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\begin{document}

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\section*{chain homology and cohomology}

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

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

[[!include homological algebra - contents]]

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{functoriality}{Functoriality}\dotfill \pageref*{functoriality} \linebreak
\noindent\hyperlink{RespectForDirectSum}{Respect for direct sums and filtered colimits}\dotfill \pageref*{RespectForDirectSum} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{InTheContextOfHomotopyTheory}{In the context of homotopy theory}\dotfill \pageref*{InTheContextOfHomotopyTheory} \linebreak
\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{eilenbergmaclane_objects}{Eilenberg-MacLane objects}\dotfill \pageref*{eilenbergmaclane_objects} \linebreak
\noindent\hyperlink{homotopy_and_cohomology}{Homotopy and cohomology}\dotfill \pageref*{homotopy_and_cohomology} \linebreak
\noindent\hyperlink{chain_homology_as_homotopy}{Chain homology as homotopy}\dotfill \pageref*{chain_homology_as_homotopy} \linebreak
\noindent\hyperlink{cohomology_of_cochain_complexes}{Cohomology of cochain complexes}\dotfill \pageref*{cohomology_of_cochain_complexes} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In the context of [[homological algebra]], for $V_\bullet \in Ch_\bullet(\mathcal{A})$ a \emph{[[chain complex]]}, its \emph{chain [[homology group]]} in degree $n$ is akin to the $n$-th [[homotopy groups]] of a topological space. It is defined to be the [[quotient]] of the $n$-[[cycles]] by the $n$-[[boundaries]] in $V_\bullet$.

[[duality|Dually]], for $V^\bullet \in Ch^\bullet(\mathcal{A})$ a [[cochain complex]], its \emph{cochain [[cohomology group]]} in degree $n$ is the quotient of the $n$-[[cocycles]] by the $n$-[[coboundaries]].

Basic examples are the [[singular homology]] and [[singular cohomology]] of a topological space, which are the (co)chain (co)homology of the [[singular complex]].

Chain homology and cochain cohomology constitute the basic invariants of (co)chain complexes. A [[quasi-isomorphism]] is a [[chain map]] between chain complexes that induces isomorphisms on all chain homology groups, akin to a [[weak homotopy equivalence]]. A [[category of chain complexes]] equipped with quasi-isomorphisms as [[weak equivalences]] is a presentation for the [[stable (infinity,1)-category]] of chain complexes.

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

Let $\mathcal{A}$ be an [[abelian category]] such as that of $R$-[[modules]] over a [[commutative ring]] $R$. For $R = \mathbb{Z}$ the [[integers]] this is the category [[Ab]] of [[abelian groups]]. For $R = k$ a [[field]], this is the category [[Vect]] of [[vector spaces]] over $k$.

Write $Ch_\bullet(\mathcal{A})$ for the [[category of chain complexes]] in $\mathcal{A}$. Write $Ch^\bullet(\mathcal{A})$ for the [[category of cochain complexes]] in $\mathcal{A}$.

We label [[differentials]] in a chain complex as follows:

\begin{displaymath}
V_\bullet
  = 
  [
   \cdots \to V_{n+1} \stackrel{\partial_n}{\to} V_n \to \cdots
  ]
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
For $V_\bullet \in Ch_\bullet(\mathcal{A})$ a [[chain complex]] and $n \in \mathbb{Z}$, the \textbf{chain homology} $H_n(V)$ of $V$ in degree $n$ is the [[abelian group]]

\begin{displaymath}
H_n(V) \coloneqq \frac{Z_n(V)}{B_n(V)} = \frac{ker(\partial_{n-1})}{im(\partial_n)}
\end{displaymath}
given by the [[quotient]] ([[cokernel]]) of the group of $n$-[[cycles]] by that of $n$-[[boundaries]] in $V_\bullet$.

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{functoriality}{}\subsubsection*{{Functoriality}}\label{functoriality}

\begin{prop}
\label{}\hypertarget{}{}
For all $n \in \mathbb{N}$ forming chain homology extends to a [[functor]] from the [[category of chain complexes]] in $\mathcal{A}$ to $\mathcal{A}$ itself

\begin{displaymath}
H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
One checks that [[chain homotopy]] (see there) respects [[cycles]] and [[boundaries]].

\end{proof}
\begin{prop}
\label{ChainHomologyRespectsDirectProduct}\hypertarget{ChainHomologyRespectsDirectProduct}{}
Chain homology commutes with [[direct product]] of chain complexes:

\begin{displaymath}
H_n(\prod_i C^{(i)})
  \simeq
  \prod_i H_n(C^{(i)})
  \,.
\end{displaymath}
Similarly for [[direct sum]].

\end{prop}
\hypertarget{RespectForDirectSum}{}\subsubsection*{{Respect for direct sums and filtered colimits}}\label{RespectForDirectSum}

\begin{prop}
\label{}\hypertarget{}{}
The chain homology functor preserves [[direct sums]]:

for $A,B \in Ch_\bullet$ and $n \in \mathbb{Z}$, the canonical morphism

\begin{displaymath}
H_n(A \oplus B) \to H_n(A) \oplus H_n(B)
\end{displaymath}
is an [[isomorphism]].

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
The chain homology functor preserves [[filtered colimits]]:

for $A \colon I \to Ch_\bullet$ a [[filtered category|filtered]] [[diagram]] and $n \in \mathbb{Z}$, the canonical morphism

\begin{displaymath}
H_n(\underset{\to_i}{\lim} A_i) \to \underset{\to_i}{\lim} H_n(A_i)
\end{displaymath}
is an [[isomorphism]].

\end{prop}
This is spelled out for instance as (\hyperlink{HopkinsMathew}{Hopkins-Mathew , prop. 23.1}).

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

\begin{itemize}%
\item For $X$ a [[topological space]] and $Sing X$ its [[singular simplicial complex]], write $N \mathbb{Z}[Sing X]$ for the [[normalized chain complex]] of the [[simplicial abelian group]] that is degreewise the [[free abelian group]] on $Sing X$. The resulting chain homology is the \emph{[[singular homology]]} of $X$

\begin{displaymath}
H_\bullet( N \mathbb{Z}[Sing X]) \simeq H_\bullet(X, \mathbb{Z})
  \,.
\end{displaymath}

\item [[Koszul homology]]


\item [[Ext]], [[Tor]]



\end{itemize}
\hypertarget{InTheContextOfHomotopyTheory}{}\subsection*{{In the context of homotopy theory}}\label{InTheContextOfHomotopyTheory}

We discuss here the notion of (co)homology of a [[chain complex]] from a more abstract point of view of [[homotopy theory]], using the [[nPOV]] on \emph{[[cohomology]]} as discussed there.

A [[chain complex]] in non-negative degree is, under the [[Dold-Kan correspondence]] a [[homological algebra]] model for a particularly nice [[topological space]] or [[∞-groupoid]]: namely one with an abelian group structure on it, a [[simplicial abelian group]]. Accordingly, an unbounded (arbitrary) [[chain complex]] is a model for a [[spectrum]] with abelian group structure.

One consequence of this embedding

\begin{displaymath}
N : Ch_+ \to \infty Grpd
\end{displaymath}
induced by the Dold-Kan [[nerve]] is that it allows to think of [[chain complexes]] as objects in the [[(∞,1)-topos]] [[∞Grpd]] or equivalently [[Top]]. Every [[(∞,1)-topos]] comes with a notion of [[homotopy]] and [[cohomology]] and so such abstract notions get induced on chain complexes.

Of course there is an independent, age-old definition of [[homology]] of chain complexes and, by dualization, of cohomology of cochain complexes.

This entry describes how these standard definition of chain homology and cohomology follow from the general [[(∞,1)-topos]] nonsense described at [[cohomology]] and [[homotopy]].

The main statement is that

\begin{itemize}%
\item the na\"i{}ve [[homology]] groups of a [[chain complex]] are really its [[homotopy groups]], in the abstract sense (i.e. with the chain complex regarded as a model for a space/$\infty$-groupoid);


\item the na\"i{}ve cohomology groups of a cochain complex are really the abstract [[cohomology groups]] of the dual [[chain complex]].



\end{itemize}
\hypertarget{preliminaries}{}\subsubsection*{{Preliminaries}}\label{preliminaries}

Before discussing chain homology and cohomology, we fix some terms and notation.

\hypertarget{eilenbergmaclane_objects}{}\paragraph*{{Eilenberg-MacLane objects}}\label{eilenbergmaclane_objects}

In a given [[(∞,1)-topos]] there is a notion of [[homotopy]] and [[cohomology]] for every (co-)coefficient object $A$ ($B$).

The particular case of [[chain complex]] [[homology]] is only the special case induced from coefficients given by the corresponding [[Eilenberg-MacLane objects]].

Assume for simplicity here and in the following that we are talking about non-negatively graded chain complexes of vector spaces over some field $k$. Then for every $n \in \mathbb{N}$ write

\begin{displaymath}
\begin{aligned}
    \mathbf{B}^n k &:=
    ( 
      \cdots
      \to 
      \mathbf{B}^n k_n
      \to  
      \cdots 
       \to \stackrel{\partial}{\to} \mathbf{B}^n k_1 
      \stackrel{\partial}{\to} \mathbf{B}^n k_0)
    \\
    &=
    (  \cdots \to k \to \cdots \to 0 \to 0 )
  \end{aligned}
\end{displaymath}
for the $n$th [[Eilenberg-MacLane object]].

Notice that this is often also denoted $k[n]$ or $k[-n]$ or $K(k,n)$.

\hypertarget{homotopy_and_cohomology}{}\paragraph*{{Homotopy and cohomology}}\label{homotopy_and_cohomology}

With the [[Dold-Kan correspondence]] understood, embedding chain complexes into [[∞-groupoids]], for any chain complexes $X_\bullet$, $A_\bullet$ and $B_\bullet$ we obtain

\begin{itemize}%
\item the $\infty$-groupoid

\begin{displaymath}
\mathbf{H}_{\infty Grpd}(X_\bullet, A_\bullet)
\end{displaymath}
whose * objects are the $A$-valued [[cocycle]]s on $X$; * morphisms are the coboundaries between these [[cocycle]]s; * 2-morphisms are the coboundaries between coboundaries * etc. and where the elements of $\pi_0 \mathbf{H}(X_\bullet,A_\bullet)$ are the cohomology classes


\item the $\infty$-groupoids

\begin{displaymath}
\mathbf{H}_{\infty Grpd}(B_\bullet, X_\bullet)
\end{displaymath}
whose * objects are the $B$-co-valued cycles on $X$; * morphisms are the boundaries between these cycles; * 2-morphisms are the boundaries between boundaries * etc. and where the elements of $\pi_0 \mathbf{H}(B_\bullet,X_\bullet)$ are the homotopy classes



\end{itemize}
\hypertarget{chain_homology_as_homotopy}{}\subsubsection*{{Chain homology as homotopy}}\label{chain_homology_as_homotopy}

For $X_\bullet := V_\bullet$ any [[chain complex]] and $H_n(V_\bullet)$ its ordinary chain [[homology]] in degree $n$, we have

\begin{displaymath}
H_n(V_\bullet) \simeq 
  \pi_0 \mathbf{H}(\mathbf{B}^n k_\bullet, V_\bullet)
  \,.
\end{displaymath}
A cycle $c : \mathbf{B}^n k_\bullet \to V_\bullet$ is a chain map

\begin{displaymath}
\itexarray{
    \cdots 
    &\stackrel{\partial}{\to}&
    0
    &\stackrel{\partial}{\to}&
    k
    &\stackrel{\partial}{\to}&
    0
    &\stackrel{\partial}{\to}&
    \cdots
    \\
    && \downarrow
    && \downarrow^{c_n}
    && \downarrow
    \\
    \cdots 
    &\stackrel{\partial}{\to}&
    V_{n+1}
    &\stackrel{\partial}{\to}&
    V_n
    &\stackrel{\partial}{\to}&
    V_{n-1}
    &\stackrel{\partial}{\to}&
    \cdots        
  }
\end{displaymath}
Such chain maps are clearly in bijection with those elements $c_n \in V_n$ that are in the kernel of $V_n \stackrel{\partial}{\to} V_{n-1}$ in that $\partial c_n = 0$.

A boundary $\lambda : c \to C'$ is a chain homotopy

\begin{displaymath}
\itexarray{
    \cdots 
    &\stackrel{\partial}{\to}&
    0
    &\stackrel{\partial}{\to}&
    k
    &\stackrel{\partial}{\to}&
    0
    &\stackrel{\partial}{\to}&
    \cdots
    \\
    && &
    {}^{\lambda_n}\swarrow
    \\
    \cdots 
    &\stackrel{\partial}{\to}&
    V_{n+1}
    &\stackrel{\partial}{\to}&
    V_n
    &\stackrel{\partial}{\to}&
    V_{n-1}
    &\stackrel{\partial}{\to}&
    \cdots        
  }
\end{displaymath}
such that $c' = c + \partial \lambda$.

(\ldots{})

\hypertarget{cohomology_of_cochain_complexes}{}\subsubsection*{{Cohomology of cochain complexes}}\label{cohomology_of_cochain_complexes}

The ordinary notion of cohomology of a [[chain complex|cochain complex]] is the special case of cohomology in the [[stable (∞,1)-category|stable (∞,1)-]] [[category of chain complexes]].

For $V^\bullet$ a cochain complex let

\begin{displaymath}
\begin{aligned}
      X &:= V_\bullet = (V^\bullet)^*
      \\
      &=
      (\cdots \to X_{n+1} \stackrel{\partial}{\to}
      X_n \stackrel{\partial}{\to} X_{n-1} \to \cdots)
      \\
      & :=
      (\cdots \to V_{n+1} \stackrel{\partial}{\to}
      V_n \stackrel{\partial}{\to} V_{n-1} \to \cdots)
  \end{aligned}
\end{displaymath}
be the corresponding dual [[chain complex]]. Let

\begin{displaymath}
\begin{aligned}
     A  &:=  \mathbf{B}^n I 
     \\
     &=
     (\cdots \to A_{n+1}  \to A_n \to A_{n-1} \to \cdots )
     \\
      & =
     (\cdots \to 0  \to I \to 0 \to \cdots )
  \end{aligned}
\end{displaymath}
be the [[chain complex]] with the tensor unit (the [[ground field]], say) in degree $n$ and trivial elsewhere. Then

\begin{displaymath}
\begin{aligned}
    \mathbf{H}(X,A)
    &=
    Ch(V_\bullet, \mathbf{B}^n I)
  \end{aligned}
\end{displaymath}
has

\begin{itemize}%
\item as objects chain morphisms $c : V_\bullet \to \mathbf{B}^n I$

\begin{displaymath}
\itexarray{
    \cdots &\to& V_{n+1} &\stackrel{\partial}{\to}& 
    V_{n} 
    &\stackrel{\partial}{\to}& V_{n-1} 
    &\to& 
    \cdots
    \\
    &&  \downarrow^{c_{n+1}}
    && \downarrow^{c_{n}}
    && \downarrow^{c_{n-1}}
    \\
    \cdots &\to& 0 &\stackrel{\partial}{\to}& 
    I 
    &\stackrel{\partial}{\to}& 0 
    &\to& 
    \cdots  
  }
  \,.
\end{displaymath}
These are in canonical bijection with the elements in the kernel of $d_{n}$ of the dual cochain complex $V^\bullet = [V_\bullet,I]$.


\item as morphism chain homotopies $\lambda : c \to c'$

\begin{displaymath}
\itexarray{
    \cdots &\to& V_{n+1} &\stackrel{\partial}{\to}& 
    V_{n} 
    &\stackrel{\partial}{\to}& V_{n-1} 
    &\to& 
    \cdots
    \\
    && 
    && 
    &{}^{\lambda}\swarrow& 
    \\
    \cdots &\to& 0 &\stackrel{\partial}{\to}& 
    I 
    &\stackrel{\partial}{\to}& 0 
    &\to& 
    \cdots  
  }
  \,.
\end{displaymath}


\end{itemize}
Comparing with the general definition of cocycles and coboudnaries from above, one confirms that

\begin{itemize}%
\item the \textbf{cocycles} are the chain maps

\begin{displaymath}
V_\bullet \to I[n]_\bullet
\end{displaymath}

\item the \textbf{coboundaries} are the chain homotopies between these chain maps.


\item the \textbf{coboundaries of coboundaries} are the second order chain homotopies between these chain homotopies.


\item etc.



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Basics are for instance in section 1.1 of

\begin{itemize}%
\item [[Charles Weibel]], \emph{[[An Introduction to Homological Algebra]]}


\item [[Michael Hopkins]] (notes by [[Akhil Mathew]]), \emph{algebraic topology -- Lectures} (\href{http://people.fas.harvard.edu/~amathew/ATnotes.pdf}{pdf})



\end{itemize}
[[!redirects chain homology]] [[!redirects cochain homology]] [[!redirects chain cohomology]] [[!redirects cochain cohomology]]

[[!redirects chain homology group]] [[!redirects chain homology groups]]



\end{document}