theory and Applications of Categories, Vol. 37, No. 29, 2021, pp. 964-978. [[TAC](http://www.tac.mta.ca/tac/volumes/37/29/37-29abs.html)]

* [[Ross Street]], _The core groupoid can suffice_, 2022. [[arXiv:2208.11332](https://arxiv.org/abs/2208.11332)]

## Notes

* [[Ross Street]], _Complete objects relative to a theory_, June 1976 ([pdf](http://maths.mq.edu.au/~street/Sketch.pdf))

* [[Ross Street]], _Gray's tensor product of 2-categories_, February 1982, ([pdf](http://maths.mq.edu.au/~street/GrayTensor.pdf))

category: people

[[!redirects R. Street]]
[[!redirects Street]]


Rota-Baxter algebra

# Rota--Baxter algebras
* table of contents
{: toc}

## Definition

Given a [[commutative unital ring]] $k$, an [[associative unital algebra|associative unital]] $k$-algebra $A$ is a __Rota--Baxter algebra__ of weight $\lambda$ if it is equipped with a __Rota--Baxter operator__ $P\colon A \to A$, which is a $k$-linear endomorphism such that

$$
P(x) P(y) = P(P(x) y) + P(x P(y)) + \lambda P(x y)
$$

## Historical sources and motivation

Rota--Baxter algebras came historically from several sources, including enumerative combinatorics problems ([[Gian-Carlo Rota]]), $q$-calculus (Jackson integral) and integrable systems (Baxter). More recently the application to the combinatorics of [[renormalization]] and Birkhoff decomposition have been found, showing that the essence of renormalization is not that crucially existing only in the presence of complex analysis.

## References

A large bibliography on Rota--Baxter algebras can be found at Li Guo's Rota--Baxter algebra [page](http://andromeda.rutgers.edu/~liguo/rba.html).

* K. Ebrahimi-Fard, L. Guo, _Rota-Baxter algebras and dendriform dialgebras_, Jour. Pure Appl. Algebra __212__ (2008), 320-339, [arXiv:math/0503647](https://arxiv.org/abs/math/0503647)

* [[Kurusch Ebrahimi-Fard]], [[Li Guo]],  _Rota--Baxter algebras in renormalization of perturbative quantum field theory_,  In  Universality and Renormalization, I. Binder and D. Kreimer, editors, Fields Institute Communicatins, v. 50, AMS 2007, 47-105, [arXiv:hep-th/0604116](https://arxiv.org/abs/hep-th/0604116).

* K. Ebrahimi-Fard, L. Guo, D. Manchon, _Birkhoff type decompositions and the Baker--Campbell--Hausdorff recursion_, Comm. Math. Physics __267__ (2006) 821-845, [arXiv:math-ph/0602004](https://arxiv.org/abs/math-ph/0602004)

* Jun Pei, Chengming Bai, Li Guo, _Rota-Baxter operators on $sl(2,C)$ and solutions of the classical Yang-Baxter equation_, [arxiv/1311.0612](http://arxiv.org/abs/1311.0612)

A generalization of Rota-Baxter operators to algebraic operads
is found in connection to certain splitting phenomenon for operads:

* Jun Pei, Chengming Bai, Li Guo, _Splitting of operads and Rota-Baxter operators on operads_, [arxiv/1306.3046](http://arxiv.org/abs/1306.3046)

There is also a [[categorification]] of the concept of Rota--Baxter algebra, namely the __Rota--Baxter category__, cf.

* Edmundo Castillo, Rafael Diaz, _Rota-Baxter categories_, [arxiv/0706.1068](https://arxiv.org/abs/0706.1068)

Among motivating results, there are also Spitzer identities from

* F. Spitzer, _A combinatorial lemma and its application to probability theory_,  Trans. Amer. Math. Soc. __82__ (1956), 323--339, [MR0079851](https://www.ams.org/mathscinet-getitem?mr=0079851), [doi](https://doi.org/10.2307/1993051)

category: algebra

[[!redirects Rota-Baxter algebra]]
[[!redirects Rota-Baxter algebras]]
[[!redirects Rota–Baxter algebra]]
[[!redirects Rota–Baxter algebras]]
[[!redirects Rota--Baxter algebra]]
[[!redirects Rota--Baxter algebras]]
[[!redirects Rota-Baxter operator]]


rotation

#Contents#
* table of contents
{:toc}

## Idea

A rotation of the [[Euclidean plane]] is an [[isometry]] which leaves exactly one point [[fixed point|fixed]] or is the [[identity map]]. This point, if unique, is called the _center of rotation_.

A rotation of an $n$-dimensional [[Euclidean space]], $n\geq 2$, is an [[isometry]] which either leaves exactly one $(n-2)$-dimensional [[hyperplane]] (e.g. a [[line]] in dimension $3$) fixed or is the identity map. This line or plane is called the _axis of rotation_.

If we fix a point, this identifies the Euclidean space  as a set with a [[vector space]] $V$ and the [[group]] of all isometries fixing this point is the [[special orthogonal group]] $SO(V)$.

Moreover, any choice of linear [[coordinates]] induces an  identification of rotations with orthogonal [[matrices]] with [[determinant]] $1$.  Therefore, in a basis, we may identify rotations with operators realizing [[action]] of a [[special orthogonal group]] $SO(n,\mathbb{R})$ on $\mathbb{R}^n$.

## Properties

Recall that $SO(n,\mathbf{R})$ is a real [[Lie group]] of dimension $\frac{n(n-1)}{2}$, in particular of dimension $3$ for $n=3$. A particular global choice of parameters (giving also smooth coordinates on an open dense subset) in 3 dimensions are the [[Euler angles]].

In 2d every nonidentity rotation sends any ray from the center of rotation to another ray from the center of rotation. The pair of the original ray and its image is constituting an ordered pair of rays, hence an oriented angle whose measure and orientation does not depend on a choice of the original ray; the corresponding signed measure is called the (oriented) angle of rotation. The rotation in 2d is determined by its center and the oriented angle of rotation.
In 2d every nonidentity rotation may be represented as a composition of two reflections around axes which intersect in the center of rotation. Conversely, a composition of two reflections is a rotation iff the intersection of axes of reflection is exactly one point, or the axes are the same (when we obtain the identity). The angle of rotation is the double of the (measure of the) angle between the axes of reflections. Any two reflections with the same center and oriented angle between their axes determine the same rotation in 2d.

In general, the orthogonal complement to the axis of rotation, passing through a given point on the $(n-2)$-dimensional fixed point axis, is a 2-plane. The restriction of a rotation to this subspace is a rotation of the plane. The measure or the angle of the $n$-dimensional rotation is the measure of the angle of the rotation in such an orthogonal section plane and it does not depend on the choice of the section plane, that is on the point on the axis. A rotation in $n$ dimensions is determined by its $(n-2)$-axis and its oriented angle of rotation. (Orientation should be discussed here.)

The group of rotations is $SO(3,\mathbf{R})$, below just called $SO(3)$.

__Euler's theorem on rotations__: a composition of rotations of a sphere is a rotation of a sphere. It is trivial that it is an isometry (when viewed as a transformation of the space), the nontrivial part of the theorem is that there is a $(n-2)$-hyperplane fixed by the composition, the statement which itself is also called the Euler's theorem. This is trivial in dimension 2 and nontrivial in dimension 3 proved by Euler and in higher dimension. Each orthogonal transformation in odd dimensional real space has an invariant (1-dimensional) axis, the statement also called the Euler's theorem (this boils down to the statement that the eigenvalues of the orthogonal matrix are on the unit circle).

## Relation to quaternions in 3d and 4d

Consider the vector space $\mathbf{R}^3\subset\mathbf{H}$ of imaginary quaternions. Let $s\in\mathbf{R}^3\backslash\{0\}$. Then $\sigma_{s^\perp}:q\mapsto -s q s^{-1}$ is a reflection with respect to the plane through origin and orthogonal to $q$. If $s\in\mathbf{H}$ is a quaternion the map $\rho'_s:q\mapsto s q s^{-1}$ leaves the space $\mathbf{R}^3$ of imaginary quaternions invariant and the restriction $\rho_s = \rho'_s|_{\mathbf{R}^3}$ is a rotation. Every rotation arises in that way and the map $s\mapsto\rho_s$ factorizes as the quotient map to the space of real rays $\mathbf{H}\to\mathbf{R}P^3$ and a homeomorphism $\mathbf{R}P^3\cong SO(3)$. The composition of rotations corresponds to the multiplication of quaternions. Components $a,b,c,d$ of quaternions involved fixing $s\in\mathbf{H}$ at the unit sphere $S^3\subset\mathbf{H}$ are called __[Euler-Rodrigues parameters](https://en.wikipedia.org/wiki/Euler%E2%80%93Rodrigues_formula)__. There is a redundancy of 1 parameter (plus the matter of 2-element kernel in the map $S^3\to\mathbf{R}P^3$), but this choice has its advantage in numerical modeling as it avoids the special role of the boundaries for the Euler angles.

There is a similar construction for [[SO(4)]] (e.g. [Berger 87, Thm 8.9.8](#Berger87)). Indeed, there is an epimorphism of Lie groups (see at *[Spin(4) -- exceptional isomorphisms](Spin4#ExceptionalIsomorphisms)*)

$$
  \tau
  \;\colonb\;
  S^3\times S^3\to SO(4),\,\,\,\,(s,r)\mapsto \{q\mapsto s q\overline{r}\}
$$
with [[kernel]] $\{(1,1),(-1,-1)\}$. The [[direct product group|direct product]] structure of the [[domain]] can be used to easily exhibit some nontrivial [[subgroups]] in $SO(4)$.

## Related concepts

* [[finite rotation group]]

* [[rigid body dynamics]], [[Euler angles]]

* [[isometry]], [[translation]], [[reflection]], [[boost]]

## Literature

* {#Berger87} [[Marcel Berger]], *Geometry I*, Springer 1987 ([doi:10.1007/978-3-540-93815-6](https://doi.org/10.1007/978-3-540-93815-6))

* Hui Cheng, K. C. Gupta, _An historical note on finite rotations_, J. Appl. Mech. Mar 1989, 56(1): 139-145 (7 pages) [doi](https://doi.org/10.1115/1.3176034)

[[!redirects rotations]]


rotation permutation

\tableofcontents

\section{Idea}

A _rotation permutation_ is, roughly speaking, a [[permutation]] in which, if we view the elements of a finite set as people standing in a circle, everybody shifts a given number of steps to the right, or everybody shifts a given number of steps to the left. It is an example of a [[cyclic permutation]].

\section{Formal definition}

Recall that a [[permutation]] of a [[finite set]] $X$ can formally be defined to be a [[bijection]] $X \rightarrow X$.

\begin{defn} Let $X$ be a finite set with $n$ elements. Fix an [[isomorphism]], that is to say a bijection, $i : X \rightarrow \{ 1, \ldots, n \}$. A _rotation permutation_ of $X$ is a permutation of $X$ which belongs to the [[permutation group]] on $X$ generated by the permutation $p: X \rightarrow X$ of $X$ such that the following diagram commutes, where $\sigma$ is the permutation of $\{ 1, \ldots, n \}$ given by $i \mapsto i+1$ for $1 \leq i \leq n-1$ and by $n \mapsto 1$.

\begin{centre}

\begin{tikzcd}
X \arrow[r, "i"] \arrow[d, swap, "p"] & \{ 1, \ldots, n \} \arrow[d, "\sigma"] \\
X \arrow[r, swap, "i"] & \{ 1, \ldots, n \}
\end{tikzcd}

\end{centre}

\end{defn}

\begin{rmk} More explicitly, for each integer $0 \leq j \leq n$, there is a rotation permutation of $\{ 1, \ldots, n \}$ given by $i \mapsto j+i$ for $1 \leq i \leq n - j$ and by $i \mapsto i - (n - j)$ for $n - j + 1 \leq i \leq n$. All rotation permutations of $\{ 1, \ldots, n \}$ are of this form. \end{rmk}

\begin{rmk} Let $X$ be a finite set with $n$ elements. Fix an isomorphism $i : X \rightarrow \{ 1, \ldots, n \}$. A rotation permutation of $X$ is a permutation $p: X \rightarrow X$ of $X$ such that the following diagram commutes, where $\sigma$ is a rotation permutation of $\{ 1, \ldots, n \}$.

\begin{centre}

\begin{tikzcd}
X \arrow[r, "i"] \arrow[d, swap, "p"] & \{ 1, \ldots, n \} \arrow[d, "\sigma"] \\
X \arrow[r, swap, "i"] & \{ 1, \ldots, n \}
\end{tikzcd}

\end{centre}

\end{rmk}

\section{Rotation of a word}

One sometimes also speaks of rotation of a [[word]] in some algebraic object.

\begin{defn} Let $w=a_{1} \cdots a_{n}$ be a word in some algebraic object, for example a free monoid or a free group. A _rotation_ of $w$ is a word in the same algebraic object of the form $w=a_{\sigma(1)} \cdots a_{\sigma(n)}$ where $\sigma$ is a rotation permutation of $\{1, \ldots, n\}$. \end{defn}

\begin{example} Let $X = \{a,b\}$ be a set. Examples of a rotation of the word $a^{2} b a b^{3} a$ in (the free monoid on) $X$ are $b a b^{3} a a^{2} = b a b^{3} a^{3}$, $a b^{3} a a^{2} b = a b^{3} a^{3} b$, and $a a^{2} b a b^{3} = a^{3} b a b^{3}$.
\end{example}

[[!redirects rotation permutations]]


rotor

(...) [[rigid body dynamics]] (...) [[spinor]] (...)

[[!redirects rotors]]


rough paths

\tableofcontents

\section{Idea}

Rough Paths are a way of defining [[Stieltjes integral|Riemann-Stieltjes type integrals]] for paths with poor $p$-variation where Ito techniques don't work in the context of solving controlled [[differential equations]].

\section{Set-Up}

We want to solve differential equations of the form:

$$
y_t=y_0+\int_0^t V(y_s)\otimes dX_s
$$

where $V=(V_1,...,V_d)$ with $V_i\in C_b^\infty(\mathbb{R}^d,\mathbb{R}^d)$ smooth bounded vector fields and $X\in C^p([0,T],\mathbb{R}^d)=\{f\colon \|f\|_{p-var}\colon = (\sup_{\mathcal{P}} |f_t-f_s|^p)^{1/p}\lt \infty\}$ is a signal with finite $p$ variation.

We want to solve this by a fixed point argument/Picard iteration. That means we must define:

$$
\int_0^t f(X_s)\otimes dX_s
$$

for smooth bounded $f$.

The classical paper ([Young 1936](#Young36)) says, in short, that one can define this integral as a Riemann-Stieltjes integral iff $p\lt 2$.

A theorem by Bichdeller-Dellecherie says, in short, that one can define this [[Itô integral|integral in Itô sense]] if and only if $X$ is a [[semimartingale]].

\section{Main Result}

Given a sequence of partitions, $\mathcal{P}_n=\{0=t_0\lt...\lt t_{n-1}\lt t\}$, if one can define $X^1_{su}\colon = \int_s^u dX_{r_1}$, $X^2_{su}\colon =\int_s^u\int_s^{r_2} dX_{r_1}\otimes dX_{r_2}$,..., $X^\ell_{su}\colon =\int_s^u \int_s^{r_\ell}...\int_s^{r_2} dX_{r_1}\otimes...\otimes dX_{r_\ell}$ where $\ell=\lfloor p \rfloor$ for $s,u\in[0,t]$ then the following "corrected Riemann sum" converges almost surely as is called the rough integral:

$$\lim_{n\to\infty}\sum_{k=0}^{n} f(X_{t_k} )X^1_{t_k t_{k+1}}+Df(X_{t_k})X^2_{t_k t_{k+1}}+...+D^{\ell-1}f(X_{t_k})X^{\ell}_{t_k t_{k+1}}\colon = \int_0^t f(X_S)\otimes d\mathbf{X}_s$$

The collection $\mathbf{X}=(X^1,...,X^\ell)$ is called a rough path of order $\ell$. We say that $(f(X),Df(X),...,D^{\ell-1}f(X))$ is controlled by $\mathbf{X}$.

\section{Example}

Fractional Brownian motion (fBm), $B_t$, is a centered, continuous Gaussian process with covariance

$$E(B_t B_s)=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right)$$

where $H\in(0,1)$ is called the Hurst parameter. When $H=1/2$ we recover regular [[Brownian motion]].

We have that $B_t\in C^p$ iff $p\gt 1/H$ and that fBm is not a semimartingale unless $H=1/2$. So SDEs driven by fBm with $H\lt 1/2$ are not solvable by classical means and rough paths must be used.

\section{References}

* {#Young36} LC Young, _An inequality of the Hölder type, connected with Stieltjes integration_, Acta Math. **67** (1936), 251--282. doi:[10.1007/BF02401743](https://doi.org/10.1007/BF02401743), ([Project Euclid](https://projecteuclid.org/euclid.acta/1485888152)).

* Peter Friz and [[Martin Hairer]], A Course on Rough Paths, (2014), doi:[10.1007/978-3-319-08332-2](https://doi.org/10.1007/978-3-319-08332-2), ([pdf](http://www.hairer.org/notes/RoughPaths.pdf)).

* Peter Friz and Nicolas Victoir, _Multidimensional Stochastic Processes as Rough Paths: Theory and Applications_ (2010) doi:[10.1017/CBO9780511845079](https://doi.org/10.1017/CBO9780511845079) ([pdf](http://page.math.tu-berlin.de/~friz/master4_May6th.pdf)).


round sphere

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[Riemannian geometry]], the [[topological space|topological]] [[n-sphere]] regarded as a [[Riemannian manifold]] in the standard way (i.e. as the [[submanifold]] of elements at constant [[distance]] from a given point in [[Euclidean space]]) is also called the _round $n$-sphere_, in order to distinguish it from other, non-[[isometry|isometric]] Riemannian manifold [[structures]] that also exists on some [[n-sphere]]. These alternatives are then also called _[[squashed spheres]]_.

## Examples of squashed $n$-spheres

[[!include coset space structure on n-spheres -- table]]

## Related concepts

* [[exotic sphere]]

* [[spherical space form]]

* [[Riemann sphere]]

[[!redirects round spheres]]

[[!redirects round n-sphere]]
[[!redirects round n-spheres]]

[[!redirects squashed sphere]]
[[!redirects squashed spheres]]

[[!redirects squashed n-sphere]]
[[!redirects squashed n-spheres]]

[[!redirects squshed n-sphere]]
[[!redirects squshed n-spheres]]


Rouquier's cocovering

[[!redirects Rouqier's cocovering]]
Let $T$ be a [[triangulated category]]. A triangulated subcategory $S$ of $T$ closed under small products is [[Bousfield localization|Bousfield]] if the inclusion $S\hookrightarrow T$ has a right adjoint. Two Bousfield subcategories $S_1, S_2$ of are said to __intersect properly__ if for any objects $I_1\in S_1$, $I_2\in S_2$ every morphism $I_1\to I_2$ or $I_2\to I_1$ factors through an object in the subcategory $S_1\cap S_2$. A finite family of Bousfield subcategories $\{S_1,\ldots, S_k\}$ of $T$ which pairwise intersect properly is a __Rouquier's cocovering__ if in addition $\cap_{i=1}^k S_i = 0$.

* Raphael Rouquier, _Dimensions of triangulated categories_, J. K-Theory 1 (2008), no. 2, 193–256.

* Daniel Murfet, _Rouquier's cocovering theorem and well-generated triangulated categories_, [arxiv:0904.2685](http://arxiv.org/abs/0904.2685)

[[!redirects Rouquier's cocovering]]
[[!redirects Rouquier cocovering]]


Roy Joshua

* [webpage](https://people.math.osu.edu/joshua.1/)

## Related entries

* [[six operations]], [[Verdier duality]]

category: people


Roy Kerr

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

## Selected writings

Introducing [[Kerr black hole spacetimes]]:

* [[Roy Kerr]], _Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics_, Phys. Rev. Lett. 11, 237 (1963) ([doi:10.1103/PhysRevLett.11.237](https://doi.org/10.1103/PhysRevLett.11.237))

Introducing [[Kerr-Schild spacetimes]]:

* [[Roy Kerr]], [[Alfred Schild]], _Some algebraically degenerate solutions of Einstein’s gravitational field equations_, Proc. Symp. Appl. Math 17, 199, 1965

Doubting the [[Penrose singularity theorem]]:

* [[Roy Kerr]], *Do Black Holes have Singularities?* [[arXiv:2312.00841](https://arxiv.org/abs/2312.00841), [[Kerr-DoBHsHaveSingularities-Nov2023.pdf:file]]]

category: people


Roy L. Crole

* [website](https://www.cs.le.ac.uk/people/rcrole/)

## Publications

On [[categorical semantics]] of [[lambda calculus]] and [[type theories]]:

* [[Roy L. Crole]], *Categories for types*, Cambridge University Press (1994) [[doi:10.1017/CBO9781139172707](https://doi.org/10.1017/CBO9781139172707)]

* [[Roy L. Crole]], *Alpha equivalence equalities*, Theoretical Computer Science, Volume 433, 18 May 2012, Pages 1-19, ([doi:10.1016/j.tcs.2012.01.030](https://doi.org/10.1016/j.tcs.2012.01.030))

category: people

[[!redirects Roy Crole]]


Roya Mohayaee

* [personal page](https://nbia.nbi.ku.dk/members/mohayaee/)

## Selected writings

On [[inhomogeneous cosmology]] as an explanation for apparent [[dark energy]]:

* {#SarkarEtAl18} Jacquaes Colin, [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], _Evidence for anisotropy of cosmic acceleration_, Astronomy & Astrophysics Letters ([arXiv:1808.04597](https://arxiv.org/abs/1808.04597))

* {#ColinMoymhayeeRamezSarkar19} Jacques Colin, [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], _A response to Rubin & Heitlauf: "Is the expansion of the universe accelerating? All signs still point to yes"_ ([arXiv:1912.04257](https://arxiv.org/abs/1912.04257))

On the evidence for [[dark energy]] in [[cosmology]]:

* {#MRS21} [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], *Do supernovae indicate an accelerating universe?* ([arXiv:2106.03119](https://arxiv.org/abs/2106.03119))

On evidence for [[inhomogeneous cosmology]] (large anisotropies) in observed radio galaxies and quasars:

* {#SHRMS22} Nathan Secrest, Sebastian von Hausegger, [[Mohamed Rameez]], [[Roya Mohayaee]], [[Subir Sarkar]], *A Challenge to the Standard Cosmological Model*, ApJ Letters (2022) [[arXiv2206.05624](https://arxiv.org/abs/2206.05624)]

category: people


Rozansky-Witten theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _Rozansky-Witten theory_ ([Rozansky-Witten 96](#RozanskyWitten96)) is a [[topological twist]] of [[D=3 N=4 super Yang-Mills theory]], or else the assignment of  [[TQFT|topological invariant]] [[quantum observables]] which such a [[perturbative quantum field theory]] assigns to a given [[3-manifold]] $\Sigma^3$, or more generally to a [[Wilson loop]] [[knot]] inside a [[3-manifold]] -- the _[[Rozansky-Witten invariants]]_.

The key result of [Rozansky-Witten 96](#RozanskyWitten96) is that, after [[gauge fixing]] and some subtle [[field (physics)|field]] identifications, the [[Feynman rules]] of RW-[[topological twist|twisted]] [[D=3 N=4 super Yang-Mills theory]] are those of [[perturbative quantization of 3d Chern-Simons theory|perturbative]] [[Chern-Simons theory]], in that the only relevant [[propagator]] is the [[Chern-Simons propagator]] and the only relevant [[Feynman diagrams]] are trivalent, _except_ that the [[Lie algebra weight system|Lie algebra weights]] assigned by [[Chern-Simons theory]] to a [[Feynman diagram|Feynman]]-[[Jacobi diagram]], are replaced by other [[weight systems]], now called _[[Rozansky-Witten weight systems]]_.

Precisely: Recalling that [[perturbative quantization of 3d Chern-Simons theory|perturbative Chern-Simons]] [[Wilson loop observables]] are given by evaluating [[Lie algebra weight systems]] on the [[Jacobi diagram]]-valued [[universal Vassiliev invariant]]

we have that [[Wilson loop]] [[Rozansky-Witten invariants]] ([Rozansky-Witten 96 (2.26)](#RozanskyWitten96)) are given by evaluating [[Rozansky-Witten invariants]] on the [[Jacobi diagram]]-valued [[universal Vassiliev invariant]]

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

These [[Rozansky-Witten weight systems]] depend only on the [[hyperkähler manifold]] $\mathcal{M}^{4n}$ which is the (classical) [[Coulomb branch]] of the RW-[[topological twist|twisted]] [[D=3 N=4 super Yang-Mills theory]], and in fact they are independent of the [[Riemannian geometry]] of $\mathcal{M}^{4n}$ and depend only on the underlying [[holomorphic symplectic manifold]] ([Kapranov 99](#Kapranov99)). Generally, they are defined for $\mathcal{M}^{4n}$ any [[hyperkähler manifold]] which is either asymptotically flat ([[ALE spaces]]) or [[compact topological space]] ([[compact hyperkähler manifolds]]).

Via the equivalent reformulation by [Kapranov 99](#Kapranov99) one finds ([Roberts-Willerton 10](#RobertsWillerton10)) that the [[Rozansky-Witten invariants]] are structurally [[Lie algebra weight systems]] themselves, but [[Lie algebra object|internal]] to the [[derived category of coherent sheaves]] of $\mathcal{M}^{4n}$ and composed with an [[integration]] over $\mathcal{M}^{4n}$ which makes the resulting [[Dolbeault cohomology]]-valued weights become [[ground field]]-valued.

\linebreak

[Konsevich 97](#Konsevich97) gives a formulation of [[Rozansky-Witten invariants]] via [[characteristic classes]] of [[foliations]] and [[Gelfand-Fuks cohomology]]. He devised a formal construction, again depending on a trivalent graph of a cohomology class of the Lie algebra of formal (in the sense of formal power series) [[Hamiltonian vector fields]] on any arbitrary finite-dimensional symplectic vector space. Characteristic classes of foliations may induce examples where this construction applies; one of the examples yields Rozansky-Witten .

## Properties

### As KK-compactification of M5-brane worldvolume

Rozansky-Witten theory may be identified with [[topological twist|topologically twisted]] [[KK-compactification]] of the [[D=6 N=(2,0) SCFT]] on the [[M5-brane]] ([Gukov-Putrov-Vafa 17, Sections 3.2 and 4.2](#GukovPutrovVafa17))

## Examples

### Rozanzky-Witten Wilson loop of unknot is $\hat A$-genus

+-- {: .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 concepts

* [[Casson invariant]]

* [[Seiberg-Witten invariant]]

* [[super Chern-Simons theory]]

## References

Original articles:

* {#RozanskyWitten96} [[Lev Rozansky]], [[Edward Witten]], _Hyper-Kähler geometry and invariants of 3-manifolds_, Selecta Math., New Ser. __3__ (1997), 401–458 ([arXiv:hep-th/9612216](https://arxiv.org/abs/hep-th/9612216), [doi:10.1007/s000290050016](https://doi.org/10.1007/s000290050016), [MR98m:57041](http://www.ams.org/mathscinet-getitem?mr=98m:57041))

* {#Konsevich97} [[Maxim Kontsevich]], _Rozansky–Witten invariants via formal geometry_, Compositio Mathematica 115: 115–127, 1999, [doi](http://dx.doi.org/10.1023/A:1000619911308), [arXiv:dg-ga/9704009](http://arxiv.org/abs/dg-ga/9704009), [MR2000h:57057](http://www.ams.org/mathscinet-getitem?mr=2000h:57057)

* {#Kapranov99} [[Mikhail Kapranov]], _Rozansky–Witten invariants via Atiyah classes_,  Compositio Math.  __115__ (1999), no. 1, 71--113 ([MR2000h:57056](http://www.ams.org/mathscinet-getitem?mr=2000h:57056), [doi](http://dx.doi.org/10.1023/A:1000664527238), [alg-geom/9704009](http://arxiv.org/abs/alg-geom/9704009))

* [[Justin Roberts]], [[Justin Sawon]], _Generalisations of Rozansky-Witten invariants_, Geom. Topol. Monogr. 4 (2002) 263-279 ([arXiv:math/0112210](https://arxiv.org/abs/math/0112210))

Unified description of [[Rozansky-Witten weight systems]] as [[Lie algebra weight systems]] for [[Lie algebra objects]] in the [[derived category of quasi-coherent sheaves]], and unified [[Wheels theorem]]:

* {#RobertsWillerton10} [[Justin Roberts]], [[Simon Willerton]], _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

Review:

* [[Justin Sawon]], _Rozansky-Witten invariants of hyperkähler manifold_, Cambridge 2000 ([arXiv:math/0404360](https://arxiv.org/abs/math/0404360))

* {#Roberts01} [[Justin Roberts]], _Rozansky-Witten theory_ ([arXiv:math/0112209](https://arxiv.org/abs/math/0112209))

Further relation to [[Chern-Simons theory]]:

* [[Anton Kapustin]], [[Natalia Saulina]], _Chern-Simons-Rozansky-Witten topological field theory_, Nucl. Phys. B823 (2009) 403-427 ([arXiv:0904.1447](https://arxiv.org/abs/0904.1447), [spire:817599/](http://inspirehep.net/record/817599/))

* Jian Qiu, Maxim Zabzine, _Odd Chern-Simons theory, Lie algebra cohomology and characteristic classes_, Commun. Math. Phys. 300:789-833, 2010
 ([arxiv/0912.1243](http://arxiv.org/abs/0912.1243))

As [[topological twist|topologically twisted]] [[KK-compactification]] of the [[D=6 N=(2,0) SCFT]] on the [[M5-brane]] (see [[D=3 N=4 super Yang-Mills theory]] and [[3d-3d correspondence]]):

* {#GukovPutrovVafa17}  [[Sergei Gukov]], Pavel Putrov, [[Cumrun Vafa]], Sections 3.2 and 3.4 of: _Fivebranes and 3-manifold homology_, J. High Energ. Phys. (2017) 2017: 71 ([arXiv:1602.05302](https://arxiv.org/abs/1602.05302))

* Cyril Closset, Heeyeon Kim, Section 6.1 of: _Comments on twisted indices in 3d supersymmetric gauge theories_, JHEP 08 (2016) 059 ([arXiv:1605.06531](https://arxiv.org/abs/1605.06531))

Discussion of [[Rozansky-Witten theory]] as a [[boundary field theory]]:

* [[Anton Kapustin]], [[Lev Rozansky]], [[Natalia Saulina]],
_Three-dimensional topological field theory and symplectic algebraic geometry I_ ([arXiv:0810.5415](https://arxiv.org/abs/0810.5415))

Relation to [[equivariant cohomology]]:

* [[Sergei Gukov]], Po-Shen Hsin, Hiraku Nakajima, Sunghyuk Park, Du Pei, Nikita Sopenko, _Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants_ ([arXiv:2005.05347](https://arxiv.org/abs/2005.05347))

* Jian Qiu, _Rozansky-Witten theory, Localised then Tilted_ ([arXiv:2011.05375](https://arxiv.org/abs/2011.05375))

Discussion of Rozansky-Witten [[defect QFT]] as an [[extended TQFT]] via the [[cobordism hypothesis]]:

* [[Ilka Brunner]], [[Nils Carqueville]], [[Daniel Roggenkamp]], *Truncated affine Rozansky--Witten models as extended TQFTs*,  Commun. Math. Phys. (2023) [[arXiv:2201.03284](https://arxiv.org/abs/2201.03284), [doi:10.1007/s00220-022-04614-4](https://doi.org/10.1007/s00220-022-04614-4)]

reviewed in:

* [[Nils Carqueville]], *Extended defect TQFTs*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#CarquevilleMar2023), [pdf](https://www.carqueville.net/nils/RW.pdf), [[Carqueville-RW.pdf:file]], video:[YT](https://www.youtube.com/watch?v=bwDJSitenVE)]

Relation to [[super Chern-Simons theory]]:

* [[Nathan Geer]], [[Matthew B. Young]], *Three dimensional topological quantum field theory from $U_q(\mathfrak{gl}(1\vert1))$ and $U(1\vert 1)$ Chern--Simons theory* [[arXiv:2210.04286](https://arxiv.org/abs/2210.04286)]

Review:

* [[Matthew B. Young]], *Non-semisimple TFT and $U(1\vert 1)$ Chern-Simons theory*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023)[[web](Center+for+Quantum+and+Topological+Systems#YoungMar2023)]

[[!redirects Rozansky-Witten class]]
[[!redirects Rozansky-Witten invariant]]
[[!redirects Rozansky-Witten invariants]]
[[!redirects Rozansky-Witten theories]]


Rozansky-Witten weight system

[[!redirects Rozansky-Witten weight systems]]

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

#Contents#
* table of contents
{:toc}

## Idea

Rozansky-Witten weight systems are [[weight systems]] on [[Jacobi diagrams]] ([[chord diagrams modulo 4T are Jacobi diagrams modulo STU|equivalently]] on [[round chord diagrams]]) given by [[Rozansky-Witten invariants]] on [[hyperkähler manifolds]], with [[coefficients]] in certain [[Dolbeault cohomology]] [[cohomology groups|groups]].

For asymptotically flat and for [[compact hyperkähler manifolds]] the induced RW-weight systems take values in the [[ground field]] and hence are actual [[weight systems]].

## Properties

Rozansky-Witten weight systems depend only on the [[hyperkähler manifold]] $\mathcal{M}^{4n}$ which is the (classical) [[Coulomb branch]] of the RW-[[topological twist|twisted]] [[D=3 N=4 super Yang-Mills theory]], and in fact they are independent of the [[Riemannian geometry]] of $\mathcal{M}^{4n}$ and depend only on the underlying holomorphic [[symplectic structure]] ([Kapranov 99](#Kapranov99)). Generally, they are defined for $\mathcal{M}^{4n}$ any [[hyperkähler manifold]] which is either asymptotically flat ([[ALE spaces]]) or [[compact topological space]] ([[compact hyperkähler manifolds]]).

Via the equivalent reformulation by [Kapranov 99](#Kapranov99) one finds ([Roberts-Willerton 10](#RobertsWillerton10)) that the [[Rozansky-Witten invariants]] are structurally [[Lie algebra weight systems]] themselves, but [[Lie algebra object|internal]] to the [[derived category of coherent sheaves]] of $\mathcal{M}^{4n}$ and composed with an [[integration]] over $\mathcal{M}^{4n}$ which makes the resulting [[Dolbeault cohomology]]-valued weights become [[ground field]]-valued.

## Examples

### Ground-field valued weight systems

In order for Rozansky-Witten weight systems to take values in the [[ground field]], hence to be actual [[weight systems]], the [[hyperkähler manifold]] has to be asymptotically flat or  [[compact topological space|compact]] (i.e. [[closed manifold|closed]]). The only known examples of [[compact hyperkähler manifolds]] are [[Hilbert schemes of points]] $X^{[n+1]}$ (for $n \in \mathbb{N}$) for $X$ either

1. a [[K3-surface]]

1. a [[4-torus]] (in which case the [[compact hyperkähler manifolds]] is really the [[fiber]] of $(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4$)

([Beauville 83](compact+hyperkähler+manifold#Beauville83))
and two exceptional examples ([O'Grady 99](compact+hyperkähler+manifold#OGrady99), [O'Grady 03](compact+hyperkähler+manifold#OGrady03)
), see [Sawon 04, Sec. 5.3](compact+hyperkähler+manifold#Sawon04).

## Related concepts

* [[D=3 N=4 super Yang-Mills theory]]

[[!include chord diagrams and weight systems -- table]]

## References

### General

Original articles:

* {#RozanskyWitten96} [[Lev Rozansky]], [[Edward Witten]], _Hyper-Kähler geometry and invariants of 3-manifolds_, Selecta Math., New Ser. __3__ (1997), 401–458 ([arXiv:hep-th/9612216](https://arxiv.org/abs/hep-th/9612216), [doi:10.1007/s000290050016](https://doi.org/10.1007/s000290050016), [MR98m:57041](http://www.ams.org/mathscinet-getitem?mr=98m:57041))

* {#Kapranov99} [[Mikhail Kapranov]], _Rozansky–Witten invariants via Atiyah classes_,  Compositio Math.  __115__ (1999), no. 1, 71--113, [MR2000h:57056](http://www.ams.org/mathscinet-getitem?mr=2000h:57056), [doi](http://dx.doi.org/10.1023/A:1000664527238), [alg-geom/9704009](http://arxiv.org/abs/alg-geom/9704009)

Unified description of [[Rozansky-Witten weight systems]] with [[Lie algebra weight systems]], and unified [[Wheels theorem]] via [[Lie algebra objects]]:

* {#RobertsWillerton10} [[Justin Roberts]], [[Simon Willerton]], _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

Review:

* {#Roberts01} [[Justin Roberts]], _Rozansky-Witten theory_ ([arXiv:math/0112209](https://arxiv.org/abs/math/0112209))

### Relation to Seiberg-Witten invariants

On relation between [[Rozansky-Witten invariants]] and [[Seiberg-Witten invariants]] of [[3-manifolds]]:

* [[Matthias Blau]], George Thompson, _On the Relationship between the Rozansky-Witten and the 3-Dimensional Seiberg-Witten Invariants_, Adv. Theor. Math. Phys. 5 (2002) 483-498 ([arXiv:hep-th/0006244](https://arxiv.org/abs/hep-th/0006244))


Rozansky-Witten Wilson loop of unknot is A-hat genus

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## Statement

+-- {: .num_prop #RozanskyWittenWilsonLoopOfUnknotIsSquareRootOfAHat}
###### Proposition
**([[Rozansky-Witten theory|Rozansky-Witten]] [[Wilson loop observable]] 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](#RobertsWillerton10), using the [[Wheels theorem]] ([Bar-Natan, Thang, Thurston 03](#BarNatanThangThurston03)) and the [[Hitchin-Sawon theorem]] ([Hitchin-Sawon 99](#HitchinSawon99)).

## References

* {#BarNatanThangThurston03} [[Dror Bar-Natan]], [[Le Tu Quoc Thang]], [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))

* {#HitchinSawon99} [[Nigel Hitchin]], [[Justin Sawon]], _Curvature and characteristic numbers of hyperkähler manifolds_, Duke Math.J. 106 (2001) 599-615 ([arXiv:math/9908114](https://arxiv.org/abs/math/9908114))

* {#RobertsWillerton10} [[Justin Roberts]], [[Simon Willerton]], _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

Review in

* {#Roberts01} [[Justin Roberts]], Section 9 of: _Rozansky-Witten theory_ ([arXiv:math/0112209](https://arxiv.org/abs/math/0112209))

[[!redirects Hitchin-Sawon theorem]]


RR field

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##  Idea

The _RR field_ or _Ramond--Ramond field_ is a [[gauge theory|gauge field]] appearing in  [[supergravity|10-dimensional type II supergravity]].

Mathematically the RR field on a space $X$ is a [[cocycle]] in [[differential K-theory]] -- or rather, in full generality, in [[twisted differential K-theory|twisted differential]] [[KR-theory]] subject to a [[self-dual higher gauge field]] constrained encoded by a quadratic form defining an [[11-dimensional Chern-Simons theory]] on twisted differential KR cocycles.

Accordingly, the [[field strength]] of the RR field, i.e. the image of the [[differential cohomology|differential K-cocycle]] in deRham cohomology, is an inhomogeneous even or odd differential form

* in [[type IIA string theory]]
  $$
    F_{RR} = R_0 + R_2 + \cdots
  $$

* in [[type IIB string theory]]
  $$
    F_{RR} = R_1 + R_3 + \cdots
  $$

The components of this are sometimes called the RR forms.

In the presence of a nontrivial [[Kalb–Ramond field]] the RR field is [[twisted cohomology|twisted]]: a cocycle in the corresponding [[twisted K-theory]].

Moreover, the RR field is constrained to be a [[self-dual higher gauge theory|self-dual]] differential K-cocycle in a suitable sense.

[[!include electric-magnetic duality -- table]]

The RR field derives its name from the way it shows up when the [[supergravity]] theory in question is derived as an effective background theory in [[string theory]]. From the [[sigma-model]] perspective of the string the RR field is the condensate of fermionic 0-mode excitations of the type II superstring for a particular choice of boundary conditons called the _Ramond_ boundary condititions. Since these boundary conditions have to be chosen for _two_ spinor components, the name appears twice.

## Related concepts

* [[RR-field tadpole]]

* [[D-brane charge]]

* [[Hanany-Witten effect]]

* [[K-theory classification of D-brane charge]]

  * [[differential K-theory]]

  * [[KK-theory]]

[[!include table of branes]]

## References

### General

Early discussion in relation to [[D-branes]]:

* [[Joseph Polchinski]], *Dirichlet-Branes and Ramond-Ramond Charges*, Phys. Rev. Lett. **75** (1995) 4724-4727 [[arXiv:hep-th/9510017](https://arxiv.org/abs/hep-th/9510017), [doi:10.1103/PhysRevLett.75.4724](https://doi.org/10.1103/PhysRevLett.75.4724)]

Review in the context of the [[K-theory classification of D-brane charge]]:

* {#Freed00} [[Daniel Freed]], example 2.10 in: *Dirac charge quantization and generalized differential cohomology_*, Surveys in Differential Geometry, Int. Press, Somerville (2000) 129-194  [[arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220), [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6), [spire:537392](https://inspirehep.net/literature/537392)]

Discussion of the [[NSR-string|NSR]] [[string scattering amplitudes|string perturbation theory]] in RR-field [[background field|background]]:

* {#BerensteinLeigh99} [[David Berenstein]], [[Robert Leigh]], _Superstring Perturbation Theory and Ramond-Ramond Backgrounds_, Phys. Rev. D **60** (1999) 106002 [[arXiv:hep-th/9904104](https://arxiv.org/abs/hep-th/9904104)]

### Self-duality and quadratic form
 {#ReferencesSelfDuality}

[[self-duality for pregeometric RR-fields -- references]]

The [[self-dual higher gauge field]] nature (see there for more) in terms of a [[quadratic form]] on [[differential K-theory]] is discussed originally around

* {#MooreWitten99} [[Gregory Moore]], [[Edward Witten]], _Self-Duality, Ramond-Ramond Fields, and K-Theory_, JHEP 0005:032,2000 ([arXiv:hep-th/9912279](http://arxiv.org/abs/hep-th/9912279))

and ([Freed 00](#Freed00)) for [[type I superstring theory]], and for [[type II superstring theory]] in

* {#Witten00} [[Edward Witten]], _Duality Relations Among Topological Effects In String Theory_, JHEP 0005:031 (2000) [[arXiv:hep-th/9912086](http://arxiv.org/abs/hep-th/9912086), [doi:10.1088/1126-6708/2000/05/032](https://doi.org/10.1088/1126-6708/2000/05/032)]

* {#FreedHopkins00} [[Daniel Freed]], [[Michael Hopkins]], *On Ramond-Ramond fields and K-theory*, JHEP 0005 (2000) 044 [[arXiv:hep-th/0002027](http://arxiv.org/abs/hep-th/0002027)]

* {#FMW00} [[Duiliu-Emanuel Diaconescu]], [[Gregory Moore]], [[Edward Witten]], *$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory*, Adv. Theor. Math. Phys. **6** (2003) 1031-1134 [[arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090)], summarised in _A Derivation of K-Theory from M-Theory_ [[arXiv:hep-th/0005091](http://arxiv.org/abs/hep-th/0005091)]

with more refined discussion in [[twisted differential K-theory|twisted differential]] [[KR-theory]] in

* {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2011) [[arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf)]

See at _[[orientifold]]_ for more on this. The relation to [[11d Chern-Simons theory]] is made manifest in

* {#BelovMooreII} Dmitriy Belov, [[Greg Moore]], _Type II Actions from 11-Dimensional Chern-Simons Theories_ [[arXiv:hep-th/0611020](http://arxiv.org/abs/hep-th/0611020)]

Review is in

* {#Szabo12} [[Richard Szabo]], section 3.6 and 4.6 of: _Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology_ [[arXiv:1209.2530](http://arxiv.org/abs/1209.2530)]

Discussion of [[Lagrangian densities]] for [[type II supergravity]] making the nature of the [[pregeometric RR-fields]] and their [[self-dual higher gauge theory|self-duality]] manifest:

* [[Gianguido Dall'Agata]], [[Kurt Lechner]], [[Mario Tonin]], *$D=10$, $N=IIB$ Supergravity: Lorentz-invariant actions and duality*, JHEP 9807:017 (1998) [[arXiv:hep-th/9806140](https://arxiv.org/abs/hep-th/9806140), [doi:10.1088/1126-6708/1998/07/017](https://doi.org/10.1088/1126-6708/1998/07/017)]

* [[Eric Bergshoeff]], [[Renata Kallosh]], [[Tomas Ortin]], [[Diederik Roest]], [[Antoine Van Proeyen]], *New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls*, Class. Quant. Grav. **18** (2001) 3359-3382 [[arXiv:hep-th/0103233](https://arxiv.org/abs/hep-th/0103233), [doi:10.1088/0264-9381/18/17/303](https://doi.org/10.1088/0264-9381/18/17/303)]

* [[Karapet Mkrtchyan]], [[Fridrich Valach]], *Democratic actions for type II supergravities*, Phys.Rev.D **107** 6 (2023) 066027 [[arXiv:2207.00626](https://arxiv.org/abs/2207.00626), [doi:10.1103/PhysRevD.107.066027](https://doi.org/10.1103/PhysRevD.107.066027)]

### Irrational RR-charge
 {#IrrationalRRCharge}

An argument that RR-charge may occur in [[irrational number|irrational]] ratios is due to

* {#BDS00} [[Constantin Bachas]], [[Michael Douglas]], [[Christoph Schweigert]], around (2.8) of _Flux Stabilization of D-branes_, JHEP 0005:048, 2000 ([arXiv:hep-th/0003037](https://arxiv.org/abs/hep-th/0003037))

In a sequence of followup articles, authors found this problematic and tried to make sense of it:

* [[Washington Taylor]], _D2-branes in B fields_, JHEP 0007 (2000) 039 ([arXiv:hep-th/0004141](https://arxiv.org/abs/hep-th/0004141))

> In this article it was argued that the D0-brane charge arising from the integral over the D2-brane of the pullback of the B field is cancelled by the bulk contributions, but in this calculation it was implicitly assumed that the gauge field $C^{(1)}$ is constant. (from [Zhou 01](#Zhou01))

* Peter Rajan, _D2-brane RR-charge on $SU(2)$_, Phys.Lett. B533 (2002) 307-312 ([arXiv:hep-th/0111245](https://arxiv.org/abs/hep-th/0111245))

* {#Zhou01} Jian-Ge Zhou, _D-branes in B Fields_, Nucl.Phys. B607 (2001) 237-246 ([arXiv:hep-th/0102178](https://arxiv.org/abs/hep-th/0102178))

Observation that this paradox is resolved under [[Hypothesis H]], at least for [[fractional D-branes]]:

* [[Simon Burton]], [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Lift of fractional D-brane charge to equivariant Cohomotopy theory|Lift of fractional D-brane charge to equivariant Cohomotopy theory]]*, J Geom  Phys **161** (2021) 104034 ([doi:10.1016/j.geomphys.2020.104034](https://doi.org/10.1016/j.geomphys.2020.104034), [arXiv:1812.09679]([arXiv:1812.09679](https://arxiv.org/abs/1812.09679))

[[!redirects RR fields]]

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[[!redirects Ramond–Ramond field]]
[[!redirects Ramond--Ramond field]]

[[!redirects Ramond-Ramond fields]]
[[!redirects Ramond–Ramond fields]]
[[!redirects Ramond--Ramond fields]]

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