group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A homotopy fiber sequence is a “long left-exact sequence” in an (∞,1)-category. (The dual concept is that of cofiber sequence.)
Traditionally fiber sequences have been considered in the context of homotopical categories such as model categories and Brown category of fibrant objects which present the (∞,1)-category in question. In particular, classically this was considered for Top itself. In these cases they are obtained in terms of homotopy pullbacks. Since, as discussed there, the homotopy fiber of a morphism may be computed as the ordinary 1-categorical fiber of any fibration resolution of this morphism, one often also speaks of fibration sequences.
Let $C$ be an (∞,1)-category with small limits and consider pointed objects of $C$, i.e. morphisms ${*} \to A$ from the terminal object ${*}$ (the point) to some object $A$. All unlabeled morphisms from the point in the following are these chosen ones and all other morphisms are taken with respect to these points.
Notice that in the case that $C$ happens to be a stable (∞,1)-category for which ${*} = 0$ all objects are canonically pointed and the notions of left- and right-exact fibration sequences coincide.
But for the notion of fibration sequence to make sense, we do not need to assume that $C$ is a stable $(\infty,1)$-category. In particular, in the context of nonabelian cohomology (see gerbe and principal 2-bundle) one considers fibration sequences in non-stable $(\infty,1)$-categories.
Now let $f : A \to B$ be a morphism in $C$.
The homotopy fiber or homotopy kernel or mapping cocone of $f$ is the pullback (which in our $(\infty,1)$-categorical context means homotopy pullback) of the point along $f$:
under construction
In (Quillen 67, section I.3) it was shown how the theory of fiber sequences and cofiber sequences arises in the abstract homotopy theory of model categories. Focusing on the fiber sequences, this perspective depends only on the category of fibrant objects inside the model category, and in fact makes sense generally in this context. This was spelled out in (Brown 73, section 4), which we review here.
In pointed objects $\mathcal{C}_f^{\ast/}$ of a category of fibrant objects $\mathcal{C}_f$, def. , consider a morphism of fiber-diagrams, hence a commuting diagram of the form
If the two vertical morphisms on the right are weak equivalences, then so is the vertical morphism in the left
(Brown 73, section 4, lemma 3)
Factor the diagram in question
through the pullback of the bottom horizontal line:
Here $f^\ast X_2 \to X_2$ is a weak equivalence by lemma and with this $X_1 \to f^\ast X_2$ is a weak equivalence by assumption and two-out-of-three.
Moreover, this diagram exhibits $fib(f_1)\to fib(\phi)$ as the base change (along $\ast \to Y_2$) of $X_1 \to f^\ast X_2$.
Hence it is now sufficient to observe that in category of fibrant objects, base change preserves weak equivalences (…).
Hence we say:
Let $\mathcal{C}$ be a model category. For $f \colon X \longrightarrow Y$ any morphism, then its homotopy fiber
is the morphism in the homotopy category $Ho(\mathcal{C})$, def. , which is represented by the fiber, def. , of any fibration resolution of $f$.
We may now state the abstract version of the statement of prop. :
Let $\mathcal{C}$ be a model category. For $f \colon X \to Y$ any morphism of pointed objects, and for $A$ a pointed object, def. , then the sequence
is exact (the sequence being the image of the homotopy fiber sequence of def. under the hom-functor of the pointed homotopy category of a model category
We may choose representatives such that $A$ is cofibrant, and $f$ is a fibration. Then we are faced with an ordinary pullback diagram
and the hom-classes are represented by genuine morphisms in $\mathcal{C}$. From this it follows immediately that $ker(p_\ast)$ includes $im(i_\ast)$. Hence it remains to show that every element in $ker(p_\ast)$ indeed comes from $im(i_\ast)$.
But an element in $ker(p_\ast)$ is represented by a morphism $\alpha \colon A \to X$ such that there is a left homotopy as in the following diagram
Now by lemma the square here has a lift $\tilde \eta$, as shown. This means that $i_1 \circ\tilde \eta$ is left homotopic to $\alpha$. But by the universal property of the fiber, $i_1 \circ \tilde \eta$ factors through $i \colon hofib(f) \to X$.
In homotopy type theory the homotopy fiber of a function term $f : A \to B$ over a function term $pt_B : * \to B$ is the type
hence the dependent sum over $A$ of the identity type on $B$ with $f(a)$ and $pt_B$ substituted. (A special case of the discussion at homotopy pullback)
For the corresponding Coq code see
A crucial difference between $\infty$-categorical fibration sequences and ordinary 1-categorical sequences is that the former are always long : in contrast to the ordinary kernel of a kernel, which is necessarily trivial, the homotopy kernel of a homotopy kernel is typically far from trivial, but is a loop space object. Due to that, each fibration sequence extend to the left by as many steps (times 3) as the objects involved have nontrivial homotopy groups.
In particular the homotopy fiber of the point ${*} \to B$ is the loop space object $\Omega B$ of $B$ (by definition):
Notice that the ordinary 1-categorical pullback of a point to itself is necessarily just the point again. Much of what makes (∞,1)-category-theory richer than ordinary category theory is this fact that the kernel of the point is not trivial, but loops. This implies in particular that the kernel of the kernel is in general nontrivial.
Namely the homotopy kernel of the morphism $ker(f) \to A$ constructed above is by definition the homotopy limit in the diagram
This is the same kind of diagram as before, just depicted after taking its mirror image along a diagonal. The point of drawing it this way is that this suggests to form the pasting diagram with the one that defines $ker(f)$
Since the $(\infty,1)$-categorical pullback satisfies the pasting law just as ordinary pullback diagrams do, it follows that the total outer square obtained this way is itself a homotopy pullback. But by definition of the loop space object $\Omega B$ this means that the kernel of the kernel is loops:
I.e. all three squares in
are (homotopy) pullback squares.
Continuing this way to the left with the pasting law, we obtain a long fiber sequence of morphisms to the left of the form
A subtlety to be aware of here is that $\Omega B$ is not quite $ker(ker(f))$, but the latter instead is $\bar \Omega f$, where $\bar \Omega$ denotes loops with reversed orientation.
A classical discussion of this in terms of computing homotopy fibers via path object fibrant replacements is e.g. in (Switzer 75, around 2.57). But let’s see it just diagrammatically:
First observe that it is indeed $\Omega f$ and not $\bar \Omega f$ that appears in the above: by “bending around” the bottom left “$\ast \to$” we get
On the other hand, if we define the homotopy fiber of any morphism $\phi$ by the diagram
then $ker(g)$ is given by the diagram
but what appears in the above pasting diagram is instead this diagram “reflected at the diagonal axis”
Here “$(-)^{-1}$” denotes the inverse of the 2-morphism (homotopies). Since it is these 2-morphisms/homotopies that become the loops in the loop space, it is here that loop reversal appears in translating between the naive iterated homotopy fiber to the construction that actually appears in the above pasting composite.
Usually, when looking at fibration sequences in 1-categorical contexts of the homotopy category of an (∞,1)-category, one doesn’t see these long fibration squences directly, but only “in cohomology”.
This can be usefully understood as follows:
recall from cohomology that for $X$ and $A$ objects in an (∞,1)-category $C$ that is an (∞,1)-topos, the $\infty$-groupoid of $A$-valued cocycle on $X$ is just $Hom_C(X,A)$, so that the corresponding cohomology classes are
where $Ho_C$ is the corresponding homotopy category of an (∞,1)-category.
The upshot being that in the right $(\infty,1)$-context cohomology is just the hom-object.
But the hom-functor has the crucial property that it is an exact functor in both arguments. This holds for $(\infty,1)$-categories just as well as for ordinary categories. For our context this means in particular that for
a homotopy pullback in $C$, for every $X \in K$ the induced diagram
is again homotopy pullback diagram (of ∞-groupoids); in particular the morphism $Hom_C(X,A \times_K B)\to Hom_C(X,A) \times_{Hom_{C}(X,K)} Hom_C(X,B)$ induced by the universal property of homotopy pullback is an equivalence.
So in particular for
a fibration sequence and for $X$ any object, there is a fibration sequence
is again a fibration sequence, now of $\infty$-groupoids. By projecting everything to connected components with $\Pi_0$ this then yields an ordinary long exact sequence of pointed sets
Due to the identitfication of cohomology with these homotopy hom-sets via $Ho_C(X,A) =: H(X,A)$, this is a “long exact sequence in cohomology”
See long exact sequence of homotopy groups.
We may read off from the non-triviality of the homotopy fiber $A$ of a morphism $f : B \to C$ to which extent $f$ fails to be an equivalence.
A morphism $f : B \to C$ in ∞Grpd is an equivalence precisely if all its homotopy fibers over every point of $C$ are contractible, i.e. are equivalent to $*$.
More generally, a morphism $f : B \to C$ in any (∞,1)-category $\mathcal{C}$ is an equivalence if for all objects $X \in \mathcal{C}$ all homotopy fibers of the morphism $\mathcal{C}(X,f) : \mathcal{C}(X,B) \to \mathcal{C}(X,C)$ are contractible.
For more on this see n-truncated object of an (∞,1)-category. Also HTT, around example 5.5.6.13. For more on homotopy fibers of hom-spaces see the section below.
We have seen that for $A \to B \stackrel{f}{\to} C$ a fiber sequence in an $(\infty,1)$-category $\mathcal{C}$, then for any other object $X$ we obtain a fiber sequence
in ∞Grpd, where the point of $Hom_{\mathcal{C}}(X,C)$ is $X \to * \to C$ with $* \to C$ the point of $C$, so that $Hom_{\mathcal{C}}(X,A)$ is the homotopy fiber over this point of the morphism given by postcomposition with $B \to C$.
Often it is important to know the homotopy fibers of $f_*$ also over other objects in $Hom_{\mathcal{C}}(X,C)$. This is notably the case when considering twisted cohomology with coefficients in $A$.
The homotopy fiber of $Hom_{\mathcal{C}}(X,B) \stackrel{f_*}{\to} Hom_{\mathcal{C}}(X,C)$ over a morphism $c : X \to C$ may be identified with the hom-object
in the over (∞,1)-category $\mathcal{C}_{/C}$.
This is HTT, prop. 5.5.5.12.
Model all (∞,1)-categories as quasi-categories.
Using the discussion at hom-object in a quasi-category, we observe that
and
under which identification the map $f_*$ is induced by the canonical
So we are done if we can show that the ordinary pullback diagram
is a homotopy pullback square, because we have an isomorphism of simplicial sets
as one checks.
Since in the above diagram all objects are Kan complexes, for the diagram to be a homotopy pullback it is sufficient that $\phi'$ is a Kan fibration for which in turn it is sufficient that it is a left fibration.
That follows by noticing that the right vertical morphism fits into the pullback diagram
But by general properties of left fibrations, the right vertical map is a left fibration. And since these are stable under pullbacks, so is $\phi'$.
Fibration sequences are familiar from the context of principal bundles.
Let $G$ be a group and let $\mathbf{B}G$ denote the corresponding one-object groupoid (in the world of ∞-groupoids) or else the classifying space $\mathcal{B}G$.
Notice that
Then that a $G$-principal bundle $P \to X$ is classified by morphism $X \to \mathbf{B}G$ means that it is the homotopy fiber of this morphism.
Indeed, as indicated at generalized universal bundle and at homotopy limit, we may compute the homotopy pullback
by first forming the standard fibrant replacement of the diagram $X \to \mathbf{B}G \leftarrow {*}$. That is given by the diagram
where $\mathbf{E}G \simeq {*}$ is the total “space” (or 2-groupoid) of the universal $G$-bundle. Once we have done this weakly equivalent replacement, the homotopy pullback may be computed as the ordinary pullback
in the ordinary 1-category of $n$-groupoids or spaces, using a replacement $\hat X \stackrel{\simeq}{\twoheadrightarrow} X$ of $X$ by an acyclic fibration (called “hypercover” in this context) (for instance the Čech groupoid associated with an open cover of $X$).
One recognizes the usual statement that principal $G$-bundles all arise as pullbacks of the universal $G$-principal bundle.
The fact that such pullbacks really are bundles whose fiber is $G$ is the statement of the long fibration sequence induced by $g$ which says that picking any point ${*} \to X$ of $X$ and then pulling back $P$ to that point (i.e. restricting it to that point) yields $\Omega \mathbf{B}G = G$:
The same logic – even the same diagrams – work for principal 2-bundles and generally for principal ∞-bundles.
Let $G$ be an ∞-group in that $\mathbf{B}G$ is an ∞-groupoid with a single object. An action of $G$ on an (∞,1)-category is an (∞,1)-functor
to (∞,1)Cat. This takes the single object of $\mathbf{B}G$ to some $(\infty,1)$-category $V$.
The action groupoid $V//G$ is the (∞,1)-categorical colimit over the action:
By the result described here this is, equivalent to the pullback of the “universal $(\infty,1)Cat$-bundle” $Z \to (\infty,1)Cat$, namely to the coCartesian fibration
classified by $\rho$ under the (∞,1)-Grothendieck construction. We obtain a fiber sequence to the left by adjoining the $(\infty,1)$-categorical pullback along the point inclusion $* \to \mathbf{B}G$
The resulting total $(\infty,1)$-pullback rectangle is the fiber of $Z \to (\infty,1)Cat$ over the $(\infty,1)$-category $C$, which is $V$ itself, as indicated.
Notice that every fibration sequence $V \to V//G \to \mathbf{B}G$ with $V//G \to \mathbf{B}G$ a coCartesian fibration arises this way, up to equivalence.
One of the most basic fibration sequences that appears all over the place in practice is the sequence of Eilenberg-MacLane objects
in $\mathbf{H} =$ ∞Grpd/Top, where $\mathbb{R}$ is the abelian group (under addition) of real number, $\mathbb{Z}$ the abelian group of integers, and where $\mathbb{R}/\mathbb{Z} = S^1 = U(1)$ is their quotient group, the circle group or unitary group $U(n)$ for $n=1$.
A quick way to see that this is indeed a fibration sequence is to realize that $\mathbb{R}/\mathbb{Z}$ is equivalent to the weak quotient action groupoid $\mathbb{R}//\mathbb{Z}$. Since everything here is in the image of the Dold-Kan correspondence
it is useful to model this fiber sequence as a sequence of chain complexes
Written this way the second morphism is evidently a degreewise surjection, hence is a fibration in the model structure on chain complexes. Therefore this being a fibration sequence is equivalent (as described at homotopy pullback) to the first morphism being equivalent to the ordinary kernel of the second, which clearly it is.
From this fiber sequence we obtain long exact sequences in cohomology, for instance in singular cohomology: let $X$ be a topological space and for $A$ an abelian group let
be its cohomology with coefficients in $A$, computed in ∞Grpd which we may present by sSet, where the fundamental ∞-groupoid $\Pi(X)$ is the singular simplicial complex $Sing X$ and we have
Then, as discussed above, the fiber sequence of coefficients $\cdots \to \mathbf{B}^{n-1} \mathbb{R}/\mathbb{Z} \to \mathbf{B}^n \mathbb{Z} \to \mathbf{B}^n \mathbb{R} \to \mathbf{B}^n \mathbb{R}/\mathbb{Z} \to \mathbf{B}^{n+1} \mathbb{Z} \to \cdots$ yields the long exact sequence in cohomology
In applications it often happens that one has a situation where the real cohomology is trivial, i.e $H^n(X,\mathbb{R}) = 0$ in some degree $n$. In that case the exactness of the sequence
implies that in this case we have an isomorphism
A Mayer-Vietoris sequence is a fiber sequence obtained from an $(\infty,1)$-pullback diagram of pointed objects:
if
is an infinity-pullback diagram in $\mathbf{H}$, then it naturally induces a fiber sequence that starts out as
This – or its associated long exact sequence of homotopy groups – is called the Mayer-Vietoris sequence of the pullback. See there for details.
The original example of Mayer-Vietoris sequences is obtained from the situtation where a homotopy pushout diagram
in $\mathbf{H} =$ ∞Grpd/Top is given (which modeled in sSet or in terms of CW-complexes in Top may be modeled by an ordinary pushout), and where $A \in \infty Grpd$ is some coefficient group object. Then applying the $(\infty,1)$-categorical hom $\mathbf{H}(-, A) : \mathbf{H}^{op} \to \infty Grpd$ yields the $\infty$-pullback diagram
By the above this is equivalent to
being a fibration 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.
The observation of long exact sequences of homotopy groups for homotopy fiber sequences originates (according to Switzer 75, p. 35) in
The first exhaustive study of these is due to
whence the terminology Puppe sequences.
Classical textbook accounts include
Robert Switzer, around 2.57 of Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Stanley Kochmann, prop. 3.2.6 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Lecture notes:
The construction in the axiomatic homotopy theory of model categories is due to
and in the context of categories of fibrant objects due to
A discussion of fiber sequences in terms of associated ∞-bundles is in
Related discussion on the behaviour of fiber sequences under left Bousfield localization of model categories is in
Alg. Geom. Topol. 13(3), 2013, pp. 1779–1813. (preprint pdf and ArXiv.)