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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{cup product} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory} [[!include (infinity,1)-topos - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{ViaTheDoldKanCorrespondence}{Via the Dold-Kan correspondence}\dotfill \pageref*{ViaTheDoldKanCorrespondence} \linebreak \noindent\hyperlink{on_moore_complexes_of_cosimplicial_algebras}{On Moore complexes of cosimplicial algebras}\dotfill \pageref*{on_moore_complexes_of_cosimplicial_algebras} \linebreak \noindent\hyperlink{InSingularCohomology}{In singular cohomology}\dotfill \pageref*{InSingularCohomology} \linebreak \noindent\hyperlink{in_abelian_sheaf_cohomology}{In abelian sheaf cohomology}\dotfill \pageref*{in_abelian_sheaf_cohomology} \linebreak \noindent\hyperlink{in_abelian_ech_cohomology}{In abelian ech cohomology}\dotfill \pageref*{in_abelian_ech_cohomology} \linebreak \noindent\hyperlink{in_echdeligne_cohomology_ordinary_differential_cohomology}{In ech-Deligne cohomology (ordinary differential cohomology)}\dotfill \pageref*{in_echdeligne_cohomology_ordinary_differential_cohomology} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{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 \begin{displaymath} H(X,A) := \pi_0 \mathbf{H}(X,A) \,. \end{displaymath} The \emph{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 \begin{displaymath} A_1 \times A_2 \longrightarrow A_3 \end{displaymath} 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 \emph{[[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 \begin{displaymath} g_1 \cdot g_2 : X \stackrel{(id,id)}{\longrightarrow} X \times X \stackrel{g_1 \times g_2}{\to} A_1 \times A_2 \to A_3 \end{displaymath} in $\mathbf{H}(X,A_3)$ obtained by combining the pairing with precomposition by the [[diagonal]] map $\Delta_X = (id_X, id_X)$. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{ViaTheDoldKanCorrespondence}{}\subsubsection*{{Via the Dold-Kan correspondence}}\label{ViaTheDoldKanCorrespondence} 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 monoidal functor|lax monoidalness]] of the Dold-Kan correspondence (see at \emph{[[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 \begin{itemize}% \item $(Ch_{\bullet \geq 0}, \otimes)$ 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]] $\otimes$; \item $(sAb, \otimes)$ for the [[category]] of [[simplicial abelian groups]], regarded as a [[symmetric monoidal category]] with the degreewise [[tensor product of abelian groups]]; \item $U \;\colon\; sAb \longrightarrow KanCplx \to sSet$ for the [[forgetful functor]] to the underlying [[simplicial sets]] (which happens to land in [[Kan complexes]]); \item $\Gamma \;\colon \; Ch_{\bullet \geq 0} \stackrel{\simeq}{\longrightarrow} sAb$ for the [[equivalence of categories]] given by the [[Dold-Kan correspondence]]; \item $DK \;\colon\; Ch_{\bullet \geq 0} \underoverset{\simeq}{\Gamma}{\longrightarrow} sAb \stackrel{F}{\longrightarrow} KanCplx \hookrightarrow sSet$ for the composite. \end{itemize} Now: \begin{itemize}% \item $\Gamma$ 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 \href{oplax+monoidal+functor#OplaxAdjointToLax}{dually} by the [[Alexander-Whitney map]]; \item $U$ is a [[strong monoidal functor]]; \item for the [[tensor product of abelian groups]] $A \otimes B$ there are canonical natural [[bilinear maps]] of underlying sets \begin{displaymath} p_{A,B} \;\colon\; U(A)\times U(B) \longrightarrow U(A \otimes B) \end{displaymath} \end{itemize} Using all this, then for \begin{displaymath} f \;\colon\; V_\bullet \otimes W_\bullet \longrightarrow Z_\bullet \end{displaymath} a given [[chain map]], this induces a map of the corresponding Kan complexes \begin{displaymath} \cup_{DK} \;\colon\; DK(V_\bullet) \times DK(W_\bullet) \longrightarrow DK(Z_\bullet) \end{displaymath} as the following [[composition|composite]] \begin{displaymath} \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) \,. \end{displaymath} 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: \begin{displaymath} \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)) \,. \end{displaymath} For example if \begin{displaymath} V_\bullet = \mathbb{Z}[n_1] \,, \;\;\; W_\bullet = \mathbb{Z}[n_2] \end{displaymath} is the [[chain complex]] concentrated in degree $n_1$ and $n_2$, respectively, on the group of [[integers]], then \begin{displaymath} DK(V_\bullet) \simeq B^{n_1} \mathbb{Z} \simeq K(\mathbb{Z},n_1) \end{displaymath} 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 \begin{displaymath} V_\bullet \otimes W_\bullet \simeq \mathbb{Z}[n_1 + n_2] \,. \end{displaymath} Hence we may take $Z_\bullet \coloneqq \mathbb{Z}[n_1 + n_2]$ and $f = id$ and we get a cup product \begin{displaymath} \cup \;\colon\; H^{n_1}(X, \mathbb{Z}) \times H^{n_2}(X, \mathbb{Z}) \to H^{n_1 + n_2}(X, \mathbb{Z}) \,. \end{displaymath} \hypertarget{on_moore_complexes_of_cosimplicial_algebras}{}\subsubsection*{{On Moore complexes of cosimplicial algebras}}\label{on_moore_complexes_of_cosimplicial_algebras} For $A = (A^\bullet)$ any [[cosimplicial algebra]], its dual [[Moore complex|Moor cochain complex]] $N^\bullet(A)$ naturally inherits the structure of a [[dg-algebra]] under the cup product. \begin{quote}% 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. \end{quote} 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 [[chain homology and cohomology|cochain cohomology]]. This suggests that the cup product should be, while not commutative, homotopy commutative in that it makes $N^\bullet(A)$ a [[commutative algebra in an (∞,1)-category|homotopy commutative monoid object]]. This in turn should mean that $N^\bullet(A)$ is an [[algebra over an operad]] for the [[E-k-operad|E-∞ operad]]. That this is indeed the case is the main statement in (\hyperlink{BergerFresse01}{Berger-Fresse 01}) \hypertarget{InSingularCohomology}{}\subsubsection*{{In singular cohomology}}\label{InSingularCohomology} 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 \emph{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 $n$-[[simplex|simplices]] in $X$ -- the [[fundamental ∞-groupoid]] of $X$. For $R$ some [[ring]], let $Maps(\Pi(X),R)^{\bullet}$ be the [[cosimplicial algebra|cosimplicial ring]] of $R$-valued functions on the spaces of $n$-simplices. The corresponding [[Moore complex|Moore cochain complex]] $C^\bullet(X)$ is the cochain complex whose [[chain homology and cohomology|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 [[poset|totally ordered set]] with $p+1$ elements. For $\mu : [p] \to [p+q]$ an injective order preserving map and $K$ some [[simplicial object|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 \textbf{cup product} \begin{displaymath} \smile : C^\bullet(X) \otimes C^\bullet(X) \to C^\bullet(X) \end{displaymath} 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 \begin{displaymath} a \smile b = (d_L^* a) \cdot (d_R^* b) \,. \end{displaymath} \begin{quote}% There is some glue missing here to connect this back to the above general definition, something involving the [[Eilenberg-Zilber map]]. \end{quote} This means that $(a \smile b)_{i_0, \cdots, i_{p+q}} = a_{i_0, \cdots, i_p} \cdot b_{i_p, \cdots, i_{p+q}}$. \begin{prop} \label{}\hypertarget{}{} This cup product enjoys the following properties: \begin{itemize}% \item 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 \begin{displaymath} d(a \smile b) = (d a) \smile b + (-1)^p a \smile (d b) \,. \end{displaymath} \item the image of the cup product on [[chain homology and cohomology|cochain cohomology]] \begin{displaymath} \smile : H^\bullet(C^\bullet(X))\otimes H^\bullet(C^\bullet(X)) \to H^\bullet(C^\bullet(X)) \end{displaymath} is associative and distributes over the addition in $H^\bullet(C^\bullet(X))$. \end{itemize} \end{prop} Accordingly, the cup product makes $H^\bullet(C^\bullet(X)) = H^\bullet(X,R)$ into a [[ring]]: the \textbf{cohomology ring} on the [[Eilenberg-MacLane spectrum|ordinary cohomology]] of $X$. See for instance section 3.2 of \begin{itemize}% \item Hatcher, \emph{Algebraic Topology} (\href{http://www.math.cornell.edu/~hatcher/AT/ATpage.html}{web} \href{http://www.math.cornell.edu/~hatcher/AT/AT.pdf}{pdf}) \end{itemize} \hypertarget{in_abelian_sheaf_cohomology}{}\subsection*{{In abelian sheaf cohomology}}\label{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 complex]]es \begin{displaymath} (A_1)_\bullet \times (A_2)_\bullet \to (A_1 \otimes A_2)_\bullet \end{displaymath} where \begin{displaymath} (A_1 \otimes A_2)_n = \oplus_p (A_1)_p \otimes (A_2)_{n-p} \,. \end{displaymath} with differential \begin{displaymath} d (a_1 \otimes a_2) = (d a_1) \otimes a_2 + (-1)^{|a_1|} a_1 \otimes d a_2 \,. \end{displaymath} \hypertarget{in_abelian_ech_cohomology}{}\subsubsection*{{In abelian ech cohomology}}\label{in_abelian_ech_cohomology} The cup product has a simple expression in abelian [[?ech cohomology]]. For $A_1$ and $A_2$ two [[chain complex]]es (of [[sheaves]] of [[abelian group]]s) construct a morphism of ech complexes \begin{displaymath} \phi : C^\bullet(\{U_i\}, A_1) \otimes C^\bullet(\{U_i\}, A_2) \to C^\bullet(\{U_i\}, A_1 \otimes A_2) \end{displaymath} by sending $\alpha \in C^p(U,A_1)_\bullet$ and $\beta \in C^q(U,A_2)_\bullet$ to \begin{displaymath} \phi(\alpha \otimes \beta)_{i_0, \cdots , i_{p + q}} \;:=\; \alpha_{i_0, \cdots, i_p} \otimes \beta_{i_p, \cdots i_{p+q}} \,. \end{displaymath} For instance (\hyperlink{Brylinski}{Brylinski, section (1.3)}) spring \hypertarget{in_echdeligne_cohomology_ordinary_differential_cohomology}{}\subsubsection*{{In ech-Deligne cohomology (ordinary differential cohomology)}}\label{in_echdeligne_cohomology_ordinary_differential_cohomology} For the case that of [[?ech cohomology|?ech]] [[hypercohomology]] with coefficients in [[Deligne complex]]es the above yields the \emph{[[Beilinson-Deligne cup-product]]} for [[ordinary differential cohomology]]. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[cap product]] \item [[intersection pairing]] \item [[functional cup product]] \item [[Baer sum]] \item [[cohomology operation]] \item [[power operation]] \item [[Massey product]] \item [[cup product in differential cohomology]] \begin{itemize}% \item [[cup product in ordinary differential cohomology]] \end{itemize} \item [[Euler class]] takes [[Whitney sum]] to [[cup product]], see \href{Euler+class#EulerClassOfWhitneySumIsCupProductOfEulerClasses}{this Prop.} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Frank Adams]], part III, sections 2 and 3 of \emph{[[Stable homotopy and generalised homology]]}, 1974 \item [[Peter May]], 18.3 and 22.3 of \emph{A concise course in algebraic topology} (\href{http://www.maths.ed.ac.uk/~aar/papers/maybook.pdf}{pdf}) \end{itemize} The cup product in ech cohomology is discussed for instance in section 1.3 of \begin{itemize}% \item [[Jean-Luc Brylinski]], \emph{Loop spaces and characteristic classes} \end{itemize} Recall from the discussion at [[models for ∞-stack (∞,1)-toposes]] that all [[hypercompletion|hypercomplete]] [[∞-stack]] [[(∞,1)-topos]]es 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 \begin{itemize}% \item [[John Frederick Jardine]], \emph{Lectures on simplicial presheaves} (\href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf}{pdf}) \end{itemize} An early treatment of cup product can be found in this classic \begin{itemize}% \item [[Hassler Whitney]], \emph{On Products in a Complex} (\href{http://www.jstor.org/pss/1968795}{JSTOR}) \end{itemize} See also \begin{itemize}% \item [[Clemens Berger]], [[Benoit Fresse]] \emph{Combinatorial operad actions on cochains}, Math. Proc. Cambridge Philos. Soc. 137 (2004), 135-174. (\href{http://arxiv.org/abs/math/0109158}{arXiv:math/0109158}) \end{itemize} [[!redirects cup products]] [[!redirects cup product in abelian Cech cohomology]] [[!redirects cup product in Cech cohomology]] \end{document}