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Special and general types
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 be an (∞,1)-category with finite (∞,1)-limits and let be pointed objects and
be any two morphisms with common codomain preserving the base points. Let be the (∞,1)-pullback
The corresponding Mayer-Vietoris sequence is the fiber sequence of the induced morphism . Often the term is used (only) for the corresponding long exact sequence of homotopy groups.
Let be a presentable (∞,1)-category.
Then , 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 is the loop space object .
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 -pullbacks it follows that with the left square in
an -pullback, so is the total outer rectangle. But again by the first statement, this is equivalent to the -pullback
which is the defining pullback for the loop space object.
Therefore the Mayer-Vietoris homotopy fiber sequence is of the form
For ∞Grpd 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).
Presentation by fibrant objects
Suppose that the (∞,1)-category is presented by a category of fibrant objects (for instance the subcategory on the fibrant objects of a model category).
Then the -pullback is presented by a homotopy pullback, and by the factorization lemma, this is given by the ordinary limit
where is a path object for . This limit coincides, up to isomorphism, with the pullback
This implies in particular that the homotopy fiber of is the loop space object , being the fiber of the path space object projection.
Over an -group
We consider now the case where carries the structure of an ∞-group (or just a grouplike H-space object) in a presentable (∞,1)-category or locally Cartesian closed (∞,1)-category .
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 in the Mayer-Vietoris sequence is itself the homotopy fiber of , hence that we have a long homotopy fiber sequence of the form
First consider two more concrete special cases.
Let be a small site and let be the (∞,1)-category of (∞,1)-sheaves on .
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 be an object with a presentation in in the image of this . We write also for this presentation, and hence for some presheaf of chain complexes .
We claim now that such satisfies the above assumption.
To see this, first notice that the evident morphism is degreewise an epimorphism, hence it is a fibration in , and since is right Quillen, so is the corresponding morphism in .
Therefore the ordinary pullback of presheaves of chain complexes
is a homotopy pullback in , as is the ordinary pullback of simplicial presheaves
Since ∞-stackification preserves finite (∞,1)-limits, this presents an (∞,1)-pullback also in .
Let be an (∞,1)-topos with a 1-site of definition (a 1-localic (∞,1)-topos).
Then (as discussed there) every ∞-group object in has a presentation by a presheaf of simplicial groups
We claim that the canonical morphism is objectwise a Kan fibration and hence a fibration in the projective model structure on simplicial presheaves.
Let be any test object. A diagram
corresponds to a -cell together with a choice of decomposition of the th horn as a difference
Since itself is a Kan complex (being a simplicial group, as discussed there) there is a filler of the horn . Define then
Since all the face maps are group homomorphisms, this is indeed a filler of :
Moreover, by construction, is a filler in
Since therefore is a projective fibration, it follows as before that the ordinary pullback
is a homotopy pullback.
For an ∞-group object as above, the (∞,1)-pullback is equivalently given by the -pullback
By prop. 1 the object is the -pullback in
By the pasting law this is equivalently given by the composite pullback of
Here the composite bottom morphism is .
Summing this up:
For an (∞,1)-sheaf (∞,1)-topos, an ∞-group-object in and and two morphisms, then there is a long homotopy fiber sequence of the form
For an (∞,1)-site of definition, there is a reflection
of into an (∞,1)-category of (∞,1)-presheaves.
By prop. 2 the statement holds in . Since embedding and reflection both preserve finite (∞,1)-limits, it hence also holds in .
Still more generally and more simply:
Let be a locally Cartesian closed (∞,1)-category. Let be an ∞-group object (or just a grouplike H-space-object). Then for 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 be a locally Cartesian closed (∞,1)-category. Let be an ∞-group object (or just a grouplike H-space-object).
Then for and two morphisms, there is a Mayer-Vietoris-type homotopy fiber sequence
Use prop. 4 with being the canonical point, i.e. the inclusion of the neutral element to find the homotopy pullback
Then use the pasting law as above.
(Co)Homology of a cover
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 equipped with an open cover .
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 ∞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 be some coefficient object, for instance an Eilenberg-MacLane object (Eilenberg-MacLane space ) for the definition of ordinary singular cohomology with coefficients in an abelian group .
Then applying the derived hom space functor 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 are the cohomology groups of with coefficients in
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 by and in -cohomology.
By duality (see universal coefficient theorem) an analogous statement holds for the homology of , and .
An original reference is
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)