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 $\mathbf{H}$. Cohomology on an object $X \in \mathbf{H}$ with coefficients in an object $A \in \mathbf{H}$ is

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

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

$$A_1 \times A_2 \longrightarrow A_3$$

in $\mathbf{H}$. In applications this is often a pairing operation with $A_1 = A_2$, i.e. $A \times A \to A'$, and typically it is the product morphism $A \times A \to A$ for a ring object structure on the coefficients $A$. (See at multiplicative cohomology theory).

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

$$g_1 \cdot g_2 \;\coloneqq\; X \stackrel{(id,id)}{\longrightarrow} X \times X \stackrel{g_1 \times g_2}{\to} A_1 \times A_2 \to A_3$$

in $\mathbf{H}(X,A_3)$ obtained by combining the pairing with precomposition by the diagonal map $\Delta_X = (id_X, id_X)$.

Examples

Via the Dold-Kan correspondence

When the coefficient object $A \in$∞Grpd is “sufficiently abelian” in that under the Dold-Kan correspondence it is represented by a chain complex then using the lax monoidalness of the Dold-Kan correspondence (see at monoidal Dold-Kan correspondence) one obtains a chain complex model for the cup product which makes the origin of the typical grading shift manifest.

Write

• $(Ch_{\bullet \geq 0}, \otimes)$ for the category of chain complexes of abelian groups, in non-negative degrees;, regarded as a symmetric monoidal category with the standard tensor product of chain complexes $\otimes$;

• $(sAb, \otimes)$ for the category of simplicial abelian groups, regarded as a symmetric monoidal category with the degreewise tensor product of abelian groups;

• $U \;\colon\; sAb \longrightarrow KanCplx \to sSet$ for the forgetful functor to the underlying simplicial sets (which happens to land in Kan complexes);

• $\Gamma \;\colon \; Ch_{\bullet \geq 0} \stackrel{\simeq}{\longrightarrow} sAb$ for the equivalence of categories given by the Dold-Kan correspondence;

• $DK \;\colon\; Ch_{\bullet \geq 0} \underoverset{\simeq}{\Gamma}{\longrightarrow} sAb \stackrel{F}{\longrightarrow} KanCplx \hookrightarrow sSet$ for the composite.

Now:

• $\Gamma$ is a lax monoidal functor, the lax monoidal structure $\gamma_{A,B} \;\colon\; \Gamma(A) \otimes \Gamma(B) \to \Gamma(A \otimes B)$ being induced dually by the Alexander-Whitney map;

• $U$ is a strong monoidal functor;

• for the tensor product of abelian groups $A \otimes B$ there are canonical natural bilinear maps of underlying sets

  $$p_{A,B} \;\colon\; U(A)\times U(B) \longrightarrow U(A \otimes B)$$

Using all this, then for

$$f \;\colon\; V_\bullet \otimes W_\bullet \longrightarrow Z_\bullet$$

a given chain map, this induces a map of the corresponding Kan complexes

$$\cup_{DK} \;\colon\; DK(V_\bullet) \times DK(W_\bullet) \longrightarrow DK(Z_\bullet)$$

as the following composite

$$\cup_{DK} \;\colon\; DK(V_\bullet)\times DK(W_\bullet) = U(\Gamma(V_\bullet)) \times U(\Gamma(W_\bullet)) \stackrel{p_{\Gamma(V_\bullet), \Gamma(W_\bullet)} }{\to} U(\Gamma(V_\bullet)\otimes \Gamma(W_\bullet)) \stackrel{U(\gamma)}{\longrightarrow} U(\Gamma(V_\bullet \otimes W_\bullet)) \stackrel{U(\Gamma(f))}{\to} U(\Gamma(Z_\bullet)) = DK(Z_\bullet) \,.$$

With this in hand then for $X$ any homotopy type, the cup product on its cohomology with coefficients in $DK(V_\bullet)$ and $DK(W_\bullet)$ is induced by just homming $X$ into this morphism:

$$\mathbf{H}(X, DK(V_\bullet)) \times \mathbf{H}(X, DK(W_\bullet)) \simeq \mathbf{H}(X, DK(V_\bullet) \times DK(W_\bullet)) \stackrel{\mathbf{H}(X,\cup_{DK})}{\longrightarrow} \mathbf{H}(X, DK(Z_\bullet)) \,.$$

For example if

$$V_\bullet = \mathbb{Z}[n_1] \,, \;\;\; W_\bullet = \mathbb{Z}[n_2]$$

is the chain complex concentrated in degree $n_1$ and $n_2$, respectively, on the group of integers, then

$$DK(V_\bullet) \simeq B^{n_1} \mathbb{Z} \simeq K(\mathbb{Z},n_1)$$

is the corresponding Eilenberg-MacLane space which classifies ordinary cohomology (singular cohomology) with integral coefficients in the given degree. By the nature of the tensor product of chain complexes one has

$$V_\bullet \otimes W_\bullet \simeq \mathbb{Z}[n_1 + n_2] \,.$$

Hence we may take $Z_\bullet \coloneqq \mathbb{Z}[n_1 + n_2]$ and $f = id$ and we get a cup product

$$\cup \;\colon\; H^{n_1}(X, \mathbb{Z}) \times H^{n_2}(X, \mathbb{Z}) \to H^{n_1 + n_2}(X, \mathbb{Z}) \,.$$

On Moore complexes of cosimplicial algebras

For $A = (A^\bullet)$ any cosimplicial algebra, its dual Moore cochain complex $N^\bullet(A)$ 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 $A^\bullet$ is functions on the singular complex of a space, which is discussed below. For the moment, see below.

This cup product operation on $N^\bullet(A)$ 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 $N^\bullet(A)$ a homotopy commutative monoid object.

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

That this is indeed the case is the main statement in (Berger-Fresse 01)

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 $\Pi(X)_\bullet := X^{\Delta_{Top}^\bullet}$ be the simplicial set of $n$-simplices in $X$ – the fundamental ∞-groupoid of $X$.

For $R$ some ring, let $Maps(\Pi(X),R)^{\bullet}$ be the cosimplicial ring of $R$-valued functions on the spaces of $n$-simplices. The corresponding Moore cochain complex $C^\bullet(X)$ 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 $p \in \mathbb{N}$, $[p] = \{0 \lt 1 \lt \cdots \lt p\}$ 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

$$\smile : C^\bullet(X) \otimes C^\bullet(X) \to C^\bullet(X)$$

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

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

There is some glue missing here to connect this back to the above general definition, something involving the Eilenberg-Zilber map.

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

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 homogeneous $a\otimes b \in C^p(X) \otimes C^q(X)$ as above we have

  $$d(a \smile b) = (d a) \smile b + (-1)^p a \smile (d b) \,.$$

• the image of the cup product on cochain cohomology

  $$\smile : H^\bullet(C^\bullet(X))\otimes H^\bullet(C^\bullet(X)) \to H^\bullet(C^\bullet(X))$$

  is associative and distributes over the addition in $H^\bullet(C^\bullet(X))$.

Accordingly, the cup product makes $H^\bullet(C^\bullet(X)) = H^\bullet(X,R)$ 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 Whitehead-generalized cohomology

For the cup product in multiplicative Whitehead-generalized cohomology theories is defined by the same recipe of “cocycles pre-composed by diagonal and post-composed by product operation”, one just has to observe that symmetric smash-?monoidal diagonals on suspension spectra

(1)$$\Sigma^\infty X \overset{ \;\; D_X \;\; }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big)$$

do exists. This is discussed at suspension spectrum – Smash-monoidal diagonalspectrum#SmashMonoidalDiagonals).

With this, given a Whitehead-generalized cohomology theory $\widetilde E$ represented by a ring spectrum

$$\big(E, 1^E, m^E \big) \;\; \in \; SymmetricMonoids \big( Ho(Spectra), \mathbb{S}, \wedge \big)$$

the smash-monoidal diagonal structure (1) on suspension spectra serves to define the cup product $(-)\cup (-)$ in the corresponding multiplicative cohomology theory structure by:

$$\begin{aligned} & \big[ \Sigma^\infty X \overset{c_i}{\longrightarrow} \Sigma^{n_i} E \big] \,\in\, {\widetilde E}{}^{n_i}(X) \\ & \Rightarrow \;\; [c_1] \cup [c_2] \, \coloneqq \, \Big[ \Sigma^\infty X \overset{ \Sigma^\infty(D_X) }{\longrightarrow} \big( \Sigma^\infty X \big) \wedge \big( \Sigma^\infty X \big) \overset{ ( c_1 \wedge c_2 ) }{\longrightarrow} \big( \Sigma^{n_1} E \big) \wedge \big( \Sigma^{n_2} E \big) \overset{ m^E }{\longrightarrow} \Sigma^{n_1 + n_2}E \Big] \;\; \in \, {\widetilde E}{}^{n_1+n_2}(X) \,. \end{aligned}$$

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 $(A_i)_\bullet$ of sheaves and the pairing that one usually considers is the tensor product of chain complexes

$$(A_1)_\bullet \times (A_2)_\bullet \to (A_1 \otimes A_2)_\bullet$$

where

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

with differential

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

In abelian Čech cohomology

The cup product has a simple expression in abelian ?ech cohomology.

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

$$\phi : C^\bullet(\{U_i\}, A_1) \otimes C^\bullet(\{U_i\}, A_2) \to C^\bullet(\{U_i\}, A_1 \otimes A_2)$$

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

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

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

• cap product

• cup-i product

• intersection pairing

• functional cup product

• Baer sum

• cohomology operation

• power operation

• Massey product

• cup product in differential cohomology

  • cup product in ordinary differential cohomology

• Euler class takes Whitney sum to cup product, see this Prop.

References

• Frank Adams, part III, sections 2 and 3 of Stable homotopy and generalised homology, 1974

• Peter May, 18.3 and 22.3 of A concise course in algebraic topology (pdf)

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

• John Frederick Jardine, Lectures on simplicial presheaves (pdf)

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

• Hassler Whitney, On Products in a Complex (JSTOR)

See also

• Clemens Berger, Benoit Fresse Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc. 137 (2004), 135-174. (arXiv:math/0109158)