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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346<?xml version="1.0" encoding="UTF-8"?>12<!-- This is an automatically generated file. -->3<Chapter Label="Chapter_Tensor_Product_and_Internal_Hom">4<Heading>Tensor Product and Internal Hom</Heading>56<P/>7<Section Label="Chapter_Tensor_Product_and_Internal_Hom_Section_Monoidal_Categories">8<Heading>Monoidal Categories</Heading>910A <Math>6</Math>-tuple <Math>( \mathbf{C}, \otimes, 1, \alpha, \lambda, \rho )</Math>11consisting of12<List>13<Item>14a category <Math>\mathbf{C}</Math>,15</Item>16<Item>17a functor <Math>\otimes: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C}</Math>,18</Item>19<Item>20an object <Math>1 \in \mathbf{C}</Math>,21</Item>22<Item>23a natural isomorphism <Math>\alpha_{a,b,c}: a \otimes (b \otimes c) \cong (a \otimes b) \otimes c</Math>,24</Item>25<Item>26a natural isomorphism <Math>\lambda_{a}: 1 \otimes a \cong a</Math>,27</Item>28<Item>29a natural isomorphism <Math>\rho_{a}: a \otimes 1 \cong a</Math>,30</Item>31</List>32is called a <Emph>monoidal category</Emph>, if33<List>34<Item>35for all objects <Math>a,b,c,d</Math>, the pentagon identity holds:36<Math>(\alpha_{a,b,c} \otimes \mathrm{id}_d) \circ \alpha_{a,b \otimes c, d} \circ ( \mathrm{id}_a \otimes \alpha_{b,c,d} ) = \alpha_{a \otimes b, c, d} \circ \alpha_{a,b,c \otimes d}</Math>,37</Item>38<Item>39for all objects <Math>a,c</Math>, the triangle identity holds:40<Math>( \rho_a \otimes \mathrm{id}_c ) \circ \alpha_{a,1,c} = \mathrm{id}_a \otimes \lambda_c</Math>.41</Item>42</List>43The corresponding GAP property is given by44<C>IsMonoidalCategory</C>.45<ManSection>46<Oper Arg="a,b" Name="TensorProductOnObjects" Label="for IsCapCategoryObject, IsCapCategoryObject"/>47<Returns>an object48</Returns>49<Description>50The arguments are two objects <Math>a, b</Math>.51The output is the tensor product <Math>a \otimes b</Math>.52</Description>53</ManSection>545556<ManSection>57<Oper Arg="C, F" Name="AddTensorProductOnObjects" Label="for IsCapCategory, IsFunction"/>58<Returns>nothing59</Returns>60<Description>61The arguments are a category <Math>C</Math> and a function <Math>F</Math>.62This operations adds the given function <Math>F</Math>63to the category for the basic operation <C>TensorProductOnObjects</C>.64<Math>F: (a,b) \mapsto a \otimes b</Math>.65</Description>66</ManSection>676869<ManSection>70<Oper Arg="alpha, beta" Name="TensorProductOnMorphisms" Label="for IsCapCategoryMorphism, IsCapCategoryMorphism"/>71<Returns>a morphism in <Math>\mathrm{Hom}(a \otimes b, a' \otimes b')</Math>72</Returns>73<Description>74The arguments are two morphisms <Math>\alpha: a \rightarrow a', \beta: b \rightarrow b'</Math>.75The output is the tensor product <Math>\alpha \otimes \beta</Math>.76</Description>77</ManSection>787980<ManSection>81<Oper Arg="s, alpha, beta, r" Name="TensorProductOnMorphismsWithGivenTensorProducts" Label="for IsCapCategoryObject, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryObject"/>82<Returns>a morphism in <Math>\mathrm{Hom}(a \otimes b, a' \otimes b')</Math>83</Returns>84<Description>85The arguments are an object <Math>s = a \otimes b</Math>,86two morphisms <Math>\alpha: a \rightarrow a', \beta: b \rightarrow b'</Math>,87and an object <Math>r = a' \otimes b'</Math>.88The output is the tensor product <Math>\alpha \otimes \beta</Math>.89</Description>90</ManSection>919293<ManSection>94<Oper Arg="C, F" Name="AddTensorProductOnMorphismsWithGivenTensorProducts" Label="for IsCapCategory, IsFunction"/>95<Returns>nothing96</Returns>97<Description>98The arguments are a category <Math>C</Math> and a function <Math>F</Math>.99This operations adds the given function <Math>F</Math>100to the category for the basic operation <C>TensorProductOnMorphismsWithGivenTensorProducts</C>.101<Math>F: ( a \otimes b, \alpha: a \rightarrow a', \beta: b \rightarrow b', a' \otimes b' ) \mapsto \alpha \otimes \beta</Math>.102</Description>103</ManSection>104105106<ManSection>107<Oper Arg="a, b, c" Name="AssociatorRightToLeft" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>108<Returns>a morphism in <Math>\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c )</Math>.109</Returns>110<Description>111The arguments are three objects <Math>a,b,c</Math>.112The output is the associator <Math>\alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c</Math>.113</Description>114</ManSection>115116117<ManSection>118<Oper Arg="s, a, b, c, r" Name="AssociatorRightToLeftWithGivenTensorProducts" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>119<Returns>a morphism in <Math>\mathrm{Hom}( a \otimes (b \otimes c), (a \otimes b) \otimes c )</Math>.120</Returns>121<Description>122The arguments are an object <Math>s = a \otimes (b \otimes c)</Math>,123three objects <Math>a,b,c</Math>,124and an object <Math>r = (a \otimes b) \otimes c</Math>.125The output is the associator <Math>\alpha_{a,(b,c)}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c</Math>.126</Description>127</ManSection>128129130<ManSection>131<Oper Arg="C, F" Name="AddAssociatorRightToLeftWithGivenTensorProducts" Label="for IsCapCategory, IsFunction"/>132<Returns>nothing133</Returns>134<Description>135The arguments are a category <Math>C</Math> and a function <Math>F</Math>.136This operations adds the given function <Math>F</Math>137to the category for the basic operation <C>AssociatorRightToLeftWithGivenTensorProducts</C>.138<Math>F: ( a \otimes (b \otimes c), a, b, c, (a \otimes b) \otimes c ) \mapsto \alpha_{a,(b,c)}</Math>.139</Description>140</ManSection>141142143<ManSection>144<Oper Arg="a, b, c" Name="AssociatorLeftToRight" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>145<Returns>a morphism in <Math>\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) )</Math>.146</Returns>147<Description>148The arguments are three objects <Math>a,b,c</Math>.149The output is the associator <Math>\alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c)</Math>.150</Description>151</ManSection>152153154<ManSection>155<Oper Arg="s, a, b, c, r" Name="AssociatorLeftToRightWithGivenTensorProducts" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>156<Returns>a morphism in <Math>\mathrm{Hom}( (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c) )</Math>.157</Returns>158<Description>159The arguments are an object <Math>s = (a \otimes b) \otimes c</Math>,160three objects <Math>a,b,c</Math>,161and an object <Math>r = a \otimes (b \otimes c)</Math>.162The output is the associator <Math>\alpha_{(a,b),c}: (a \otimes b) \otimes c \rightarrow a \otimes (b \otimes c)</Math>.163</Description>164</ManSection>165166167<ManSection>168<Oper Arg="C, F" Name="AddAssociatorLeftToRightWithGivenTensorProducts" Label="for IsCapCategory, IsFunction"/>169<Returns>nothing170</Returns>171<Description>172The arguments are a category <Math>C</Math> and a function <Math>F</Math>.173This operations adds the given function <Math>F</Math>174to the category for the basic operation <C>AssociatorLeftToRightWithGivenTensorProducts</C>.175<Math>F: (( a \otimes b ) \otimes c, a, b, c, a \otimes (b \otimes c )) \mapsto \alpha_{(a,b),c}</Math>.176</Description>177</ManSection>178179180<ManSection>181<Attr Arg="C" Name="TensorUnit" Label="for IsCapCategory"/>182<Returns>an object183</Returns>184<Description>185The argument is a category <Math>\mathbf{C}</Math>.186The output is the tensor unit <Math>1</Math> of <Math>\mathbf{C}</Math>.187</Description>188</ManSection>189190191<ManSection>192<Oper Arg="C, F" Name="AddTensorUnit" Label="for IsCapCategory, IsFunction"/>193<Returns>nothing194</Returns>195<Description>196The arguments are a category <Math>C</Math> and a function <Math>F</Math>.197This operations adds the given function <Math>F</Math>198to the category for the basic operation <C>TensorUnit</C>.199<Math>F: ( ) \mapsto 1</Math>.200</Description>201</ManSection>202203204<ManSection>205<Attr Arg="a" Name="LeftUnitor" Label="for IsCapCategoryObject"/>206<Returns>a morphism in <Math>\mathrm{Hom}(1 \otimes a, a )</Math>207</Returns>208<Description>209The argument is an object <Math>a</Math>.210The output is the left unitor <Math>\lambda_a: 1 \otimes a \rightarrow a</Math>.211</Description>212</ManSection>213214215<ManSection>216<Oper Arg="a, s" Name="LeftUnitorWithGivenTensorProduct" Label="for IsCapCategoryObject, IsCapCategoryObject"/>217<Returns>a morphism in <Math>\mathrm{Hom}(1 \otimes a, a )</Math>218</Returns>219<Description>220The arguments are an object <Math>a</Math> and an object <Math>s = 1 \otimes a</Math>.221The output is the left unitor <Math>\lambda_a: 1 \otimes a \rightarrow a</Math>.222</Description>223</ManSection>224225226<ManSection>227<Oper Arg="C, F" Name="AddLeftUnitorWithGivenTensorProduct" Label="for IsCapCategory, IsFunction"/>228<Returns>nothing229</Returns>230<Description>231The arguments are a category <Math>C</Math> and a function <Math>F</Math>.232This operations adds the given function <Math>F</Math>233to the category for the basic operation <C>LeftUnitorWithGivenTensorProduct</C>.234<Math>F: (a, 1 \otimes a) \mapsto \lambda_a</Math>.235</Description>236</ManSection>237238239<ManSection>240<Attr Arg="a" Name="LeftUnitorInverse" Label="for IsCapCategoryObject"/>241<Returns>a morphism in <Math>\mathrm{Hom}(a, 1 \otimes a)</Math>242</Returns>243<Description>244The argument is an object <Math>a</Math>.245The output is the inverse of the left unitor <Math>\lambda_a^{-1}: a \rightarrow 1 \otimes a</Math>.246</Description>247</ManSection>248249250<ManSection>251<Oper Arg="a, r" Name="LeftUnitorInverseWithGivenTensorProduct" Label="for IsCapCategoryObject, IsCapCategoryObject"/>252<Returns>a morphism in <Math>\mathrm{Hom}(a, 1 \otimes a)</Math>253</Returns>254<Description>255The argument is an object <Math>a</Math> and an object <Math>r = 1 \otimes a</Math>.256The output is the inverse of the left unitor <Math>\lambda_a^{-1}: a \rightarrow 1 \otimes a</Math>.257</Description>258</ManSection>259260261<ManSection>262<Oper Arg="C, F" Name="AddLeftUnitorInverseWithGivenTensorProduct" Label="for IsCapCategory, IsFunction"/>263<Returns>nothing264</Returns>265<Description>266The arguments are a category <Math>C</Math> and a function <Math>F</Math>.267This operations adds the given function <Math>F</Math>268to the category for the basic operation <C>LeftUnitorInverseWithGivenTensorProduct</C>.269<Math>F: (a, 1 \otimes a) \mapsto \lambda_a^{-1}</Math>.270</Description>271</ManSection>272273274<ManSection>275<Attr Arg="a" Name="RightUnitor" Label="for IsCapCategoryObject"/>276<Returns>a morphism in <Math>\mathrm{Hom}(a \otimes 1, a )</Math>277</Returns>278<Description>279The argument is an object <Math>a</Math>.280The output is the right unitor <Math>\rho_a: a \otimes 1 \rightarrow a</Math>.281</Description>282</ManSection>283284285<ManSection>286<Oper Arg="a, s" Name="RightUnitorWithGivenTensorProduct" Label="for IsCapCategoryObject, IsCapCategoryObject"/>287<Returns>a morphism in <Math>\mathrm{Hom}(a \otimes 1, a )</Math>288</Returns>289<Description>290The arguments are an object <Math>a</Math> and an object <Math>s = a \otimes 1</Math>.291The output is the right unitor <Math>\rho_a: a \otimes 1 \rightarrow a</Math>.292</Description>293</ManSection>294295296<ManSection>297<Oper Arg="C, F" Name="AddRightUnitorWithGivenTensorProduct" Label="for IsCapCategory, IsFunction"/>298<Returns>nothing299</Returns>300<Description>301The arguments are a category <Math>C</Math> and a function <Math>F</Math>.302This operations adds the given function <Math>F</Math>303to the category for the basic operation <C>RightUnitorWithGivenTensorProduct</C>.304<Math>F: (a, a \otimes 1) \mapsto \rho_a</Math>.305</Description>306</ManSection>307308309<ManSection>310<Attr Arg="a" Name="RightUnitorInverse" Label="for IsCapCategoryObject"/>311<Returns>a morphism in <Math>\mathrm{Hom}( a, a \otimes 1 )</Math>312</Returns>313<Description>314The argument is an object <Math>a</Math>.315The output is the inverse of the right unitor <Math>\rho_a^{-1}: a \rightarrow a \otimes 1</Math>.316</Description>317</ManSection>318319320<ManSection>321<Oper Arg="a, r" Name="RightUnitorInverseWithGivenTensorProduct" Label="for IsCapCategoryObject, IsCapCategoryObject"/>322<Returns>a morphism in <Math>\mathrm{Hom}( a, a \otimes 1 )</Math>323</Returns>324<Description>325The arguments are an object <Math>a</Math> and an object <Math>r = a \otimes 1</Math>.326The output is the inverse of the right unitor <Math>\rho_a^{-1}: a \rightarrow a \otimes 1</Math>.327</Description>328</ManSection>329330331<ManSection>332<Oper Arg="C, F" Name="AddRightUnitorInverseWithGivenTensorProduct" Label="for IsCapCategory, IsFunction"/>333<Returns>nothing334</Returns>335<Description>336The arguments are a category <Math>C</Math> and a function <Math>F</Math>.337This operations adds the given function <Math>F</Math>338to the category for the basic operation <C>RightUnitorInverseWithGivenTensorProduct</C>.339<Math>F: (a, a \otimes 1) \mapsto \rho_a^{-1}</Math>.340</Description>341</ManSection>342343344<ManSection>345<Oper Arg="a, L" Name="LeftDistributivityExpanding" Label="for IsCapCategoryObject, IsList"/>346<Returns>a morphism in <Math>\mathrm{Hom}( a \otimes (b_1 \oplus \dots \oplus b_n), (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) )</Math>347</Returns>348<Description>349The arguments are an object <Math>a</Math>350and a list of objects <Math>L = (b_1, \dots, b_n)</Math>.351The output is the left distributivity morphism352<Math>a \otimes (b_1 \oplus \dots \oplus b_n) \rightarrow (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)</Math>.353</Description>354</ManSection>355356357<ManSection>358<Oper Arg="s, a, L, r" Name="LeftDistributivityExpandingWithGivenObjects" Label="for IsCapCategoryObject, IsCapCategoryObject, IsList, IsCapCategoryObject"/>359<Returns>a morphism in <Math>\mathrm{Hom}( s, r )</Math>360</Returns>361<Description>362The arguments are an object <Math>s = a \otimes (b_1 \oplus \dots \oplus b_n)</Math>,363an object <Math>a</Math>,364a list of objects <Math>L = (b_1, \dots, b_n)</Math>,365and an object <Math>r = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)</Math>.366The output is the left distributivity morphism367<Math>s \rightarrow r</Math>.368</Description>369</ManSection>370371372<ManSection>373<Oper Arg="C, F" Name="AddLeftDistributivityExpandingWithGivenObjects" Label="for IsCapCategory, IsFunction"/>374<Returns>nothing375</Returns>376<Description>377The arguments are a category <Math>C</Math> and a function <Math>F</Math>.378This operations adds the given function <Math>F</Math>379to the category for the basic operation <C>LeftDistributivityExpandingWithGivenObjects</C>.380<Math>F: (a \otimes (b_1 \oplus \dots \oplus b_n), a, L, (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)) \mapsto \mathrm{LeftDistributivityExpandingWithGivenObjects}(a,L)</Math>.381</Description>382</ManSection>383384385<ManSection>386<Oper Arg="a, L" Name="LeftDistributivityFactoring" Label="for IsCapCategoryObject, IsList"/>387<Returns>a morphism in <Math>\mathrm{Hom}( (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), a \otimes (b_1 \oplus \dots \oplus b_n) )</Math>388</Returns>389<Description>390The arguments are an object <Math>a</Math>391and a list of objects <Math>L = (b_1, \dots, b_n)</Math>.392The output is the left distributivity morphism393<Math>(a \otimes b_1) \oplus \dots \oplus (a \otimes b_n) \rightarrow a \otimes (b_1 \oplus \dots \oplus b_n)</Math>.394</Description>395</ManSection>396397398<ManSection>399<Oper Arg="s, a, L, r" Name="LeftDistributivityFactoringWithGivenObjects" Label="for IsCapCategoryObject, IsCapCategoryObject, IsList, IsCapCategoryObject"/>400<Returns>a morphism in <Math>\mathrm{Hom}( s, r )</Math>401</Returns>402<Description>403The arguments are an object <Math>s = (a \otimes b_1) \oplus \dots \oplus (a \otimes b_n)</Math>,404an object <Math>a</Math>,405a list of objects <Math>L = (b_1, \dots, b_n)</Math>,406and an object <Math>r = a \otimes (b_1 \oplus \dots \oplus b_n)</Math>.407The output is the left distributivity morphism408<Math>s \rightarrow r</Math>.409</Description>410</ManSection>411412413<ManSection>414<Oper Arg="C, F" Name="AddLeftDistributivityFactoringWithGivenObjects" Label="for IsCapCategory, IsFunction"/>415<Returns>nothing416</Returns>417<Description>418The arguments are a category <Math>C</Math> and a function <Math>F</Math>.419This operations adds the given function <Math>F</Math>420to the category for the basic operation <C>LeftDistributivityFactoringWithGivenObjects</C>.421<Math>F: ((a \otimes b_1) \oplus \dots \oplus (a \otimes b_n), a, L, a \otimes (b_1 \oplus \dots \oplus b_n)) \mapsto \mathrm{LeftDistributivityFactoringWithGivenObjects}(a,L)</Math>.422</Description>423</ManSection>424425426<ManSection>427<Oper Arg="L, a" Name="RightDistributivityExpanding" Label="for IsList, IsCapCategoryObject"/>428<Returns>a morphism in <Math>\mathrm{Hom}( (b_1 \oplus \dots \oplus b_n) \otimes a, (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) )</Math>429</Returns>430<Description>431The arguments are a list of objects <Math>L = (b_1, \dots, b_n)</Math>432and an object <Math>a</Math>.433The output is the right distributivity morphism434<Math>(b_1 \oplus \dots \oplus b_n) \otimes a \rightarrow (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)</Math>.435</Description>436</ManSection>437438439<ManSection>440<Oper Arg="s, L, a, r" Name="RightDistributivityExpandingWithGivenObjects" Label="for IsCapCategoryObject, IsList, IsCapCategoryObject, IsCapCategoryObject"/>441<Returns>a morphism in <Math>\mathrm{Hom}( s, r )</Math>442</Returns>443<Description>444The arguments are an object <Math>s = (b_1 \oplus \dots \oplus b_n) \otimes a</Math>,445a list of objects <Math>L = (b_1, \dots, b_n)</Math>,446an object <Math>a</Math>,447and an object <Math>r = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)</Math>.448The output is the right distributivity morphism449<Math>s \rightarrow r</Math>.450</Description>451</ManSection>452453454<ManSection>455<Oper Arg="C, F" Name="AddRightDistributivityExpandingWithGivenObjects" Label="for IsCapCategory, IsFunction"/>456<Returns>nothing457</Returns>458<Description>459The arguments are a category <Math>C</Math> and a function <Math>F</Math>.460This operations adds the given function <Math>F</Math>461to the category for the basic operation <C>RightDistributivityExpandingWithGivenObjects</C>.462<Math>F: ((b_1 \oplus \dots \oplus b_n) \otimes a, L, a, (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)) \mapsto \mathrm{RightDistributivityExpandingWithGivenObjects}(L,a)</Math>.463</Description>464</ManSection>465466467<ManSection>468<Oper Arg="L, a" Name="RightDistributivityFactoring" Label="for IsList, IsCapCategoryObject"/>469<Returns>a morphism in <Math>\mathrm{Hom}( (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), (b_1 \oplus \dots \oplus b_n) \otimes a)</Math>470</Returns>471<Description>472The arguments are a list of objects <Math>L = (b_1, \dots, b_n)</Math>473and an object <Math>a</Math>.474The output is the right distributivity morphism475<Math>(b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a) \rightarrow (b_1 \oplus \dots \oplus b_n) \otimes a </Math>.476</Description>477</ManSection>478479480<ManSection>481<Oper Arg="s, L, a, r" Name="RightDistributivityFactoringWithGivenObjects" Label="for IsCapCategoryObject, IsList, IsCapCategoryObject, IsCapCategoryObject"/>482<Returns>a morphism in <Math>\mathrm{Hom}( s, r )</Math>483</Returns>484<Description>485The arguments are an object <Math>s = (b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a)</Math>,486a list of objects <Math>L = (b_1, \dots, b_n)</Math>,487an object <Math>a</Math>,488and an object <Math>r = (b_1 \oplus \dots \oplus b_n) \otimes a</Math>.489The output is the right distributivity morphism490<Math>s \rightarrow r</Math>.491</Description>492</ManSection>493494495<ManSection>496<Oper Arg="C, F" Name="AddRightDistributivityFactoringWithGivenObjects" Label="for IsCapCategory, IsFunction"/>497<Returns>nothing498</Returns>499<Description>500The arguments are a category <Math>C</Math> and a function <Math>F</Math>.501This operations adds the given function <Math>F</Math>502to the category for the basic operation <C>RightDistributivityFactoringWithGivenObjects</C>.503<Math>F: ((b_1 \otimes a) \oplus \dots \oplus (b_n \otimes a), L, a, (b_1 \oplus \dots \oplus b_n) \otimes a) \mapsto \mathrm{RightDistributivityFactoringWithGivenObjects}(L,a)</Math>.504</Description>505</ManSection>506507508</Section>509510511<Section Label="Chapter_Tensor_Product_and_Internal_Hom_Section_Braided_Monoidal_Categories">512<Heading>Braided Monoidal Categories</Heading>513514A monoidal category <Math>\mathbf{C}</Math> equipped with a natural isomorphism515<Math>B_{a,b}: a \otimes b \cong b \otimes a</Math>516is called a <Emph>braided monoidal category</Emph>517if518<List>519<Item>520<Math>\lambda_a \circ B_{a,1} = \rho_a</Math>,521</Item>522<Item>523<Math>(B_{c,a} \otimes \mathrm{id}_b) \circ \alpha_{c,a,b} \circ B_{a \otimes b,c} = \alpha_{a,c,b} \circ ( \mathrm{id}_a \otimes B_{b,c}) \circ \alpha^{-1}_{a,b,c}</Math>,524</Item>525<Item>526<Math>( \mathrm{id}_b \otimes B_{c,a} ) \circ \alpha^{-1}_{b,c,a} \circ B_{a,b \otimes c} = \alpha^{-1}_{b,a,c} \circ (B_{a,b} \otimes \mathrm{id}_c) \circ \alpha_{a,b,c}</Math>.527</Item>528</List>529The corresponding GAP property is given by530<C>IsBraidedMonoidalCategory</C>.531<ManSection>532<Oper Arg="a,b" Name="Braiding" Label="for IsCapCategoryObject, IsCapCategoryObject"/>533<Returns>a morphism in <Math>\mathrm{Hom}( a \otimes b, b \otimes a )</Math>.534</Returns>535<Description>536The arguments are two objects <Math>a,b</Math>.537The output is the braiding <Math> B_{a,b}: a \otimes b \rightarrow b \otimes a</Math>.538</Description>539</ManSection>540541542<ManSection>543<Oper Arg="s,a,b,r" Name="BraidingWithGivenTensorProducts" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>544<Returns>a morphism in <Math>\mathrm{Hom}( a \otimes b, b \otimes a )</Math>.545</Returns>546<Description>547The arguments are an object <Math>s = a \otimes b</Math>,548two objects <Math>a,b</Math>,549and an object <Math>r = b \otimes a</Math>.550The output is the braiding <Math> B_{a,b}: a \otimes b \rightarrow b \otimes a</Math>.551</Description>552</ManSection>553554555<ManSection>556<Oper Arg="C, F" Name="AddBraidingWithGivenTensorProducts" Label="for IsCapCategory, IsFunction"/>557<Returns>nothing558</Returns>559<Description>560The arguments are a category <Math>C</Math> and a function <Math>F</Math>.561This operations adds the given function <Math>F</Math>562to the category for the basic operation <C>BraidingWithGivenTensorProducts</C>.563<Math>F: (a \otimes b, a, b, b \otimes a) \rightarrow B_{a,b}</Math>.564</Description>565</ManSection>566567568<ManSection>569<Oper Arg="a,b" Name="BraidingInverse" Label="for IsCapCategoryObject, IsCapCategoryObject"/>570<Returns>a morphism in <Math>\mathrm{Hom}( b \otimes a, a \otimes b )</Math>.571</Returns>572<Description>573The arguments are two objects <Math>a,b</Math>.574The output is the inverse of the braiding <Math> B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b</Math>.575</Description>576</ManSection>577578579<ManSection>580<Oper Arg="s,a,b,r" Name="BraidingInverseWithGivenTensorProducts" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>581<Returns>a morphism in <Math>\mathrm{Hom}( b \otimes a, a \otimes b )</Math>.582</Returns>583<Description>584The arguments are an object <Math>s = b \otimes a</Math>,585two objects <Math>a,b</Math>,586and an object <Math>r = a \otimes b</Math>.587The output is the braiding <Math> B_{a,b}^{-1}: b \otimes a \rightarrow a \otimes b</Math>.588</Description>589</ManSection>590591592<ManSection>593<Oper Arg="C, F" Name="AddBraidingInverseWithGivenTensorProducts" Label="for IsCapCategory, IsFunction"/>594<Returns>nothing595</Returns>596<Description>597The arguments are a category <Math>C</Math> and a function <Math>F</Math>.598This operations adds the given function <Math>F</Math>599to the category for the basic operation <C>BraidingInverseWithGivenTensorProducts</C>.600<Math>F: (b \otimes a, a, b, a \otimes b) \rightarrow B_{a,b}^{-1}</Math>.601</Description>602</ManSection>603604605</Section>606607608<Section Label="Chapter_Tensor_Product_and_Internal_Hom_Section_Symmetric_Monoidal_Categories">609<Heading>Symmetric Monoidal Categories</Heading>610611A braided monoidal category <Math>\mathbf{C}</Math> is called <Emph>symmetric monoidal category</Emph>612if <Math>B_{a,b}^{-1} = B_{b,a}</Math>.613The corresponding GAP property is given by614<C>IsSymmetricMonoidalCategory</C>.615</Section>616617618<Section Label="Chapter_Tensor_Product_and_Internal_Hom_Section_Symmetric_Closed_Monoidal_Categories">619<Heading>Symmetric Closed Monoidal Categories</Heading>620621A symmetric monoidal category <Math>\mathbf{C}</Math>622which has for each functor <Math>- \otimes b: \mathbf{C} \rightarrow \mathbf{C}</Math>623a right adjoint (denoted by <Math>\underline{\mathrm{Hom}}(b,-)</Math>)624is called a <Emph>symmetric closed monoidal category</Emph>.625The corresponding GAP property is given by626<C>IsSymmetricClosedMonoidalCategory</C>.627<ManSection>628<Oper Arg="a, b" Name="InternalHomOnObjects" Label="for IsCapCategoryObject, IsCapCategoryObject"/>629<Returns>an object630</Returns>631<Description>632The arguments are two objects <Math>a,b</Math>.633The output is the internal hom object <Math>\underline{\mathrm{Hom}}(a,b)</Math>.634</Description>635</ManSection>636637638<ManSection>639<Oper Arg="C, F" Name="AddInternalHomOnObjects" Label="for IsCapCategory, IsFunction"/>640<Returns>nothing641</Returns>642<Description>643The arguments are a category <Math>C</Math> and a function <Math>F</Math>.644This operations adds the given function <Math>F</Math>645to the category for the basic operation <C>InternalHomOnObjects</C>.646<Math>F: (a,b) \mapsto \underline{\mathrm{Hom}}(a,b)</Math>.647</Description>648</ManSection>649650651<ManSection>652<Oper Arg="alpha, beta" Name="InternalHomOnMorphisms" Label="for IsCapCategoryMorphism, IsCapCategoryMorphism"/>653<Returns>a morphism in <Math>\mathrm{Hom}( \underline{\mathrm{Hom}}(a',b), \underline{\mathrm{Hom}}(a,b') )</Math>654</Returns>655<Description>656The arguments are two morphisms <Math>\alpha: a \rightarrow a', \beta: b \rightarrow b'</Math>.657The output is the internal hom morphism658<Math>\underline{\mathrm{Hom}}(\alpha,\beta): \underline{\mathrm{Hom}}(a',b) \rightarrow \underline{\mathrm{Hom}}(a,b')</Math>.659</Description>660</ManSection>661662663<ManSection>664<Oper Arg="s, alpha, beta, r" Name="InternalHomOnMorphismsWithGivenInternalHoms" Label="for IsCapCategoryObject, IsCapCategoryMorphism, IsCapCategoryMorphism, IsCapCategoryObject"/>665<Returns>a morphism in <Math>\mathrm{Hom}( \underline{\mathrm{Hom}}(a',b), \underline{\mathrm{Hom}}(a,b') )</Math>666</Returns>667<Description>668The arguments are an object <Math>s = \underline{\mathrm{Hom}}(a',b)</Math>,669two morphisms <Math>\alpha: a \rightarrow a', \beta: b \rightarrow b'</Math>,670and an object <Math>r = \underline{\mathrm{Hom}}(a,b')</Math>.671The output is the internal hom morphism672<Math>\underline{\mathrm{Hom}}(\alpha,\beta): \underline{\mathrm{Hom}}(a',b) \rightarrow \underline{\mathrm{Hom}}(a,b')</Math>.673</Description>674</ManSection>675676677<ManSection>678<Oper Arg="C, F" Name="AddInternalHomOnMorphismsWithGivenInternalHoms" Label="for IsCapCategory, IsFunction"/>679<Returns>nothing680</Returns>681<Description>682The arguments are a category <Math>C</Math> and a function <Math>F</Math>.683This operations adds the given function <Math>F</Math>684to the category for the basic operation <C>InternalHomOnMorphismsWithGivenInternalHoms</C>.685<Math>F: (\underline{\mathrm{Hom}}(a',b), \alpha: a \rightarrow a', \beta: b \rightarrow b', \underline{\mathrm{Hom}}(a,b') ) \mapsto \underline{\mathrm{Hom}}(\alpha,\beta)</Math>.686</Description>687</ManSection>688689690<ManSection>691<Oper Arg="a,b" Name="EvaluationMorphism" Label="for IsCapCategoryObject, IsCapCategoryObject"/>692<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes a, b )</Math>.693</Returns>694<Description>695The arguments are two objects <Math>a, b</Math>.696The output is the evaluation morphism <Math>\mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b</Math>, i.e.,697the counit of the tensor hom adjunction.698</Description>699</ManSection>700701702<ManSection>703<Oper Arg="a,b, s" Name="EvaluationMorphismWithGivenSource" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>704<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes a, b )</Math>.705</Returns>706<Description>707The arguments are two objects <Math>a,b</Math> and an object <Math>s = \mathrm{\underline{Hom}}(a,b) \otimes a</Math>.708The output is the evaluation morphism <Math>\mathrm{ev}_{a,b}: \mathrm{\underline{Hom}}(a,b) \otimes a \rightarrow b</Math>, i.e.,709the counit of the tensor hom adjunction.710</Description>711</ManSection>712713714<ManSection>715<Oper Arg="C, F" Name="AddEvaluationMorphismWithGivenSource" Label="for IsCapCategory, IsFunction"/>716<Returns>nothing717</Returns>718<Description>719The arguments are a category <Math>C</Math> and a function <Math>F</Math>.720This operations adds the given function <Math>F</Math>721to the category for the basic operation <C>EvaluationMorphismWithGivenSource</C>.722<Math>F: (a, b, \mathrm{\underline{Hom}}(a,b) \otimes a) \mapsto \mathrm{ev}_{a,b}</Math>.723</Description>724</ManSection>725726727<ManSection>728<Oper Arg="a,b" Name="CoevaluationMorphism" Label="for IsCapCategoryObject, IsCapCategoryObject"/>729<Returns>a morphism in <Math>\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b, a \otimes b) )</Math>.730</Returns>731<Description>732The arguments are two objects <Math>a,b</Math>.733The output is the coevaluation morphism <Math>\mathrm{coev}_{a,b}: a \rightarrow \mathrm{\underline{Hom}(b, a \otimes b)}</Math>, i.e.,734the unit of the tensor hom adjunction.735</Description>736</ManSection>737738739<ManSection>740<Oper Arg="a,b,r" Name="CoevaluationMorphismWithGivenRange" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>741<Returns>a morphism in <Math>\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b, a \otimes b) )</Math>.742</Returns>743<Description>744The arguments are two objects <Math>a,b</Math> and an object <Math>r = \mathrm{\underline{Hom}(b, a \otimes b)}</Math>.745The output is the coevaluation morphism <Math>\mathrm{coev}_{a,b}: a \rightarrow \mathrm{\underline{Hom}(b, a \otimes b)}</Math>, i.e.,746the unit of the tensor hom adjunction.747</Description>748</ManSection>749750751<ManSection>752<Oper Arg="C, F" Name="AddCoevaluationMorphismWithGivenRange" Label="for IsCapCategory, IsFunction"/>753<Returns>nothing754</Returns>755<Description>756The arguments are a category <Math>C</Math> and a function <Math>F</Math>.757This operations adds the given function <Math>F</Math>758to the category for the basic operation <C>CoevaluationMorphismWithGivenRange</C>.759<Math>F: (a, b, \mathrm{\underline{Hom}}(b, a \otimes b)) \mapsto \mathrm{coev}_{a,b}</Math>.760</Description>761</ManSection>762763764<ManSection>765<Oper Arg="a, b, f" Name="TensorProductToInternalHomAdjunctionMap" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryMorphism"/>766<Returns>a morphism in <Math>\mathrm{Hom}( a, \mathrm{\underline{Hom}}(b,c) )</Math>.767</Returns>768<Description>769The arguments are objects <Math>a,b</Math> and a morphism <Math>f: a \otimes b \rightarrow c</Math>.770The output is a morphism <Math>g: a \rightarrow \mathrm{\underline{Hom}}(b,c)</Math>771corresponding to <Math>f</Math> under the tensor hom adjunction.772</Description>773</ManSection>774775776<ManSection>777<Oper Arg="C, F" Name="AddTensorProductToInternalHomAdjunctionMap" Label="for IsCapCategory, IsFunction"/>778<Returns>nothing779</Returns>780<Description>781The arguments are a category <Math>C</Math> and a function <Math>F</Math>.782This operations adds the given function <Math>F</Math>783to the category for the basic operation <C>TensorProductToInternalHomAdjunctionMap</C>.784<Math>F: (a, b, f: a \otimes b \rightarrow c) \mapsto ( g: a \rightarrow \mathrm{\underline{Hom}}(b,c) )</Math>.785</Description>786</ManSection>787788789<ManSection>790<Oper Arg="b, c, g" Name="InternalHomToTensorProductAdjunctionMap" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryMorphism"/>791<Returns>a morphism in <Math>\mathrm{Hom}(a \otimes b, c)</Math>.792</Returns>793<Description>794The arguments are objects <Math>b,c</Math> and a morphism <Math>g: a \rightarrow \mathrm{\underline{Hom}}(b,c)</Math>.795The output is a morphism <Math>f: a \otimes b \rightarrow c</Math> corresponding to <Math>g</Math> under the796tensor hom adjunction.797</Description>798</ManSection>799800801<ManSection>802<Oper Arg="C, F" Name="AddInternalHomToTensorProductAdjunctionMap" Label="for IsCapCategory, IsFunction"/>803<Returns>nothing804</Returns>805<Description>806The arguments are a category <Math>C</Math> and a function <Math>F</Math>.807This operations adds the given function <Math>F</Math>808to the category for the basic operation <C>InternalHomToTensorProductAdjunctionMap</C>.809<Math>F: (b, c, g: a \rightarrow \mathrm{\underline{Hom}}(b,c)) \mapsto ( g: a \otimes b \rightarrow c )</Math>.810</Description>811</ManSection>812813814<ManSection>815<Oper Arg="a,b,c" Name="MonoidalPreComposeMorphism" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>816<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) )</Math>.817</Returns>818<Description>819The arguments are three objects <Math>a,b,c</Math>.820The output is the precomposition morphism821<Math>\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c)</Math>.822</Description>823</ManSection>824825826<ManSection>827<Oper Arg="s,a,b,c,r" Name="MonoidalPreComposeMorphismWithGivenObjects" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>828<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c), \mathrm{\underline{Hom}}(a,c) )</Math>.829</Returns>830<Description>831The arguments are832an object <Math>s = \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c)</Math>,833three objects <Math>a,b,c</Math>,834and an object <Math>r = \mathrm{\underline{Hom}}(a,c)</Math>.835The output is the precomposition morphism836<Math>\mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c) \rightarrow \mathrm{\underline{Hom}}(a,c)</Math>.837</Description>838</ManSection>839840841<ManSection>842<Oper Arg="C, F" Name="AddMonoidalPreComposeMorphismWithGivenObjects" Label="for IsCapCategory, IsFunction"/>843<Returns>nothing844</Returns>845<Description>846The arguments are a category <Math>C</Math> and a function <Math>F</Math>.847This operations adds the given function <Math>F</Math>848to the category for the basic operation <C>MonoidalPreComposeMorphismWithGivenObjects</C>.849<Math>F: (\mathrm{\underline{Hom}}(a,b) \otimes \mathrm{\underline{Hom}}(b,c),a,b,c,\mathrm{\underline{Hom}}(a,c)) \mapsto \mathrm{MonoidalPreComposeMorphismWithGivenObjects}_{a,b,c}</Math>.850</Description>851</ManSection>852853854<ManSection>855<Oper Arg="a,b,c" Name="MonoidalPostComposeMorphism" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>856<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) )</Math>.857</Returns>858<Description>859The arguments are three objects <Math>a,b,c</Math>.860The output is the postcomposition morphism861<Math>\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c)</Math>.862</Description>863</ManSection>864865866<ManSection>867<Oper Arg="s,a,b,c,r" Name="MonoidalPostComposeMorphismWithGivenObjects" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>868<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b), \mathrm{\underline{Hom}}(a,c) )</Math>.869</Returns>870<Description>871The arguments are872an object <Math>s = \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b)</Math>,873three objects <Math>a,b,c</Math>,874and an object <Math>r = \mathrm{\underline{Hom}}(a,c)</Math>.875The output is the postcomposition morphism876<Math>\mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}: \mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b) \rightarrow \mathrm{\underline{Hom}}(a,c)</Math>.877</Description>878</ManSection>879880881<ManSection>882<Oper Arg="C, F" Name="AddMonoidalPostComposeMorphismWithGivenObjects" Label="for IsCapCategory, IsFunction"/>883<Returns>nothing884</Returns>885<Description>886The arguments are a category <Math>C</Math> and a function <Math>F</Math>.887This operations adds the given function <Math>F</Math>888to the category for the basic operation <C>MonoidalPostComposeMorphismWithGivenObjects</C>.889<Math>F: (\mathrm{\underline{Hom}}(b,c) \otimes \mathrm{\underline{Hom}}(a,b),a,b,c,\mathrm{\underline{Hom}}(a,c)) \mapsto \mathrm{MonoidalPostComposeMorphismWithGivenObjects}_{a,b,c}</Math>.890</Description>891</ManSection>892893894<ManSection>895<Attr Arg="a" Name="DualOnObjects" Label="for IsCapCategoryObject"/>896<Returns>an object897</Returns>898<Description>899The argument is an object <Math>a</Math>.900The output is its dual object <Math>a^{\vee}</Math>.901</Description>902</ManSection>903904905<ManSection>906<Oper Arg="C, F" Name="AddDualOnObjects" Label="for IsCapCategory, IsFunction"/>907<Returns>nothing908</Returns>909<Description>910The arguments are a category <Math>C</Math> and a function <Math>F</Math>.911This operations adds the given function <Math>F</Math>912to the category for the basic operation <C>DualOnObjects</C>.913<Math>F: a \mapsto a^{\vee}</Math>.914</Description>915</ManSection>916917918<ManSection>919<Attr Arg="alpha" Name="DualOnMorphisms" Label="for IsCapCategoryMorphism"/>920<Returns>a morphism in <Math>\mathrm{Hom}( b^{\vee}, a^{\vee} )</Math>.921</Returns>922<Description>923The argument is a morphism <Math>\alpha: a \rightarrow b</Math>.924The output is its dual morphism <Math>\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}</Math>.925</Description>926</ManSection>927928929<ManSection>930<Oper Arg="s,alpha,r" Name="DualOnMorphismsWithGivenDuals" Label="for IsCapCategoryObject, IsCapCategoryMorphism, IsCapCategoryObject"/>931<Returns>a morphism in <Math>\mathrm{Hom}( b^{\vee}, a^{\vee} )</Math>.932</Returns>933<Description>934The argument is an object <Math>s = b^{\vee}</Math>,935a morphism <Math>\alpha: a \rightarrow b</Math>,936and an object <Math>r = a^{\vee}</Math>.937The output is the dual morphism <Math>\alpha^{\vee}: b^{\vee} \rightarrow a^{\vee}</Math>.938</Description>939</ManSection>940941942<ManSection>943<Oper Arg="C, F" Name="AddDualOnMorphismsWithGivenDuals" Label="for IsCapCategory, IsFunction"/>944<Returns>nothing945</Returns>946<Description>947The arguments are a category <Math>C</Math> and a function <Math>F</Math>.948This operations adds the given function <Math>F</Math>949to the category for the basic operation <C>DualOnMorphismsWithGivenDuals</C>.950<Math>F: (b^{\vee},\alpha,a^{\vee}) \mapsto \alpha^{\vee}</Math>.951</Description>952</ManSection>953954955<ManSection>956<Attr Arg="a" Name="EvaluationForDual" Label="for IsCapCategoryObject"/>957<Returns>a morphism in <Math>\mathrm{Hom}( a^{\vee} \otimes a, 1 )</Math>.958</Returns>959<Description>960The argument is an object <Math>a</Math>.961The output is the evaluation morphism <Math>\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1</Math>.962</Description>963</ManSection>964965966<ManSection>967<Oper Arg="s,a,r" Name="EvaluationForDualWithGivenTensorProduct" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>968<Returns>a morphism in <Math>\mathrm{Hom}( a^{\vee} \otimes a, 1 )</Math>.969</Returns>970<Description>971The arguments are an object <Math>s = a^{\vee} \otimes a</Math>,972an object <Math>a</Math>,973and an object <Math>r = 1</Math>.974The output is the evaluation morphism <Math>\mathrm{ev}_{a}: a^{\vee} \otimes a \rightarrow 1</Math>.975</Description>976</ManSection>977978979<ManSection>980<Oper Arg="C, F" Name="AddEvaluationForDualWithGivenTensorProduct" Label="for IsCapCategory, IsFunction"/>981<Returns>nothing982</Returns>983<Description>984The arguments are a category <Math>C</Math> and a function <Math>F</Math>.985This operations adds the given function <Math>F</Math>986to the category for the basic operation <C>EvaluationForDualWithGivenTensorProduct</C>.987<Math>F: (a^{\vee} \otimes a, a, 1) \mapsto \mathrm{ev}_{a}</Math>.988</Description>989</ManSection>990991992<ManSection>993<Attr Arg="a" Name="CoevaluationForDual" Label="for IsCapCategoryObject"/>994<Returns>a morphism in <Math>\mathrm{Hom}(1,a \otimes a^{\vee})</Math>.995</Returns>996<Description>997The argument is an object <Math>a</Math>.998The output is the coevaluation morphism <Math>\mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}</Math>.999</Description>1000</ManSection>100110021003<ManSection>1004<Oper Arg="s,a,r" Name="CoevaluationForDualWithGivenTensorProduct" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>1005<Returns>a morphism in <Math>\mathrm{Hom}(1,a \otimes a^{\vee})</Math>.1006</Returns>1007<Description>1008The arguments are an object <Math>s = 1</Math>,1009an object <Math>a</Math>,1010and an object <Math>r = a \otimes a^{\vee}</Math>.1011The output is the coevaluation morphism <Math>\mathrm{coev}_{a}:1 \rightarrow a \otimes a^{\vee}</Math>.1012</Description>1013</ManSection>101410151016<ManSection>1017<Oper Arg="C, F" Name="AddCoevaluationForDualWithGivenTensorProduct" Label="for IsCapCategory, IsFunction"/>1018<Returns>nothing1019</Returns>1020<Description>1021The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1022This operations adds the given function <Math>F</Math>1023to the category for the basic operation <C>CoevaluationForDualWithGivenTensorProduct</C>.1024<Math>F: (1, a, a \otimes a^{\vee}) \mapsto \mathrm{coev}_{a}</Math>.1025</Description>1026</ManSection>102710281029<ManSection>1030<Attr Arg="a" Name="MorphismToBidual" Label="for IsCapCategoryObject"/>1031<Returns>a morphism in <Math>\mathrm{Hom}(a, (a^{\vee})^{\vee})</Math>.1032</Returns>1033<Description>1034The argument is an object <Math>a</Math>.1035The output is the morphism to the bidual <Math>a \rightarrow (a^{\vee})^{\vee}</Math>.1036</Description>1037</ManSection>103810391040<ManSection>1041<Oper Arg="a, r" Name="MorphismToBidualWithGivenBidual" Label="for IsCapCategoryObject, IsCapCategoryObject"/>1042<Returns>a morphism in <Math>\mathrm{Hom}(a, (a^{\vee})^{\vee})</Math>.1043</Returns>1044<Description>1045The arguments are an object <Math>a</Math>,1046and an object <Math>r = (a^{\vee})^{\vee}</Math>.1047The output is the morphism to the bidual <Math>a \rightarrow (a^{\vee})^{\vee}</Math>.1048</Description>1049</ManSection>105010511052<ManSection>1053<Oper Arg="C, F" Name="AddMorphismToBidualWithGivenBidual" Label="for IsCapCategory, IsFunction"/>1054<Returns>nothing1055</Returns>1056<Description>1057The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1058This operations adds the given function <Math>F</Math>1059to the category for the basic operation <C>MorphismToBidualWithGivenBidual</C>.1060<Math>F: (a, (a^{\vee})^{\vee}) \mapsto (a \rightarrow (a^{\vee})^{\vee})</Math>.1061</Description>1062</ManSection>106310641065<ManSection>1066<Oper Arg="a,a',b,b'" Name="TensorProductInternalHomCompatibilityMorphism" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>1067<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'))</Math>.1068</Returns>1069<Description>1070The arguments are four objects <Math>a, a', b, b'</Math>.1071The output is the natural morphism1072<Math>\mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')</Math>.1073</Description>1074</ManSection>107510761077<ManSection>1078<Oper Arg="a,a',b,b',L" Name="TensorProductInternalHomCompatibilityMorphismWithGivenObjects" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsList"/>1079<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'))</Math>.1080</Returns>1081<Description>1082The arguments are four objects <Math>a, a', b, b'</Math>,1083and a list <Math>L = [ \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') ]</Math>.1084The output is the natural morphism1085<Math>\mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b') \rightarrow \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b')</Math>.1086</Description>1087</ManSection>108810891090<ManSection>1091<Oper Arg="C, F" Name="AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects" Label="for IsCapCategory, IsFunction"/>1092<Returns>nothing1093</Returns>1094<Description>1095The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1096This operations adds the given function <Math>F</Math>1097to the category for the basic operation <C>TensorProductInternalHomCompatibilityMorphismWithGivenObjects</C>.1098<Math>F: ( a,a',b,b', [ \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') ]) \mapsto \mathrm{TensorProductInternalHomCompatibilityMorphismWithGivenObjects}_{a,a',b,b'}</Math>.1099</Description>1100</ManSection>110111021103<ManSection>1104<Oper Arg="a,b" Name="TensorProductDualityCompatibilityMorphism" Label="for IsCapCategoryObject, IsCapCategoryObject"/>1105<Returns>a morphism in <Math>\mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} )</Math>.1106</Returns>1107<Description>1108The arguments are two objects <Math>a,b</Math>.1109The output is the natural morphism1110<Math>\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}</Math>.1111</Description>1112</ManSection>111311141115<ManSection>1116<Oper Arg="s,a,b,r" Name="TensorProductDualityCompatibilityMorphismWithGivenObjects" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>1117<Returns>a morphism in <Math>\mathrm{Hom}( a^{\vee} \otimes b^{\vee}, (a \otimes b)^{\vee} )</Math>.1118</Returns>1119<Description>1120The arguments are an object <Math>s = a^{\vee} \otimes b^{\vee}</Math>,1121two objects <Math>a,b</Math>,1122and an object <Math>r = (a \otimes b)^{\vee}</Math>.1123The output is the natural morphism1124<Math>\mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}_{a,b}: a^{\vee} \otimes b^{\vee} \rightarrow (a \otimes b)^{\vee}</Math>.1125</Description>1126</ManSection>112711281129<ManSection>1130<Oper Arg="C, F" Name="AddTensorProductDualityCompatibilityMorphismWithGivenObjects" Label="for IsCapCategory, IsFunction"/>1131<Returns>nothing1132</Returns>1133<Description>1134The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1135This operations adds the given function <Math>F</Math>1136to the category for the basic operation <C>TensorProductDualityCompatibilityMorphismWithGivenObjects</C>.1137<Math>F: ( a^{\vee} \otimes b^{\vee}, a, b, (a \otimes b)^{\vee} ) \mapsto \mathrm{TensorProductDualityCompatibilityMorphismWithGivenObjects}_{a,b}</Math>.1138</Description>1139</ManSection>114011411142<ManSection>1143<Oper Arg="a,b" Name="MorphismFromTensorProductToInternalHom" Label="for IsCapCategoryObject, IsCapCategoryObject"/>1144<Returns>a morphism in <Math>\mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) )</Math>.1145</Returns>1146<Description>1147The arguments are two objects <Math>a,b</Math>.1148The output is the natural morphism <Math>\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)</Math>.1149</Description>1150</ManSection>115111521153<ManSection>1154<Oper Arg="s,a,b,r" Name="MorphismFromTensorProductToInternalHomWithGivenObjects" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>1155<Returns>a morphism in <Math>\mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) )</Math>.1156</Returns>1157<Description>1158The arguments are an object <Math>s = a^{\vee} \otimes b</Math>,1159two objects <Math>a,b</Math>,1160and an object <Math>r = \mathrm{\underline{Hom}}(a,b)</Math>.1161The output is the natural morphism <Math>\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)</Math>.1162</Description>1163</ManSection>116411651166<ManSection>1167<Oper Arg="C, F" Name="AddMorphismFromTensorProductToInternalHomWithGivenObjects" Label="for IsCapCategory, IsFunction"/>1168<Returns>nothing1169</Returns>1170<Description>1171The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1172This operations adds the given function <Math>F</Math>1173to the category for the basic operation <C>MorphismFromTensorProductToInternalHomWithGivenObjects</C>.1174<Math>F: ( a^{\vee} \otimes b, a, b, \mathrm{\underline{Hom}}(a,b) ) \mapsto \mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}_{a,b}</Math>.1175</Description>1176</ManSection>117711781179<ManSection>1180<Oper Arg="a,b" Name="IsomorphismFromTensorProductToInternalHom" Label="for IsCapCategoryObject, IsCapCategoryObject"/>1181<Returns>a morphism in <Math>\mathrm{Hom}( a^{\vee} \otimes b, \mathrm{\underline{Hom}}(a,b) )</Math>.1182</Returns>1183<Description>1184The arguments are two objects <Math>a,b</Math>.1185The output is the natural morphism <Math>\mathrm{IsomorphismFromTensorProductToInternalHom}_{a,b}: a^{\vee} \otimes b \rightarrow \mathrm{\underline{Hom}}(a,b)</Math>.1186</Description>1187</ManSection>118811891190<ManSection>1191<Oper Arg="C, F" Name="AddIsomorphismFromTensorProductToInternalHom" Label="for IsCapCategory, IsFunction"/>1192<Returns>nothing1193</Returns>1194<Description>1195The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1196This operations adds the given function <Math>F</Math>1197to the category for the basic operation <C>IsomorphismFromTensorProductToInternalHom</C>.1198<Math>F: ( a, b ) \mapsto \mathrm{IsomorphismFromTensorProductToInternalHom}_{a,b}</Math>.1199</Description>1200</ManSection>120112021203<ManSection>1204<Oper Arg="a,b" Name="MorphismFromInternalHomToTensorProduct" Label="for IsCapCategoryObject, IsCapCategoryObject"/>1205<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b )</Math>.1206</Returns>1207<Description>1208The arguments are two objects <Math>a,b</Math>.1209The output is the inverse of <Math>\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}</Math>, namely1210<Math>\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b</Math>.1211</Description>1212</ManSection>121312141215<ManSection>1216<Oper Arg="s,a,b,r" Name="MorphismFromInternalHomToTensorProductWithGivenObjects" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>1217<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b )</Math>.1218</Returns>1219<Description>1220The arguments are an object <Math>s = \mathrm{\underline{Hom}}(a,b)</Math>,1221two objects <Math>a,b</Math>,1222and an object <Math>r = a^{\vee} \otimes b</Math>.1223The output is the inverse of <Math>\mathrm{MorphismFromTensorProductToInternalHomWithGivenObjects}</Math>, namely1224<Math>\mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b</Math>.1225</Description>1226</ManSection>122712281229<ManSection>1230<Oper Arg="C, F" Name="AddMorphismFromInternalHomToTensorProductWithGivenObjects" Label="for IsCapCategory, IsFunction"/>1231<Returns>nothing1232</Returns>1233<Description>1234The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1235This operations adds the given function <Math>F</Math>1236to the category for the basic operation <C>MorphismFromInternalHomToTensorProductWithGivenObjects</C>.1237<Math>F: ( \mathrm{\underline{Hom}}(a,b),a,b,a^{\vee} \otimes b ) \mapsto \mathrm{MorphismFromInternalHomToTensorProductWithGivenObjects}_{a,b}</Math>.1238</Description>1239</ManSection>124012411242<ManSection>1243<Oper Arg="a,b" Name="IsomorphismFromInternalHomToTensorProduct" Label="for IsCapCategoryObject, IsCapCategoryObject"/>1244<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a,b), a^{\vee} \otimes b )</Math>.1245</Returns>1246<Description>1247The arguments are two objects <Math>a,b</Math>.1248The output is the inverse of <Math>\mathrm{IsomorphismFromTensorProductToInternalHom}</Math>, namely1249<Math>\mathrm{IsomorphismFromInternalHomToTensorProduct}_{a,b}: \mathrm{\underline{Hom}}(a,b) \rightarrow a^{\vee} \otimes b</Math>.1250</Description>1251</ManSection>125212531254<ManSection>1255<Oper Arg="C, F" Name="AddIsomorphismFromInternalHomToTensorProduct" Label="for IsCapCategory, IsFunction"/>1256<Returns>nothing1257</Returns>1258<Description>1259The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1260This operations adds the given function <Math>F</Math>1261to the category for the basic operation <C>IsomorphismFromInternalHomToTensorProduct</C>.1262<Math>F: ( a,b ) \mapsto \mathrm{IsomorphismFromInternalHomToTensorProduct}_{a,b}</Math>.1263</Description>1264</ManSection>126512661267<ManSection>1268<Attr Arg="alpha" Name="TraceMap" Label="for IsCapCategoryMorphism"/>1269<Returns>a morphism in <Math>\mathrm{Hom}(1,1)</Math>.1270</Returns>1271<Description>1272The argument is a morphism <Math>\alpha</Math>.1273The output is the trace morphism <Math>\mathrm{trace}_{\alpha}: 1 \rightarrow 1</Math>.1274</Description>1275</ManSection>127612771278<ManSection>1279<Oper Arg="C, F" Name="AddTraceMap" Label="for IsCapCategory, IsFunction"/>1280<Returns>nothing1281</Returns>1282<Description>1283The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1284This operations adds the given function <Math>F</Math>1285to the category for the basic operation <C>TraceMap</C>.1286<Math>F: \alpha \mapsto \mathrm{trace}_{\alpha}</Math>1287</Description>1288</ManSection>128912901291<ManSection>1292<Attr Arg="a" Name="RankMorphism" Label="for IsCapCategoryObject"/>1293<Returns>a morphism in <Math>\mathrm{Hom}(1,1)</Math>.1294</Returns>1295<Description>1296The argument is an object <Math>a</Math>.1297The output is the rank morphism <Math>\mathrm{rank}_a: 1 \rightarrow 1</Math>.1298</Description>1299</ManSection>130013011302<ManSection>1303<Oper Arg="C, F" Name="AddRankMorphism" Label="for IsCapCategory, IsFunction"/>1304<Returns>nothing1305</Returns>1306<Description>1307The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1308This operations adds the given function <Math>F</Math>1309to the category for the basic operation <C>RankMorphism</C>.1310<Math>F: a \mapsto \mathrm{rank}_{a}</Math>1311</Description>1312</ManSection>131313141315<ManSection>1316<Attr Arg="a" Name="IsomorphismFromDualToInternalHom" Label="for IsCapCategoryObject"/>1317<Returns>a morphism in <Math>\mathrm{Hom}(a^{\vee}, \mathrm{Hom}(a,1))</Math>.1318</Returns>1319<Description>1320The argument is an object <Math>a</Math>.1321The output is the isomorphism1322<Math>\mathrm{IsomorphismFromDualToInternalHom}_{a}: a^{\vee} \rightarrow \mathrm{Hom}(a,1)</Math>.1323</Description>1324</ManSection>132513261327<ManSection>1328<Oper Arg="C, F" Name="AddIsomorphismFromDualToInternalHom" Label="for IsCapCategory, IsFunction"/>1329<Returns>nothing1330</Returns>1331<Description>1332The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1333This operations adds the given function <Math>F</Math>1334to the category for the basic operation <C>IsomorphismFromDualToInternalHom</C>.1335<Math>F: a \mapsto \mathrm{IsomorphismFromDualToInternalHom}_{a}</Math>1336</Description>1337</ManSection>133813391340<ManSection>1341<Attr Arg="a" Name="IsomorphismFromInternalHomToDual" Label="for IsCapCategoryObject"/>1342<Returns>a morphism in <Math>\mathrm{Hom}(\mathrm{Hom}(a,1), a^{\vee})</Math>.1343</Returns>1344<Description>1345The argument is an object <Math>a</Math>.1346The output is the isomorphism1347<Math>\mathrm{IsomorphismFromInternalHomToDual}_{a}: \mathrm{Hom}(a,1) \rightarrow a^{\vee}</Math>.1348</Description>1349</ManSection>135013511352<ManSection>1353<Oper Arg="C, F" Name="AddIsomorphismFromInternalHomToDual" Label="for IsCapCategory, IsFunction"/>1354<Returns>nothing1355</Returns>1356<Description>1357The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1358This operations adds the given function <Math>F</Math>1359to the category for the basic operation <C>IsomorphismFromInternalHomToDual</C>.1360<Math>F: a \mapsto \mathrm{IsomorphismFromInternalHomToDual}_{a}</Math>1361</Description>1362</ManSection>136313641365<ManSection>1366<Oper Arg="t, a, alpha" Name="UniversalPropertyOfDual" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryMorphism"/>1367<Returns>a morphism in <Math>\mathrm{Hom}(t, a^{\vee})</Math>.1368</Returns>1369<Description>1370The arguments are two objects <Math>t,a</Math>,1371and a morphism <Math>\alpha: t \otimes a \rightarrow 1</Math>.1372The output is the morphism <Math>t \rightarrow a^{\vee}</Math>1373given by the universal property of <Math>a^{\vee}</Math>.1374</Description>1375</ManSection>137613771378<ManSection>1379<Oper Arg="C, F" Name="AddUniversalPropertyOfDual" Label="for IsCapCategory, IsFunction"/>1380<Returns>nothing1381</Returns>1382<Description>1383The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1384This operations adds the given function <Math>F</Math>1385to the category for the basic operation <C>UniversalPropertyOfDual</C>.1386<Math>F: ( t,a,\alpha: t \otimes a \rightarrow 1 ) \mapsto ( t \rightarrow a^{\vee} )</Math>.1387</Description>1388</ManSection>138913901391<ManSection>1392<Attr Arg="alpha" Name="LambdaIntroduction" Label="for IsCapCategoryMorphism"/>1393<Returns>a morphism in <Math>\mathrm{Hom}( 1, \mathrm{\underline{Hom}}(a,b) )</Math>.1394</Returns>1395<Description>1396The argument is a morphism <Math>\alpha: a \rightarrow b</Math>.1397The output is the corresponding morphism <Math>1 \rightarrow \mathrm{\underline{Hom}}(a,b)</Math>1398under the tensor hom adjunction.1399</Description>1400</ManSection>140114021403<ManSection>1404<Oper Arg="C, F" Name="AddLambdaIntroduction" Label="for IsCapCategory, IsFunction"/>1405<Returns>nothing1406</Returns>1407<Description>1408The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1409This operations adds the given function <Math>F</Math>1410to the category for the basic operation <C>LambdaIntroduction</C>.1411<Math>F: ( \alpha: a \rightarrow b ) \mapsto ( 1 \rightarrow \mathrm{\underline{Hom}}(a,b) )</Math>.1412</Description>1413</ManSection>141414151416<ManSection>1417<Oper Arg="a,b,alpha" Name="LambdaElimination" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryMorphism"/>1418<Returns>a morphism in <Math>\mathrm{Hom}(a,b)</Math>.1419</Returns>1420<Description>1421The arguments are two objects <Math>a,b</Math>,1422and a morphism <Math>\alpha: 1 \rightarrow \mathrm{\underline{Hom}}(a,b)</Math>.1423The output is a morphism <Math>a \rightarrow b</Math> corresponding to <Math>\alpha</Math>1424under the tensor hom adjunction.1425</Description>1426</ManSection>142714281429<ManSection>1430<Oper Arg="C, F" Name="AddLambdaElimination" Label="for IsCapCategory, IsFunction"/>1431<Returns>nothing1432</Returns>1433<Description>1434The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1435This operations adds the given function <Math>F</Math>1436to the category for the basic operation <C>LambdaElimination</C>.1437<Math>F: ( a,b,\alpha: 1 \rightarrow \mathrm{\underline{Hom}}(a,b) ) \mapsto ( a \rightarrow b )</Math>.1438</Description>1439</ManSection>144014411442<ManSection>1443<Attr Arg="a" Name="IsomorphismFromObjectToInternalHom" Label="for IsCapCategoryObject"/>1444<Returns>a morphism in <Math>\mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a))</Math>.1445</Returns>1446<Description>1447The argument is an object <Math>a</Math>.1448The output is the natural isomorphism <Math>a \rightarrow \mathrm{\underline{Hom}}(1,a)</Math>.1449</Description>1450</ManSection>145114521453<ManSection>1454<Oper Arg="a,r" Name="IsomorphismFromObjectToInternalHomWithGivenInternalHom" Label="for IsCapCategoryObject, IsCapCategoryObject"/>1455<Returns>a morphism in <Math>\mathrm{Hom}(a, \mathrm{\underline{Hom}}(1,a))</Math>.1456</Returns>1457<Description>1458The argument is an object <Math>a</Math>,1459and an object <Math>r = \mathrm{\underline{Hom}}(1,a)</Math>.1460The output is the natural isomorphism <Math>a \rightarrow \mathrm{\underline{Hom}}(1,a)</Math>.1461</Description>1462</ManSection>146314641465<ManSection>1466<Oper Arg="C, F" Name="AddIsomorphismFromObjectToInternalHomWithGivenInternalHom" Label="for IsCapCategory, IsFunction"/>1467<Returns>nothing1468</Returns>1469<Description>1470The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1471This operations adds the given function <Math>F</Math>1472to the category for the basic operation <C>IsomorphismFromObjectToInternalHomWithGivenInternalHom</C>.1473<Math>F: ( a, \mathrm{\underline{Hom}}(1,a) ) \mapsto ( a \rightarrow \mathrm{\underline{Hom}}(1,a) )</Math>.1474</Description>1475</ManSection>147614771478<ManSection>1479<Attr Arg="a" Name="IsomorphismFromInternalHomToObject" Label="for IsCapCategoryObject"/>1480<Returns>a morphism in <Math>\mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a)</Math>.1481</Returns>1482<Description>1483The argument is an object <Math>a</Math>.1484The output is the natural isomorphism <Math>\mathrm{\underline{Hom}}(1,a) \rightarrow a</Math>.1485</Description>1486</ManSection>148714881489<ManSection>1490<Oper Arg="a,s" Name="IsomorphismFromInternalHomToObjectWithGivenInternalHom" Label="for IsCapCategoryObject, IsCapCategoryObject"/>1491<Returns>a morphism in <Math>\mathrm{Hom}(\mathrm{\underline{Hom}}(1,a),a)</Math>.1492</Returns>1493<Description>1494The argument is an object <Math>a</Math>,1495and an object <Math>s = \mathrm{\underline{Hom}}(1,a)</Math>.1496The output is the natural isomorphism <Math>\mathrm{\underline{Hom}}(1,a) \rightarrow a</Math>.1497</Description>1498</ManSection>149915001501<ManSection>1502<Oper Arg="C, F" Name="AddIsomorphismFromInternalHomToObjectWithGivenInternalHom" Label="for IsCapCategory, IsFunction"/>1503<Returns>nothing1504</Returns>1505<Description>1506The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1507This operations adds the given function <Math>F</Math>1508to the category for the basic operation <C>IsomorphismFromInternalHomToObjectWithGivenInternalHom</C>.1509<Math>F: ( a, \mathrm{\underline{Hom}}(1,a) ) \mapsto ( \mathrm{\underline{Hom}}(1,a) \rightarrow a )</Math>.1510</Description>1511</ManSection>151215131514</Section>151515161517<Section Label="Chapter_Tensor_Product_and_Internal_Hom_Section_Rigid_Symmetric_Closed_Monoidal_Categories">1518<Heading>Rigid Symmetric Closed Monoidal Categories</Heading>15191520A symmetric closed monoidal category <Math>\mathbf{C}</Math> satisfying1521<List>1522<Item>1523the natural morphism1524<Math>\underline{\mathrm{Hom}}(a_1,b_1) \otimes \underline{\mathrm{Hom}}(a_2,b_2) \rightarrow \underline{\mathrm{Hom}}(a_1 \otimes a_2,b_1 \otimes b_2)</Math>1525is an isomorphism,1526</Item>1527<Item>1528the natural morphism1529<Math>a \rightarrow \underline{\mathrm{Hom}}(\underline{\mathrm{Hom}}(a, 1), 1)</Math>1530is an isomorphism1531</Item>1532</List>1533is called a <Emph>rigid symmetric closed monoidal category</Emph>.1534<ManSection>1535<Oper Arg="a,a',b,b'" Name="TensorProductInternalHomCompatibilityMorphismInverse" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject"/>1536<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'))</Math>.1537</Returns>1538<Description>1539The arguments are four objects <Math>a, a', b, b'</Math>.1540The output is the natural morphism1541<Math>\mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')</Math>.1542</Description>1543</ManSection>154415451546<ManSection>1547<Oper Arg="a,a',b,b',L" Name="TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects" Label="for IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsCapCategoryObject, IsList"/>1548<Returns>a morphism in <Math>\mathrm{Hom}( \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b'), \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'))</Math>.1549</Returns>1550<Description>1551The arguments are four objects <Math>a, a', b, b'</Math>,1552and a list <Math>L = [ \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') ]</Math>.1553The output is the natural morphism1554<Math>\mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}: \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') \rightarrow \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b')</Math>.1555</Description>1556</ManSection>155715581559<ManSection>1560<Oper Arg="C, F" Name="AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects" Label="for IsCapCategory, IsFunction"/>1561<Returns>nothing1562</Returns>1563<Description>1564The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1565This operations adds the given function <Math>F</Math>1566to the category for the basic operation <C>TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects</C>.1567<Math>F: ( a,a',b,b', [ \mathrm{\underline{Hom}}(a,a') \otimes \mathrm{\underline{Hom}}(b,b'), \mathrm{\underline{Hom}}(a \otimes b,a' \otimes b') ]) \mapsto \mathrm{TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects}_{a,a',b,b'}</Math>.1568</Description>1569</ManSection>157015711572<ManSection>1573<Attr Arg="a" Name="MorphismFromBidual" Label="for IsCapCategoryObject"/>1574<Returns>a morphism in <Math>\mathrm{Hom}((a^{\vee})^{\vee},a)</Math>.1575</Returns>1576<Description>1577The argument is an object <Math>a</Math>.1578The output is the inverse of the morphism to the bidual <Math>(a^{\vee})^{\vee} \rightarrow a</Math>.1579</Description>1580</ManSection>158115821583<ManSection>1584<Oper Arg="a, s" Name="MorphismFromBidualWithGivenBidual" Label="for IsCapCategoryObject, IsCapCategoryObject"/>1585<Returns>a morphism in <Math>\mathrm{Hom}((a^{\vee})^{\vee},a)</Math>.1586</Returns>1587<Description>1588The argument is an object <Math>a</Math>,1589and an object <Math>s = (a^{\vee})^{\vee}</Math>.1590The output is the inverse of the morphism to the bidual <Math>(a^{\vee})^{\vee} \rightarrow a</Math>.1591</Description>1592</ManSection>159315941595<ManSection>1596<Oper Arg="C, F" Name="AddMorphismFromBidualWithGivenBidual" Label="for IsCapCategory, IsFunction"/>1597<Returns>nothing1598</Returns>1599<Description>1600The arguments are a category <Math>C</Math> and a function <Math>F</Math>.1601This operations adds the given function <Math>F</Math>1602to the category for the basic operation <C>MorphismFromBidualWithGivenBidual</C>.1603<Math>F: (a, (a^{\vee})^{\vee}) \mapsto ((a^{\vee})^{\vee} \rightarrow a)</Math>.1604</Description>1605</ManSection>160616071608</Section>160916101611<P/>1612</Chapter>1613161416151616