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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*{characteristic class} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - 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{characteristic_classes_of_principal_bundles}{Characteristic classes of principal bundles}\dotfill \pageref*{characteristic_classes_of_principal_bundles} \linebreak \noindent\hyperlink{of_line_bundles}{Of line bundles}\dotfill \pageref*{of_line_bundles} \linebreak \noindent\hyperlink{of_linear_representations}{Of linear representations}\dotfill \pageref*{of_linear_representations} \linebreak \noindent\hyperlink{chern_character}{Chern character}\dotfill \pageref*{chern_character} \linebreak \noindent\hyperlink{invariants}{$k$-Invariants}\dotfill \pageref*{invariants} \linebreak \noindent\hyperlink{level_in_chernsimons_theory}{Level in $\infty$-Chern-Simons theory}\dotfill \pageref*{level_in_chernsimons_theory} \linebreak \noindent\hyperlink{of_lagrangian_submanifolds}{Of Lagrangian submanifolds}\dotfill \pageref*{of_lagrangian_submanifolds} \linebreak \noindent\hyperlink{classes_in_the_sense_of_fuks}{Classes in the sense of Fuks}\dotfill \pageref*{classes_in_the_sense_of_fuks} \linebreak \noindent\hyperlink{fukss_definition}{Fuks's definition}\dotfill \pageref*{fukss_definition} \linebreak \noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \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} In the general context of [[cohomology]], as described there, a [[cocycle]] representing a cohomology class on an object $X$ with coefficients in an object $A$ is a [[morphism]] $c : X \to A$ in a given ambient [[(∞,1)-topos]] $\mathbf{H}$. The same applies with the object $A$ taken as the domain object: for $B$ yet another object, the $B$-valued cohomology of $A$ is similarly $H(A,B) = \pi_0 \mathbf{H}(A,B)$. For $[k] \in H(A,B)$ any cohomology class in there, we obtain an [[∞-functor]] \begin{displaymath} [k(-)] : \mathbf{H}(X,A) \to \mathbf{H}(X,B) \end{displaymath} from the $A$-valued cohomology of $X$ to its $B$-valued cohomology, simply from the composition operation \begin{displaymath} \mathbf{H}(X,A) \times \mathbf{H}(A,B) \to \mathbf{H}(X,B) \,. \end{displaymath} Quite generally, for $[c] \in H(X,A)$ an $A$-cohomology class, its image $[k(c)] \in H(X,B)$ is the corresponding \textbf{characteristic class}. Notice that if $A = \mathbf{B}G$ is [[connected]], an $A$-cocycle on $X$ is a $G$-[[principal ∞-bundle]]. Hence characteristic classes are equivalently characteristic classes of principal $\infty$-bundles. From the [[nPOV]], where [[cocycle]]s are elements in an [[derived hom space|(∞,1)-categorical hom-space]], forming \textbf{characteristic classes} is nothing but the \emph{composition} of cocycles. In practice one is interested in this notion for particularly simple objects $B$, notably for $B$ an [[Eilenberg-MacLane object]] $\mathbf{B}^n K$ for some component $K$ of a [[spectrum object]]. This serves to \textbf{characterize} cohomology with coefficients in a complicated object $A$ by a collection of cohomology classes with simpler coefficients. Historically the name \emph{characteristic class} came a little different way about, however (see also [[historical note on characteristic classes]]). In that case, with the usual notation $H^n(X,K) := H(X, \mathbf{B}^n K)$, a given characteristic class in degree $n$ assigns \begin{displaymath} [k(-)] : \mathbf{H}(X,A) \to H^n(X,K) \,. \end{displaymath} Moreover, recall from the discussion at [[cohomology]] that to every [[cocycle]] $c : X \to A$ is associated the object $P \to X$ that it classifies -- its [[homotopy fiber]] -- which may be thought of as an $A$-[[principal ∞-bundle]] over $X$ with classifying map $X \to A$. One typically thinks of the characteristic class $[k(c)]$ as characterizing this [[principal ∞-bundle]] $P$. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{characteristic_classes_of_principal_bundles}{}\subsubsection*{{Characteristic classes of principal bundles}}\label{characteristic_classes_of_principal_bundles} This is the archetypical example: let $\mathbf{H} =$ [[Top]] $\simeq$ [[∞Grpd]], the canonical [[(∞,1)-topos]] of [[discrete ∞-groupoid]]s, or more generally let $\mathbf{H} =$ [[ETop∞Grpd]], the [[cohesive (∞,1)-topos]] of [[Euclidean-topological ∞-groupoid]]s. For $G$ [[topological group]] write $B G$ for its [[classifying space]]: the ([[geometric realization]] of its) [[delooping]]. For $A$ any other [[abelian group|abelian]] [[topological group]], similarly write $B^n A$ for its $n$-fold [[delooping]]. If $A$ is a [[discrete group]] then this is the [[Eilenberg-MacLane space]] $K(A,n)$. Generally, \begin{displaymath} H^n(B G, A) = \pi_0 \mathbf{H}(B G, B^n A) \end{displaymath} is the [[cohomology]] of $B G$ with coefficients in $A$. Every [[cocycle]] $k : B G \to B^n A$ represents a characteristic class $[k]$ on $B G$ with coefficients in $A$. A $G$-[[principal bundle]] $P \to X$ is classified by some map $c: X \to B G$. For any $k \in H^n(B G, A)$ a degree $n$ cohomology class of the classifying space, the corresponding composite map $X \stackrel{c}{\to} B G \stackrel{k}{\to} B^n A$ represents a class $[k(c)] \in H^n(X,A)$. This is the corresponding characteristic class of the bundle. Notable families of examples include: \begin{itemize}% \item for $G = O$ the [[orthogonal group]]: [[Pontryagin classes]], [[Stiefel-Whitney classes]], [[Wu classes]]; [[one-loop anomaly polynomial I8]] \item for $G = SO$ the [[special orthogonal group]]: [[Euler classes]]; \item for $G = U$ the [[unitary group]]: [[Chern classes]], [[Conner-Floyd Chern classes]]; [[Todd class]]; \item for $G = \mathbf{B}U(1)$ the [[circle n-group|circle 2-group]]: the [[Dixmier-Douady class]]. \end{itemize} \hypertarget{of_line_bundles}{}\subsubsection*{{Of line bundles}}\label{of_line_bundles} \begin{itemize}% \item [[canonical characteristic class]], [[Theta characteristic]] \end{itemize} \hypertarget{of_linear_representations}{}\subsubsection*{{Of linear representations}}\label{of_linear_representations} \begin{itemize}% \item [[characteristic class of a linear representation]] \end{itemize} \hypertarget{chern_character}{}\subsubsection*{{Chern character}}\label{chern_character} The [[Chern character]] is a natural characteristic class with values in real cohomology. See there for more details. \hypertarget{invariants}{}\subsubsection*{{$k$-Invariants}}\label{invariants} \begin{itemize}% \item [[k-invariants]] \end{itemize} \hypertarget{level_in_chernsimons_theory}{}\subsubsection*{{Level in $\infty$-Chern-Simons theory}}\label{level_in_chernsimons_theory} \begin{itemize}% \item [[level (Chern-Simons theory)]] \end{itemize} \hypertarget{of_lagrangian_submanifolds}{}\subsubsection*{{Of Lagrangian submanifolds}}\label{of_lagrangian_submanifolds} A characteristic class of [[Lagrangian submanifolds]] is the \emph{[[Maslov index]]}. \hypertarget{classes_in_the_sense_of_fuks}{}\subsubsection*{{Classes in the sense of Fuks}}\label{classes_in_the_sense_of_fuks} In (\hyperlink{Fuks}{Fuks (1987), section 7}) an axiomatization of characteristic classes is proposed. We review the definition and discuss how it is a special case of the one given above. \hypertarget{fukss_definition}{}\paragraph*{{Fuks's definition}}\label{fukss_definition} Fuks considers a base category $\mathcal{T}$ of ``spaces'' and a category $\mathcal{S}$ of spaces with a structure (for example, space together with a [[vector bundle]] on it), this category should be a category over $\mathcal{T}$, i.e. at least equipped with a [[functor]] $U : \mathcal{S}\to\mathcal{T}$. A morphism of categories with structures is a morphism in the [[overcategory]] [[Cat]]$/\mathcal{T}$, i.e. a morphism $U\to U'$ is a functor $F: dom(U)\to dom(U')$ such that $U' F = U$. Suppose now the category $\mathcal{T}$ is equipped with a [[cohomology theory]] which is, for purposes of this definition, a functor of the form $H : \mathcal{T}^{op} \to A$ where $A$ is some [[concrete category]], typically category of [[algebra over an algebraic theory|T-algebras]] for some [[algebraic theory]] in [[Set]], e.g. the category of [[abelian group]]s. Define $\mathcal{H} = \mathcal{H}_H$ as a category whose objects are pairs $(X,a)$ where $X$ is a space (= object in $\mathcal{T}$) and $a\in H(X)$. This makes sense as $A$ is a [[concrete category]]. A morphism $(X,a)\to (Y,b)$ is a morphism $f: X\to Y$ such that $H(f)(b) = a$. We also denote $f^* = H(f)$, hence $f^*(b) = a$. A \textbf{characteristic class of structures of type $\mathcal{S}$ with values in} $H$ in the sense of (\hyperlink{Fuks}{Fuks}) is a morphism of structures $h: \mathcal{S}\to\mathcal{H}_H$ over $\mathcal{T}$. In other words, to each structure $S$ of the type $\mathcal{S}$ over a space $X$ in $\mathcal{T}$ it assigns an element $h(S)$ in $H(X)$ such that for a morphism $t: S\to T$ in $\mathcal{S}$ the homomorphism $(U(t))^* : H(Y)\to H(X)$, where $Y = U(T)$, sends $h(S)$ to $h(T)$. \hypertarget{discussion}{}\paragraph*{{Discussion}}\label{discussion} Notice that $\mathcal{H}_H \to \mathcal{T}$ in the above is nothing but the [[fibered category]] that under the [[Grothendieck construction]] is an equivalent incarnation of the [[presheaf]] $H$. In fact, since $A$ in the above is assume to be just a 1-category of sets with structure, $\mathcal{H}_H$ is just its [[category of elements]] of $H$. Similarly in all applications that arise in practice (for instance for the structure of [[vector bundle]]s) that was mentioned, the functor $\mathcal{S} \to \mathcal{T}$ is a [[fibered category]], too, corresponding under the inverse of the [[Grothendieck construction]] to a [[prestack]] $F_{\mathcal{S}}$. Therefore morphisms of fibered categories over $\mathcal{T}$ \begin{displaymath} c : \mathcal{S} \to \mathcal{H}_H \end{displaymath} are equivalently morphisms of [[prestack|(pre)]][[stack]]s \begin{displaymath} c : F_{\mathcal{S}} \to H \,. \end{displaymath} In either picture, these are morphism in a [[2-topos]] over the [[site]] $\mathcal{T}$. So, as before, for $X \in \mathcal{T}$ some [[space]], a $\mathcal{S}$-structure on $X$ (for instance a [[vector bundle]]) is a moprhism in the topos \begin{displaymath} g : X \to F_{\mathcal{S}} \end{displaymath} (in this setup simply by the [[2-Yoneda lemma]]) and the characteristic class $[c(g)]$ of that bundle is the bullback of that [[universal characteristic class|universal class]] $c$, hence the class represented by the composite \begin{displaymath} c(g) : X \stackrel{g}{\to} F_{\mathcal{S}} \stackrel{c}{\to} H \,. \end{displaymath} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[universal characteristic class]], [[universal characteristic map]]; \item [[differential characteristic class]] \item [[characteristic class of a linear representation]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Textbook accounts include \begin{itemize}% \item [[John Milnor]], [[Jim Stasheff]], \emph{Characteristic classes}, Princeton Univ. Press (1974) \item [[Stanley Kochmann]], section 2.3 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996 \item [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], \emph{[[Basic Bundle Theory and K-Cohomology Invariants]]}, Lecture Notes in Physics, Springer 2008 (\href{http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf}{pdf}) \item [[Peter May]], chapter 23 of \emph{A concise course in algebraic topology} (\href{http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf}{pdf}) \end{itemize} Exposition with motivation from [[mathematical physics]] includes \begin{itemize}% \item [[Yang Zhang]], \emph{A brief introduction to characteristic classes from the differentiable viewpoint}, 2011 (\href{http://www.nbi.dk/~zhang/notes/A%20brief%20introduction%20to%20characteristic%20classes%20from%20the%20differentiable%20viewpoint.pdf}{pdf}) \end{itemize} Further texts include \begin{itemize}% \item Jean-Pierre Schneiders, \emph{Introduction to characteristic classes and index theory} (book), Lisboa (Lisbon) 2000 \item [[Johan Dupont]], \emph{Fibre bundles and Chern-Weil theory}, Lecture Notes Series \textbf{69}, Dept. of Math., University of Aarhus, 2003, 115 pp. \href{http://data.imf.au.dk/publications/ln/2003/imf-ln-2003-69.pdf}{pdf} \item Shigeyuki Morita, \emph{Geometry of characteristic classes}, Transl. Math. Mon. \textbf{199}, AMS 2001 \item [[Raoul Bott]], L. W. Tu, \emph{Differential forms in algebraic topology}, GTM \textbf{82}, Springer 1982. \item D. B. Fuks, \emph{ } , appendix to the Russian translation of K. S. Brown, \emph{Cohomology of groups}, Moskva, Mir 1987. \end{itemize} [[!redirects characteristic class]] [[!redirects characteristic classes]] \end{document}