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Theorem List for Metamath Proof Explorer - 13801-13900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxpcco 13801 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp       comp       comp

Theoremxpcco1st 13802 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp                                          comp

Theoremxpcco2nd 13803 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      comp                                          comp

Theoremxpchom2 13804 Value of the set of morphisms in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpcco2 13805 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                                                comp       comp       comp

Theoremxpccatid 13806* The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpcid 13807 The identity morphism in the product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theoremxpccat 13808 The product of two categories is a category. (Contributed by Mario Carneiro, 11-Jan-2017.)
c

Theorem1stfval 13809* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem1stf1 13810 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem1stf2 13811 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem2ndfval 13812* Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem2ndf1 13813 Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem2ndf2 13814 Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                                    F

Theorem1stfcl 13815 The first projection functor is a functor onto the left argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      F

Theorem2ndfcl 13816 The second projection functor is a functor onto the right argument. (Contributed by Mario Carneiro, 11-Jan-2017.)
c                      F

Theoremprfval 13817* Value of the pairing functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprf1 13818 Value of the pairing functor on objects. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprf2fval 13819* Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprf2 13820 Value of the pairing functor on morphisms. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F

Theoremprfcl 13821 The pairing of functors and is a functor . (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F        c

Theoremprf1st 13822 Cancellation of pairing with first projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F                      F func

Theoremprf2nd 13823 Cancellation of pairing with second projection. (Contributed by Mario Carneiro, 12-Jan-2017.)
⟨,⟩F                      F func

Theorem1st2ndprf 13824 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 13825 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 13826 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 13827 Extend class notation with the evaluation functor.
evalF

Syntaxccurf 13828 Extend class notation with the currying of a functor.
curryF

Syntaxcuncf 13829 Extend class notation with the uncurrying of a functor.
uncurryF

Syntaxcdiag 13830 Extend class notation to include the diagonal functor.
Δfunc

Definitiondf-evlf 13831* 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 13832* Define the curry functor, which maps a functor to curryF . (Contributed by Mario Carneiro, 11-Jan-2017.)
curryF

Definitiondf-uncf 13833* 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 13834* 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 13835* Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF                                    comp       Nat

Theoremevlf2 13836* Value of the evaluation functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF                                    comp       Nat

Theoremevlf2val 13837 Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF                                    comp       Nat

Theoremevlf1 13838 Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF

Theoremevlfcllem 13839 Lemma for evlfcl 13840. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF        FuncCat                      Nat                                           comp c comp

Theoremevlfcl 13840 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 13841* Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf1fval 13842* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf1 13843* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf11 13844 Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf12 13845 The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf1cl 13846 The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c

Theoremcurf2 13847* Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c

Theoremcurf2val 13848 Value of a component of the curry functor natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c

Theoremcurf2cl 13849 The curry functor at a morphism is a natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c                                                         Nat

Theoremcurfcl 13850 The curry functor of a functor is a functor curryF . (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF        FuncCat                      c

Theoremcurfpropd 13851 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 13852 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 13853 The uncurry operation takes a functor to a functor uncurryF . (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat        c

Theoremuncf1 13854 Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat

Theoremuncf2 13855 Value of the uncurry functor on a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat                                                                              comp

Theoremcurfuncf 13856 Cancellation of curry with uncurry. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat        curryF

Theoremuncfcurf 13857 Cancellation of uncurry with curry. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                      c        uncurryF

Theoremdiagval 13858 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 13859 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 13860 The constant functor of is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc

Theoremdiag11 13861 Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc

Theoremdiag12 13862 Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc

Theoremdiag2 13863 Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)
Δfunc

Theoremdiag2cl 13864 The diagonal functor at a morphism is a natural transformation between constant functors. (Contributed by Mario Carneiro, 7-Jan-2017.)
Δfunc                                                               Nat

Theoremcurf2ndf 13865 As shown in diagval 13858, 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 13866 Extend class notation with the Hom functor.
HomF

Syntaxcyon 13867 Extend class notation with the Yoneda embedding.
Yon

Definitiondf-hof 13868* Define the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from oppCat to , whose object part is the hom-function , and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
HomF f comp comp

Definitiondf-yon 13869 Define the Yoneda embedding, which is the currying of the (opposite) Hom functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
Yon oppCat curryF HomFoppCat

Theoremhofval 13870* Value of the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from oppCat to , whose object part is the hom-function , and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF                            comp       f

Theoremhof1fval 13871 The object part of the Hom functor is the f operation, which is just a functionalized version of . That is, it is a two argument function, which maps to the set of morphisms from to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF              f

Theoremhof1 13872 The object part of the Hom functor maps to the set of morphisms from to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF

Theoremhof2fval 13873* The morphism part of the Hom functor, for morphisms (which since the first argument is contravariant means morphisms and ), yields a function (a morphism of ) mapping to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF                                                        comp

Theoremhof2val 13874* The morphism part of the Hom functor, for morphisms (which since the first argument is contravariant means morphisms and ), yields a function (a morphism of ) mapping to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF                                                        comp

Theoremhof2 13875 The morphism part of the Hom functor, for morphisms (which since the first argument is contravariant means morphisms and ), yields a function (a morphism of ) mapping to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF                                                        comp

Theoremhofcllem 13876 Lemma for hofcl 13877. (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF       oppCat                            f                                                                                            comp comp comp

Theoremhofcl 13877 Closure of the Hom functor. Note that the codomain is the category for any universe which contains each Hom-set. This corresponds to the assertion that be locally small (with respect to ). (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF       oppCat                            f        c

Theoremoppchofcl 13878 Closure of the opposite Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
oppCat       HomF                            f        c

Theoremyonval 13879 Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon              oppCat       HomF       curryF

Theoremyoncl 13880 The Yoneda embedding is a functor from the category to the category of presheaves on . (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon              oppCat              FuncCat               f

Theoremyon1cl 13881 The Yoneda embedding at an object of is a presheaf on , also known as the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon                            oppCat                     f

Theoremyon11 13882 Value of the Yoneda embedding at an object. The partially evaluated Yoneds embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon

Theoremyon12 13883 Value of the Yoneda embedding at a morphism. The partially evaluated Yoneds embedding is also the contravariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon                                          comp

Theoremyon2 13884 Value of the Yoneda embedding at a morphism. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon                                          comp

Theoremhofpropd 13885 If two categories have the same set of objects, morphisms, and compositions, then they have the same Hom functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf                     HomF HomF

Theoremyonpropd 13886 If two categories have the same set of objects, morphisms, and compositions, then they have the same Yoneda functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
f f        compf compf                     Yon Yon

Theoremoppcyon 13887 Value of the opposite Yoneda embedding. (Contributed by Mario Carneiro, 26-Jan-2017.)
oppCat       Yon       HomF              curryF

Theoremoyoncl 13888 The opposite Yoneda embedding is a functor from oppCat to the functor category . (Contributed by Mario Carneiro, 26-Jan-2017.)
oppCat       Yon                            f        FuncCat

Theoremoyon1cl 13889 The opposite Yoneda embedding at an object of is a functor from to Set, also known as the covariant Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
oppCat       Yon                            f

Theoremyonedalem1 13890 Lemma for yoneda 13901. (Contributed by Mario Carneiro, 28-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        c c

Theoremyonedalem21 13891 Lemma for yoneda 13901. (Contributed by Mario Carneiro, 28-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                      Nat

Theoremyonedalem3a 13892* Lemma for yoneda 13901. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                      Nat        Nat

Theoremyonedalem4a 13893* Lemma for yoneda 13901. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f

Theoremyonedalem4b 13894* Lemma for yoneda 13901. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f

Theoremyonedalem4c 13895* Lemma for yoneda 13901. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                                    Nat

Theoremyonedalem22 13896 Lemma for yoneda 13901. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                                    Nat

Theoremyonedalem3b 13897* Lemma for yoneda 13901. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                                    Nat               Nat        comp comp

Theoremyonedalem3 13898* Lemma for yoneda 13901. (Contributed by Mario Carneiro, 28-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        Nat        c Nat

Theoremyonedainv 13899* The Yoneda Lemma with explicit inverse. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        Nat        Inv

Theoremyonffthlem 13900* Lemma for yonffth 13902. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        Nat        Inv              Full Faith

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