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\begin{document}

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\section*{Mayer-Vietoris sequence}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

\noindent\hyperlink{ideas}{Ideas}\dotfill \pageref*{ideas} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{GeneralProperties}{General}\dotfill \pageref*{GeneralProperties} \linebreak
\noindent\hyperlink{PresentationByFibrantObjects}{Presentation by fibrant objects}\dotfill \pageref*{PresentationByFibrantObjects} \linebreak
\noindent\hyperlink{OverAGroupObject}{Over an $\infty$-group}\dotfill \pageref*{OverAGroupObject} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{CohomologyOfACover}{(Co)Homology of a cover}\dotfill \pageref*{CohomologyOfACover} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{ideas}{}\subsection*{{Ideas}}\label{ideas}

\emph{Mayer-Vietoris sequence} is the term for the [[fiber sequence]] -- or often for the corresponding [[long exact sequence of homotopy groups]] -- induced from an [[(∞,1)-pullback]] (or for a [[homotopy pullback]] presenting it).

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

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-limits]] and let $X, Y, B$ be [[pointed objects]] and

\begin{displaymath}
f : X \to B
\end{displaymath}
and

\begin{displaymath}
g : Y \to B
\end{displaymath}
be any two [[morphisms]] with common [[codomain]] preserving the base points. Let $X \times_B Y$ be the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    X \times_B Y &\to& Y
    \\
    \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{g}}
    \\
    X &\stackrel{f}{\to}& B
  }
  \,.
\end{displaymath}
The corresponding \textbf{Mayer-Vietoris sequence} is the [[fiber sequence]] of the induced morphism $X \times_B Y \to X \times Y$. Often the term is used (only) for the corresponding [[long exact sequence of homotopy groups]].

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

\hypertarget{GeneralProperties}{}\subsubsection*{{General}}\label{GeneralProperties}

\begin{prop}
\label{SequenceFromDiagonal}\hypertarget{SequenceFromDiagonal}{}
Let $\mathcal{C}$ be a [[presentable (∞,1)-category]].

Then $X \times_B Y$, which by definition sits in

\begin{displaymath}
\itexarray{
    X \times_B Y &\to& Y
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{g}}
    \\
    X  &\stackrel{f}{\to}& B
  }
\end{displaymath}
is equivalently also the following [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    X \times_B Y &\to& B
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}}
    \\
    X \times Y &\stackrel{(f,g)}{\to}& B \times B
  }
  \,,
\end{displaymath}
where the right vertical morphism is the [[diagonal]].

Moreover, the [[homotopy fiber]] of $X \times_B Y \to X \times Y$ is the [[loop space object]] $\Omega B$.

\end{prop}
See also at \emph{[[homotopy pullback]]} \href{homotopy+pullback#HomotopyPullbackByFactorizationLemma}{this corollary}.

\begin{proof}
The first statement one checks for instance by choosing a presentation by a [[combinatorial model category]] and then proceeding as below in the discussion \emph{\hyperlink{PresentationByFibrantObjects}{Presentation by fibrant objects}}. Then by the [[pasting law]] for $(\infty,1)$-pullbacks it follows that with the left square in

\begin{displaymath}
\itexarray{
    \Omega B &\to& X \times_B Y &\to & B
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    * &\to& X \times Y &\stackrel{(f,g)}{\to}& B \times B
  }
\end{displaymath}
an $(\infty,1)$-pullback, so is the total outer rectangle. But again by the first statement, this is equivalent to the $(\infty,1)$-pullback

\begin{displaymath}
\itexarray{
    \Omega B &\to& * 
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    * &\to& B
  }
  \,,
\end{displaymath}
which is the defining pullback for the [[loop space object]].

\end{proof}
Therefore the Mayer-Vietoris [[homotopy fiber sequence]] is of the form

\begin{displaymath}
\Omega B \to X \times_B Y \to X \times Y
  \,.
\end{displaymath}
For $\mathcal{C} =$ [[∞Grpd]] $\simeq L_{whe}$ [[Top]], this point of view is amplified in (\hyperlink{DyerRoitberg80}{Dyer-Roitberg 80}).

\begin{cor}
\label{LongExactSequenceOfHomotopyGroups}\hypertarget{LongExactSequenceOfHomotopyGroups}{}
The corresponding [[long exact sequence of homotopy groups]] is of the form

\begin{displaymath}
\cdots \to \pi_{n+1} B \to \pi_n X \times_B Y \stackrel{(f_*, g_*)}{\to} \pi_n X \oplus \pi_n Y \stackrel{f_* - g_*}{\to} \pi_n B \to \cdots
\end{displaymath}
\begin{displaymath}
\cdots \to \pi_2 B \to \pi_1 X \times_B Y \stackrel{(f_+, g_*)}{\to}  \pi_1 X \times \pi_1 Y \stackrel{f_* \cdot g_*^{-1}}{\to} \pi_1 B \to \pi_0 (X \times_B Y) \to \pi_0 (X \times Y)
  \,.
\end{displaymath}
\end{cor}
This is what has historically been the definition of Mayer-Vietories sequences (\hyperlink{EckmannHilton}{Eckmann-Hilton 64}).

\hypertarget{PresentationByFibrantObjects}{}\subsubsection*{{Presentation by fibrant objects}}\label{PresentationByFibrantObjects}

Suppose that the [[(∞,1)-category]] $\mathcal{C}$ is [[simplicial localization|presented]] by a [[category of fibrant objects]] $C$ (for instance the [[subcategory]] on the fibrant objects of a [[model category]]).

Then the $(\infty,1)$-pullback $X \times_B Y$ is presented by a [[homotopy pullback]], and by the [[factorization lemma]], this is given by the ordinary [[limit]]

\begin{displaymath}
\itexarray{
    X \times^h_B Y &\to& &\to& Y
    \\
    \downarrow && && \downarrow^{\mathrlap{g}}
    \\
    && B^I &\to& B 
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{f}{\to}& B
  }
  \,,
\end{displaymath}
where $B \stackrel{\simeq}{\to} B^I \to B \times B$ is a [[path object]] for $B$. This limit coincides, up to [[isomorphism]], with the [[pullback]]

\begin{displaymath}
\itexarray{
    X \times_B^h Y &\to& B^I
    \\
    \downarrow && \downarrow
    \\
    X \times Y &\stackrel{(f,g)}{\to}& B \times B
  }
  \,.
\end{displaymath}
This implies in particular that the [[homotopy fiber]] of $X \times_B^h Y \to X \times Y$ is the [[loop space object]] $\Omega B$, being the [[fiber]] of the [[path space object]] projection.

\hypertarget{OverAGroupObject}{}\subsubsection*{{Over an $\infty$-group}}\label{OverAGroupObject}

We consider now the case where $B$ carries the structure of an [[∞-group]] (or just a grouplike [[H-space]] object) in a [[presentable (∞,1)-category]] or [[locally Cartesian closed (∞,1)-category]] $\mathcal{C}$.

In this case (as discussed in a moment), we have an [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    B &\to& *
    \\
    \downarrow^{\mathrlap{\Delta_B}} 
    &\swArrow_{\simeq}& 
    \downarrow^{\mathrlap{e}}
    \\
    B \times B
    &\stackrel{(-)\cdot (-)^{-1}}{\to}&
    B
  }
  \,,
\end{displaymath}
where the bottom horizontal morphism is the composite

\begin{displaymath}
(-)\cdot (-)^{-1} : B \times B \stackrel{(id, (-)^{-1})}{\to} B \times B \stackrel{\cdot}{\to} B
\end{displaymath}
of a morphism that sends the second argument to its inverse with the group composition operation.

It then follows by the [[pasting law]] and prop. \ref{SequenceFromDiagonal} that in this case the morphism $X \times_B Y \to X \times Y$ in the Mayer-Vietoris sequence is itself the homotopy fiber of $X \times Y \stackrel{f \cdot g^{-1}}{\longrightarrow} B$, hence that we have a long homotopy fiber sequence of the form

\begin{displaymath}
\Omega B \longrightarrow X \times_B Y \longrightarrow X \times Y
  \stackrel{f \cdot g^{-1}}{\longrightarrow} B
  \,.
\end{displaymath}
First consider two more concrete special cases.

\begin{example}
\label{}\hypertarget{}{}
Let $S$ be a small [[site]] and let $\mathcal{C} = Sh_{(\infty,1)}(S)$ be the [[(∞,1)-category of (∞,1)-sheaves]] on $S$.

This is [[presentable (∞,1)-category|presented]] by the projective [[model structure on simplicial presheaves]]

\begin{displaymath}
\mathcal{C} \simeq ([S^{op}, sSet]_{proj, loc})^\circ
  \,.
\end{displaymath}
As discussed there, the [[Dold-Kan correspondence]] prolongs to a [[Quillen adjunction]] on [[presheaves]] whose [[right adjoint]] is

\begin{displaymath}
\Xi : 
  [S^{op}, Ch_{\bullet \leq 0}(Ab)]_{proj}
  \to 
  [S^{op}, sAb]_{proj}
  \to
  [S^{op}, sSet]_{proj}
  \,.
\end{displaymath}
Let then $B \in \mathcal{C}$ be an object with a presentation in $[S^{op}, sSet]$ in the image of this $\Xi$. We write $B$ also for this presentation, and hence $B = \Xi(\tilde B)$ for some presheaf of chain complexes $\tilde B$.

We claim now that such $B$ satisfies the above assumption.

To see this, first notice that the evident morphism $- : \tilde B \times \tilde B \to \tilde B$ is degreewise an [[epimorphism]], hence it is a fibration in $[S^{op}, Ch_{\bullet \geq 0}(Ab)]_{proj}$, and since $\Xi$ is [[Quillen adjunction|right Quillen]], so is the corresponding morphism $- : B \times B \to B$ in $[S^{op}, sSet]_{proj}$.

Therefore the ordinary pullback of presheaves of chain complexes

\begin{displaymath}
\itexarray{
    \tilde B &\to& *
    \\
    \downarrow^{\mathrlap{\Delta_{\tilde B}}} && \downarrow^{\mathrlap{0}}
    \\
    \tilde B \times \tilde B &\stackrel{-}{\to}& \tilde B
  }
\end{displaymath}
is a [[homotopy pullback]] in $[S^{op}, Ch_{\bullet \geq 0}(Ab)]_{proj}$, as is the ordinary pullback of simplicial presheaves

\begin{displaymath}
\itexarray{
    B &\to& *
    \\
    \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{\mathrlap{0}}
    \\
    B \times B &\stackrel{-}{\to}& B
  }
\end{displaymath}
in $[S^{op}, sSet]_{proj}$.

Since [[∞-stackification]] preserves [[finite (∞,1)-limits]], this presents an [[(∞,1)-pullback]] also in $\mathcal{C}$.

\end{example}
\begin{example}
\label{SimplicialGroupObjectAsBase}\hypertarget{SimplicialGroupObjectAsBase}{}
Let $\mathcal{C}$ be an [[(∞,1)-topos]] with a 1-[[site]] $S$ of definition (a [[n-localic (∞,1)-topos|1-localic (∞,1)-topos]]).

Then (as discussed there) every [[∞-group]] object in $\mathcal{C}$ has a presentation by a presheaf of [[simplicial groups]]

\begin{displaymath}
B \in [S^{op}, sGrp]_{proj} \to [S^{op}, sSet]_{proj}
  \,.
\end{displaymath}
We claim that the canonical morphism $- : B \times B \to B$ is objectwise a [[Kan fibration]] and hence a fibration in the projective [[model structure on simplicial presheaves]].

Let $U \in S$ be any test object. A diagram

\begin{displaymath}
\itexarray{ 
    \Lambda[k]^i &\stackrel{(ha, hb)}{\to}& B(U) \times B(U)
    \\
    \downarrow^{\mathrlap{j}} && \downarrow
    \\
    \Delta[k] &\stackrel{\sigma}{\to}& B(U)
  }
\end{displaymath}
corresponds to a $k$-cell $\sigma \in B(U)$ together with a choice of decomposition of the $i$th [[horn]] $j^* \sigma$ as a difference

\begin{displaymath}
(j^* \sigma)_l = ha_l \cdot hb_l^{-1}
  \,.
\end{displaymath}
Since $B(U)$ itself is a [[Kan complex]] (being a [[simplicial group]], as discussed there) there is a filler $b \colon \Delta[k] \to B(U)$ of the [[horn]] $hb \colon \Lambda[k]^i \to B(U)$. Define then

\begin{displaymath}
a \coloneqq \sigma \cdot b
  \,.
\end{displaymath}
Since all the face maps are group homomorphisms, this is indeed a filler of $ha$:

\begin{displaymath}
\begin{aligned}
    \delta_l(a) 
      & = \delta_l(\sigma \cdot b)
    \\
      & = \delta_l(\sigma) \cdot \delta_l(b)
    \\   
      & = \delta_l(\sigma) \cdot hb_l
    \\
      & = ha_l
  \end{aligned}
  \,.
\end{displaymath}
Moreover, by construction, $(a,b)$ is a filler in

\begin{displaymath}
\itexarray{ 
    \Lambda[k]^i &\stackrel{(ha, hb)}{\to}& B(U) \times B(U)
    \\
    \downarrow^{\mathrlap{i}} 
    &{}^{(a,b)}\nearrow& 
    \downarrow
    \\
    \Delta[k] &\stackrel{\sigma}{\to}& B(U)
  }
  \,.
\end{displaymath}
Since therefore $- \colon B \times B \to B$ is a projective fibration, it follows as before that the ordinary pullback

\begin{displaymath}
\itexarray{
    B &\to& *
    \\
    \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{e}
    \\
    B \times B &\stackrel{-}{\to}& B
  }
\end{displaymath}
is a [[homotopy pullback]].

\end{example}
\begin{prop}
\label{SequenceOverGroupObjectIn1LocalicSituation}\hypertarget{SequenceOverGroupObjectIn1LocalicSituation}{}
For $B$ an [[∞-group]] object as above, the [[(∞,1)-pullback]] $X \times_B Y$ is equivalently given by the $(\infty,1)$-pullback

\begin{displaymath}
\itexarray{
   X \times_B Y &\to& * 
   \\
   \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{0}}
   \\ 
   X \times Y &\stackrel{f \cdot g^{-1}}{\to}& B
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By prop. \ref{SequenceFromDiagonal} the object $X \times_B Y$ is the $(\infty,1)$-pullback in

\begin{displaymath}
\itexarray{
    X \times_B Y &\to& B
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}}
    \\
    X \times Y &\stackrel{(f,g)}{\to}& B \times B
  }
  \,.
\end{displaymath}
By the [[pasting law]] this is equivalently given by the composite pullback of

\begin{displaymath}
\itexarray{
    X \times_B Y &\to& B &\to& *
    \\
    \downarrow 
    &\swArrow_{\simeq}& 
    \downarrow^{\mathrlap{\Delta_B}}
    &\swArrow_{\simeq}& 
    \downarrow^{\mathrlap{0}}
    \\
    X \times Y &\stackrel{(f,g)}{\to}& B \times B
    &\stackrel{-}{\to}& B
  }
  \,.
\end{displaymath}
Here the composite bottom morphism is $(f - g)$.

\end{proof}
Summing this up:

\begin{prop}
\label{}\hypertarget{}{}
For $\mathbf{H}$ an [[(∞,1)-sheaf]] [[(∞,1)-topos]], $B$ an [[∞-group]]-object in $\mathbf{H}$ and $f\colon X \to B$ and $g \colon Y\to B$ two morphisms, then there is a long [[homotopy fiber sequence]] of the form

\begin{displaymath}
\Omega B \longrightarrow X \times_B Y \longrightarrow X \times Y
  \stackrel{f \cdot g^{-1}}{\longrightarrow} B
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
For $\mathcal{C}$ an [[(∞,1)-site]] of definition, there is a [[reflective sub-(infinity,1)-category|reflection]]

\begin{displaymath}
\mathbf{H} \stackrel{\longleftarrow}{\hookrightarrow}
  [C^{op},\infty Grpd]
\end{displaymath}
of $\mathbf{H}$ into an [[(∞,1)-category of (∞,1)-presheaves]].

By prop. \ref{SequenceOverGroupObjectIn1LocalicSituation} the statement holds in $[C^{op},\infty Grpd]$. Since embedding and reflection both preserve [[finite (∞,1)-limits]], it hence also holds in $\mathbf{H}$.

\end{proof}
Still more generally and more simply:

\begin{prop}
\label{HTTArgumentForPullback}\hypertarget{HTTArgumentForPullback}{}
Let $\mathcal{C}$ be a [[locally Cartesian closed (∞,1)-category]]. Let $G$ be an [[∞-group]] object (or just a grouplike [[H-space]]-object). Then for $\phi \colon D \longrightarrow G$ any morphism we have a [[homotopy pullback]] square of the form

\begin{displaymath}
\itexarray{
     G \times D &\longrightarrow& D
     \\
     \downarrow && \downarrow^{\mathrlap{\phi}}
     \\
     G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G
  }
  \,.
\end{displaymath}
\end{prop}
(\href{http://nforum.mathforge.org/discussion/6403/generality-of-mayervietoris-in-an-infinitytopos-/?Focus=51417#Comment_51417}{nForum discussion})

\begin{proof}
By \href{locally+cartesian+closed+infinity,1-category#InternalLogic}{this discussion} we may use [[homotopy type theory]] reasoning. Starting out with the discussion at \emph{\href{homotopy+pullback#InHomotopyTypeTheory}{homotopy pullback -- In homotopy type theory}} we obtain

\begin{displaymath}
\begin{aligned}
  D\times_G  (G\times G)
  &= \sum_{d:D} \sum_{g_1:G} \sum_{g_2:G} (g_1\cdot g_2^{-1} = \phi(d)) \\
  &= \sum_{d:D} \sum_{g_1:G} \sum_{g_2:G} (g_1 = \phi(d)\cdot g_2) \\
  &= \sum_{d:D} \sum_{g_2:G} \sum_{g_1:G} (g_1 = \phi(d)\cdot g_2) \\
  &= \sum_{d:D} \sum_{g_2:G} \mathbf{1}\\
  &= D\times G
\end{aligned}
  \,,
\end{displaymath}
where the second but last step consists of observing a contractible based [[path space object]] (see the discussion at [[factorization lemma]]).

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a [[locally Cartesian closed (∞,1)-category]]. Let $G$ be an [[∞-group]] object (or just a grouplike [[H-space]]-object).

Then for $f \colon X \to G$ and $g \colon Y \to G$ two morphisms, there is a Mayer-Vietoris-type [[homotopy fiber sequence]]

\begin{displaymath}
\cdots \to \Omega G
  \longrightarrow 
  X \times_G Y
  \longrightarrow
  X \times Y
  \stackrel{f \cdot (g^{-1})}{\longrightarrow}
  G
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Use prop. \ref{HTTArgumentForPullback} with $\phi$ being the canonical point, i.e. the inclusion $e \colon \ast \to G$ of [[generalized the|the]] neutral element to find the homotopy pullback

\begin{displaymath}
\itexarray{
     G  &\longrightarrow& \ast
     \\
     \downarrow^{\mathrlap{\Delta}} && \downarrow^{\mathrlap{e}}
     \\
     G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G
  }
  \,.
\end{displaymath}
Then use the [[pasting law]] as above.

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

\hypertarget{CohomologyOfACover}{}\subsubsection*{{(Co)Homology of a cover}}\label{CohomologyOfACover}

A special case of the general Mayer-Vietoris sequence, corollary \ref{LongExactSequenceOfHomotopyGroups} -- which historically was the first case considered -- applies to the [[cohomology]]/[[homology]] of a [[topological space]] $X$ equipped with an [[open cover]] $\{U_1, U_2 \to X\}$.

Being a [[cover]] means (see [[effective epimorphism in an (∞,1)-category]]) that there is a [[homotopy pushout]] [[diagram]] of the form

\begin{displaymath}
\itexarray{
    U \cap V &\hookrightarrow& U
    \\
    \downarrow && \downarrow
    \\
    V &\to& X
  }
\end{displaymath}
in the [[(∞,1)-topos]] $\mathbf{H} =$ [[∞Grpd]]/[[Top]].

When this is [[presentable (∞,1)-category|presented]] by the standard [[model structure on simplicial sets]] or in terms of [[CW-complex]]es by the [[model structure on topological spaces]], it is given by an ordinary [[pushout]].

Let then $A \in \infty Grpd \simeq Top$ be some coefficient object, for instance an [[Eilenberg-MacLane object]] $\mathbf{B}^n G$ ([[Eilenberg-MacLane space]] $\cdots \simeq K(G,n)$) for the definition of ordinary [[singular cohomology]] with coefficients in an [[abelian group]] $G$.

Then applying the [[derived hom space]] functor $\mathbf{H}(-, A) : \mathbf{H}^{op} \to \infty Grpd$ yields the [[(∞,1)-pullback]] [[diagram]]

\begin{displaymath}
\itexarray{
    \mathbf{H}(X, A) &\to& \mathbf{H}(U,A)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(V,A) &\to& \mathbf{H}(U \cap V, A)
  }
\end{displaymath}
to which we can apply the homotopical Mayer-Vietoris sequence.

Notice that (as discussed in detail at [[cohomology]]) the [[homotopy groups]] of the [[∞-groupoid]] $\mathbf{H}(X,\mathbf{B}^n G)$ are the [[cohomology groups]] of $X$ with coefficients in $G$

\begin{displaymath}
\pi_k \mathbf{H}(X, \mathbf{B}^n G)
  \simeq
  H^{n-k}(X, G)
  \,.
\end{displaymath}
By the above \href{GeneralProperties}{general properties} the above homotopy pullback is equivalent to

\begin{displaymath}
\mathbf{H}(X,A) \to \mathbf{H}(U,A) \times \mathbf{H}(V,A)
  \to \mathbf{H}(U \cap V, A)
\end{displaymath}
being a [[fiber sequence]]. The corresponding long exact sequence in cohomology (as discussed above) is what is traditionally called the Mayer-Vietoris sequence of the cover of $X$ by $U$ and $V$ in $A$-cohomology.

By duality (see [[universal coefficient theorem]]) an analogous statement holds for the [[homology]] of $X$, $U$ and $V$.

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

An original reference is

\begin{itemize}%
\item [[Beno Eckmann]] and [[Peter Hilton]], \emph{Unions and intersections in homotopy theory}, Comment. Math. Helv. 3 (1964),2 93-307, \href{http://dx.doi.org/10.1007/BF02566918}{doi}

\end{itemize}
A more modern review that emphasizes the role of [[homotopy fiber sequences]] is in

\begin{itemize}%
\item Eldon Dyer, Joseph Roitberg, \emph{Note on sequence of Mayer-Vietoris type}, Proceedings of the AMS, volume 80, number 4 (1980) (\href{http://www.ams.org/journals/proc/1980-080-04/S0002-9939-1980-0587950-8/S0002-9939-1980-0587950-8.pdf}{pdf})

\end{itemize}
Discussion in the context of [[stable model categories]] includes

\begin{itemize}%
\item [[Peter May]], lemma 5.7 of \emph{The additivity of traces in triangulated categories}, Adv. Math., 163(1):34-73, 2001 (\href{http://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf}{pdf})

\end{itemize}
Discussion in the context of [[homotopy type theory]] includes

\begin{itemize}%
\item E Cavallo et al, \emph{Exactness of the Mayer-Vietoris Sequence in Homotopy Type Theory} (\href{http://www.contrib.andrew.cmu.edu/~ecavallo/works/mayer-vietoris.pdf}{pdf})

\end{itemize}
[[!redirects Mayer-Vietoris sequences]] [[!redirects Mayer-Vietoris long exact sequence]]



\end{document}