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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcoafval 15901* The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  A  =  (Nat `  C )   &    |-  .xb  =  (comp `  C )   =>    |- 
 .x.  =  ( g  e.  A ,  f  e. 
 { h  e.  A  |  (coda `  h )  =  (domA `  g
 ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  (
 ( 2nd `  g )
 ( <. (domA `  f ) ,  (domA `  g )
 >.  .xb  (coda `  g ) ) ( 2nd `  f )
 ) >. )
 
Theoremeldmcoa 15902 A pair  <. G ,  F >. is in the domain of the arrow composition, if the domain of  G equals the codomain of  F. (In this case we say  G and  F are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  A  =  (Nat `  C )   =>    |-  ( G dom  .x.  F  <->  ( F  e.  A  /\  G  e.  A  /\  (coda `  F )  =  (domA `  G ) ) )
 
Theoremdmcoass 15903 The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  A  =  (Nat `  C )   =>    |- 
 dom  .x.  C_  ( A  X.  A )
 
Theoremhomdmcoa 15904 If  F : X --> Y and  G : Y --> Z, then  G and  F are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  H  =  (Homa `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   =>    |-  ( ph  ->  G dom  .x. 
 F )
 
Theoremcoaval 15905 Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  H  =  (Homa `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   &    |-  .xb 
 =  (comp `  C )   =>    |-  ( ph  ->  ( G  .x.  F )  = 
 <. X ,  Z ,  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb  Z ) ( 2nd `  F )
 ) >. )
 
Theoremcoa2 15906 The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  H  =  (Homa `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   &    |-  .xb 
 =  (comp `  C )   =>    |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb  Z ) ( 2nd `  F )
 ) )
 
Theoremcoahom 15907 The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  H  =  (Homa `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   =>    |-  ( ph  ->  ( G  .x.  F )  e.  ( X H Z ) )
 
Theoremcoapm 15908 Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 .x.  =  (compa `  C )   &    |-  A  =  (Nat `  C )   =>    |- 
 .x.  e.  ( A  ^pm  ( A  X.  A ) )
 
Theoremarwlid 15909 Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |- 
 .x.  =  (compa `  C )   &    |-  .1.  =  (Ida `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( (  .1.  `  Y )  .x.  F )  =  F )
 
Theoremarwrid 15910 Right identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |- 
 .x.  =  (compa `  C )   &    |-  .1.  =  (Ida `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   =>    |-  ( ph  ->  ( F  .x.  (  .1.  `  X ) )  =  F )
 
Theoremarwass 15911 Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |-  H  =  (Homa `  C )   &    |- 
 .x.  =  (compa `  C )   &    |-  .1.  =  (Ida `  C )   &    |-  ( ph  ->  F  e.  ( X H Y ) )   &    |-  ( ph  ->  G  e.  ( Y H Z ) )   &    |-  ( ph  ->  K  e.  ( Z H W ) )   =>    |-  ( ph  ->  (
 ( K  .x.  G )  .x.  F )  =  ( K  .x.  ( G  .x.  F ) ) )
 
8.3  Examples of categories
 
8.3.1  The category of sets
 
Syntaxcsetc 15912 Extend class notation to include the category Set.
 class  SetCat
 
Definitiondf-setc 15913* Definition of the category Set, relativized to a subset  u. Example 3.3(1) of [Adamek] p. 22. This is the category of all sets in  u and functions between these sets. Generally, we will take  u to be a weak universe or Grothendieck 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.)
 |-  SetCat  =  ( u  e. 
 _V  |->  { <. ( Base `  ndx ) ,  u >. , 
 <. ( Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( y  ^m  x ) ) >. , 
 <. (comp `  ndx ) ,  ( v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( z 
 ^m  ( 2nd `  v
 ) ) ,  f  e.  ( ( 2nd `  v
 )  ^m  ( 1st `  v ) )  |->  ( g  o.  f ) ) ) >. } )
 
Theoremsetcval 15914* Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( y 
 ^m  x ) ) )   &    |-  ( ph  ->  .x. 
 =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  (
 ( 2nd `  v )  ^m  ( 1st `  v
 ) )  |->  ( g  o.  f ) ) ) )   =>    |-  ( ph  ->  C  =  { <. ( Base `  ndx ) ,  U >. , 
 <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } )
 
Theoremsetcbas 15915 Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  U  =  ( Base `  C ) )
 
Theoremsetchomfval 15916* Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( y  ^m  x ) ) )
 
Theoremsetchom 15917 Set of arrows of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X H Y )  =  ( Y  ^m  X ) )
 
Theoremelsetchom 15918 A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( F  e.  ( X H Y )  <->  F : X --> Y ) )
 
Theoremsetccofval 15919* Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   =>    |-  ( ph  ->  .x. 
 =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( z  ^m  ( 2nd `  v ) ) ,  f  e.  (
 ( 2nd `  v )  ^m  ( 1st `  v
 ) )  |->  ( g  o.  f ) ) ) )
 
Theoremsetcco 15920 Composition in the category of sets. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  F : X --> Y )   &    |-  ( ph  ->  G : Y --> Z )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  =  ( G  o.  F ) )
 
Theoremsetccatid 15921* Lemma for setccat 15922. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   =>    |-  ( U  e.  V  ->  ( C  e.  Cat  /\  ( Id `  C )  =  ( x  e.  U  |->  (  _I  |`  x ) ) ) )
 
Theoremsetccat 15922 The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   =>    |-  ( U  e.  V  ->  C  e.  Cat )
 
Theoremsetcid 15923 The identity arrow in the category of sets is the identity function. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  .1.  =  ( Id `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   =>    |-  ( ph  ->  (  .1.  `  X )  =  (  _I  |`  X ) )
 
Theoremsetcmon 15924 A monomorphism of sets is an injection. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  M  =  (Mono `  C )   =>    |-  ( ph  ->  ( F  e.  ( X M Y )  <->  F : X -1-1-> Y ) )
 
Theoremsetcepi 15925 An epimorphism of sets is a surjection. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  E  =  (Epi `  C )   &    |-  ( ph  ->  2o  e.  U )   =>    |-  ( ph  ->  ( F  e.  ( X E Y )  <->  F : X -onto-> Y ) )
 
Theoremsetcsect 15926 A section in the category of sets, written out. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  S  =  (Sect `  C )   =>    |-  ( ph  ->  ( F ( X S Y ) G  <->  ( F : X
 --> Y  /\  G : Y
 --> X  /\  ( G  o.  F )  =  (  _I  |`  X ) ) ) )
 
Theoremsetcinv 15927 An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  N  =  (Inv `  C )   =>    |-  ( ph  ->  ( F ( X N Y ) G  <->  ( F : X
 -1-1-onto-> Y  /\  G  =  `' F ) ) )
 
Theoremsetciso 15928 An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  I  =  ( 
 Iso  `  C )   =>    |-  ( ph  ->  ( F  e.  ( X I Y )  <->  F : X -1-1-onto-> Y ) )
 
Theoremresssetc 15929 The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the  SetCat `
 U categories for different  U are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  D  =  (
 SetCat `  V )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  V  C_  U )   =>    |-  ( ph  ->  (
 ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
 
Theoremfuncsetcres2 15930 A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  C  =  ( SetCat `  U )   &    |-  D  =  (
 SetCat `  V )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  V  C_  U )   =>    |-  ( ph  ->  ( E  Func  D )  C_  ( E  Func  C ) )
 
8.3.2  The category of categories
 
Syntaxccatc 15931 Extend class notation to include the category Cat.
 class CatCat
 
Definitiondf-catc 15932* Definition of the category Cat, which consists of all categories in the universe  u (i.e. "small categories", see definition 3.44. of [Adamek] p. 39), with functors as the morphisms. Definition 3.47 of [Adamek] p. 40. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |- CatCat  =  ( u  e.  _V  |->  [_ ( u  i^i  Cat )  /  b ]_ { <. (
 Base `  ndx ) ,  b >. ,  <. ( Hom  `  ndx ) ,  ( x  e.  b ,  y  e.  b  |->  ( x  Func  y )
 ) >. ,  <. (comp `  ndx ) ,  ( v  e.  ( b  X.  b ) ,  z  e.  b  |->  ( g  e.  ( ( 2nd `  v )  Func  z
 ) ,  f  e.  (  Func  `  v ) 
 |->  ( g  o.func  f )
 ) ) >. } )
 
Theoremcatcval 15933* Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  B  =  ( U  i^i  Cat ) )   &    |-  ( ph  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x  Func  y ) ) )   &    |-  ( ph  ->  .x.  =  (
 v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v )  Func  z ) ,  f  e.  (  Func  `  v
 )  |->  ( g  o.func  f
 ) ) ) )   =>    |-  ( ph  ->  C  =  { <. ( Base `  ndx ) ,  B >. , 
 <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } )
 
Theoremcatcbas 15934 Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
 
Theoremcatchomfval 15935* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  H  =  ( x  e.  B ,  y  e.  B  |->  ( x  Func  y ) ) )
 
Theoremcatchom 15936 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X H Y )  =  ( X  Func  Y ) )
 
Theoremcatccofval 15937* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   =>    |-  ( ph  ->  .x. 
 =  ( v  e.  ( B  X.  B ) ,  z  e.  B  |->  ( g  e.  ( ( 2nd `  v
 )  Func  z ) ,  f  e.  (  Func  `  v )  |->  ( g  o.func  f ) ) ) )
 
Theoremcatcco 15938 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  F  e.  ( X  Func  Y ) )   &    |-  ( ph  ->  G  e.  ( Y  Func  Z ) )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  =  ( G  o.func 
 F ) )
 
Theoremcatccatid 15939* Lemma for catccat 15941. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   =>    |-  ( U  e.  V  ->  ( C  e.  Cat  /\  ( Id `  C )  =  ( x  e.  B  |->  (idfunc `  x ) ) ) )
 
Theoremcatcid 15940 The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  I  =  (idfunc `  X )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (  .1.  `  X )  =  I )
 
Theoremcatccat 15941 The category of categories is a category, see remark 3.48 in [Adamek] p. 40. (Clearly it cannot be an element of itself, hence it is "large" with respect to  U.) (Contributed by Mario Carneiro, 3-Jan-2017.)
 |-  C  =  (CatCat `  U )   =>    |-  ( U  e.  V  ->  C  e.  Cat )
 
Theoremresscatc 15942 The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat `  U categories for different  U are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  D  =  (CatCat `  V )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  V 
 C_  U )   =>    |-  ( ph  ->  ( ( Hom f  `  ( Cs  V ) )  =  ( Hom f  `  D )  /\  (compf `  ( Cs  V ) )  =  (compf `  D ) ) )
 
Theoremcatcisolem 15943* Lemma for catciso 15944. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  R  =  (
 Base `  X )   &    |-  S  =  ( Base `  Y )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  (Inv `  C )   &    |-  H  =  ( x  e.  S ,  y  e.  S  |->  `' ( ( `' F `  x ) G ( `' F `  y ) ) )   &    |-  ( ph  ->  F ( ( X Full  Y )  i^i  ( X Faith  Y ) ) G )   &    |-  ( ph  ->  F : R
 -1-1-onto-> S )   =>    |-  ( ph  ->  <. F ,  G >. ( X I Y ) <. `' F ,  H >. )
 
Theoremcatciso 15944 A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. Remark 3.28(2) in [Adamek] p. 34. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  R  =  (
 Base `  X )   &    |-  S  =  ( Base `  Y )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  I  =  (  Iso  `  C )   =>    |-  ( ph  ->  ( F  e.  ( X I Y )  <->  ( F  e.  ( ( X Full  Y )  i^i  ( X Faith  Y ) )  /\  ( 1st `  F ) : R -1-1-onto-> S ) ) )
 
Theoremcatcoppccl 15945 The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  O  =  (oppCat `  X )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  O  e.  B )
 
Theoremcatcfuccl 15946 The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
 |-  C  =  (CatCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  Q  =  ( X FuncCat  Y )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  Q  e.  B )
 
8.3.3  The category of extensible structures

The "category of extensible structures" ExtStrCat is the category of all sets in a universe regarded as extensible structures and the functions between their base sets, see df-estrc 15950.

Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are all sets in a universe  u, which can be an arbitrary set, see estrcbas 15952. Generally, we will take  u to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. If a set is not a real extensible structure, it is regarded as extensible structure with an empty base set. Because of bascnvimaeqv 15948 we do not need to restrict the universe to sets which "have a base". The morphisms (or arrows) between two objects, i.e. sets from the universe, are the mappings between their base sets, see estrchomfval 15953, whereas the composition is the ordinary composition of functions, see estrccofval 15956 and estrcco 15957.

It is shown that the category of extensible structures ExtStrCat is actually a category, see estrccat 15960 with the identity function as identity arrow, see estrcid 15961.

In the following, some background information about the category of extensible structures is given, taken from the discussion in Github issue #1507 (see https://github.com/metamath/set.mm/issues/1507):

At the beginning, the categories of non-unital rings RngCat and unital rings RingCat were defined separately (as unordered triples of ordereds pairs, see dfrngc2 38722 and dfringc2 38768, but with special compositions). With this definitions, however, theorem rngcresringcat 38780 could not be proven, because the compositions were not compatible. Unfortunately, no precise definition of the composition within the category of rings could be found in the literature. In section 3.3 EXAMPLES, paragraph (2) of [Adamek] p. 22, however, a definition is given for "Grp", the category of groups: "The following constructs; i.e., categories of structured sets and structure-preserving functions between them (o will always be the composition of functions and idA will always be the identity function on A): ... (b) Grp with objects all groups and morphisms all homomorphisms between them." Therefore, the compositions should have been harmonized by using the composition of the category of sets  SetCat, see df-setc 15913, which is the ordinary composition of functions. Analogously, categories of Rngs (and Rings) could have been shown to be restrictions resp. subcategories of the category of sets.

BJ and MC observed, however, that "...  |`cat [cannot be used] to restrict the category Set to Ring, because the homs are different. Although Ring is a concrete category, a hom between rings R and S is a function (Base`R) --> (Base`S) with certain properties, unlike in Set where it is a function R --> S.". Therefore, MC suggested that "we could have an alternative version of the Set category consisting of extensible structures (in U) together with (A Hom B) := (Base`A) --> (Base`B). This category is not isomorphic to Set because different extensible structures can have the same base set, but it is equivalent to Set; the relevant functors are (U`A) = (Base`A), the forgetful functor, and (F`A) = { <. (Base`ndx), A >. }". This led to the current definition of ExtStrCat, see df-estrc 15950. The claimed equivalence is proven by equivestrcsetc 15979. Having a definition of a category of extensible structures, the categories of non-unital and unital rings can be defined as appropriate restrictions of the category of extensible structures, see df-rngc 38709 and df-ringc 38755.

In the same way, more subcategories could be provided, resulting in the following "inclusion chain" by proving theorems like rngcresringcat 38780, although the morphisms of the shown categories are different ( "->" means "is subcategory of"):

RingCat-> RngCat-> GrpCat -> MndCat -> MgmCat -> ExtStrCat

According to MC, "If we generalize from subcategories to embeddings, then we can even fit  SetCat into the chain, equivalent to ExtStrCat at the end." As mentioned before, the equivalence of  SetCat and ExtStrCat is proven by equivestrcsetc 15979. Furthermore, it can be shown that  SetCat is embedded into ExtStrCat, see embedsetcestrc 15994.

Remark: equivestrcsetc 15979 as well as embedsetcestrc 15994 require that the index of the base set extractor is contained within the considered universe. This is assured by assuming that the natural numbers are contained within the considered universe:  om  e.  U (see wunndx 15091), but it would be currently sufficient to assume that  1  e.  U, because the index value of the base set extractor is hard-coded as  1, see basendx 15127.

Some people, however, feel uncomfortable to say that a ring "is a" group (without mentioning the restriction to the addition, which is usually found in the literature, e.g. the definition of a ring in [Herstein] p. 126: "... Note that so far all we have said is that R is an abelian group under +.". The main argument against a ring being a group is the number of components/slots: usually, a group consists of (exactly!) two components (a base set and an operation), whereas a ring consists of (exactly!) three components (a base set and two operations). According to this "definition", a ring cannot be a group.

This is also an (unfortunately informal) argument for the category of rings not being a subcategory of the category of abelian groups in "Categories and Functors", Bodo Pareigis, Academic Press, New York, London, 1970: "A category A is called a subcategory of a category B if Ob(A) C_ Ob(B) and MorA(X,Y) C_ MorB(X,Y) for all X,Y e. Ob(A), if the composition of morphisms in A coincides with the composition of the same morphisms in B and if the identity of an object in A is also the identity of the same object viewed as an object in B. Then there is a forgetful functor from A to B. We note that Ri [the category of rings] is not a subcategory of Ab [the category of abelian groups]. In fact, Ob(Ri) C_ Ob(Ab) is not true, although every ring can also be regarded as an abelian group. The corresponding abelian groups of two rings may coincide even if the rings do not coincide. The multiplication may be defined differently.".

As long as we define Rings, Groups, etc. in a way that  A  e.  Ring  ->  A  e.  Grp is valid (see ringgrp 17711) the corresponding categories are in a subcategory relation. If we do not want Rings to be Groups (then the category of rings would not be a subcategory of the category of groups, as observed by Pareigis), we would have to change the definitions of Magmas, Monoids, Groups, Rings etc. to restrict them to have exactly the required number of slots, so that the following holds

 g  e.  Grp  ->  g Struct  <. ( Base `  ndx ) ,  ( +g  `  ndx ) >.

 r  e.  Ring 
->  r Struct  <. ( Base `  ndx ) ,  ( +g  ` ndx ) , ( .r  ndx ) >.

 
Theoremfncnvimaeqv 15947 The inverse images of the universal class  _V under functions on the universal class  _V are the universal class  _V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.)
 |-  ( F  Fn  _V  ->  ( `' F " _V )  =  _V )
 
Theorembascnvimaeqv 15948 The inverse image of the universal class  _V under the base function is the universal class  _V itself. (Proposed by Mario Carneiro, 7-Mar-2020.) (Contributed by AV, 7-Mar-2020.)
 |-  ( `' Base " _V )  =  _V
 
Syntaxcestrc 15949 Extend class notation to include the category ExtStr.
 class ExtStrCat
 
Definitiondf-estrc 15950* Definition of the category ExtStr of extensible structures. This is the category whose objects are all sets in a universe  u regarded as extensible structures and whose morphisms are the functions between their base sets. If a set is not a real extensible structure, it is regarded as extensible structure with an empty base set. Because of bascnvimaeqv 15948 we do not need to restrict the universe to sets which "have a base". Generally, we will take  u to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Proposed by Mario Carneiro, 5-Mar-2020.) (Contributed by AV, 7-Mar-2020.)
 |- ExtStrCat  =  ( u  e.  _V  |->  {
 <. ( Base `  ndx ) ,  u >. ,  <. ( Hom  `  ndx ) ,  ( x  e.  u ,  y  e.  u  |->  ( ( Base `  y )  ^m  ( Base `  x )
 ) ) >. ,  <. (comp `  ndx ) ,  (
 v  e.  ( u  X.  u ) ,  z  e.  u  |->  ( g  e.  ( (
 Base `  z )  ^m  ( Base `  ( 2nd `  v ) ) ) ,  f  e.  (
 ( Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) ) 
 |->  ( g  o.  f
 ) ) ) >. } )
 
Theoremestrcval 15951* Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( ( Base `  y )  ^m  ( Base `  x ) ) ) )   &    |-  ( ph  ->  .x. 
 =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( ( Base `  z
 )  ^m  ( Base `  ( 2nd `  v
 ) ) ) ,  f  e.  ( (
 Base `  ( 2nd `  v
 ) )  ^m  ( Base `  ( 1st `  v
 ) ) )  |->  ( g  o.  f ) ) ) )   =>    |-  ( ph  ->  C  =  { <. ( Base ` 
 ndx ) ,  U >. ,  <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } )
 
Theoremestrcbas 15952 Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  U  =  ( Base `  C )
 )
 
Theoremestrchomfval 15953* Set of morphisms ("arrows") of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  H  =  ( x  e.  U ,  y  e.  U  |->  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) )
 
Theoremestrchom 15954 The morphisms between extensible structures are mappings between their base sets. (Contributed by AV, 7-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  A  =  (
 Base `  X )   &    |-  B  =  ( Base `  Y )   =>    |-  ( ph  ->  ( X H Y )  =  ( B  ^m  A ) )
 
Theoremelestrchom 15955 A morphism between extensible structures is a function between their base sets. (Contributed by AV, 7-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  A  =  (
 Base `  X )   &    |-  B  =  ( Base `  Y )   =>    |-  ( ph  ->  ( F  e.  ( X H Y )  <->  F : A --> B ) )
 
Theoremestrccofval 15956* Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   =>    |-  ( ph  ->  .x.  =  ( v  e.  ( U  X.  U ) ,  z  e.  U  |->  ( g  e.  ( (
 Base `  z )  ^m  ( Base `  ( 2nd `  v ) ) ) ,  f  e.  (
 ( Base `  ( 2nd `  v ) )  ^m  ( Base `  ( 1st `  v ) ) ) 
 |->  ( g  o.  f
 ) ) ) )
 
Theoremestrcco 15957 Composition in the category of extensible structures. (Contributed by AV, 7-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  .x.  =  (comp `  C )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  ( ph  ->  Z  e.  U )   &    |-  A  =  (
 Base `  X )   &    |-  B  =  ( Base `  Y )   &    |-  D  =  ( Base `  Z )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : B --> D )   =>    |-  ( ph  ->  ( G ( <. X ,  Y >.  .x.  Z ) F )  =  ( G  o.  F ) )
 
Theoremestrcbasbas 15958 An element of the base set of the base set of the category of extensible structures (i.e. the base set of an extensible structure) belongs to the considered weak universe. (Contributed by AV, 22-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  B  =  ( Base `  C )   &    |-  ( ph  ->  U  e. WUni )   =>    |-  ( ( ph  /\  E  e.  B )  ->  ( Base `  E )  e.  U )
 
Theoremestrccatid 15959* Lemma for estrccat 15960. (Contributed by AV, 8-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   =>    |-  ( U  e.  V  ->  ( C  e.  Cat  /\  ( Id `  C )  =  ( x  e.  U  |->  (  _I  |`  ( Base `  x ) ) ) ) )
 
Theoremestrccat 15960 The category of extensible structures is a category. (Contributed by AV, 8-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   =>    |-  ( U  e.  V  ->  C  e.  Cat )
 
Theoremestrcid 15961 The identity arrow in the category of extensible structures is the identity function of base sets. (Contributed by AV, 8-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |- 
 .1.  =  ( Id `  C )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  X  e.  U )   =>    |-  ( ph  ->  (  .1.  `  X )  =  (  _I  |`  ( Base `  X ) ) )
 
Theoremestrchomfn 15962 The Hom-set operation in the category of extensible structures (in a universe) is a function. (Contributed by AV, 8-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  H  Fn  ( U  X.  U ) )
 
Theoremestrchomfeqhom 15963 The functionalized Hom-set operation equals the Hom-set operation in the category of extensible structures (in a universe). (Contributed by AV, 8-Mar-2020.)
 |-  C  =  (ExtStrCat `  U )   &    |-  ( ph  ->  U  e.  V )   &    |-  H  =  ( Hom  `  C )   =>    |-  ( ph  ->  ( Hom f  `  C )  =  H )
 
Theoremestrreslem1 15964 Lemma 1 for estrres 15966. (Contributed by AV, 14-Mar-2020.)
 |-  ( ph  ->  C  =  { <. ( Base `  ndx ) ,  B >. , 
 <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } )   &    |-  ( ph  ->  B  e.  V )   =>    |-  ( ph  ->  B  =  ( Base `  C )
 )
 
Theoremestrreslem2 15965 Lemma 2 for estrres 15966. (Contributed by AV, 14-Mar-2020.)
 |-  ( ph  ->  C  =  { <. ( Base `  ndx ) ,  B >. , 
 <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  .x. 
 e.  Y )   =>    |-  ( ph  ->  (
 Base `  ndx )  e. 
 dom  C )
 
Theoremestrres 15966 Any restriction of a category (as an extensible structure which is an unordered triple of ordered pairs) is an unordered triple of ordered pairs. (Contributed by AV, 15-Mar-2020.)
 |-  ( ph  ->  C  =  { <. ( Base `  ndx ) ,  B >. , 
 <. ( Hom  `  ndx ) ,  H >. , 
 <. (comp `  ndx ) , 
 .x.  >. } )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  .x. 
 e.  Y )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  A 
 C_  B )   =>    |-  ( ph  ->  ( ( Cs  A ) sSet  <. ( Hom  `  ndx ) ,  G >. )  =  { <. (
 Base `  ndx ) ,  A >. ,  <. ( Hom  `  ndx ) ,  G >. ,  <. (comp `  ndx ) ,  .x.  >. } )
 
Theoremfuncestrcsetclem1 15967* Lemma 1 for funcestrcsetc 15976. (Contributed by AV, 22-Mar-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   =>    |-  ( ( ph  /\  X  e.  B )  ->  ( F `  X )  =  ( Base `  X )
 )
 
Theoremfuncestrcsetclem2 15968* Lemma 2 for funcestrcsetc 15976. (Contributed by AV, 22-Mar-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   =>    |-  ( ( ph  /\  X  e.  B )  ->  ( F `  X )  e.  U )
 
Theoremfuncestrcsetclem3 15969* Lemma 3 for funcestrcsetc 15976. (Contributed by AV, 22-Mar-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   =>    |-  ( ph  ->  F : B --> C )
 
Theoremfuncestrcsetclem4 15970* Lemma 4 for funcestrcsetc 15976. (Contributed by AV, 22-Mar-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   =>    |-  ( ph  ->  G  Fn  ( B  X.  B ) )
 
Theoremfuncestrcsetclem5 15971* Lemma 5 for funcestrcsetc 15976. (Contributed by AV, 23-Mar-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   &    |-  M  =  ( Base `  X )   &    |-  N  =  (
 Base `  Y )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X G Y )  =  (  _I  |`  ( N  ^m  M ) ) )
 
Theoremfuncestrcsetclem6 15972* Lemma 6 for funcestrcsetc 15976. (Contributed by AV, 23-Mar-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   &    |-  M  =  ( Base `  X )   &    |-  N  =  (
 Base `  Y )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  H  e.  ( N  ^m  M ) ) 
 ->  ( ( X G Y ) `  H )  =  H )
 
Theoremfuncestrcsetclem7 15973* Lemma 7 for funcestrcsetc 15976. (Contributed by AV, 23-Mar-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   =>    |-  ( ( ph  /\  X  e.  B )  ->  (
 ( X G X ) `  ( ( Id
 `  E ) `  X ) )  =  ( ( Id `  S ) `  ( F `  X ) ) )
 
Theoremfuncestrcsetclem8 15974* Lemma 8 for funcestrcsetc 15976. (Contributed by AV, 15-Feb-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X G Y ) : ( X ( Hom  `  E ) Y ) --> ( ( F `  X ) ( Hom  `  S ) ( F `  Y ) ) )
 
Theoremfuncestrcsetclem9 15975* Lemma 9 for funcestrcsetc 15976. (Contributed by AV, 23-Mar-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( H  e.  ( X ( Hom  `  E ) Y )  /\  K  e.  ( Y ( Hom  `  E ) Z ) ) )  ->  (
 ( X G Z ) `  ( K (
 <. X ,  Y >. (comp `  E ) Z ) H ) )  =  ( ( ( Y G Z ) `  K ) ( <. ( F `  X ) ,  ( F `  Y ) >. (comp `  S ) ( F `
  Z ) ) ( ( X G Y ) `  H ) ) )
 
Theoremfuncestrcsetc 15976* The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 23-Mar-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   =>    |-  ( ph  ->  F ( E  Func  S ) G )
 
Theoremfthestrcsetc 15977* The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is faithful. (Contributed by AV, 2-Apr-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   =>    |-  ( ph  ->  F ( E Faith  S ) G )
 
Theoremfullestrcsetc 15978* The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is full. (Contributed by AV, 2-Apr-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   =>    |-  ( ph  ->  F ( E Full  S ) G )
 
Theoremequivestrcsetc 15979* The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is an equivalence. According to definition 3.33 (1) of [Adamek] p. 36, "A functor F : A -> B is called an equivalence provided that it is full, faithful, and isomorphism-dense in the sense that for any B-object B' there exists some A-object A' such that F(A') is isomorphic to B'.". Therefore, the category of sets and the category of extensible structures are equivalent, according to definition 3.33 (2) of [Adamek] p. 36, "Categories A and B are called equivalent provided that there is an equivalence from A to B.". (Contributed by AV, 2-Apr-2020.)
 |-  E  =  (ExtStrCat `  U )   &    |-  S  =  ( SetCat `  U )   &    |-  B  =  (
 Base `  E )   &    |-  C  =  ( Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  F  =  ( x  e.  B  |->  (
 Base `  x ) ) )   &    |-  ( ph  ->  G  =  ( x  e.  B ,  y  e.  B  |->  (  _I  |`  ( (
 Base `  y )  ^m  ( Base `  x )
 ) ) ) )   &    |-  ( ph  ->  ( Base ` 
 ndx )  e.  U )   =>    |-  ( ph  ->  ( F ( E Faith  S ) G  /\  F ( E Full  S ) G 
 /\  A. b  e.  C  E. a  e.  B  E. i  i :
 b
 -1-1-onto-> ( F `  a ) ) )
 
Theoremsetc1strwun 15980 A constructed one-slot structure with the objects of the category of sets as base set in a weak universe. (Contributed by AV, 27-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   =>    |-  ( ( ph  /\  X  e.  C )  ->  { <. (
 Base `  ndx ) ,  X >. }  e.  U )
 
Theoremfuncsetcestrclem1 15981* Lemma 1 for funcsetcestrc 15991. (Contributed by AV, 27-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   =>    |-  ( ( ph  /\  X  e.  C )  ->  ( F `  X )  =  { <. ( Base `  ndx ) ,  X >. } )
 
Theoremfuncsetcestrclem2 15982* Lemma 2 for funcsetcestrc 15991. (Contributed by AV, 27-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   =>    |-  ( ( ph  /\  X  e.  C )  ->  ( F `  X )  e.  U )
 
Theoremfuncsetcestrclem3 15983* Lemma 3 for funcsetcestrc 15991. (Contributed by AV, 27-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  E  =  (ExtStrCat `  U )   &    |-  B  =  ( Base `  E )   =>    |-  ( ph  ->  F : C --> B )
 
Theoremembedsetcestrclem 15984* Lemma for embedsetcestrc 15994. (Contributed by AV, 31-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  E  =  (ExtStrCat `  U )   &    |-  B  =  ( Base `  E )   =>    |-  ( ph  ->  F : C -1-1-> B )
 
Theoremfuncsetcestrclem4 15985* Lemma 4 for funcsetcestrc 15991. (Contributed by AV, 27-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   =>    |-  ( ph  ->  G  Fn  ( C  X.  C ) )
 
Theoremfuncsetcestrclem5 15986* Lemma 5 for funcsetcestrc 15991. (Contributed by AV, 27-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  C  /\  Y  e.  C )
 )  ->  ( X G Y )  =  (  _I  |`  ( Y  ^m  X ) ) )
 
Theoremfuncsetcestrclem6 15987* Lemma 6 for funcsetcestrc 15991. (Contributed by AV, 27-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   =>    |-  ( ( ph  /\  ( X  e.  C  /\  Y  e.  C )  /\  H  e.  ( Y 
 ^m  X ) ) 
 ->  ( ( X G Y ) `  H )  =  H )
 
Theoremfuncsetcestrclem7 15988* Lemma 7 for funcsetcestrc 15991. (Contributed by AV, 27-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   &    |-  E  =  (ExtStrCat `  U )   =>    |-  ( ( ph  /\  X  e.  C )  ->  (
 ( X G X ) `  ( ( Id
 `  S ) `  X ) )  =  ( ( Id `  E ) `  ( F `  X ) ) )
 
Theoremfuncsetcestrclem8 15989* Lemma 8 for funcsetcestrc 15991. (Contributed by AV, 28-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   &    |-  E  =  (ExtStrCat `  U )   =>    |-  ( ( ph  /\  ( X  e.  C  /\  Y  e.  C )
 )  ->  ( X G Y ) : ( X ( Hom  `  S ) Y ) --> ( ( F `  X ) ( Hom  `  E ) ( F `  Y ) ) )
 
Theoremfuncsetcestrclem9 15990* Lemma 9 for funcsetcestrc 15991. (Contributed by AV, 28-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   &    |-  E  =  (ExtStrCat `  U )   =>    |-  ( ( ph  /\  ( X  e.  C  /\  Y  e.  C  /\  Z  e.  C )  /\  ( H  e.  ( X ( Hom  `  S ) Y )  /\  K  e.  ( Y ( Hom  `  S ) Z ) ) )  ->  (
 ( X G Z ) `  ( K (
 <. X ,  Y >. (comp `  S ) Z ) H ) )  =  ( ( ( Y G Z ) `  K ) ( <. ( F `  X ) ,  ( F `  Y ) >. (comp `  E ) ( F `
  Z ) ) ( ( X G Y ) `  H ) ) )
 
Theoremfuncsetcestrc 15991* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only, preserving the morphisms as mappings between the corresponding base sets. (Contributed by AV, 28-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   &    |-  E  =  (ExtStrCat `  U )   =>    |-  ( ph  ->  F ( S  Func  E ) G )
 
Theoremfthsetcestrc 15992* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is faithful. (Contributed by AV, 31-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   &    |-  E  =  (ExtStrCat `  U )   =>    |-  ( ph  ->  F ( S Faith  E ) G )
 
Theoremfullsetcestrc 15993* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is full. (Contributed by AV, 1-Apr-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   &    |-  E  =  (ExtStrCat `  U )   =>    |-  ( ph  ->  F ( S Full  E ) G )
 
Theoremembedsetcestrc 15994* The "embedding functor" from the category of sets into the category of extensible structures which sends each set to an extensible structure consisting of the base set slot only is an embedding. According to definition 3.27 (1) of [Adamek] p. 34, a functor "F is called an embedding provided that F is injective on morphisms", or according to remark 3.28 (1) in [Adamek] p. 34, "a functor is an embedding if and only if it is faithful and injective on objects". (Contributed by AV, 31-Mar-2020.)
 |-  S  =  ( SetCat `  U )   &    |-  C  =  (
 Base `  S )   &    |-  ( ph  ->  F  =  ( x  e.  C  |->  {
 <. ( Base `  ndx ) ,  x >. } ) )   &    |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   &    |-  ( ph  ->  G  =  ( x  e.  C ,  y  e.  C  |->  (  _I  |`  ( y  ^m  x ) ) ) )   &    |-  E  =  (ExtStrCat `  U )   &    |-  B  =  (
 Base `  E )   =>    |-  ( ph  ->  ( F ( S Faith  E ) G  /\  F : C -1-1-> B ) )
 
8.4  Categorical constructions
 
8.4.1  Product of categories
 
Syntaxcxpc 15995 Extend class notation with the product of two categories.
 class  X.c
 
Syntaxc1stf 15996 Extend class notation with the first projection functor.
 class  1stF
 
Syntaxc2ndf 15997 Extend class notation with the second projection functor.
 class  2ndF
 
Syntaxcprf 15998 Extend class notation with the functor pairing operation.
 class ⟨,⟩F
 
Definitiondf-xpc 15999* 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.)
 |- 
 X.c 
 =  ( r  e. 
 _V ,  s  e. 
 _V  |->  [_ ( ( Base `  r )  X.  ( Base `  s ) ) 
 /  b ]_ [_ ( u  e.  b ,  v  e.  b  |->  ( ( ( 1st `  u ) ( Hom  `  r
 ) ( 1st `  v
 ) )  X.  (
 ( 2nd `  u )
 ( Hom  `  s ) ( 2nd `  v
 ) ) ) ) 
 /  h ]_ { <. (
 Base `  ndx ) ,  b >. ,  <. ( Hom  `  ndx ) ,  h >. ,  <. (comp `  ndx ) ,  ( x  e.  ( b  X.  b
 ) ,  y  e.  b  |->  ( g  e.  ( ( 2nd `  x ) h y ) ,  f  e.  ( h `
  x )  |->  <.
 ( ( 1st `  g
 ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) ) >. (comp `  r ) ( 1st `  y ) ) ( 1st `  f )
 ) ,  ( ( 2nd `  g )
 ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
 >. (comp `  s )
 ( 2nd `  y )
 ) ( 2nd `  f
 ) ) >. ) )
 >. } )
 
Definitiondf-1stf 16000* Define the first projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
 |- 
 1stF  =  ( r  e.  Cat ,  s  e.  Cat  |->  [_ (
 ( Base `  r )  X.  ( Base `  s )
 )  /  b ]_ <. ( 1st  |`  b ) ,  ( x  e.  b ,  y  e.  b  |->  ( 1st  |`  ( x ( Hom  `  (
 r  X.c  s ) ) y ) ) ) >. )
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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 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39291
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