Home Metamath Proof ExplorerTheorem List (p. 142 of 324) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22341) Hilbert Space Explorer (22342-23864) Users' Mathboxes (23865-32387)

Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfuccoval 14101 Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat               comp       comp

Theoremfuccocl 14102 The composition of two natural transformations is a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfucidcl 14103 The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat

Theoremfuclid 14104 Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfucrid 14105 Right identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfucass 14106 Associativity of natural transformation composition. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat        Nat        comp

Theoremfuccatid 14107* The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat

Theoremfuccat 14108 The functor category is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat

Theoremfucid 14109 The identity morphism in the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
FuncCat

Theoremfucsect 14110* Two natural transformations are in a section iff all the components are in a section relation. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat                      Sect       Sect

Theoremfucinv 14111* Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat                      Inv       Inv

Theoreminvfuc 14112* If is an inverse to for each , and is a natural transformation, then is also a natural transformation, and they are inverse in the functor category. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat                      Inv       Inv

Theoremfuciso 14113* A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat

Theoremnatpropd 14114 If two categories have the same set of objects, morphisms, and compositions, then they have the same natural transformations. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf       f f        compf compf                                   Nat Nat

Theoremfucpropd 14115 If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf       f f        compf compf                                   FuncCat FuncCat

8.2  Arrows (disjointified hom-sets)

Syntaxcdoma 14116 Extend class notation to include the domain extractor for an arrow.

Syntaxccoda 14117 Extend class notation to include the codomain extractor for an arrow.
coda

Syntaxcarw 14118 Extend class notation to include the collection of all arrows of a category.
Nat

Syntaxchoma 14119 Extend class notation to include the set of all arrows with a specific domain and codomain.
Homa

Definitiondf-doma 14120 Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)

Definitiondf-coda 14121 Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
coda

Definitiondf-homa 14122* Definition of the hom-set extractor for arrows, which tags the morphisms of the underlying hom-set with domain and codomain, which can then be extracted using df-doma 14120 and df-coda 14121. (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
Homa

Definitiondf-arw 14123 Definition of the set of arrows of a category. We will use the term "arrow" to denote a morphism tagged with its domain and codomain, as opposed to , which allows hom-sets for distinct objects to overlap. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat Homa

Theoremhomarcl 14124 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomafval 14125* Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomaf 14126 Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomaval 14127 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhoma 14128 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhomai 14129 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhomai2 14130 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomarcl2 14131 Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomarel 14132 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhoma1 14133 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomahom2 14134 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomahom 14135 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomadm 14136 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomacd 14137 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       coda

Theoremhomadmcd 14138 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremarwval 14139 The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa

Theoremarwrcl 14140 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremarwhoma 14141 An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa       coda

Theoremhomarw 14142 A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa

Theoremarwdm 14143 The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremarwcd 14144 The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremdmaf 14145 The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremcdaf 14146 The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremarwhom 14147 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremarwdmcd 14148 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       coda

8.2.1  Identity and composition for arrows

Syntaxcida 14149 Extend class notation to include identity for arrows.
Ida

Syntaxccoa 14150 Extend class notation to include composition for arrows.
compa

Definitiondf-ida 14151* Definition of the identity arrow, which is just the identity morphism tagged with its domain and codomain. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
Ida

Definitiondf-coa 14152* Definition of the composition of arrows. Since arrows are tagged with domain and codomain, this does not need to be a 5-ary operation like the regular composition in a category comp. Instead, it is a partial binary operation on arrows, which is defined when the domain of the first arrow matches the codomain of the second. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa Nat Nat coda coda compcoda

Theoremidafval 14153* Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidaval 14154 Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremida2 14155 Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidahom 14156 Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            Homa

Theoremidadm 14157 Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidacd 14158 Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            coda

Theoremidaf 14159 The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                     Nat

Theoremcoafval 14160* The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat       comp       coda coda coda

Theoremeldmcoa 14161 A pair is in the domain of the arrow composition, if the domain of equals the codomain of . (In this case we say and are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat       coda

Theoremdmcoass 14162 The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat

Theoremhomdmcoa 14163 If and , then and are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa

Theoremcoaval 14164 Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa                     comp

Theoremcoa2 14165 The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa                     comp

Theoremcoahom 14166 The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa

Theoremcoapm 14167 Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat

Theoremarwlid 14168 Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

Theoremarwrid 14169 Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

Theoremarwass 14170 Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

8.3  Examples of categories

8.3.1  The category of sets

Syntaxcsetc 14171 Extend class notation to include the category Set.

Definitiondf-setc 14172* Definition of the category Set, relativized to a subset . This is the category of all sets in and functions between these sets. Generally, we will take to be a weak universe or Grothendieck's universe, because these sets have closure properties as good as the real thing. (Contributed by FL, 8-Nov-2013.) (Revised by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcval 14173* Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcbas 14174 Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetchomfval 14175* Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetchom 14176 Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremelsetchom 14177 A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetccofval 14178* Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcco 14179 Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetccatid 14180* Lemma for setccat 14181. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetccat 14181 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetcid 14182 The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetcmon 14183 A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono

Theoremsetcepi 14184 An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi

Theoremsetcsect 14185 A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
Sect

Theoremsetcinv 14186 An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoremsetciso 14187 An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremresssetc 14188 The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the categories for different are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
f s f compfs compf

Theoremfuncsetcres2 14189 A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)

8.3.2  The category of categories

Syntaxccatc 14190 Extend class notation to include the category Cat.
CatCat

Definitiondf-catc 14191* Definition of the category Cat, which consists of all categories in the universe , with functors as the morphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat comp func

Theoremcatcval 14192* Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                            func        comp

Theoremcatcbas 14193 Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchomfval 14194* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchom 14195 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatccofval 14196* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp       func

Theoremcatcco 14197 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp                                          func

Theoremcatccatid 14198* Lemma for catccat 14200. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat              idfunc

Theoremcatcid 14199 The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     idfunc

Theoremcatccat 14200 The category of categories is a category. (Clearly it cannot be an element of itself, hence it is "large" with respect to .) (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32387
 Copyright terms: Public domain < Previous  Next >