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Theorem List for Metamath Proof Explorer - 14301-14400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-diag 14301* 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 14302* Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF                                    comp       Nat

Theoremevlf2 14303* Value of the evaluation functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF                                    comp       Nat

Theoremevlf2val 14304 Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF                                    comp       Nat

Theoremevlf1 14305 Value of the evaluation functor at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF

Theoremevlfcllem 14306 Lemma for evlfcl 14307. (Contributed by Mario Carneiro, 12-Jan-2017.)
evalF        FuncCat                      Nat                                           comp c comp

Theoremevlfcl 14307 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 14308* Value of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf1fval 14309* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf1 14310* Value of the object part of the curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf11 14311 Value of the double evaluated curry functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf12 14312 The partially evaluated curry functor at a morphism. (Contributed by Mario Carneiro, 12-Jan-2017.)
curryF                             c

Theoremcurf1cl 14313 The partially evaluated curry functor is a functor. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c

Theoremcurf2 14314* Value of the curry functor at a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c

Theoremcurf2val 14315 Value of a component of the curry functor natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c

Theoremcurf2cl 14316 The curry functor at a morphism is a natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                             c                                                         Nat

Theoremcurfcl 14317 The curry functor of a functor is a functor curryF . (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF        FuncCat                      c

Theoremcurfpropd 14318 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 14319 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 14320 The uncurry operation takes a functor to a functor uncurryF . (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat        c

Theoremuncf1 14321 Value of the uncurry functor on an object. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat

Theoremuncf2 14322 Value of the uncurry functor on a morphism. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat                                                                              comp

Theoremcurfuncf 14323 Cancellation of curry with uncurry. (Contributed by Mario Carneiro, 13-Jan-2017.)
uncurryF                      FuncCat        curryF

Theoremuncfcurf 14324 Cancellation of uncurry with curry. (Contributed by Mario Carneiro, 13-Jan-2017.)
curryF                      c        uncurryF

Theoremdiagval 14325 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 14326 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 14327 The constant functor of is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc

Theoremdiag11 14328 Value of the constant functor at an object. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc

Theoremdiag12 14329 Value of the constant functor at a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Δfunc

Theoremdiag2 14330 Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)
Δfunc

Theoremdiag2cl 14331 The diagonal functor at a morphism is a natural transformation between constant functors. (Contributed by Mario Carneiro, 7-Jan-2017.)
Δfunc                                                               Nat

Theoremcurf2ndf 14332 As shown in diagval 14325, 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 14333 Extend class notation with the Hom functor.
HomF

Syntaxcyon 14334 Extend class notation with the Yoneda embedding.
Yon

Definitiondf-hof 14335* 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 14336 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 14337* 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 14338 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 14339 The object part of the Hom functor maps to the set of morphisms from to . (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF

Theoremhof2fval 14340* 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 14341* 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 14342 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 14343 Lemma for hofcl 14344. (Contributed by Mario Carneiro, 15-Jan-2017.)
HomF       oppCat                            f                                                                                            comp comp comp

Theoremhofcl 14344 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 14345 Closure of the opposite Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
oppCat       HomF                            f        c

Theoremyonval 14346 Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon              oppCat       HomF       curryF

Theoremyoncl 14347 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 14348 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 14349 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 14350 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 14351 Value of the Yoneda embedding at a morphism. (Contributed by Mario Carneiro, 17-Jan-2017.)
Yon                                          comp

Theoremhofpropd 14352 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 14353 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 14354 Value of the opposite Yoneda embedding. (Contributed by Mario Carneiro, 26-Jan-2017.)
oppCat       Yon       HomF              curryF

Theoremoyoncl 14355 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 14356 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 14357 Lemma for yoneda 14368. (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 14358 Lemma for yoneda 14368. (Contributed by Mario Carneiro, 28-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                      Nat

Theoremyonedalem3a 14359* Lemma for yoneda 14368. (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 14360* Lemma for yoneda 14368. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f

Theoremyonedalem4b 14361* Lemma for yoneda 14368. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f

Theoremyonedalem4c 14362* Lemma for yoneda 14368. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                                    Nat

Theoremyonedalem22 14363 Lemma for yoneda 14368. (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f                                    Nat

Theoremyonedalem3b 14364* Lemma for yoneda 14368. (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 14365* Lemma for yoneda 14368. (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 14366* 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 14367* Lemma for yonffth 14369. (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

Theoremyoneda 14368* The Yoneda Lemma. There is a natural isomorphism between the functors and , where is the natural transformations from Yon to , and is the evaluation functor. Here we need two universes to state the claim: the smaller universe is used for forming the functor category op , which itself does not (necessarily) live in but instead is an element of the larger universe . (If is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set in this case.) (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon                     oppCat                     FuncCat        HomF       c FuncCat        evalF        func tpos func F ⟨,⟩F F                      f        f        Nat

Theoremyonffth 14369 The Yoneda Lemma. The Yoneda embedding, the curried Hom functor, is full and faithful, and hence is a representation of the category as a full subcategory of the category of presheaves on . (Contributed by Mario Carneiro, 29-Jan-2017.)
Yon       oppCat              FuncCat                      f        Full Faith

Theoremyoniso 14370* If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)
Yon       oppCat              CatCat                     FuncCat        s                             f

PART 9  BASIC ORDER THEORY

9.1  Presets and directed sets using extensible structures

Syntaxcpreset 14371 Extend class notation with the class of all presets.

Syntaxcdrs 14372 Extend class notation with the class of all directed sets.
Dirset

Definitiondf-preset 14373* Define the class of preordered sets (presets). A preset is a set equipped with a transitive and reflexive relation.

Preorders are a natural generalization of order for sets where there is a well-defined ordering, but it in some sense "fails to capture the whole story", in that there may be pairs of elements which are indistinguishable under the order. Two elements which are not equal but are less-or-equal to each other behave the same under all order operations and may be thought of as "tied".

A preorder can naturally be strengthened by requiring that there are no ties, resulting in a partial order, or by stating that all comparable pairs of elements are tied, resulting in an equivalence relation. Every preorder naturally factors into these two types; the tied relation on a preorder is an equivalence relation and the quotient under that relation is a partial order. (Contributed by FL, 17-Nov-2014.) (Revised by Stefan O'Rear, 31-Jan-2015.)

Definitiondf-drs 14374* Define the class of directed sets. A directed set is a nonempty preordered set where every pair of elements have some upper bound. Note that it is not required that there exist a least upper bound.

There is no consensus in the literature over whether directed sets are allowed to be empty. It is slightly more convenient for us if they are not. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Dirset

Theoremisprs 14375* Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Theoremprslem 14376 Lemma for prsref 14377 and prstr 14378. (Contributed by Mario Carneiro, 1-Feb-2015.)

Theoremprsref 14377 Less-or-equal is reflexive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremprstr 14378 Less-or-equal is transitive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremisdrs 14379* Property of being a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsdir 14380* Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsprs 14381 A directed set is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsbn0 14382 The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremdrsdirfi 14383* Any finite number of elements in a directed set have a common upper bound. Here is where the non-emptiness constraint in df-drs 14374 first comes into play; without it we would need an additional constraint that not be empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

Theoremisdrs2 14384* Directed sets may be defined in terms of finite subsets. Again, without nonemptiness we would need to restrict to nonempty subsets here. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Dirset

9.2  Posets and lattices using extensible structures

9.2.1  Posets

Syntaxcpo 14385 Extend class notation with the class of posets.

Syntaxcplt 14386 Extend class notation with less-than for posets.

Syntaxclub 14387 Extend class notation with poset least upper bound.

Syntaxcglb 14388 Extend class notation with poset greatest lower bound.

Syntaxcjn 14389 Extend class notation with poset join.

Syntaxcmee 14390 Extend class notation with poset meet.

Definitiondf-poset 14391* Define the class of posets. Definition of poset in [Crawley] p. 1. Note that Crawley-Dilworth require that a poset base set be nonempty, but we follow the convention of most authors who don't make this a requirement.

The quantifiers provide a notational shorthand to allow us to refer to the base and ordering relation as and the definition rather than having to repeat and throughout. These quantifiers can be eliminated with ceqsex2v 2985 and related theorems. (Contributed by NM, 18-Oct-2012.)

Theoremispos 14392* The predicate "is a poset." (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)

Theoremispos2 14393* A poset is an antisymmetric preset.

EDITORIAL: could become the definition of poset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremposprs 14394 A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Theoremposi 14395 Lemma for poset properties. (Contributed by NM, 11-Sep-2011.)

Theoremposref 14396 A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.)

Theoremposasymb 14397 A poset ordering is asymetric. (Contributed by NM, 21-Oct-2011.)

Theorempostr 14398 A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)

Theorem0pos 14399 Technical lemma to simplify the statement of ipopos 14574. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 13493) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)

Theoremisposd 14400* Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.)

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