|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[category theory]], an abstract branch of [[mathematics]], [[distributive law]]s between monads are a way to express abstractly that two algebraic structures distribute one over the other one.
| | Hello and welcome. My title is Irwin and I completely dig that name. In her professional lifestyle she is a payroll clerk but she's always needed her personal business. To gather coins is what his family and him appreciate. South Dakota is where me and my husband live.<br><br>Feel free to surf to my site: [http://kitakeluarga.com/index.php?do=/profile-12707/info/ at home std testing] |
| | |
| Suppose that <math>(S,\mu^S,\eta^S)</math> and <math>(T,\mu^T,\eta^T)</math> are two [[monad (category theory)|monads]] on a [[category theory|category]] '''C'''. In general, there is no natural monad structure on the composite functor ''ST''. On the other hand, there is a natural monad structure on the functor ''ST'' if there is a distributive law of the monad ''S'' over the monad ''T''.
| |
| | |
| Formally, a '''distributive law''' of the monad ''S'' over the monad ''T'' is a [[natural transformation]]
| |
| :<math>l:TS\to ST</math>
| |
| such that the diagrams
| |
| :[[Image:Distributive law monads mult1.png]] [[Image:Distributive law monads mult2.png]]
| |
| :[[Image:Distributive law monads unit1.png]] and [[Image:Distributive law monads unit2.png]]
| |
| commute.
| |
| | |
| This law induces a composite monad ''ST'' with
| |
| * as multiplication: <math>S\mu^T\cdot\mu^STT\cdot SlT</math>,
| |
| * as unit: <math>\eta^ST\cdot\eta^T</math>.
| |
| | |
| == See also ==
| |
| * [[distributive law]]
| |
| | |
| == References ==
| |
| * {{cite journal
| |
| | author = Jon Beck
| |
| | authorlink = Jonathan Mock Beck
| |
| | year = 1969
| |
| | title = Distributive laws
| |
| | journal = Lecture Notes in Mathematics
| |
| | volume = 80
| |
| | pages = 119–140
| |
| | doi = 10.1007/BFb0083084
| |
| | series = Lecture Notes in Mathematics
| |
| | isbn = 978-3-540-04601-1
| |
| }}
| |
| | |
| * {{cite book
| |
| | author = [[Michael Barr (mathematician)|Michael Barr]] and Charles Wells
| |
| | title = Toposes, Triples and Theories
| |
| | url = http://www.case.edu/artsci/math/wells/pub/pdf/ttt.pdf
| |
| | publisher = [[Springer-Verlag]]
| |
| | year = 1985
| |
| | isbn = 0-387-96115-1
| |
| }}
| |
| * {{nlab|id=distributive+law|title=Distributive law}}
| |
| | |
| * G. Böhm, Internal bialgebroids, entwining structures and corings, AMS Contemp. Math. 376 (2005) 207–226; [http://arxiv.org/abs/math.QA/0311244 arXiv:math.QA/0311244]
| |
| | |
| * T. Brzeziński, S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), no. 2, 467–492 [http://arxiv.org/abs/q-alg/9602022 arXiv].
| |
| | |
| * T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.
| |
| | |
| * T. F. Fox, M. Markl, Distributive laws, bialgebras, and cohomology. Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), 167–205, Contemp. Math. 202, AMS 1997.
| |
| | |
| * S. Lack, Composing PROPS, [http://www.tac.mta.ca/tac/volumes/13/9/13-09abs.html Theory Appl. Categ.] 13 (2004), No. 9, 147–163.
| |
| | |
| * S. Lack, R. Street, The formal theory of monads II, Special volume celebrating the 70th birthday of Professor Max Kelly. J. Pure Appl. Algebra 175 (2002), no. 1-3, 243–265.
| |
| | |
| * M. Markl, Distributive laws and Koszulness. Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 307–323 ([http://www.numdam.org/numdam-bin/fitem?id=AIF_1996__46_2_307_0 numdam])
| |
| | |
| * R. Street, The formal theory of monads, J. Pure Appl. Alg. 2, 149–168 (1972)
| |
| | |
| * Z. Škoda, Distributive laws for monoidal categories ([http://front.math.ucdavis.edu/math.CT/0406310 arXiv:0406310]); Equivariant monads and equivariant lifts versus a 2-category of distributive laws ([http://front.math.ucdavis.edu/0707.1609 arXiv:0707.1609]); Bicategory of entwinings [http://arxiv.org/abs/0805.4611 arXiv:0805.4611]
| |
| | |
| * Z. Škoda, Some equivariant constructions in noncommutative geometry, Georgian Math. J. 16 (2009) 1; 183–202 [http://front.math.ucdavis.edu/0811.4770 arXiv:0811.4770]
| |
| | |
| * R. Wisbauer, Algebras versus coalgebras. Appl. Categ. Structures 16 (2008), no. 1-2, 255–295.
| |
| | |
| [[Category:Adjoint functors]]
| |
| | |
| | |
| {{categorytheory-stub}}
| |
Hello and welcome. My title is Irwin and I completely dig that name. In her professional lifestyle she is a payroll clerk but she's always needed her personal business. To gather coins is what his family and him appreciate. South Dakota is where me and my husband live.
Feel free to surf to my site: at home std testing