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\begin{document}

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\section*{cocycle}

\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{realizations_of_cocycles}{Realizations of cocycles}\dotfill \pageref*{realizations_of_cocycles} \linebreak
\noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{CochainCocycles}{Cochain cocycles}\dotfill \pageref*{CochainCocycles} \linebreak
\noindent\hyperlink{Classification}{Objects classified by cocycles}\dotfill \pageref*{Classification} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

From the [[nPOV]], a [[cohomology]] set $H(X,A)$ of cohomology classes

\begin{enumerate}%
\item on a [[domain]] [[object]] $X$


\item with [[coefficients]] in an [[object]] $A$


\item in an [[(∞,1)-topos]] $\mathbf{H}$



\end{enumerate}
is the [[connected components]]/[[decategorification]] of

\begin{itemize}%
\item the [[derived hom space|hom ∞-groupoid]] $\mathbf{H}(X,A)$ -- the [[cocycle space]].

\end{itemize}
The [[objects]] in this [[cocycle space]] [[∞-groupoid]] $\mathbf{H}(X,A)$ are called \textbf{cocycles}; the [[morphisms]] are called \emph{[[coboundaries]]}.

For suitable choices of $\mathbf{H}$ this applies notably to cocycles in

\begin{itemize}%
\item general [[abelian sheaf cohomology]]


\item general [[nonabelian cohomology]]



\end{itemize}
and in particular to

\begin{itemize}%
\item [[chain homology and cohomology|chain cohomology]]


\item [[group cohomology]]


\item [[nonabelian group cohomology]].



\end{itemize}
It also encompasses variations such as cocycles in

\begin{itemize}%
\item [[equivariant cohomology]]


\item [[twisted cohomology]]


\item [[differential cohomology]].



\end{itemize}
For a detailed discussion of this and how it relates to various familiar realizations of cocycles, see [[cohomology]] and the links provided there.

\hypertarget{realizations_of_cocycles}{}\subsection*{{Realizations of cocycles}}\label{realizations_of_cocycles}

The objects in the [[derived hom space|hom ∞-groupoid]] $\mathbf{H}(X,A)$ are often expressed in terms of various 1-categorical models for $\mathbf{H}$, such as a [[homotopical category]] $C$ equipped with

\begin{itemize}%
\item the extra structure of a [[calculus of fractions]],


\item or with the structure of a [[category of fibrant objects]]


\item or even with the full structure of a [[model category]].



\end{itemize}
In all of these cases, cocycles $c$ on $X$ with coefficients in $A$ may be modeled by [[span]]s of the form

\begin{displaymath}
\itexarray{
    X &\stackrel{\in FW}{\leftarrow}& \tilde X \to A
  }
\end{displaymath}
in the ordinary [[category]] $C$, where the morphism on the left is taken from a special class of morphisms (for instance from the class of acyclic fibrations in the case that $C$ is a [[category of fibrant objects]]). In each case the relevant [[hom-set]] in the [[homotopy category]] $H(X,A) = \pi_0 \mathbf{H}(X,A)$ is given by the collection of cocycles module an [[equivalence relation]] given by coboundaries. In formulas

\begin{displaymath}
H(X,A) = colim_{\tilde X \stackrel{\in FW}{\to} X} Hom_{\pi C}(X,A)
  \,.
\end{displaymath}
For details see the respective discussion at [[homotopy category]], [[calculus of fractions]] and [[category of fibrant objects]].

\hypertarget{terminology}{}\subsection*{{Terminology}}\label{terminology}

In the existing literature on [[localization]]s, the spans $X \stackrel{\in FW}{\leftarrow} \tilde X \to X$ are often not called by a dedicated special term. On the other hand, in the existing literature that explicitly uses the term ``cocycle'', often more pedestrian definitions are used and it is not made explicit that morphisms in a [[homotopy category]] are being represented.

An notable exception to this is the article

\begin{itemize}%
\item Jardine, \emph{Cocycle categories} (\href{http://www.math.uiuc.edu/K-theory/0782/}{web})

\end{itemize}
that makes both the abstract concept and the terminology of cocycles explicit and manifest. The author is mainly motivated from the [[model structure on simplicial presheaves]] and its variants, which in particular models cocycles and cohomology of [[abelian sheaf cohomology]]. But more generally it models [[nonabelian cohomology]]. Notably when the underlying space is the point, it models ordinary [[chain homology and cohomology|chain cohomology]] as well as [[group cohomology]] and [[nonabelian group cohomology]].

Notice that this article chooses to work with the full structure of a [[model category]] but presents constructions for cocycles entirely analogous to and in fact inspired by those used in a [[category of fibrant objects]] or in one equipped with a [[calculus of fractions]]. The author emphasizes that he can give a definition where the left leg of the cocycle spans are not required to be acyclic fibrations, but can be any weak equivalences. But all this is just a technical question of how exactly to model a cocycle, not a question of principle of concept. For instance in this context every cocycle defined with respect to a weak equivalence over its domain is cohomologous to one defined with respect to an acyclic fibration over its domain.

In the special case that the category $C$ is [[Cat]] equipped with the [[folk model structure on Cat]], cocycles out of acyclic fibrations -- which are [[k-surjective functor]]s for all $k$ in this case -- have been considered in

\begin{itemize}%
\item Makkai, \emph{\href{http://www.math.mcgill.ca/makkai/anafun/}{Avoiding the axiom of choice in general category theory}} .

\end{itemize}
There they are called \textbf{[[anafunctor]]s}.

One could take this as a suggestion to find a dedicated term for spans as above and call generally such a span an \textbf{anamorphism}. An anamorphism would be effectively the same as a cocycle, but the term \emph{[[morphism]]} in it would amplify the nature of cocycles as morphisms.

So

\begin{itemize}%
\item a \emph{cocycle} on $X$ with coefficients in $A$

\end{itemize}
would correspond to

\begin{itemize}%
\item an \emph{anamorphism} from $X$ to $Y$.

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{CochainCocycles}{}\subsubsection*{{Cochain cocycles}}\label{CochainCocycles}

The archetypical example of a notion of cocycles is that of [[chain homology and cohomology|chain cohomology]]:

a non-negatively graded [[chain complex]] $V_\bullet = (V_0 \stackrel{\partial_V}{\leftarrow} V_1 \stackrel{\partial_V}{\leftarrow} V_2 \stackrel{}{\leftarrow} \cdots)$ is given. An element $v \in V_n$ is a \textbf{chain}. A linear dual $\omega : V_n \to k$ on its elements is a \textbf{cochain}. The cochains arrange into the [[cochain complex]] $V^\bullet = (V^0 \stackrel{d_V}{\to} V^1 \to \stackrel{d_V}{\to} V^2 \to \cdots) := (V_0^* \stackrel{(\partial_V)^*}{\to} V_1^* \to \stackrel{(\partial_V)^*}{\to} V_2^* \to \cdots)$

A cochain $\omega \in V^n$ is a \textbf{cocycle} if it is closed with respect to the [[differential]] $d_V$ in that

\begin{displaymath}
d_V \omega = 0
  \,.
\end{displaymath}
One sees that such cocycles are in bijection to [[morphism]]s of [[chain complex]]es

\begin{displaymath}
\omega : V_\bullet \to (\mathbf{B}^n k)_\bullet
  \,,
\end{displaymath}
where on the right we have the [[Eilenberg-MacLane object]] of the ground field $k$, which is the chain complex trivial everywhere except in degree $n$, where it is $k$:

\begin{displaymath}
\mathbf{B}^n k = (0 \leftarrow \cdots \leftarrow 0 \leftarrow k \leftarrow 0 \leftarrow \cdots)
  \,.
\end{displaymath}
By definition such a morphism is a collection of morphisms $\omega_r : V_r \to (\mathbf{B}^n k)_r$, of which by definition only $\omega_n \in V_n^* = V^n$ can be nontrivial.

For the collection of these maps to be a morphism of chain complexes they have to make all squares in sight commute. The only nontrivial one in this case is the one

\begin{displaymath}
\itexarray{
    V_{n+1} &\to& 0
    \\
    {}^{\mathllap{\partial_V}}\downarrow && \downarrow
    \\
    V_n &\stackrel{\omega}{\to}& k
  }
  \,.
\end{displaymath}
Its commutativity means in formulas that

\begin{displaymath}
\omega \circ \partial_V = d_V \omega = 0
  \,,
\end{displaymath}
which is the cocycle condition from above.

In most cases the morphism $\omega : V_\bullet \to \mathbf{B}^n k$ defined this way is already a morphism in the relevant [[(∞,1)-category]] $\mathbf{H}_{Ch_\bullet}$ of chain complexes: this is modeled for instance by the projective [[model structure on chain complexes]]. In this every object is fibrant, and the cofibrant objects are those consisting of projective $k$-[[module]]s. If we assume that all our modules are projective (for instance in the archetypical case that our modules are simply [[vector space]]s), then $\omega : V_\bullet \to \mathbf{B}^n k$ is a cocycle in $\mathbf{H}_{Ch_\bullet}$ from the above abstract nonsense point of view. For its cohomology class we may write

\begin{displaymath}
[\omega] \in H^n(V_\bullet,k) = \pi_0 \mathbf{H}_{Ch_\bullet}(V_\bullet, \mathbf{B}^n k)
  \,.
\end{displaymath}
\hypertarget{Classification}{}\subsection*{{Objects classified by cocycles}}\label{Classification}

One says for $\omega \in \mathbf{H}(X,A)$ a cocycle, that the object \textbf{classified by the cocycle} is its [[homotopy fiber]] $P \to X$ regarded as an object in the [[overcategory]] over $X$.

This homotopy fiber may be thought of as the internal [[principal ∞-bundle]] in $\mathbf{H}$ with classifying map $\omega$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include chains and cochains - table]]

[[!redirects cocycles]]



\end{document}