singularHomologyFunctorZeroOfTotallyDisconnectedSpace

English

The nth singular homology functor with coefficients in C is defined as a functor from C to TopCat ⥤ C.

Русский

n-ая гомологическая функция задается как функтор из C в TopCat ⥤ C.

LaTeX

$$$$ \mathrm{singularHomologyFunctor} : C \to \mathrm{TopCat} \;\to \; C $$$$

Lean4

/-- The zeroth singular homology of a totally disconnected space is the
free `R`-module generated by elements of `X`. -/
noncomputable def singularHomologyFunctorZeroOfTotallyDisconnectedSpace :
    ((singularHomologyFunctor C 0).obj R).obj X ≅ ∐ fun _ : X ↦ R :=
  have := hasCoproducts_shrink.{0, w} (C := C)
  have : HasZeroObject C := ⟨_, initialIsInitial.isZero⟩
  (HomologicalComplex.homologyFunctor _ _ 0).mapIso (singularChainComplexFunctorIsoOfTotallyDisconnectedSpace C R X) ≪≫
    ChainComplex.alternatingConstHomologyZero _

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