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Theorem List for Metamath Proof Explorer - 25101-25200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrdmob 25101 The range of is the class of the objects. (Contributed by FL, 10-Jan-2008.)

Theoremrcmob 25102 The range of is the class of the objects. (Contributed by FL, 10-Jan-2008.)

Theoremaidm2 25103 The underlying directed multi graph of a deductive system. (Contributed by FL, 20-Sep-2009.)

Theoremdmrngcmp 25104 Domain and range of the domain of the composition. (Contributed by FL, 5-Oct-2009.)

16.12.48  Categories

SyntaxccatOLD 25105 Extend class notation with the class of categories.

Definitiondf-catOLD 25106* A category is a deductive system where composition is associative, and identities are neutral elements. A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated to those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.)

TheoremiscatOLD 25107* The predicate "is a category". (Contributed by FL, 24-Oct-2007.)

Theoremcati 25108* Definitional properties of a category. (Contributed by FL, 24-Oct-2007.)

Theorem0alg 25109 Lemma for 0ded 25110. (Contributed by FL, 10-Jan-2008.)

Theorem0ded 25110 A deductive system with no object and no morphism. (Contributed by FL, 10-Jan-2008.)

Theorem0catOLD 25111 Category has no object and no morphism. (Contributed by FL, 10-Jan-2008.)

Theorem1cat 25112 Category has one object and one morphism. (Contributed by FL, 30-Oct-2007.)

Theoremstrcat 25113 Structure of a category. (Contributed by FL, 26-Oct-2007.)

Theoremrelcat 25114 A category is a relation. (Contributed by FL, 26-Oct-2007.)

Theoremreldcat 25115 The domain of a category is a relation. (Contributed by FL, 26-Oct-2007.)

Theoremrelrcat 25116 The range of a category is a relation. (Contributed by FL, 26-Oct-2007.)

Theoremcatded 25117 A category is a deductive system. (Contributed by FL, 26-Oct-2007.)

Theoremdomc 25118 The 1st "axiom" of a category: is a mapping from the morphisms of to the objects of . (Contributed by FL, 2-Jan-2008.)

Theoremcodc 25119 The 2nd "axiom" of a category is a mapping from the morphisms of to the objects of . (Contributed by FL, 2-Jan-2008.)

Theoremidc 25120 The 3rd "axiom" of a category is a mapping from the objects of to the morphisms of . (Contributed by FL, 5-Dec-2007.)

Theoremcmppfc 25121 The 4th "axiom" of a category: is a partial operation from the morphisms of to the morphisms of . (Contributed by FL, 10-Mar-2008.)

Theoremidosc 25122 The 5th "axiom" of a category: identities are morphisms whose domains and codomains are equal. (Contributed by FL, 5-Dec-2007.)

Theoremcmppfcd 25123 The 6th "axiom" of a category: is only defined when the domain of equals the codomain of . (Contributed by FL, 10-Mar-2008.)

Theoremdomcmpc 25124 The 7th "axiom" of a category: when is defined its domain is the domain of . (Contributed by FL, 10-Mar-2008.)

Theoremcodcmpc 25125 The 8th "axiom" of a category: when is defined its codomain is the codomain of . (Contributed by FL, 10-Jan-2008.)

Theoremcmpasso 25126 The 9th "axiom" of a category: is associative. (Contributed by FL, 29-Oct-2007.)

Theoremcmpida 25127 The 10th "axiom" of a category: is a left neutral element. (Contributed by FL, 29-Oct-2007.)

Theoremcmpidb 25128 The 11th "axiom" of a category: is a right neutral element. (Contributed by FL, 24-Oct-2007.)

Theoremdmo 25129 The domain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)

Theoremcdmo 25130 The codomain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)

Theoremjdmo 25131 An identity is a morphism. (Contributed by FL, 10-Jan-2008.)

Theoremcmpmorp 25132 Conditions for a composite to be a morphism. (Contributed by FL, 10-Mar-2008.)

Theoremmorcat 25133 Two ways to define the set of the morphisms of a category. (Contributed by FL, 19-Sep-2009.)

Theoremcmppfc1 25134 Composition is a function. (Contributed by FL, 5-Oct-2009.)

Theoremdualalg 25135 The dual of a is a . (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdualded 25136 The dual of a deductive system is a deductive system. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdualcat2 25137 The dual of a category is a category. Joy of cats 3.5 (Contributed by FL, 4-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
tpos

16.12.49  Homsets

SyntaxchomOLD 25138 Extend class notation with the function returning all the morphisms between two objects.

Definitiondf-homOLD 25139* is a function which returns for each pair of objects the morphisms whose domain is and codomain . JFM CAT1 def. 6 (Contributed by FL, 6-May-2007.)

Theoremishoma 25140* Definition of . (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremishomb 25141* The homset . (Contributed by FL, 18-May-2007.)

Theoremishomc 25142 The predicate JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)

Theoremishomd 25143 The predicate JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)

Theoremehm 25144 The elements of a homset are morphisms. JFM CAT1 th. 21. (Contributed by FL, 5-Dec-2007.)

Theoremdehm 25145 Domain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)

Theoremcehm 25146 Codomain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)

Theoremmrdmcd 25147 A morphism belongs to the homset between its domain and its codomain. JFM CAT1 th. 22. (Contributed by FL, 1-Nov-2007.)

Theoremeqidob 25148 When the identities are equal, the objects are equal. JFM CAT1 th. 45. (Contributed by FL, 24-Apr-2007.)

Theoremhomib 25149 The homset which belongs to. JFM CAT1 th. 55. (Contributed by FL, 5-Dec-2007.)

Theoremhine 25150 The homset is not empty. JFM CAT1 th. 56. (Contributed by FL, 3-Jan-2008.)

Theoremcmphmia 25151 Composite of the member of a homset with the identity. JFM CAT1 th. 57 (Contributed by FL, 5-Dec-2007.)

Theoremcmphmib 25152 Composite of a member of a homset with the identity. JFM CAT1 th. 58 (Contributed by FL, 5-Dec-2007.)

Theoremiri 25153 Composite of an identity with itself. JFM CAT1 th. 59 (Contributed by FL, 5-Dec-2007.)

Theoremcmpassoh 25154 is associative. Homset-based version of cmpasso 25126. (Contributed by FL, 10-Mar-2008.)

Theoremhomgrf 25155 Homset of a composite. JFM CAT1 th. 51 (Contributed by FL, 10-Mar-2008.)

16.12.50  Monomorphisms, Epimorphisms, Isomorphisms

SyntaxcepiOLD 25156 Extend class notation with the class of all epimorphisms.
Epic

SyntaxcmonOLD 25157 Extend class notation with the class of all monomorphisms.
MonoOLD

SyntaxcisoOLD 25158 Extend class notation with the class of all isomorphisms.

Definitiondf-monOLD 25159* Function returning the monomorphisms of the category . JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.)
MonoOLD

Definitiondf-epiOLD 25160* Function returning the epimorphisms of the category . JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.)
Epic

Definitiondf-isoc 25161* Function returning the isomorphisms of the category . The Joy of Cats p. 28. (Contributed by FL, 9-Jun-2014.)

Theoremismona 25162* Monomorphisms of a category. (Contributed by FL, 5-Dec-2007.)
MonoOLD

Theoremismonb 25163* The predicate "is a monomorphism". (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremismonb1 25164* The predicate "is a monomorphism". (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremismonb2 25165 A monomorphism is a left-cancelable morphism. (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremimonclem 25166* Lemma for ismonc 25167. (Contributed by FL, 1-Jan-2008.)

Theoremismonc 25167* The predicate "is a monomorphism" when the morphism belongs to a homset. (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremcmpmon 25168 The composite of two monomorphisms is a monomorphism. JFM CAT1 th. 61 (Contributed by FL, 10-Mar-2008.)
MonoOLD MonoOLD MonoOLD

Theoremicmpmon 25169 If is a monomorphism then is a monomorphism. JFM CAT1 th. 62 (Contributed by FL, 17-Mar-2008.)
MonoOLD MonoOLD

Theoremidmon 25170 If there exists such as then F is a monomorphism. JFM CAT1 th. 63. (Contributed by FL, 5-May-2008.)
MonoOLD

Theoremimmon 25171 A morphism identity is a monomorphism. JFM CAT1 th. 64. (Contributed by FL, 5-May-2008.)
MonoOLD

Theoremisepia 25172* Epimorphisms of a category . (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
Epic

Theoremisepib 25173* The predicate "is an epimorphism". (Contributed by FL, 8-Aug-2008.)
Epic

Theoremisepib1 25174* The predicate "is an epimorphism". (Contributed by FL, 10-Aug-2008.)
Epic

Theoremisepib2 25175 An epimorphism is a right-cancelable morphism. (Contributed by FL, 10-Aug-2008.)
Epic

Theoremiepiclem 25176* Lemma for isepic 25177. (Contributed by FL, 6-Oct-2008.)

Theoremisepic 25177* The predicate "is an epimorphism" when the morphism belongs to a homset. (Contributed by FL, 27-Oct-2008.)
Epic

Theoremisiso 25178* Isomorphisms of a category. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 11-Sep-2015.)

Syntaxccinv 25179 Extend class notation to include a function that returns the inverses of a morphism.

Definitiondf-cinv 25180* Function ( indexed by the category ) returning the inverses of a morphism . (Contributed by FL, 22-Dec-2008.)

Theoremcinvlem1 25181* The set of the inverses of all the morphisms . (Contributed by FL, 22-Dec-2008.)

Theoremcinvlem2 25182* The set of the inverses of the morphism . (Contributed by FL, 22-Dec-2008.)

Theoremcinvlem3 25183 The set of the inverses of the morphism . (Contributed by FL, 22-Dec-2008.)

16.12.51  Functors

SyntaxcfuncOLD 25184 Extend class notation with the class of all functors.

Syntaxcifunc 25185 Extend class notation with the class of all isomorphisms.

Definitiondf-funcOLD 25186* Function returning all the functors from a category to a category . Intuitively a functor associates any morphism of to a morphism of , any object of to an object of , and respects the identity, the composition, the domain and the codomain. Here to capture the idea that a functor associates any object of to an object of we write it associates any identity of to an identity of which simplifies the definition. (Contributed by FL, 10-Feb-2008.)

Theoremisfuna 25187* The class of functors between the categories and . (Contributed by FL, 10-Feb-2008.)

Theoremisfunb 25188* The predicate "is a functor" . (Contributed by FL, 10-Feb-2008.)

Theoremfmamo 25189 A functor is a mapping between morphisms. (Contributed by FL, 10-Feb-2008.)

Theoremvidfunid 25190* The functor associates every object of to an object in . For the identification of objects with the identities see df-funcOLD 25186. JFM CAT1 th. 97. (Contributed by FL, 10-Feb-2008.)

Theoremiddvvidd 25191* Functors preserve domains. JFM CAT1 th. 98. (Contributed by FL, 5-May-2008.)

Theoremidcvvidc 25192* Functors preserve codomains. JFM CAT1 th. 98. (Contributed by FL, 5-May-2008.)

Theoremfmp 25193* Functors preserve morphisms composition. JFM CAT1 th. 99. (Contributed by FL, 2-Aug-2009.)

Theoremidfisf 25194 The identity functor is a functor. (Contributed by FL, 15-Jul-2008.)

Definitiondf-isof 25195* Class of isomorphisms. (Contributed by FL, 21-May-2012.)

16.12.52  Subcategories

Syntaxcsubcat 25196 Extend class notation with a function returning all the subcategories of a given category.

Definitiondf-subcat 25197 is the set of all the subcategories of the category . All the objects and all the morphisms of the subcategory belong to the supercategory. The identity of an object, the domain and the codomain of a morphism are the same in the subcategory and the supercategory. The composition of the subcategory is a restriction of the composition of the supercategory. (Contributed by FL, 17-Sep-2009.)

Theoremissubcat 25198 The set of all the subcategories of . (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremissubcata 25199 The predicate "is a subcategory of" . (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremissubcatb 25200 The predicate "is a subcategory of" . (Contributed by FL, 17-Sep-2009.)

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