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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-ded 24901* Definition of a deductive system. Lambeck and Scott. Introduction to higher order categorical logic. p. 47. 1986. Unformally we can say a deductive system is a directed multi graph where for each object a specific morphism called identity of the object exists and where for some pairs of morphisms the composite exists. Deductive system are named so because morphisms may be interpreted as logical deductions, objects as sets of formulas and compositions as inferences. (Contributed by FL, 24-Oct-2007.)

Theoremisded 24902* The predicate "is a deductive system". (Contributed by FL, 24-Oct-2007.)

Theoremdedi 24903* Properties of a deductive system. (Contributed by FL, 24-Oct-2007.)

Theorem1ded 24904 Category is a deductive system. We can think of the morphism of Category as corresponding to . (Contributed by FL, 30-Oct-2007.)

Theoremstrded 24905 Structure of a deductive system. (Contributed by FL, 28-Oct-2007.)

Theoremrelded 24906 A deductive system is a relation. (Contributed by FL, 28-Oct-2007.)

Theoremreldded 24907 The domain of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)

Theoremrelrded 24908 The range of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)

Theoremdedalg 24909 A deductive system is an "algebra". (Contributed by FL, 28-Oct-2007.)

Theoremidosd 24910 The identity is a morphism which has the same object as its domain and its codomain. (Contributed by FL, 28-Oct-2007.)

Theoremcmppfd 24911 is only defined when the domain of is the codomain of . (Contributed by FL, 29-Oct-2007.)

Theoremdomcmpd 24912 When is defined its domain is the domain of . (Contributed by FL, 29-Oct-2007.)

Theoremcodcmpd 24913 When is defined its codomain is the codomain of . (Contributed by FL, 29-Oct-2007.)

Theoremrdmob 24914 The range of is the class of the objects. (Contributed by FL, 10-Jan-2008.)

Theoremrcmob 24915 The range of is the class of the objects. (Contributed by FL, 10-Jan-2008.)

Theoremaidm2 24916 The underlying directed multi graph of a deductive system. (Contributed by FL, 20-Sep-2009.)

Theoremdmrngcmp 24917 Domain and range of the domain of the composition. (Contributed by FL, 5-Oct-2009.)

16.11.48  Categories

SyntaxccatOLD 24918 Extend class notation with the class of categories.

Definitiondf-catOLD 24919* 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 24920* The predicate "is a category". (Contributed by FL, 24-Oct-2007.)

Theoremcati 24921* Definitional properties of a category. (Contributed by FL, 24-Oct-2007.)

Theorem0alg 24922 Lemma for 0ded 24923. (Contributed by FL, 10-Jan-2008.)

Theorem0ded 24923 A deductive system with no object and no morphism. (Contributed by FL, 10-Jan-2008.)

Theorem0catOLD 24924 Category has no object and no morphism. (Contributed by FL, 10-Jan-2008.)

Theorem1cat 24925 Category has one object and one morphism. (Contributed by FL, 30-Oct-2007.)

Theoremstrcat 24926 Structure of a category. (Contributed by FL, 26-Oct-2007.)

Theoremrelcat 24927 A category is a relation. (Contributed by FL, 26-Oct-2007.)

Theoremreldcat 24928 The domain of a category is a relation. (Contributed by FL, 26-Oct-2007.)

Theoremrelrcat 24929 The range of a category is a relation. (Contributed by FL, 26-Oct-2007.)

Theoremcatded 24930 A category is a deductive system. (Contributed by FL, 26-Oct-2007.)

Theoremdomc 24931 The 1st "axiom" of a category: is a mapping from the morphisms of to the objects of . (Contributed by FL, 2-Jan-2008.)

Theoremcodc 24932 The 2nd "axiom" of a category is a mapping from the morphisms of to the objects of . (Contributed by FL, 2-Jan-2008.)

Theoremidc 24933 The 3rd "axiom" of a category is a mapping from the objects of to the morphisms of . (Contributed by FL, 5-Dec-2007.)

Theoremcmppfc 24934 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 24935 The 5th "axiom" of a category: identities are morphisms whose domains and codomains are equal. (Contributed by FL, 5-Dec-2007.)

Theoremcmppfcd 24936 The 6th "axiom" of a category: is only defined when the domain of equals the codomain of . (Contributed by FL, 10-Mar-2008.)

Theoremdomcmpc 24937 The 7th "axiom" of a category: when is defined its domain is the domain of . (Contributed by FL, 10-Mar-2008.)

Theoremcodcmpc 24938 The 8th "axiom" of a category: when is defined its codomain is the codomain of . (Contributed by FL, 10-Jan-2008.)

Theoremcmpasso 24939 The 9th "axiom" of a category: is associative. (Contributed by FL, 29-Oct-2007.)

Theoremcmpida 24940 The 10th "axiom" of a category: is a left neutral element. (Contributed by FL, 29-Oct-2007.)

Theoremcmpidb 24941 The 11th "axiom" of a category: is a right neutral element. (Contributed by FL, 24-Oct-2007.)

Theoremdmo 24942 The domain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)

Theoremcdmo 24943 The codomain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)

Theoremjdmo 24944 An identity is a morphism. (Contributed by FL, 10-Jan-2008.)

Theoremcmpmorp 24945 Conditions for a composite to be a morphism. (Contributed by FL, 10-Mar-2008.)

Theoremmorcat 24946 Two ways to define the set of the morphisms of a category. (Contributed by FL, 19-Sep-2009.)

Theoremcmppfc1 24947 Composition is a function. (Contributed by FL, 5-Oct-2009.)

Theoremdualalg 24948 The dual of a is a . (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdualded 24949 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 24950 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

16.11.49  Homsets

SyntaxchomOLD 24951 Extend class notation with the function returning all the morphisms between two objects.

Definitiondf-homOLD 24952* 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 24953* Definition of . (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremishomb 24954* The homset . (Contributed by FL, 18-May-2007.)

Theoremishomc 24955 The predicate JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)

Theoremishomd 24956 The predicate JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)

Theoremehm 24957 The elements of a homset are morphisms. JFM CAT1 th. 21. (Contributed by FL, 5-Dec-2007.)

Theoremdehm 24958 Domain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)

Theoremcehm 24959 Codomain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)

Theoremmrdmcd 24960 A morphism belongs to the homset between its domain and its codomain. JFM CAT1 th. 22. (Contributed by FL, 1-Nov-2007.)

Theoremeqidob 24961 When the identities are equal, the objects are equal. JFM CAT1 th. 45. (Contributed by FL, 24-Apr-2007.)

Theoremhomib 24962 The homset which belongs to. JFM CAT1 th. 55. (Contributed by FL, 5-Dec-2007.)

Theoremhine 24963 The homset is not empty. JFM CAT1 th. 56. (Contributed by FL, 3-Jan-2008.)

Theoremcmphmia 24964 Composite of the member of a homset with the identity. JFM CAT1 th. 57 (Contributed by FL, 5-Dec-2007.)

Theoremcmphmib 24965 Composite of a member of a homset with the identity. JFM CAT1 th. 58 (Contributed by FL, 5-Dec-2007.)

Theoremiri 24966 Composite of an identity with itself. JFM CAT1 th. 59 (Contributed by FL, 5-Dec-2007.)

Theoremcmpassoh 24967 is associative. Homset-based version of cmpasso 24939. (Contributed by FL, 10-Mar-2008.)

Theoremhomgrf 24968 Homset of a composite. JFM CAT1 th. 51 (Contributed by FL, 10-Mar-2008.)

16.11.50  Monomorphisms, Epimorphisms, Isomorphisms

SyntaxcepiOLD 24969 Extend class notation with the class of all epimorphisms.
Epic

SyntaxcmonOLD 24970 Extend class notation with the class of all monomorphisms.
MonoOLD

SyntaxcisoOLD 24971 Extend class notation with the class of all isomorphisms.

Definitiondf-monOLD 24972* Function returning the monomorphisms of the category . JFM CAT1 def. 10. (Contributed by FL, 5-Dec-2007.)
MonoOLD

Definitiondf-epiOLD 24973* Function returning the epimorphisms of the category . JFM CAT1 def. 11. (Contributed by FL, 8-Aug-2008.)
Epic

Definitiondf-isoc 24974* Function returning the isomorphisms of the category . The Joy of Cats p. 28. (Contributed by FL, 9-Jun-2014.)

Theoremismona 24975* Monomorphisms of a category. (Contributed by FL, 5-Dec-2007.)
MonoOLD

Theoremismonb 24976* The predicate "is a monomorphism". (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremismonb1 24977* The predicate "is a monomorphism". (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremismonb2 24978 A monomorphism is a left-cancelable morphism. (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremimonclem 24979* Lemma for ismonc 24980. (Contributed by FL, 1-Jan-2008.)

Theoremismonc 24980* The predicate "is a monomorphism" when the morphism belongs to a homset. (Contributed by FL, 2-Jan-2008.)
MonoOLD

Theoremcmpmon 24981 The composite of two monomorphisms is a monomorphism. JFM CAT1 th. 61 (Contributed by FL, 10-Mar-2008.)
MonoOLD MonoOLD MonoOLD

Theoremicmpmon 24982 If is a monomorphism then is a monomorphism. JFM CAT1 th. 62 (Contributed by FL, 17-Mar-2008.)
MonoOLD MonoOLD

Theoremidmon 24983 If there exists such as then F is a monomorphism. JFM CAT1 th. 63. (Contributed by FL, 5-May-2008.)
MonoOLD

Theoremimmon 24984 A morphism identity is a monomorphism. JFM CAT1 th. 64. (Contributed by FL, 5-May-2008.)
MonoOLD

Theoremisepia 24985* Epimorphisms of a category . (Contributed by FL, 8-Aug-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
Epic

Theoremisepib 24986* The predicate "is an epimorphism". (Contributed by FL, 8-Aug-2008.)
Epic

Theoremisepib1 24987* The predicate "is an epimorphism". (Contributed by FL, 10-Aug-2008.)
Epic

Theoremisepib2 24988 An epimorphism is a right-cancelable morphism. (Contributed by FL, 10-Aug-2008.)
Epic

Theoremiepiclem 24989* Lemma for isepic 24990. (Contributed by FL, 6-Oct-2008.)

Theoremisepic 24990* The predicate "is an epimorphism" when the morphism belongs to a homset. (Contributed by FL, 27-Oct-2008.)
Epic

Theoremisiso 24991* Isomorphisms of a category. (Contributed by FL, 9-Jun-2014.) (Revised by Mario Carneiro, 11-Sep-2015.)

Syntaxccinv 24992 Extend class notation to include a function that returns the inverses of a morphism.

Definitiondf-cinv 24993* Function ( indexed by the category ) returning the inverses of a morphism . (Contributed by FL, 22-Dec-2008.)

Theoremcinvlem1 24994* The set of the inverses of all the morphisms . (Contributed by FL, 22-Dec-2008.)

Theoremcinvlem2 24995* The set of the inverses of the morphism . (Contributed by FL, 22-Dec-2008.)

Theoremcinvlem3 24996 The set of the inverses of the morphism . (Contributed by FL, 22-Dec-2008.)

16.11.51  Functors

SyntaxcfuncOLD 24997 Extend class notation with the class of all functors.

Syntaxcifunc 24998 Extend class notation with the class of all isomorphisms.

Definitiondf-funcOLD 24999* 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 25000* The class of functors between the categories and . (Contributed by FL, 10-Feb-2008.)

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