English
The HomComplex is the cochain complex of degreewise morphisms from F to G with the natural horizontal differentials.
Русский
HomComplex есть когайн-комплекс степенной последовательности морфизмов F→G с естественными вертикальными дифференциалами.
LaTeX
$$$ \\text{HomComplex}(F,G)\\_i = \\text{Cochain}(F,G,i),\\; d_i = δ_{hom}(F,G,i,i+1) $$$
Lean4
/-- The cochain complex of homomorphisms between two cochain complexes `F` and `G`.
In degree `n : ℤ`, it consists of the abelian group `HomComplex.Cochain F G n`. -/
@[simps! X d_hom_apply]
def HomComplex : CochainComplex AddCommGrpCat ℤ
where
X i := AddCommGrpCat.of (Cochain F G i)
d i j := AddCommGrpCat.ofHom (δ_hom ℤ F G i j)
shape _ _ hij := by ext; simp [δ_shape _ _ hij]
d_comp_d' _ _ _ _ _ := by ext; simp [δ_δ]
