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

Theoremsetciso 14201 An isomorphism in the category of sets is a bijection. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremresssetc 14202 The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the categories for different are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
f s f compfs compf

Theoremfuncsetcres2 14203 A functor into a smaller category of sets is a functor into the larger category. (Contributed by Mario Carneiro, 28-Jan-2017.)

8.3.2  The category of categories

Syntaxccatc 14204 Extend class notation to include the category Cat.
CatCat

Definitiondf-catc 14205* Definition of the category Cat, which consists of all categories in the universe , with functors as the morphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat comp func

Theoremcatcval 14206* Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                            func        comp

Theoremcatcbas 14207 Set of objects of the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchomfval 14208* Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatchom 14209 Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremcatccofval 14210* Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp       func

Theoremcatcco 14211 Composition in the category of categories. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     comp                                          func

Theoremcatccatid 14212* Lemma for catccat 14214. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat              idfunc

Theoremcatcid 14213 The identity arrow in the category of categories is the identity functor. (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat                     idfunc

Theoremcatccat 14214 The category of categories is a category. (Clearly it cannot be an element of itself, hence it is "large" with respect to .) (Contributed by Mario Carneiro, 3-Jan-2017.)
CatCat

Theoremresscatc 14215 The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat categories for different are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
CatCat       CatCat                     f s f compfs compf

Theoremcatcisolem 14216* Lemma for catciso 14217. (Contributed by Mario Carneiro, 29-Jan-2017.)
CatCat                                                 Inv              Full Faith

Theoremcatciso 14217 A functor is an isomorphism of categories if and only if it is full and faithful, and is a bijection on the objects. (Contributed by Mario Carneiro, 29-Jan-2017.)
CatCat                                                        Full Faith

Theoremcatcoppccl 14218 The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              oppCat       WUni

Theoremcatcfuccl 14219 The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              FuncCat        WUni

8.4  Categorical constructions

8.4.1  Product of categories

Syntaxcxpc 14220 Extend class notation with the product of two categories.
c

Syntaxc1stf 14221 Extend class notation with the first projection functor.
F

Syntaxc2ndf 14222 Extend class notation with the second projection functor.
F

Syntaxcprf 14223 Extend class notation with the functor pairing operation.
⟨,⟩F

Definitiondf-xpc 14224* Define the binary product of categories, which has objects for each pair of objects of the factors, and morphisms for each pair of morphisms of the factors. Composition is componentwise. (Contributed by Mario Carneiro, 10-Jan-2017.)
c comp comp comp

Definitiondf-1stf 14225* Define the first projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
F c

Definitiondf-2ndf 14226* Define the second projection functor out of the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
F c

Definitiondf-prf 14227* Define the pairing operation for functors (which takes two functors and and produces ⟨,⟩F c ). (Contributed by Mario Carneiro, 11-Jan-2017.)
⟨,⟩F

Theoremfnxpc 14228 The binary product of categories is a two-argument function. (Contributed by Mario Carneiro, 10-Jan-2017.)
c

Theoremxpcval 14229* Value of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
c                                    comp       comp                                          comp

Theoremxpcbas 14230 Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)
c

Theoremxpchomfval 14231* Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpchom 14232 Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremrelxpchom 14233 A hom-set in the binary product of categories is a relation. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpccofval 14234* Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp       comp       comp

Theoremxpcco 14235 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp       comp       comp

Theoremxpcco1st 14236 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp                                          comp

Theoremxpcco2nd 14237 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp                                          comp

Theoremxpchom2 14238 Value of the set of morphisms in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpcco2 14239 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                                                comp       comp       comp

Theoremxpccatid 14240* The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpcid 14241 The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpccat 14242 The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theorem1stfval 14243* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem1stf1 14244 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem1stf2 14245 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem2ndfval 14246* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem2ndf1 14247 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem2ndf2 14248 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem1stfcl 14249 The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      F

Theorem2ndfcl 14250 The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      F

Theoremprfval 14251* Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprf1 14252 Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprf2fval 14253* Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprf2 14254 Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprfcl 14255 The pairing of functors and is a functor . (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F        c

Theoremprf1st 14256 Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F                      F func

Theoremprf2nd 14257 Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F                      F func

Theorem1st2ndprf 14258 Break a functor into a product category into first and second projections. (Contributed by Mario Carneiro, 12-Jan-2017.)
c                             F func ⟨,⟩F F func

Theoremcatcxpccl 14259 The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
CatCat              c        WUni

Theoremxpcpropd 14260 If two categories have the same set of objects, morphisms, and compositions, then they have the same product category. (Contributed by Mario Carneiro, 17-Jan-2017.)
f f        compf compf       f f        compf compf                                   c c

8.4.2  Functor evaluation

Syntaxcevlf 14261 Extend class notation with the evaluation functor.
evalF

Syntaxccurf 14262 Extend class notation with the currying of a functor.
curryF

Syntaxcuncf 14263 Extend class notation with the uncurrying of a functor.
uncurryF

Syntaxcdiag 14264 Extend class notation to include the diagonal functor.
Δfunc

Definitiondf-evlf 14265* Define the evaluation functor, which is the extension of the evaluation map of functors, to a functor . (Contributed by Mario Carneiro, 11-Jan-2017.)
evalF Nat comp

Definitiondf-curf 14266* Define the curry functor, which maps a functor to curryF . (Contributed by Mario Carneiro, 11-Jan-2017.)
curryF

Definitiondf-uncf 14267* Define the uncurry functor, which can be defined equationally using evalF. Strictly speaking, the third category argument is not needed, since the resulting functor is extensionally equal regardless, but it is used in the equational definition and is too much work to remove. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF evalF func func F ⟨,⟩F F

Definitiondf-diag 14268* Define the diagonal functor, which is the functor whose object part is . The value of the functor at an object is the constant functor which maps all objects in to and all morphisms to . The morphism part is a natural transformation between these functors, which takes to the natural transformation with every component equal to . (Contributed by Mario Carneiro, 6-Jan-2017.)
Δfunc curryF F

Theoremevlfval 14269* Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF                                    comp       Nat

Theoremevlf2 14270* Value of the evaluation functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF                                    comp       Nat

Theoremevlf2val 14271 Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF                                    comp       Nat

Theoremevlf1 14272 Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF

Theoremevlfcllem 14273 Lemma for evlfcl 14274. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF        FuncCat                      Nat                                           comp c comp

Theoremevlfcl 14274 The evaluation functor is a bifunctor (a two-argument functor) with the first parameter taking values in the set of functors , and the second parameter in . (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF        FuncCat                      c

Theoremcurfval 14275* Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf1fval 14276* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf1 14277* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf11 14278 Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf12 14279 The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf1cl 14280 The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c

Theoremcurf2 14281* Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c

Theoremcurf2val 14282 Value of a component of the curry functor natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c

Theoremcurf2cl 14283 The curry functor at a morphism is a natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c                                                         Nat

Theoremcurfcl 14284 The curry functor of a functor is a functor curryF . (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF        FuncCat                      c

Theoremcurfpropd 14285 If two categories have the same set of objects, morphisms, and compositions, then they curry the same functor to the same result. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf       f f        compf compf                                   c        curryF curryF

Theoremuncfval 14286 Value of the uncurry functor, which is the reverse of the curry functor, taking to uncurryF . (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat        evalF func func F ⟨,⟩F F

Theoremuncfcl 14287 The uncurry operation takes a functor to a functor uncurryF . (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat        c

Theoremuncf1 14288 Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat

Theoremuncf2 14289 Value of the uncurry functor on a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat                                                                              comp

Theoremcurfuncf 14290 Cancellation of curry with uncurry. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat        curryF

Theoremuncfcurf 14291 Cancellation of uncurry with curry. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                      c        uncurryF

Theoremdiagval 14292 Define the diagonal functor, which is the functor whose object part is . We can define this equationally as the currying of the first projection functor, and by expressing it this way we get a quick proof of functoriality. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc                     curryF F

Theoremdiagcl 14293 The diagonal functor is a functor from the base category to the functor category. Another way of saying this is that the constant functor is a construction that is natural in (and covariant). (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc                     FuncCat

Theoremdiag1cl 14294 The constant functor of is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc

Theoremdiag11 14295 Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc

Theoremdiag12 14296 Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc

Theoremdiag2 14297 Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)
Δfunc

Theoremdiag2cl 14298 The diagonal functor at a morphism is a natural transformation between constant functors. (Contributed by Mario Carneiro, 7-Jan-2017.)
Δfunc                                                               Nat

Theoremcurf2ndf 14299 As shown in diagval 14292, the currying of the first projection is the diagonal functor. On the other hand, the currying of the second projection is , which is a constant functor of the identity functor at . (Contributed by Mario Carneiro, 15-Jan-2017.)
FuncCat                      curryF F Δfuncidfunc

8.4.3  Hom functor

Syntaxchof 14300 Extend class notation with the Hom functor.
HomF

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