English
There is an isomorphism between the product of fundamental groupoids and the fundamental groupoid of the product space.
Русский
Существует изоморфизм между произведением фундаментальных grupaодов и фундаментальной grupaодой произведения пространств.
LaTeX
$$$$ \pi_x A \\times \\pi_x B \\cong \\pi_x (TopCat.of (A \\times B)). $$$$
Lean4
theorem eq_diag_path :
(πₘ (TopCat.ofHom f)).map p ≫ ⟦H.evalAt x₁⟧ = H.diagonalPath' p ∧
(⟦H.evalAt x₀⟧ ≫ (πₘ (TopCat.ofHom g)).map p : fromTop (f x₀) ⟶ fromTop (g x₁)) = H.diagonalPath' p :=
by
rw [H.apply_zero_path, H.apply_one_path, H.evalAt_eq]
erw [H.evalAt_eq]
dsimp only [prodToProdTopI]
constructor
· slice_lhs 2 4 =>
rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this
slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]
rfl
· slice_lhs 2 4 =>
rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this
slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]
rfl
