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\begin{document}

%-------------------------------------------------------------------


\section*{homology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{algebraic_topology}{}\paragraph*{{Algebraic topology}}\label{algebraic_topology}

[[!include algebraic topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{of_chain_complexes}{Of chain complexes}\dotfill \pageref*{of_chain_complexes} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{generalized_homology}{Generalized homology}\dotfill \pageref*{generalized_homology} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{of_chain_complexes}{}\subsection*{{Of chain complexes}}\label{of_chain_complexes}

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

Under the [[Dold-Kan correspondence]], [[∞-groupoids]] with strict abelian group structure (modeled by [[Kan complexes]] that are [[simplicial object|simplicial]] abelian groups) are identified with non-negatively graded [[chain complexes]] of abelian groups

\begin{displaymath}
N_\bullet : SimpAb \stackrel{\simeq}{\to} Ch_+
  \,.
\end{displaymath}
The \textbf{homology groups} of a [[chain complex]] of abelian groups are the image under this identification of the [[homotopy groups]] of the corresponding [[∞-groupoids]]. More details on this are at [[chain homology and cohomology]].

So at least for the case of chain complexes of abelian groups we have the slogan

\textbf{homology} = [[homotopy]] under [[Dold-Kan correspondence]]

Of course historically the development of concepts was precisely the opposite: chain homology is an old fundamental concept in [[homological algebra]] that is simpler to deal with than [[simplicial homotopy groups]]. The computational simplification for chain complexes is what makes the [[Dold-Kan correspondence]] useful after all.

Conceptually, however, it can be useful to understand homology as a special kind of [[homotopy]]. This is maybe most vivid in the [[duality|dual]] picture: [[cohomology]] derives its name from that fact that [[chain homology and cohomology]] are dual concepts. But later generalizations of [[cohomology]] to [[generalized (Eilenberg-Steenrod) cohomology]] and further to [[nonabelian cohomology]] showed that the restricted notion of homology is an insufficient dual model for cohomology: what cohomology is really dual to is the more general concept of [[homotopy]]. More on this is at [[cohomotopy]] and [[Eckmann-Hilton duality]].

\hypertarget{definition}{}\subsubsection*{{Definition}}\label{definition}

The category of abelian groups is in particular an [[abelian category]]. We can define [[chain complexes]] and their homology in any [[abelian category]] $C$.

Let $C$ be an [[abelian category]] and let

\begin{displaymath}
V_\bullet = ( \cdots  \to V_{n+1} \stackrel{\delta_n}{\to}
  V_n \stackrel{\delta_{n-1}}{\to}
  V_{n-1} \to \cdots
  )
\end{displaymath}
be a [[chain complex]] in $C$. For each integer $n \in \mathbb{N}$ this induces the following diagram of [[kernel]]s, [[cokernel]]s and [[image]]s

\begin{displaymath}
\itexarray{
    && im \delta_n &&\to&& ker \delta_{n-1}
    \\
    & \nearrow && \searrow && \swarrow
    \\
    V_{n+1}
    &&\stackrel{\delta_n}{\to}&&
    V_n
    &&\stackrel{\delta_{n-1}}{\to}&&
    V_{n-1}
    \\
    & &&
    \swarrow && \searrow && \nearrow
    \\
    && coker \delta_n &&\stackrel{}{\to}&&
    im \delta_{n-1}
  }
\end{displaymath}
the \textbf{homology} $H_n(V)$ of $V$ in degree $n$ is the object

\begin{displaymath}
\begin{aligned}
    im(ker \delta_{n-1} \to  V_n \to coker \delta_{n}) 
    & \simeq
    coker(im \delta_n \to ker \delta_{n-1})
    \\
    & \simeq
    coker(V_{n+1} \to ker \delta_{n-1})
    \\
    & \simeq
    ker(coker \delta_n \to im \delta_{n-1})
    \\
    & \simeq
    ker(coker \delta_n \to V_{n-1})
  \end{aligned}
\end{displaymath}
\begin{itemize}%
\item If $H_n(V) \simeq 0$ then one says that the complex $V$ is [[exact sequence|exact]] in degree $n$.

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

In the special case that $C$ is the category of abelian groups, or of vector spaces, this definition reduces to the more familiar simpler statement:

the $n$-th homology group of the [[chain complex]] $V_\bullet$ is the quotient group

\begin{displaymath}
H_n(V) = ker(\partial_n) / im(\partial_{n+1})
  \,.
\end{displaymath}
\hypertarget{generalized_homology}{}\subsection*{{Generalized homology}}\label{generalized_homology}

By the [[Brown representability theorem]] every [[spectrum]] $A$ induces a [[generalized (Eilenberg-Steenrod) cohomology]] theory, and dually a \textbf{generalized homology theory}.

For $X$ a [[topological space]] and $A$ a [[spectrum]], the generalized homology of spectrum of $X$ with coefficients in $A$ is

\begin{displaymath}
X \wedge A := \Sigma^\infty(X)\wedge A
  \,,
\end{displaymath}
where on the right we have the [[smash product of spectra]] with the [[suspension spectrum]] of $X$ and on the left we abbreviate this to the [[(∞,1)-colimit|(∞,1)-tensoring]] of [[Spec]] over [[Top]].

The corresponding [[homology group]]s are the [[homotopy group]]s of this spectrum:

\begin{displaymath}
E_n(X,A) := \pi_n(X \wedge A) := [\Sigma^n \mathbb{S}, X \wedge A ]
  \,.
\end{displaymath}
where $\mathbb{S}$ is the [[sphere spectrum]]. For more see [[generalized homology]].

\hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2}

\begin{itemize}%
\item [[chain homology]]

\begin{itemize}%
\item [[relative homology]]

\end{itemize}

\item [[simplicial homology]]


\item [[singular homology]]

\begin{itemize}%
\item [[cellular homology]]

\end{itemize}

\item [[generalized homology]]


\item [[factorization homology]]


\item [[Kan-Thurston Theorem]]



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

\begin{itemize}%
\item [[effective homology]]


\item [[persistent homology]]



\end{itemize}
The relation between homology, cohomology and homotopy:

[[!include homotopy-homology-cohomology]]

The ingredients of homology and cohomology:

[[!include chains and cochains - table]]

[[!redirects homologies]] [[!redirects homology group]] [[!redirects homology groups]]



\end{document}