[Merged by Bors] - refactor(CategoryTheory/SmallObject): generalization of the definitions#20256
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[Merged by Bors] - refactor(CategoryTheory/SmallObject): generalization of the definitions#20256
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…iterations of functors in two cases
Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>
Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
… small-object-8-bis-refactor
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Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
Co-authored-by: Markus Himmel <markus@himmel-villmar.de>
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…ns (#20256) The ongoing definition of the iteration of a natural transformation `ε : 𝟭 C ⟶ F` (with `F : C ⥤ C`) is generalized to "successor structures" (which shall become a mathlibism), i.e. in a category `D`, this consists of a zeroth object `X₀`, a successor application `succ : D → D` and, for all `X : D`, a map `toSucc X : X → succ X` (which does not have to be natural: it is not always so in some applications). For such a `Φ : SuccStruct D`, if `J` is a well-ordered type, we define the `J`-th iteration of `Φ`. (In the case `J := ℕ`, this is the colimit of `succ (succ (succ (succ ... X₀)))`.) The iteration of a functor is a particular case of this constructor with `D := C ⥤ C`. As `toSucc` does not have to be natural in `X`, the caveat is that the proofs make extensive use of equalities of objects in `C` and `Arrow C`, while my previous construction used comparison isomorphisms. Nevertheless, the proofs look much more clean now. One of the reasons is that in the inductive construction (file `Iteration.Nonempty`), in the terms of data, we only need to provide a functor, and then all the fields are in `Prop`. (In the downstream API, we shall obviously use isomorphisms instead of equalities...) This PR supersedes #19264. The results are used in #20245 in order to get functorial factorizations in the small object argument. After refactoring my code, I found that this approach had already been used in 2018 by Reid Barton in his pioneering formalization work in Lean 3 towards the model category structure on topological spaces. Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr>
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The ongoing definition of the iteration of a natural transformation
ε : 𝟭 C ⟶ F(withF : C ⥤ C) is generalized to "successor structures" (which shall become a mathlibism), i.e. in a categoryD, this consists of a zeroth objectX₀, a successor applicationsucc : D → Dand, for allX : D, a maptoSucc X : X → succ X(which does not have to be natural: it is not always so in some applications). For such aΦ : SuccStruct D, ifJis a well-ordered type, we define theJ-th iteration ofΦ. (In the caseJ := ℕ, this is the colimit ofsucc (succ (succ (succ ... X₀))).)The iteration of a functor is a particular case of this constructor with
D := C ⥤ C.As
toSuccdoes not have to be natural inX, the caveat is that the proofs make extensive use of equalities of objects inCandArrow C, while my previous construction used comparison isomorphisms. Nevertheless, the proofs look much more clean now. One of the reasons is that in the inductive construction (fileIteration.Nonempty), in the terms of data, we only need to provide a functor, and then all the fields are inProp. (In the downstream API, we shall obviously use isomorphisms instead of equalities...)This PR supersedes #19264. The results are used in #20245 in order to get functorial factorizations in the small object argument.
After refactoring my code, I found that this approach had already been used in 2018 by Reid Barton in his pioneering formalization work in Lean 3 towards the model category structure on topological spaces.