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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfucinv 14201* 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 14202* 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 14203* A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
FuncCat               Nat

Theoremnatpropd 14204 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 14205 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 14206 Extend class notation to include the domain extractor for an arrow.

Syntaxccoda 14207 Extend class notation to include the codomain extractor for an arrow.
coda

Syntaxcarw 14208 Extend class notation to include the collection of all arrows of a category.
Nat

Syntaxchoma 14209 Extend class notation to include the set of all arrows with a specific domain and codomain.
Homa

Definitiondf-doma 14210 Definition of the domain extractor for an arrow. (Contributed by FL, 24-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)

Definitiondf-coda 14211 Definition of the codomain extractor for an arrow. (Contributed by FL, 26-Oct-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
coda

Definitiondf-homa 14212* 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 14210 and df-coda 14211. (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 11-Jan-2017.)
Homa

Definitiondf-arw 14213 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 14214 Reverse closure for an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomafval 14215* Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomaf 14216 Functionality of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomaval 14217 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhoma 14218 Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhomai 14219 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremelhomai2 14220 Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomarcl2 14221 Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomarel 14222 An arrow is an ordered pair. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhoma1 14223 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomahom2 14224 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomahom 14225 The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomadm 14226 The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremhomacd 14227 The codomain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       coda

Theoremhomadmcd 14228 Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa

Theoremarwval 14229 The set of arrows is the union of all the disjointified hom-sets. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa

Theoremarwrcl 14230 The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremarwhoma 14231 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 14232 A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat       Homa

Theoremarwdm 14233 The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremarwcd 14234 The codomain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremdmaf 14235 The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat

Theoremcdaf 14236 The codomain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
Nat              coda

Theoremarwhom 14237 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 14238 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 14239 Extend class notation to include identity for arrows.
Ida

Syntaxccoa 14240 Extend class notation to include composition for arrows.
compa

Definitiondf-ida 14241* 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 14242* 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 14243* Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidaval 14244 Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremida2 14245 Morphism part of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidahom 14246 Domain and codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            Homa

Theoremidadm 14247 Domain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida

Theoremidacd 14248 Codomain of the identity arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                            coda

Theoremidaf 14249 The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Ida                     Nat

Theoremcoafval 14250* The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat       comp       coda coda coda

Theoremeldmcoa 14251 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 14252 The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat

Theoremhomdmcoa 14253 If and , then and are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa

Theoremcoaval 14254 Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa                     comp

Theoremcoa2 14255 The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa                     comp

Theoremcoahom 14256 The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Homa

Theoremcoapm 14257 Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
compa       Nat

Theoremarwlid 14258 Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

Theoremarwrid 14259 Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
Homa       compa       Ida

Theoremarwass 14260 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 14261 Extend class notation to include the category Set.

Definitiondf-setc 14262* 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 14263* Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcbas 14264 Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetchomfval 14265* Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetchom 14266 Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremelsetchom 14267 A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetccofval 14268* Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetcco 14269 Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
comp

Theoremsetccatid 14270* Lemma for setccat 14271. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetccat 14271 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetcid 14272 The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremsetcmon 14273 A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono

Theoremsetcepi 14274 An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi

Theoremsetcsect 14275 A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
Sect

Theoremsetcinv 14276 An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv

Theoremsetciso 14277 An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremresssetc 14278 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 14279 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 14280 Extend class notation to include the category Cat.
CatCat

Definitiondf-catc 14281* 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 14282* Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                            func        comp

Theoremcatcbas 14283 Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchomfval 14284* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchom 14285 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatccofval 14286* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp       func

Theoremcatcco 14287 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp                                          func

Theoremcatccatid 14288* Lemma for catccat 14290. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat              idfunc

Theoremcatcid 14289 The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     idfunc

Theoremcatccat 14290 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

Theoremresscatc 14291 The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat categories for different are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
CatCat       CatCat                     f s f compfs compf

Theoremcatcisolem 14292* Lemma for catciso 14293. (Contributed by Mario Carneiro, 29-Jan-2017.)
CatCat                                                 Inv              Full Faith

Theoremcatciso 14293 A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. (Contributed by Mario Carneiro, 29-Jan-2017.)
CatCat                                                        Full Faith

Theoremcatcoppccl 14294 The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              oppCat       WUni

Theoremcatcfuccl 14295 The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              FuncCat        WUni

8.4  Categorical constructions

8.4.1  Product of categories

Syntaxcxpc 14296 Extend class notation with the product of two categories.
c

Syntaxc1stf 14297 Extend class notation with the first projection functor.
F

Syntaxc2ndf 14298 Extend class notation with the second projection functor.
F

Syntaxcprf 14299 Extend class notation with the functor pairing operation.
⟨,⟩F

Definitiondf-xpc 14300* Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)
c comp comp comp

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