The Alexander–Whitney map #
Let C be a preadditive monoidal category with w-small coproducts, and let R and S be
objects of C. For simplicial sets K and L, the Alexander–Whitney map is the morphism
of chain complexes SSet.alexanderWhitney K L R S from (K ⊗ L).chainComplex (R ⊗ S), the
simplicial chains of the product K × L, to K.chainComplex R ⊗ L.chainComplex S, the tensor
product of the simplicial chains of the factors. On an n-simplex (x, y) of K × L it is
∑_{p + q = n} x|[0, …, p] ⊗ y|[p, …, n],
the front p-face of x tensored with the back q-face of y, the faces being Mathlib's
SimplexCategory.subinterval. The tensor product of chain complexes is Mathlib's monoidal
structure on ChainComplex C ℕ, whose differential is d (a ⊗ b) = d a ⊗ b + (-1)^p a ⊗ d b
for a of degree p.
The Alexander–Whitney map is one half of the Eilenberg–Zilber comparison between chains on a
product and tensor products of chains, and composing it with the diagonal gives the cup product
of cochains. For the usual coefficients take C := ModuleCat k and R = S = k.
Main definitions and results #
SSet.alexanderWhitney: the Alexander–Whitney chain map.SSet.ιChainComplex_alexanderWhitney_f: its value on a simplex.SSet.alexanderWhitney_naturality: it is natural in both simplicial sets.SSet.alexanderWhitney_coefficient_naturality: it is natural in both coefficient objects.
References #
- S. Eilenberg and J. A. Zilber, On products of complexes, Amer. J. Math. 75 (1953).
- C. Weibel, An Introduction to Homological Algebra, Section 8.5.
The Alexander–Whitney map C(K × L; R ⊗ S) ⟶ C(K; R) ⊗ C(L; S), sending an n-simplex
(x, y) of K × L to ∑_{p + q = n} x|[0, …, p] ⊗ y|[p, …, n], the front p-face of x
tensored with the back q-face of y (SSet.ιChainComplex_alexanderWhitney_f). It is a
morphism of chain complexes for the Koszul sign convention on the tensor product.
Equations
- K.alexanderWhitney L R S = { f := SSet.alexanderWhitneyX✝ K L R S, comm' := ⋯ }
Instances For
The Alexander–Whitney map on the summand of an n-simplex x of K × L: the sum over
p ≤ n of the front p-face of x.1 tensored with the back (n - p)-face of x.2.
The Alexander–Whitney map on the summand of an n-simplex x of K × L: the sum over
p ≤ n of the front p-face of x.1 tensored with the back (n - p)-face of x.2.
The Alexander–Whitney map is natural in both simplicial sets.
The Alexander–Whitney map is natural in both simplicial sets.
The Alexander–Whitney map is natural in both coefficient objects.
The Alexander–Whitney map is natural in both coefficient objects.