group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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).
Let $\mathcal{C}$ be an (∞,1)-category with finite (∞,1)-limits and let $X, Y, B$ be pointed objects and
and
be any two morphisms with common codomain preserving the base points. Let $X \times_B Y$ be the (∞,1)-pullback
The corresponding 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.
Let $\mathcal{C}$ be a presentable (∞,1)-category.
Then $X \times_B Y$ is equivalently given by the (∞,1)-pullback
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 $\Omega B$.
The first statement one checks for instance by choosing a presentation by a combinatorial model category and then proceeding as below in the discussion Presentation by fibrant objects. Then by the pasting law for $(\infty,1)$-pullbacks it follows that with the left square in
an $(\infty,1)$-pullback, so is the total outer rectangle. But again by the first statement, this is equivalent to the $(\infty,1)$-pullback
which is the defining pullback for the loop space object.
Therefore the Mayer-Vietoris fiber sequence is of the form
The corresponding long exact sequence of homotopy groups is of the form
This has historically been the definition of Mayer-Vietories sequences (Eckmann-Hilton).
Suppose that the (∞,1)-category $\mathcal{C}$ is 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
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
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.
We consider the special case where $B$ is an abelian ∞-group in a presentable (∞,1)-category $\mathcal{C}$.
In this case we have an (∞,1)-pullback
where the bottom horizontal morphism is the composite
of a morphism that sends the second argument to its inverse with the group composition operation.
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 presented by the projective model structure on simplicial presheaves
As discussed there, the Dold-Kan correspondence prolongs to a Quillen adjunction on presheaves whose right adjoint is
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 right Quillen, so is the corresponding morphism $- : B \times B \to B$ in $[S^{op}, sSet]_{proj}$.
Therefore the ordiary pullback of presheaves of chain complexes
is a homotopy pullback in $[S^{op}, Ch_{\bullet \geq 0}(Ab)]_{proj}$, as is the ordinary pullback of simplicial presheaves
in $[S^{op}, sSet]_{proj}$.
Since ∞-stackification preserves finite (∞,1)-limits, this presents an (∞,1)-pullback also in $\mathcal{C}$.
More generally:
Let $\mathcal{C}$ be an (∞,1)-topos with a 1-site $S$ of definition (a 1-localic (∞,1)-topos).
Then (as discussed there) every ∞-group object in $\mathcal{C}$ has a presentation by a presheaf of simplicial groups
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
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
Since $B(U)$ itself is a Kan complex (being a simplicial group, as discussed there) there is a filler $b : \Delta[k] \to B(U)$ of the horn $hb : \Lambda[k]^i \to B(U)$. Define then
Since all the face maps are group homomorphisms, this is indeed a filler of $ha$:
Moreover, by construction, $(a,b)$ is a filler in
Since therefore $- : B \times B \to B$ is a projective fibration, it follows as before that the ordinary pullback
is a homotopy pullback.
For $B$ an ∞-group object as above, the (∞,1)-pullback $X \times_B Y$ is equivalently given by the $(\infty,1)$-pullback
By prop. 1 the object $X \times_B Y$ is the $(\infty,1)$-pullback in
By the pasting law this is equivalently given by the composite pullback of
Here the composite bottom morphism is $(f - g)$.
A special case of the general Mayer-Vietoris sequence, corollary 1 – 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
in the (∞,1)-topos $\mathbf{H} =$ ∞Grpd/Top.
When this is presented by the standard model structure on simplicial sets or in terms of CW-complexes 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
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 $G$
By the above general properties the above homotopy pullback is equivalent to
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 $V$.
An original reference is
A more modern review that emphasizes the role of fiber sequences is in