Contents

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

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

in $H$. In applications this is often 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

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, which is discussed below. For the moment, see below.

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

This suggests that the cup product should be, while not commutative, 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

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

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

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

where

with differential

In abelian Čech cohomology

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

For $A_1$ and $A_2$ two chain complexes (of sheaves of abelian groups) construct a morphism of Čech complexes

by sending $\alpha \in C^p(U,A_1{)}_{•}$ and $\beta \in C^q(U,A_2{)}_{•}$ to

For instance (Brylinski, section (1.3)) spring

In Čech-Deligne cohomology (ordinary differential cohomology)

For the case that of Čech hypercohomology with coefficients in Deligne complexes the above yields the Beilinson-Deligne cup-product for ordinary differential cohomology.

Related concepts

• intersection pairing

• Baer sum

• cohomology operation

• Massey product

References

The cup product in Čech cohomology is discussed for instance in section 1.3 of

• Jean-Luc Brylinski, Loop spaces and characteristic classes

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 towards the most general description. For a bit of discussion of this see around page 19 of

• Rick 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)