The Alexander–Whitney map of simplicial objects #
Let C be a preadditive monoidal category with finite coproducts, and let A and B be
simplicial objects in C. Their tensor product A ⊗ B is taken degreewise,
(A ⊗ B)ₙ = Aₙ ⊗ Bₙ, with the tensor product of the structure maps. The Alexander–Whitney map
CategoryTheory.SimplicialObject.alexanderWhitney A B is the morphism of chain complexes from
the alternating face map complex of A ⊗ B to the tensor product of the alternating face map
complexes of A and of B. In degree n it is
∑_{p + q = n} A(front p-face) ⊗ B(back q-face),
where the front p-face of ⦋n⦌ is the inclusion of the vertices 0, …, p and the back q-face
is the inclusion of the vertices p, …, n, both given by 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 simplicial chains of a product of simplicial sets are the case where A and B are free on
the simplices of the factors (SSet.alexanderWhitney). Singular chains with coefficients in a
local coefficient system form a simplicial module that is not free on the simplices with a fixed
coefficient object, which is why the construction is stated for arbitrary simplicial objects: the
Alexander–Whitney diagonal of chains with local coefficients, along which cup and cap products
with local coefficients are taken, is an instance of it.
The Alexander–Whitney map is coassociative and counital, with counit the identity of 𝟙_ C in
degree zero, so it makes the alternating face map complex an oplax monoidal functor from
simplicial objects with the degreewise tensor product to chain complexes. On normalized chains,
the mate of this structure along the Dold–Kan correspondence is the lax monoidal structure of the
Dold–Kan functor from chain complexes to simplicial objects (J. Lurie, Higher Algebra,
Construction 1.3.1.13), which underlies the simplicial enrichment of a differential graded
category.
Main definitions and results #
CategoryTheory.SimplicialObject.alexanderWhitney: the Alexander–Whitney chain map.CategoryTheory.SimplicialObject.alexanderWhitney_f: its degree-ncomponent.CategoryTheory.SimplicialObject.alexanderWhitney_naturality: it is natural in both simplicial objects.CategoryTheory.SimplicialObject.alexanderWhitney_coassoc: it is coassociative.CategoryTheory.SimplicialObject.alternatingFaceMapComplexOplaxMonoidal: the instance(alternatingFaceMapComplex C).OplaxMonoidal, with comultiplication the Alexander–Whitney map (CategoryTheory.SimplicialObject.alternatingFaceMapComplex_δ) and counit the identity of𝟙_ Cin degree zero (CategoryTheory.SimplicialObject.alternatingFaceMapComplex_η_f_zero).
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.
- J. Lurie, Higher Algebra, Construction 1.3.1.13.
The Alexander–Whitney map of two simplicial objects, from the alternating face map
complex of their degreewise tensor product to the tensor product of their alternating face map
complexes. In degree n it is the sum over p + q = n of the front p-face of A tensored
with the back q-face of B (CategoryTheory.SimplicialObject.alexanderWhitney_f). It is a
morphism of chain complexes for the Koszul sign convention on the tensor product.
Instances For
The degree-n component of the Alexander–Whitney map: the sum over p ≤ n of the front
p-face of A tensored with the back (n - p)-face of B.
The Alexander–Whitney map is natural in both simplicial objects.
The Alexander–Whitney map is natural in both simplicial objects.
Coassociativity of the Alexander–Whitney map: the two ways of splitting a simplex into a front, a middle and a back face agree, up to the associators of simplicial objects and of chain complexes.
Coassociativity of the Alexander–Whitney map: the two ways of splitting a simplex into a front, a middle and a back face agree, up to the associators of simplicial objects and of chain complexes.
The Alexander–Whitney oplax monoidal structure on the alternating face map complex
functor: the comultiplication is the Alexander–Whitney map
(CategoryTheory.SimplicialObject.alternatingFaceMapComplex_δ), and the counit is the identity
of 𝟙_ C in degree zero (CategoryTheory.SimplicialObject.alternatingFaceMapComplex_η_f_zero).
Coassociativity is CategoryTheory.SimplicialObject.alexanderWhitney_coassoc.
The comultiplication of the alternating face map complex functor is the Alexander–Whitney map.
The counit of the alternating face map complex functor is the identity of 𝟙_ C in degree
zero; in positive degrees its target is zero.