CategoryTheory.Limits.Types.descSet_spec

theorem CategoryTheory.Limits.Types.descSet_spec : ∀ {J : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} J] {K : CategoryTheory.Functor J (Type u)} {c : CategoryTheory.Limits.Cocone K} (hc : CategoryTheory.Limits.IsColimit c) {lc : CategoryTheory.Limits.Cocone (K.comp CategoryTheory.uliftFunctor.{v, u})} (s : Set c.pt) (ls : Set lc.pt), CategoryTheory.Limits.Types.descSet hc ls = s ↔ ∀ (j : J) (x : K.obj j), lc.ι.app j { down := x } ∈ ls ↔ c.ι.app j x ∈ s

This theorem, `CategoryTheory.Limits.Types.descSet_spec`, describes a property of the map, `descSet hc`, in the context of category theory and cocones. It states that for any category `J`, a functor from `J` to `Type u` denoted by `K`, a cocone `c` over `K` that is a colimit, another cocone `lc` over the functor composition of `K` and the `uliftFunctor`, and sets `s` and `ls` of points of `c` and `lc` respectively, the set `descSet hc ls` equals `s` if and only if the following condition holds: for all objects `j` in `J` and all elements `x` in the functor `K` applied to `j`, the element `{ down := x }` under the natural transformation of `lc` applied to `j` lies in `ls` if and only if the element `x` under the natural transformation of `c` applied to `j` lies in `s`. Here, `descSet hc ls` is a mapping that associates to each element in a vertex of the original diagram its image in the cocone point. It captures the idea of mapping elements from the original diagram to the cocone point and then assessing their membership in different sets.

More concisely: For any category J, functor K : J -> Type u, colimit cocone c over K, another cocone lc over the composition of K and uliftFunctor, and sets s of points in c and ls of points in lc, the sets s and descSet hc ls are equal if and only if, for all j in J and all x in K j, the element x under the natural transformation of lc applied to j is in ls if and only if the element x under the natural transformation of c applied to j is in s.