Contents

Idea

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.

Definition

In higher category theory

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 $(\mathrm{\infty },1)$-category. In particular, in the context of nonabelian cohomology (see gerbe and principal 2-bundle) one considers fibration sequences in non-stable $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-categorical context means homotopy pullback) of the point along $f$:

In homotopy type theory

In homotopy type theory the homotopy fiber of a function term $f:A\to B$ over a function term ${\mathrm{pt}}_B:*\to B$ is the type

hence the dependent sum over $A$ of the identity type on $B$ with $f(a)$ and ${\mathrm{pt}}_B$ substituted. (A special case of the discussion at homotopy pullback)

For the corresponding Coq code see

• Vladimir Voevodsky, Foundations/Generalities/uuo.v

Long fibration sequences

A crucial difference between $\mathrm{\infty }$-categorical fibration sequences and ordinary 1-categorical sequences is that the former are always long : in contrast to the ordinary kernel of a kernel, wich 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.

Kernel of a kernel: loop objects

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 $(\mathrm{\infty },1)$-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 te 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.

Long fibration sequences

Continuing this way to the left, we obtain a long sequence of morphisms to the left

Here the $\overline{\Omega }$ indicates that the map involves reversing the direction of loops. This comes about by looking closely at the pullback diagrams that this comes from

Again, all squares and all pasting squares appearing here are homotopy pullback squares. If I had labeled to two morphisms to the point out of the loop object one would see that $\overline{\Omega }f$ indeed reverses orientation of loops.

Long exact sequences in cohomology

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 $\mathrm{\infty }$-groupoid of $A$-valued cocycle on $X$ is just ${\mathrm{Hom}}_C(X,A)$, so that the corresponding cohomology classes are

where ${\mathrm{Ho}}_C$ is the corresponding homotopy category of an (∞,1)-category.

The upshot being that in the right $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-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 ${\mathrm{Hom}}_C(X,A{\times }_{K}B)\to {\mathrm{Hom}}_C(X,A){\times }_{{\mathrm{Hom}}_C(X,K)}{\mathrm{Hom}}_C(X,B)$ 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 $\mathrm{\infty }$-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 ${\mathrm{Ho}}_C(X,A)=:H(X,A)$, this is a “long exact sequence in cohomology”

Long exact sequences of homotopy groups

See long exact sequence of homotopy groups.

Examples

Characterization of equivalences

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.

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.

Fiber sequences of $(\mathrm{\infty },1)$-functor categories

We have seen that for $A\to B\stackrel{f}{\to }C$ a fiber sequence in an $(\mathrm{\infty },1)$-category $𝒞$, then for any other object $X$ we obtain a fiber sequence

in ∞Grpd, where the point of ${\mathrm{Hom}}_{𝒞}(X,C)$ is $X\to *\to C$ with $*\to C$ the point of $C$, so that ${\mathrm{Hom}}_{𝒞}(X,A)$ 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 ${\mathrm{Hom}}_{𝒞}(X,C)$. This is notably the case when considering twisted cohomology with coefficients in $A$.

This is HTT, prop. 5.5.5.12.

Principal $\mathrm{\infty }$-bundles

Fibration sequences are familiar from the context of principal bundles.

Let $G$ be a group and let $BG$ denote the corresponding one-object groupoid (in the world of ∞-groupoids) or else the classifying space $ℬG$.

Notice that

Then that a $G$-principal bundle $P\to X$ is classified by morphism $X\to BG$ 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 $X\to BG\leftarrow *$. That is given by the diagram

where $EG\simeq *$ 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 $\Omega BG=G$:

The same logic – even the same diagrams – work for principal 2-bundles and generally for principal ∞-bundles.

Action groupoids

Let $G$ be an ∞-group in that $BG$ 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 $BG$ to some $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)\mathrm{Cat}$-bundle” $Z\to (\mathrm{\infty },1)\mathrm{Cat}$, namely to the coCartesian fibration

classified by $\rho$ under the (∞,1)-Grothendieck construction. We obtain a fiber sequence to the left by adjoining the $(\mathrm{\infty },1)$-categorical pullback along the point inclusion $*\to BG$

The resulting total $(\mathrm{\infty },1)$-pullback rectangle is the fiber of $Z\to (\mathrm{\infty },1)\mathrm{Cat}$ over the $(\mathrm{\infty },1)$-category $C$, which is $V$ itself, as indicated.

Notice that every fibration sequence $V\to V//G\to BG$ with $V//G\to BG$ a coCartesian fibration arises this way, up to equivalence.

Integral versus real cohomology

One of the most basic fibration sequences that appears all over the place in practice is the sequence of Eilenberg-MacLane objects

in $H=$ ∞Grpd/Top, where $ℝ$ is the abelian group (under addition) of real number, $\mathbb{Z}$ the abelian group of integers, and where $ℝ/\mathbb{Z}=S^1=U(1)$ 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 $ℝ/\mathbb{Z}$ is equivalent to the weak quotient action groupoid $ℝ//\mathbb{Z}$. 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 $\mathrm{Sing}X$ and we have

Then, as discussed above, the fiber sequence of coefficients $\cdots \to B^{n-1}ℝ/\mathbb{Z}\to B^n\mathbb{Z}\to B^nℝ\to B^nℝ/\mathbb{Z}\to B^{n+1}\mathbb{Z}\to \cdots$ 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 $H^n(X,ℝ)=0$ in some degree $n$. In that case the exactness of the sequence

implies that in this case we have an isomorphism

Mayer-Vietoris sequences

A Mayer-Vietoris sequence is a fiber sequence obtained from an $(\mathrm{\infty },1)$-pullback diagram of pointed objects:

if

is an infinity-pullback diagram in $H$, 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.

Of a cover

The original example of Mayer-Vietoris sequences is obtained from the situtation where a homotopy pushout diagram

in $H=$ ∞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 $A\in \mathrm{\infty }\mathrm{Grpd}$ is some coefficient group object. Then applying the $(\mathrm{\infty },1)$-categorical hom $H(-,A):H^{\mathrm{op}}\to \mathrm{\infty }\mathrm{Grpd}$ yields the $\mathrm{\infty }$-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.

References

A discussion of fiber sequences in terms of associated ∞-bundles is in

• Matthias Wendt, Classifying spaces and fibrations of simplicial sheaves (arXiv)

Related discussion on the behaviour of fiber sequences under left Bousfield localization of model categories is in

• Matthias Wendt, Classifying spaces and fibrations of simplicial sheaves (pdf)