|
|
| Line 1: |
Line 1: |
| In [[category theory]], a '''traced monoidal category''' is a category with some extra structure which gives a reasonable notion of feedback.
| | Hi there. My title is Sophia Meagher even though it is not the title on my birth certification. For a whilst I've been in Alaska but I will have to transfer in a year or two. To play domino is some thing I really appreciate doing. Distributing manufacturing has been his profession for some time.<br><br>Here is my page - psychic readings ([http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review Our Web Page]) |
| | |
| A '''traced symmetric monoidal category''' is a [[symmetric monoidal category]] '''C''' together with a family of functions
| |
| :<math>\mathrm{Tr}^U_{X,Y}:\mathbf{C}(X\otimes U,Y\otimes U)\to\mathbf{C}(X,Y)</math>
| |
| called a ''trace'', satisfying the following conditions:
| |
| * naturality in ''X'': for every <math>f:X\otimes U\to Y\otimes U</math> and <math>g:X'\to X</math>,
| |
| ::<math>\mathrm{Tr}^U_{X,Y}(f)g=\mathrm{Tr}^U_{X',Y}(f(g\otimes U))</math>
| |
| | |
| [[Image:Trace diagram naturality 1.svg|thumb|center|400px|Naturality in X]]
| |
| | |
| * naturality in ''Y'': for every <math>f:X\otimes U\to Y\otimes U</math> and <math>g:Y\to Y'</math>,
| |
| ::<math>g\mathrm{Tr}^U_{X,Y}(f)=\mathrm{Tr}^U_{X,Y'}((g\otimes U)f)</math>
| |
| | |
| [[Image:Trace diagram naturality 2.svg|thumb|center|400px|Naturality in Y]]
| |
| | |
| * dinaturality in ''U'': for every <math>f:X\otimes U\to Y\otimes U'</math> and <math>g:U'\to U</math>
| |
| ::<math>\mathrm{Tr}^U_{X,Y}((Y\otimes g)f)=\mathrm{Tr}^{U'}_{X,Y}(f(X\otimes g))</math>
| |
| | |
| [[Image:Trace diagram dinaturality.svg|thumb|center|400px|Dinaturality in U]]
| |
| | |
| * vanishing I: for every <math>f:X\otimes I\to Y\otimes I</math>,
| |
| ::<math>\mathrm{Tr}^I_{X,Y}(f)=f</math>
| |
| | |
| [[Image:Trace diagram vanishing.svg|thumb|center|400px|Vanishing I]]
| |
| | |
| * vanishing II: for every <math>f:X\otimes U\otimes V\to Y\otimes U\otimes V</math>
| |
| ::<math>\mathrm{Tr}^{U\otimes V}_{X,Y}(f)=\mathrm{Tr}^U_{X,Y}(\mathrm{Tr}^V_{X\otimes U,Y\otimes U}(f))</math>
| |
| | |
| [[Image:Trace diagram associativity.svg|thumb|center|400px|Vanishing II]]
| |
| | |
| * superposing: for every <math>f:X\otimes U\to Y\otimes U</math> and <math>g:W\to Z</math>,
| |
| ::<math>g\otimes \mathrm{Tr}^U_{X,Y}(f)=\mathrm{Tr}^U_{W\otimes X,Z\otimes Y}(g\otimes f)</math>
| |
| | |
| [[Image:Trace diagram superposition.svg|thumb|center|400px|Superposing]]
| |
| | |
| * yanking:
| |
| ::<math>\mathrm{Tr}^U_{U,U}(\gamma_{U,U})=U</math>
| |
| (where <math>\gamma</math> is the symmetry of the monoidal category).
| |
| | |
| [[Image:Trace diagram yanking.svg|thumb|center|400px|Yanking]]
| |
| | |
| == Properties ==
| |
| * Every [[compact closed category]] admits a trace.
| |
| | |
| * Given a traced monoidal category '''C''', the ''Int construction'' generates the free (in some bicategorical sense) compact closure Int('''C''') of '''C'''.
| |
| | |
| == References ==
| |
| * {{cite journal
| |
| | author = [[André Joyal]], [[Ross Street]], [[Dominic Verity]]
| |
| | year = 1996
| |
| | title = Traced monoidal categories
| |
| | journal = Mathematical Proceedings of the Cambridge Philosophical Society
| |
| | volume = 3
| |
| | pages = 447–468
| |
| | doi = 10.1017/S0305004100074338
| |
| }}
| |
| | |
| [[Category:Monoidal categories]]
| |
| | |
| | |
| {{categorytheory-stub}}
| |
Hi there. My title is Sophia Meagher even though it is not the title on my birth certification. For a whilst I've been in Alaska but I will have to transfer in a year or two. To play domino is some thing I really appreciate doing. Distributing manufacturing has been his profession for some time.
Here is my page - psychic readings (Our Web Page)