nLab Mayer-Vietoris sequence

Ideas

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).

Definition

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

$$f : X \to B$$

and

$$g : Y \to B$$

be any two morphisms with common codomain preserving the base points. Let $X \times_B Y$ be the (∞,1)-pullback

$$\array{ X \times_B Y &\to& Y \\ \downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{g}} \\ X &\stackrel{f}{\to}& B } \,.$$

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.

Properties

General

Proposition

Let $\mathcal{C}$ be a presentable (∞,1)-category.

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

$$\array{ X \times_B Y &\to& Y \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{g}} \\ X &\stackrel{f}{\to}& B }$$

is equivalently also the following (∞,1)-pullback

$$\array{ X \times_B Y &\to& B \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}} \\ X \times Y &\stackrel{(f,g)}{\to}& B \times B } \,,$$

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.

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 Presentation by fibrant objects. Then by the pasting law for $(\infty,1)$-pullbacks it follows that with the left square in

$$\array{ \Omega B &\to& X \times_B Y &\to & B \\ \downarrow &\swArrow_{\simeq}& \downarrow &\swArrow_{\simeq}& \downarrow \\ * &\to& X \times Y &\stackrel{(f,g)}{\to}& B \times B }$$

an $(\infty,1)$-pullback, so is the total outer rectangle. But again by the first statement, this is equivalent to the $(\infty,1)$-pullback

$$\array{ \Omega B &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ * &\to& B } \,,$$

which is the defining pullback for the loop space object.

Therefore the Mayer-Vietoris homotopy fiber sequence is of the form

$$\Omega B \to X \times_B Y \to X \times Y \,.$$

For $\mathcal{C} =$ ∞Grpd $\simeq L_{whe}$ Top, this point of view is amplified in (Dyer-Roitberg 80).

Corollary

The corresponding long exact sequence of homotopy groups is of the form

$$\cdots \to \pi_{n+1} B \to \pi_n X \times_B Y \stackrel{(f_*, g_*)}{\to} \pi_n X \oplus \pi_n Y \stackrel{f_* - g_*}{\to} \pi_n B \to \cdots$$

$$\cdots \to \pi_2 B \to \pi_1 X \times_B Y \stackrel{(f_+, g_*)}{\to} \pi_1 X \times \pi_1 Y \stackrel{f_* \cdot g_*^{-1}}{\to} \pi_1 B \to \pi_0 (X \times_B Y) \to \pi_0 (X \times Y) \,.$$

This is what has historically been the definition of Mayer-Vietories sequences (Eckmann-Hilton 64).

Presentation by fibrant objects

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

$$\array{ X \times^h_B Y &\to& &\to& Y \\ \downarrow && && \downarrow^{\mathrlap{g}} \\ && B^I &\to& B \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& B } \,,$$

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

$$\array{ X \times_B^h Y &\to& B^I \\ \downarrow && \downarrow \\ X \times Y &\stackrel{(f,g)}{\to}& B \times B } \,.$$

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.

Over an $\infty$-group

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

$$\array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} &\swArrow_{\simeq}& \downarrow^{\mathrlap{e}} \\ B \times B &\stackrel{(-)\cdot (-)^{-1}}{\to}& B } \,,$$

where the bottom horizontal morphism is the composite

$$(-)\cdot (-)^{-1} : B \times B \stackrel{(id, (-)^{-1})}{\to} B \times B \stackrel{\cdot}{\to} B$$

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. 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

$$\Omega B \longrightarrow X \times_B Y \longrightarrow X \times Y \stackrel{f \cdot g^{-1}}{\longrightarrow} B \,.$$

First consider two more concrete special cases.

Example

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

$$\mathcal{C} \simeq ([S^{op}, sSet]_{proj, loc})^\circ \,.$$

As discussed there, the Dold-Kan correspondence prolongs to a Quillen adjunction on presheaves whose right adjoint is

$$\Xi : [S^{op}, Ch_{\bullet \leq 0}(Ab)]_{proj} \to [S^{op}, sAb]_{proj} \to [S^{op}, sSet]_{proj} \,.$$

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

$$\array{ \tilde B &\to& * \\ \downarrow^{\mathrlap{\Delta_{\tilde B}}} && \downarrow^{\mathrlap{0}} \\ \tilde B \times \tilde B &\stackrel{-}{\to}& \tilde B }$$

is a homotopy pullback in $[S^{op}, Ch_{\bullet \geq 0}(Ab)]_{proj}$, as is the ordinary pullback of simplicial presheaves

$$\array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{\mathrlap{0}} \\ B \times B &\stackrel{-}{\to}& B }$$

in $[S^{op}, sSet]_{proj}$.

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

Example

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

$$B \in [S^{op}, sGrp]_{proj} \to [S^{op}, sSet]_{proj} \,.$$

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

$$\array{ \Lambda[k]^i &\stackrel{(ha, hb)}{\to}& B(U) \times B(U) \\ \downarrow^{\mathrlap{j}} && \downarrow \\ \Delta[k] &\stackrel{\sigma}{\to}& B(U) }$$

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

$$(j^* \sigma)_l = ha_l \cdot hb_l^{-1} \,.$$

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

$$a \coloneqq \sigma \cdot b \,.$$

Since all the face maps are group homomorphisms, this is indeed a filler of $ha$:

$$\begin{aligned} \delta_l(a) & = \delta_l(\sigma \cdot b) \\ & = \delta_l(\sigma) \cdot \delta_l(b) \\ & = \delta_l(\sigma) \cdot hb_l \\ & = ha_l \end{aligned} \,.$$

Moreover, by construction, $(a,b)$ is a filler in

$$\array{ \Lambda[k]^i &\stackrel{(ha, hb)}{\to}& B(U) \times B(U) \\ \downarrow^{\mathrlap{i}} &{}^{(a,b)}\nearrow& \downarrow \\ \Delta[k] &\stackrel{\sigma}{\to}& B(U) } \,.$$

Since therefore $- \colon B \times B \to B$ is a projective fibration, it follows as before that the ordinary pullback

$$\array{ B &\to& * \\ \downarrow^{\mathrlap{\Delta_B}} && \downarrow^{e} \\ B \times B &\stackrel{-}{\to}& B }$$

is a homotopy pullback.

Proposition

For $B$ an ∞-group object as above, the (∞,1)-pullback $X \times_B Y$ is equivalently given by the $(\infty,1)$-pullback

$$\array{ X \times_B Y &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{0}} \\ X \times Y &\stackrel{f \cdot g^{-1}}{\to}& B } \,.$$

Proof

By prop. the object $X \times_B Y$ is the $(\infty,1)$-pullback in

$$\array{ X \times_B Y &\to& B \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}} \\ X \times Y &\stackrel{(f,g)}{\to}& B \times B } \,.$$

By the pasting law this is equivalently given by the composite pullback of

$$\array{ X \times_B Y &\to& B &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\Delta_B}} &\swArrow_{\simeq}& \downarrow^{\mathrlap{0}} \\ X \times Y &\stackrel{(f,g)}{\to}& B \times B &\stackrel{-}{\to}& B } \,.$$

Here the composite bottom morphism is $(f - g)$.

Summing this up:

Proposition

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

$$\Omega B \longrightarrow X \times_B Y \longrightarrow X \times Y \stackrel{f \cdot g^{-1}}{\longrightarrow} B \,.$$

Proof

For $\mathcal{C}$ an (∞,1)-site of definition, there is a reflection

$$\mathbf{H} \stackrel{\longleftarrow}{\hookrightarrow} [C^{op},\infty Grpd]$$

of $\mathbf{H}$ into an (∞,1)-category of (∞,1)-presheaves.

By prop. 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:

Proposition

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

$$\array{ G \times D &\longrightarrow& D \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G } \,.$$

(nForum discussion)

Proof

By this discussion we may use homotopy type theory reasoning. Starting out with the discussion at homotopy pullback – In homotopy type theory we obtain

$$\begin{aligned} D\times_G (G\times G) &= \sum_{d:D} \sum_{g_1:G} \sum_{g_2:G} (g_1\cdot g_2^{-1} = \phi(d)) \\ &= \sum_{d:D} \sum_{g_1:G} \sum_{g_2:G} (g_1 = \phi(d)\cdot g_2) \\ &= \sum_{d:D} \sum_{g_2:G} \sum_{g_1:G} (g_1 = \phi(d)\cdot g_2) \\ &= \sum_{d:D} \sum_{g_2:G} \mathbf{1}\\ &= D\times G \end{aligned} \,,$$

where the second but last step consists of observing a contractible based path space object (see the discussion at factorization lemma).

Corollary

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

$$\cdots \to \Omega G \longrightarrow X \times_G Y \longrightarrow X \times Y \stackrel{f \cdot (g^{-1})}{\longrightarrow} G \,.$$

Proof

Use prop. with $\phi$ being the canonical point, i.e. the inclusion $e \colon \ast \to G$ of the neutral element to find the homotopy pullback

$$\array{ G &\longrightarrow& \ast \\ \downarrow^{\mathrlap{\Delta}} && \downarrow^{\mathrlap{e}} \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G } \,.$$

Then use the pasting law as above.

Examples

(Co)Homology of a cover

A special case of the general Mayer-Vietoris sequence, corollary – 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

$$\array{ U \cap V &\hookrightarrow& U \\ \downarrow && \downarrow \\ V &\to& X }$$

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

$$\array{ \mathbf{H}(X, A) &\to& \mathbf{H}(U,A) \\ \downarrow && \downarrow \\ \mathbf{H}(V,A) &\to& \mathbf{H}(U \cap V, A) }$$

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 $G$

$$\pi_k \mathbf{H}(X, \mathbf{B}^n G) \simeq H^{n-k}(X, G) \,.$$

By the above general properties the above homotopy pullback is equivalent to

$$\mathbf{H}(X,A) \to \mathbf{H}(U,A) \times \mathbf{H}(V,A) \to \mathbf{H}(U \cap V, A)$$

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 $V$.

Related concepts

• diamond principle

• Brown-Gersten property

References

An original reference is

• Beno Eckmann and Peter Hilton, Unions and intersections in homotopy theory, Comment. Math. Helv. 3 (1964),2 93-307, doi

A more modern review that emphasizes the role of homotopy fiber sequences is in

• Eldon Dyer, Joseph Roitberg, Note on sequence of Mayer-Vietoris type, Proceedings of the AMS, volume 80, number 4 (1980) (pdf)

Discussion in the context of stable model categories includes

• Peter May, lemma 5.7 of The additivity of traces in triangulated categories, Adv. Math., 163(1):34-73, 2001 (pdf)

Discussion in the context of homotopy type theory includes

• E Cavallo et al, Exactness of the Mayer-Vietoris Sequence in Homotopy Type Theory (pdf)