English
Reiterates that the strict Segal structure provides IsStrictSegal by taking spineEquiv to be bijective for all m,h.
Русский
Повторяет, что строгая структура Segal обеспечивает IsStrictSegal, поскольку spineEquiv является биективным для всех m,h.
LaTeX
$$IsStrictSegal X with segal m h := (spineEquiv m h).bijective$$
Lean4
/-- If we take the path along the spine of the `j`th face of a `spineToSimplex`,
the common vertices will agree with those of the original path `f`. In particular,
a vertex `i` with `i < j` can be identified with the same vertex in `f`. -/
theorem spine_δ_vertex_lt (hij : i.castSucc < j) :
(X.spine m _ (X.map (tr (δ j)).op (sx.spineToSimplex (m + 1) _ f))).vertex i = f.vertex i.castSucc :=
by
rw [spine_vertex, ← FunctorToTypes.map_comp_apply, ← op_comp, ← tr_comp, SimplexCategory.const_comp,
spineToSimplex_vertex]
dsimp only [δ, len_mk, mkHom, Hom.toOrderHom_mk, Fin.succAboveOrderEmb_apply, OrderEmbedding.toOrderHom_coe]
rw [Fin.succAbove_of_castSucc_lt j i hij]
