nLab
cup product

Idea

Recall from the discussion at cohomology that every notion of cohomology (e.g. group cohomology, abelian sheaf cohomology, etc) is given by Hom-spaces in an (∞,1)-topos $H$. Cohomology on an object $X\in H$ with coefficients in an object $A\in H$ is

$$H(X,A):={\pi }_{0}H(X,A).$$H(X,A) := \pi_0 \mathbf{H}(X,A) \,.

The cup product is an operation on cocycles with coefficients $A_1$ and $A_2$ that is induced from a pairing morphism

$$A_1\times A_2\to A_3$$A_1 \times A_2 \to A_3

in $H$. In applications this is often this is a pairing operation with $A_1=A_2$, i.e. $A\times A\to A\prime$

If $g_1:X\to A_1$ and $g_2:X\to A_2$ are two cocycles in $H(X,A_1)$ and $H(X,A_2)$, respectively, then their cup product with respect to this pairing is the cocycle

$$g_1\cdot g_2:X\stackrel{\mathrm{Id}\times \mathrm{Id}}{\to }X\times X\stackrel{g_1\times g_2}{\to }{A}_1\times A_2\to A_3$$g_1 \cdot g_2 : X \stackrel{Id \times Id}{\to} X \times X \stackrel{g_1 \times g_2}{\to} A_1 \times A_2 \to A_3

in $H(X,A_3)$.

On Moore complexes of cosimplicial algebras

For $A=(A^{•})$ any cosimplicial algebra, its dual Moor cochain complex $N^{•}(A)$ naturally inherits the structure of a dg-algebra under the cup product.

The general formula is literally the same as that for the case where $A^{•}$ is functions on the singular complex of a space, wich is discussed below. For the moment, see below.

This cup product operation on $N^{•}(A)$ is not in general commutative. However, it is a standarrd fact that it becomes commutative after passing to cochain cohomology.

This suggests that the cup product should be, while not commutative, be homotopy commutative in that it makes $N^{•}(A)$ a homotopy commutative monoid object.

This in turn should mean that $N^{•}(A)$ is an algebra over an operad for the E-∞ operad.

That this is indeed the case is the main statement in

• Clemens Berger, Benoit Fresse? Combinatorial operad actions on cochains (arXiv)

In singular cohomology

A special case of the cup product on Moore complexes is the complex of singular cohomology, which is the Moore complex of the cosimplicial algebra of functions on the singular simplicial set of a topological space.

Often in the literature by cup product is meant specifically the realization of the cup product on singular cohomology.

For $X$ a topological space, let $\Pi (X{)}_{•}:=X^{{\Delta }_{\mathrm{Top}}^{•}}$ be the simplicial set of $n$-simplices in $X$ – the fundamental ∞-groupoid of $X$.

For $R$ some ring, let $\mathrm{Maps}(\Pi (X),R{)}^{•}$ be the cosimplicial ring of $R$-valued functions on the spaces of $n$-simplices. The corresponding Moore cochain complex $C^{•}(X)$ is the cochain complex whose cochain cohomology is the singular cohomology of the space $X$: a homogeneous element ${\omega }_p\in C^p(X)$ is a function on $p$-simplices in $X$.

Write, as usual, for $p\in \mathbb{N}$, $[p]=\{0<1<\cdots <p\}$ for the totally ordered set with $p+1$ elements. For $\mu :[p]\to [p+q]$ an injective order preserving map and $K$ some cosimplicial object, write $d_{\mu }^{*}K:K^p\to K^{p+q}$ for the image of this map under $K$.

Specifically, for $p,q\in \mathbb{N}$ let $L:[p]\to [p+q]$ be the map that sends $i\in [p]$ to $i\in [p+q]$ and let $R:[q]\to [p+q]$ be the map that sends $i\in [q]$ to $i+q\in [p+q]$.

Then the cup product

$$⌣:C^{•}(X)\otimes C^{•}(X)\to C^{•}(X)$$\smile : C^\bullet(X) \otimes C^\bullet(X) \to C^\bullet(X)

is the cochain map that on homogeneous elements $a\otimes b\in C^p(X)\otimes C^q(X)\subset C^{•}(X)\otimes C^{•}(X)$ is defined by the formula

$$a⌣b=(d_L^{*}a)\cdot (d_R^{*}b).$$a \smile b = (d_L^* a) \cdot (d_R^* b) \,.

This means that $(a⌣b{)}_{i_0,\cdots ,i_{p+q}}=a_{i_0,\cdots ,i_p}\cdot b_{i_p,\cdots ,i_{p+q}}$.

Proposition

This cup product enjoys the following properties:

• it is indeed a cochain complex morphism as claimed, in that it respects the differential: for any homogeneosu $a\otimes b\in C^p(X)\otimes C^q(X)$ as above we have

  $$d(a⌣b)=(da)⌣b+(-1{)}^{p}a⌣(db).$$d(a \smile b) = (d a) \smile b + (-1)^p a \smile (d b) \,.

• the image of the cup product on cochain cohomology

  $$⌣:H^{•}(C^{•}(X))\otimes H^{•}(C^{•}(X))\to H^{•}(C^{•}(X))$$\smile : H^\bullet(C^\bullet(X))\otimes H^\bullet(C^\bullet(X)) \to H^\bullet(C^\bullet(X))

  is associative and distributes over the additon in $H^{•}(C^{•}(X))$.

Accordingly, the cup product makes $H^{•}(C^{•}(X))=H^{•}(X,R)$ into a ring: the cohomology ring on the ordinary cohomology of $X$.

See for instance section 3.2 of

• Hatcher, Algebraic Topology (web pdf)

In abelian sheaf cohomology

Traditionally the cup product is considered for abelian cohomology, such as generalized (Eilenberg-Steenrod) cohomology and more generally abelian sheaf cohomology.

In that case all coefficient objects $A_i$ are complexes $(A_i{)}_{•}$ of sheaves and the pairing that one usually considers is the tensor product of chain complexes

$$(A_1{)}_{•}\times (A_2{)}_{•}\to (A_1\otimes A_2{)}_{•}$$(A_1)_\bullet \times (A_2)_\bullet \to (A_1 \otimes A_2)_\bullet

where

$$(A_1\otimes A_2{)}_n={\oplus }_p(A_1{)}_p\otimes (A_2{)}_{n-p}.$$(A_1 \otimes A_2)_n = \oplus_p (A_1)_p \otimes (A_2)_{n-p} \,.

with differential

$$d(a_1\otimes a_2)=(d{a}_1)\otimes a_2+(-1{)}^{\mid a_1\mid }{a}_1\otimes d{a}_2.$$d (a_1 \otimes a_2) = (d a_1) \otimes a_2 + (-1)^{|a_1|} a_1 \otimes d a_2 \,.

In abelian Čech cohomology

The cup product has a simple expression in abelian Čech cohomology.

For $\alpha \in C(U,A_1{)}_{•}$ and $\beta \in C(U,A_2{)}_{•}$ two Čech cocycles with respect to the same cover $\{U_i\to X\}$ of $X$, their cup product is simply the cocycle in $C(U,A_1\otimes A_2)$ given by

$$(\alpha \cdot \beta {)}_{i_0,\cdots ,i_{p+q}}:={\alpha }_{i_0,\cdots ,i_p}\otimes {\beta }_{i_p,\cdots i_{p+q}}.$$(\alpha \cdot \beta)_{i_0, \cdots , i_{p + q}} \;:=\; \alpha_{i_0, \cdots, i_p} \otimes \beta_{i_p, \cdots i_{p+q}} \,.

References

Recall from the discussion at models for ∞-stack (∞,1)-toposes that all hypercomplete ∞-stack (∞,1)-toposes are modeled by the model structure on simplicial presheaves. Accordingly understanding the cup product on simplicial presheaves goes a long way to wards the most general description. For a bit of discussion of this see around page 19 of

• Jardine Lectures on simplicial presheaves (pdf)

An early treatment of cup product can be found in this classic

• Whitney, On Products in a Complex (JSTOR)