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Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremoppcinv 14001 An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat                            Inv       Inv

Theoremoppciso 14002 An isomorphism in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
oppCat

Theoremsectmon 14003 If is a section of , then is a monomorphism. A monomorphism that arises from a section is also known as a split monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono       Sect

Theoremmonsect 14004 If is a monomorphism and is a section of , then is an inverse of and they are both isomorphisms. This is also stated as "a monomorphism which is also a split epimorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
Mono       Sect                            Inv

Theoremsectepi 14005 If is a section of , then is an epimorphism. An epimorphism that arises from a section is also known as a split epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi       Sect

Theoremepisect 14006 If is an epimorphism and is a section of , then is an inverse of and they are both isomorphisms. This is also stated as "an epimorphism which is also a split monomorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
Epi       Sect                            Inv

8.1.5  Subcategories

Syntaxcssc 14007 Extend class notation to include the subset relation for subcategories.
cat

Syntaxcresc 14008 Extend class notation to include category restriction (which is like structure restriction but also allows limiting the collection of morphisms).
cat

Syntaxcsubc 14009 Extend class notation to include the collection of subcategories of a category.
Subcat

Definitiondf-ssc 14010* Define the subset relation for subcategories. Despite the name, this is not really a "category-aware" definition, which is to say it makes no explicit references to homsets or composition; instead this is a subset-like relation on the functions that are used as subcategory specifications in df-subc 14012, which makes it play an analogous role to the subset relation applied to the subgroups of a group. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Definitiondf-resc 14011* Define the restriction of a category to a given set of arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat s sSet

Definitiondf-subc 14012* Subcat is the set of all the subcategory specifications of the category . Like df-subg 14941, this is not actually a collection of categories, but only sets which when given operations from the base category (using df-resc 14011) form a 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.) (Revised by Mario Carneiro, 4-Jan-2017.)
Subcat cat f comp

Theoremsscrel 14013 The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theorembrssc 14014* The subcategory subset relation is a relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscpwex 14015* An analogue of pwex 4382 for the subcategory subset relation: The collection of subcategory subsets of a given set is a set. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsubcrcl 14016 Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat

Theoremsscfn1 14017 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscfn2 14018 The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssclem 14019 Lemma for ssc1 14021 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremisssc 14020* Value of the subcategory subset relation when the arguments are known functions. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssc1 14021 Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssc2 14022 Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscres 14023 Any function restricted to a square domain is a subcategory subset of the original. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremsscid 14024 The subcategory subset relation is reflexive. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat

Theoremssctr 14025 The subcategory subset relation is transitive. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat cat cat

Theoremssceq 14026 The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat cat

Theoremrescval 14027 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        s sSet

Theoremrescval2 14028 Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat                             s sSet

Theoremrescbas 14029 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat

Theoremreschom 14030 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat

Theoremreschomf 14031 Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat                                    f

Theoremrescco 14032 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat                                    comp       comp

Theoremrescabs 14033 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat cat cat

Theoremrescabs2 14034 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.)
s cat cat

Theoremissubc 14035* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
f               comp                     Subcat cat

Theoremissubc2 14036* Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
f               comp                     Subcat cat

Theoremsubcssc 14037 An element in the set of subcategories is a subset of the category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat       f        cat

Theoremsubcfn 14038 An element in the set of subcategories is a binary function. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubcss1 14039 The objects of a subcategory are a subset of the objects of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubcss2 14040 The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubcidcl 14041 The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat

Theoremsubccocl 14042 A subcategory is closed under composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Subcat                     comp

Theoremsubccatid 14043* A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        Subcat

Theoremsubcid 14044 The identity in a subcategory is the same as the original category. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        Subcat

Theoremsubccat 14045 A subcategory is a category. (Contributed by Mario Carneiro, 4-Jan-2017.)
cat        Subcat

Theoremissubc3 14046* Alternate definition of a subcategory, as a subset of the category which is itself a category. The assumption that the identity be closed is necessary just as in the case of a monoid, issubm2 14749, for the same reasons, since categories are a generalization of monoids. (Contributed by Mario Carneiro, 6-Jan-2017.)
f               cat                      Subcat cat

Theoremfullsubc 14047 The full subcategory generated by a subset of objects is the category with these objects and the same morphisms as the original. The result is always a subcategory (and it is full, meaning that all morphisms of the original category between objects in the subcategory is also in the subcategory). (Contributed by Mario Carneiro, 4-Jan-2017.)
f                      Subcat

Theoremfullresc 14048 The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)
f                      s        cat        f f compf compf

Theoremresscat 14049 A category restricted to a smaller set of objects is a category. (Contributed by Mario Carneiro, 6-Jan-2017.)
s

Theoremsubsubc 14050 A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
cat        Subcat Subcat Subcat cat

8.1.6  Functors

Syntaxcfunc 14051 Extend class notation with the class of all functors.

Syntaxcidfu 14052 Extend class notation with identity functor.
idfunc

Syntaxccofu 14053 Extend class notation with functor composition.
func

Syntaxcresf 14054 Extend class notation to include restriction of a functor to a subcategory.
f

Definitiondf-func 14055* 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.) (Revised by Mario Carneiro, 2-Jan-2017.)
comp comp

Definitiondf-idfu 14056* Define the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Definitiondf-cofu 14057* Define the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Definitiondf-resf 14058* Define the restriction of a functor to a subcategory (analogue of df-res 4890). (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremrelfunc 14059 The set of functors is a relation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncrcl 14060 Reverse closure for a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)

Theoremisfunc 14061* Value of the set of functors between two categories. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       comp

Theoremisfuncd 14062* Deduce that an operation is a functor of categories. (Contributed by Mario Carneiro, 4-Jan-2017.)
comp       comp

Theoremfuncf1 14063 The object part of a functor is a function on objects. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncixp 14064* The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncf2 14065 The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncfn2 14066 The morphism part of a functor is a function. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremfuncid 14067 A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremfuncco 14068 A functor maps composition in the source category to composition in the target. (Contributed by Mario Carneiro, 2-Jan-2017.)
comp       comp

Theoremfuncsect 14069 The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
Sect       Sect

Theoremfuncinv 14070 The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
Inv       Inv

Theoremfunciso 14071 The image of an isomorphism under a functor is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremfuncoppc 14072 A functor on categories yields a functor on the opposite categories (in the same direction). (Contributed by Mario Carneiro, 4-Jan-2017.)
oppCat       oppCat              tpos

Theoremidfuval 14073* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfu2nd 14074 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfu2 14075 Value of the morphism part of the identity functor. (Contributed by Mario Carneiro, 28-Jan-2017.)
idfunc

Theoremidfu1st 14076 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfu1 14077 Value of the object part of the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremidfucl 14078 The identity functor is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc

Theoremcofuval 14079* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofu1st 14080 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofu1 14081 Value of the object part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
func

Theoremcofu2nd 14082 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofu2 14083 Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-Jan-2017.)
func

Theoremcofuval2 14084* Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofucl 14085 The composition of two functors is a functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
func

Theoremcofuass 14086 Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)
func func func func

Theoremcofulid 14087 The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc       func

Theoremcofurid 14088 The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
idfunc       func

Theoremresfval 14089* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremresfval2 14090* Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremresf1st 14091 Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremresf2nd 14092 Value of the functor restriction operator on morphisms. (Contributed by Mario Carneiro, 6-Jan-2017.)
f

Theoremfuncres 14093 A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat       f cat

Theoremfuncres2b 14094* Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat                            cat

Theoremfuncres2 14095 A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Subcat cat

Theoremwunfunc 14096 A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
WUni

Theoremfuncpropd 14097 If two categories have the same set of objects, morphisms, and compositions, then they have the same functors. (Contributed by Mario Carneiro, 17-Jan-2017.)
f f        compf compf       f f        compf compf

Theoremfuncres2c 14098 Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)
s

8.1.7  Full & faithful functors

Syntaxcful 14099 Extend class notation with the class of all full functors.
Full

Syntaxcfth 14100 Extend class notation with the class of all faithful functors.
Faith

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