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Theorems · Definition · algebraic topology

FundamentalGroupoidFunctor.piIso

{I : Type u} →
  (X : I → TopCat) →
    CategoryTheory.Grpd.of ((i : I) → ↑(FundamentalGroupoid.fundamentalGroupoidFunctor.obj (X i))) ≅
      FundamentalGroupoid.fundamentalGroupoidFunctor.obj (TopCat.of ((i : I) → ↑(X i)))

Shows piToPiTop is an isomorphism, whose inverse is precisely the pi product of the induced projections. This shows that fundamentalGroupoidFunctor preserves products.

Defined in
Mathlib.AlgebraicTopology.FundamentalGroupoid.Product
Cited by
2 results in Mathlib
Foundations
Depth 144 from the axioms · uses propext, Classical.choice, Quot.sound
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FundamentalGroupoidFunctor.piIso_hom · cited by 0FundamentalGroupoidFuncto…FundamentalGroupoidFunctor.piIso_inv · cited by 0FundamentalGroupoidFuncto…CategoryTheory.Functor.obj · cited by 19642Functor.objCategoryTheory.Iso · cited by 3963CategoryTheory.IsoTopCat.carrier · cited by 3184TopCat.carrierTopCat · cited by 1889TopCatCategoryTheory.Bundled.α · cited by 736Bundled.αCategoryTheory.Groupoid · cited by 182CategoryTheory.GroupoidCategoryTheory.Grpd · cited by 30CategoryTheory.GrpdCategoryTheory.Functor.pi' · cited by 23Functor.pi'FundamentalGroupoid.fundamentalGroupoidFunctor · cited by 21FundamentalGroupoid.funda…CategoryTheory.Grpd.of · cited by 9Grpd.ofFundamentalGroupoidFunctor.proj · cited by 3FundamentalGroupoidFuncto…FundamentalGroupoidFunctor.piToPiTop · cited by 3FundamentalGroupoidFuncto…FundamentalGroupoidFunctor.pi…CITED BYCITES

Cites12

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