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 $X \times_B Y$, which by definition sits in
is equivalently also the following (∞,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 $\Omega B$.
See also at homotopy pullback this corollary.
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 homotopy fiber sequence is of the form
For $\mathcal{C} =$ ∞Grpd $\simeq L_{whe}$ Top, this point of view is amplified in (Dyer-Roitberg 80).
The corresponding long exact sequence of homotopy groups is of the form
This is what has historically been the definition of Mayer-Vietories sequences (Eckmann-Hilton 64).
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 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
where the bottom horizontal morphism is the composite
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. 1 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
First consider two more concrete special cases.
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 ordinary 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 $\mathcal{C}$.
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 $b \colon \Delta[k] \to B(U)$ of the horn $hb \colon \Lambda[k]^i \to B(U)$. 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 $- \colon B \times B \to B$ 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 $(f - g)$.
Summing this up:
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
For $\mathcal{C}$ an (∞,1)-site of definition, there is a reflection
of $\mathbf{H}$ into an (∞,1)-category of (∞,1)-presheaves.
By prop. 2 the statement holds in $[C^{op},\infty Grpd]$. Since embedding and reflection both preserve finite (∞,1)-limits, it hence also holds in $\mathbf{H}$.
Still more generally and more simply:
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
By this discussion we may use homotopy type theory reasoning. Starting out with the discussion at homotopy pullback – In homotopy type theory we obtain
where the second but last step consists of observing a contractible based path space object (see the discussion at factorization lemma).
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
Use prop. 4 with $\phi$ being the canonical point, i.e. the inclusion $e \colon \ast \to G$ of the neutral element to find the homotopy pullback
Then use the pasting law as above.
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 homotopy fiber sequences is in
Discussion in the context of stable model categories includes
Discussion in the context of homotopy type theory includes