(also nonabelian homological algebra)
The notation “$\underset{\longleftarrow}{\lim}^1$” is common notation for the first derived functor $R^1 \underset{\longleftarrow}{\lim}$ of the limit functor.
Here we consider the case of limits over sequential diagrams of abelian groups (prop. 2 below). In good cases, this is the only obstruction to a naive limit of homotopy sets being the homotopy classes of the correct homotopy limit. Such a situation is expressed by a short exact Milnor sequence (below).
Given a tower $A_\bullet$ of abelian groups
write
for the homomorphism given by
The limit of a sequence as in def. 1 – hence the group $\underset{\longleftarrow}{\lim}_n A_n$ universally equipped with morphisms $\underset{\longleftarrow}{\lim}_n A_n \overset{p_n}{\to} A_n$ such that all
commute – is equivalently the kernel of the morphism $\partial$ in def. 1.
Given a tower $A_\bullet$ of abelian groups
then $\underset{\longleftarrow}{\lim}^1 A_\bullet$ is the cokernel of the map $\partial$ in def. 1, hence the group that makes a long exact sequence of the form
There is a generalization to groups (not necessarily abelian). (Bousfield-Kan 72, IX.2.1)
The functor $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. 2) satisfies
for every short exact sequence $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0 \;\;\; \in Ab^{(\mathbb{N}, \geq)}$ then the induced sequence
is a long exact sequence of abelian groups;
if $A_\bullet$ is a tower such that all maps are surjections, then $\underset{\longleftarrow}{\lim}^1_n A_n \simeq 0$.
(e.g. Bousfield-Kan 72, ch IX, prop. 2.3 and 2.4, Switzer 75, prop. 7.63, Goerss-Jardine 96, section VI. lemma 2.11)
For the first property: Given $A_\bullet$ a tower of abelian groups, write
for the homomorphism from def. 1 regarded as the single non-trivial differential in a cochain complex of abelian groups. Then by remark 1 and def. 2 we have $H^0(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim} A_\bullet$ and $H^1(L(A_\bullet)) \simeq \underset{\longleftarrow}{\lim}^1 A_\bullet$.
With this, then for a short exact sequence of towers $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$ the long exact sequence in question is the long exact sequence in homology of the corresponding short exact sequence of complexes
For the second statement: If all the $f_k$ are surjective, then inspection shows that the homomorphism $\partial$ in def. 1 is surjective. Hence its cokernel vanishes.
The category $Ab^{(\mathbb{N}, \geq)}$ of towers of abelian groups has enough injectives.
The functor $(-)_n \colon Ab^{(\mathbb{N}, \geq)} \to Ab$ that picks the $n$-th component of the tower has a right adjoint $r_n$, which sends an abelian group $A$ to the tower
Since $(-)_n$ itself is evidently an exact functor, its right adjoint preserves injective objects (prop.).
So with $A_\bullet \in Ab^{(\mathbb{N}, \geq)}$, let $A_n \hookrightarrow \tilde A_n$ be an injective resolution of the abelian group $A_n$, for each $n \in \mathbb{N}$. Then
is an injective resolution for $A_\bullet$.
The functor $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. 2) is the first right derived functor of the limit functor $\underset{\longleftarrow}{\lim} \colon Ab^{(\mathbb{N},\geq)} \longrightarrow Ab$.
(Bousfield-Kan 72, chapter IX.2, remark 2.6)
By lemma 1 there are enough injectives in $Ab^{(\mathbb{N}, \geq)}$. So for $A_\bullet \in Ab^{(\mathbb{N}, \geq)}$ the given tower of abelian groups, let
be an injective resolution. We need to show that
Since limits preserve kernels, this is equivalently
Now observe that each injective $J^q_\bullet$ is a tower of epimorphism. This follows by the defining right lifting property applied against the monomorphisms of towers of the following form
Therefore by the second item of prop. 1 the long exact sequence from the first item of prop. 1 applied to the short exact sequence
becomes
Exactness of this sequence gives the desired identification $\underset{\longleftarrow}{\lim}^1 A_\bullet \simeq (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1)) \,.$
The functor $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. 2) is in fact the unique functor, up to natural isomorphism, satisfying the conditions in prop. 3.
The proof of prop. 2 only used the conditions from prop. 1, hence any functor satisfying these conditions is the first right derived functor of $\underset{\longleftarrow}{\lim}$, up to natural isomorphism.
A tower $A_\bullet$ of abelian groups
is said to satisfy the Mittag-Leffler condition if for all $k$ there exists $i \geq k$ such that for all $j \geq i \geq k$ the image of the homomorphism $A_i \to A_k$ equals that of $A_j \to A_k$
(e.g. Switzer 75, def. 7.74)
The Mittag-Leffler condition, def. 3, is satisfied in particular when all morphisms $A_{i+1}\to A_i$ are epimorphisms (hence surjections of the underlying sets).
If a tower $A_\bullet$ satisfies the Mittag-Leffler condition, def. 3, then its $\underset{\leftarrow}{\lim}^1$ vanishes:
e.g. (Switzer 75, theorem 7.75, Kochmann 96, prop. 4.2.3, Weibel 94, prop. 3.5.7)
One needs to show that with the Mittag-Leffler condition, then the cokernel of $\partial$ in def. 1 vanishes, hence that $\partial$ is an epimorphism in this case, hence that every $(a_n)_{n \in \mathbb{N}} \in \underset{n}{\prod} A_n$ has a preimage under $\partial$. So use the Mittag-Leffler condition to find pre-images of $a_n$ by induction over $n$.
Given a cotower
of abelian groups, then for every abelian group $B \in Ab$ there is a short exact sequence of the form
where $Hom(-,-)$ denotes the hom-group, $Ext^1(-,-)$ denotes the first Ext-group (and so $Hom(-,-) = Ext^0(-,-)$).
Consider the homomorphism
which sends $a_n \in A_n$ to $a_n - f_n(a_n)$. Its cokernel is the colimit over the cotower, but its kernel is trivial (in contrast to the otherwise formally dual situation in remark 1). Hence (as opposed to the long exact sequence in def. 2) there is a short exact sequence of the form
Every short exact sequence gives rise to a long exact sequence of derived functors (prop.) which in the present case starts out as
where we used that direct sum is the coproduct in abelian groups, so that homs out of it yield a product, and where the morphism $\partial$ is the one from def. 1 corresponding to the tower
Hence truncating this long sequence by forming kernel and cokernel of $\partial$, respectively, it becomes the short exact sequence in question.
(Milnor exact sequence for homotopy groups)
Let
be a tower of fibrations, for instance a tower of simplicial sets with each map a Kan fibration (and $X_0$, hence each $X_n$ a Kan complex), or a tower of topological spaces with each map a Serre fibration. Then for each $q \in \mathbb{N}$ there is a short exact sequence
for $\pi_\bullet$ the homotopy group-functor (exact as pointed sets for $i = 0$, as groups for $i \geq 1$) which says that
e.g. (Bousfield-Kan 72, chapter IX, theorem 3.1, Goerss-Jardine 96, section VI. prop. 2.15)
With respect to the classical model structure on simplicial sets or the classical model structure on topological spaces, a tower of fibrations as stated is a fibrant object in the injective model structure on functors $[(\mathbb{N},\geq), sSet]_{inj}$ ($[(\mathbb{N},\geq), Top]_{inj}$) (prop). Hence the plain limit over this diagram represents the homotopy limit. By the discussion there, up to weak equivalence that homotopy limit is also the pullback in
where on the right we have the product over all the canonical fibrations out of the path space objects. Hence also the left vertical morphism is a fibration, and so by taking its fiber over a basepoint, the pasting law gives a homotopy fiber sequence
The long exact sequence of homotopy groups of this fiber sequence goes
Chopping that off by forming kernel and cokernel yields the claim for positive $q$. For $q = 0$ it follows by inspection.
Let
be a tower of chain complexes (of abelian groups) such that it satisfies degree-wise the Mittag-Leffler condition, def. 3, and write
for its limit. Then for each $q \in \mathbb{Z}$ the chain homology $H_q(-)$ of the limit sits in a short exact sequence with the ordinary $\underset{\longleftarrow}{\lim}$ and the $\underset{\longleftarrow}{\lim}^1$ of the chain homologies:
(e.g. Weibel 94, prop. 3.5.8)
(Milnor exact sequence for generalized cohomology)
Let $X$ be a pointed CW-complex, $X = \underset{\longrightarrow}{\lim}_n X_n$ and let $\tilde E^\bullet$ an additive reduced cohomology theory.
Then the canonical morphisms make a short exact sequence
saying that
the failure of the canonical comparison map $\tilde E^\bullet(X) \to \underset{\longleftarrow}{\lim} \tilde E^\bullet(X_n)$ to the limit of the cohomology groups on the finite stages to be an isomorphism is at most in a non-vanishing kernel;
this kernel is precisely the $\lim^1$ (def. 2) of the cohomology groups at the finite stages in one degree lower.
e.g. (Switzer 75, prop. 7.66, Kochmann 96, prop. 4.2.2)
For
the sequence of stages of the (pointed) CW-complex $X = \underset{\longleftarrow}{\lim}_n X_n$, write
for the disjoint unions of the cylinders over all the stages in even and all those in odd degree, respectively.
These come with canonical inclusion maps into the mapping telescope $Tel(X_\bullet)$ (def.), which we denote by
Observe that
$A_X \cup B_X \simeq Tel(X_\bullet)$;
$A_X \cap B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_n$;
and that there are homotopy equivalences
$A_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n+1}$
$B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n}$
$Tel(X_\bullet) \simeq X$.
The first two are obvious, the third is this proposition.
This implies that the Mayer-Vietoris sequence (prop.) for $\tilde E^\bullet$ on the cover $A \sqcup B \to X$ is isomorphic to the bottom horizontal sequence in the following diagram:
hence that the bottom sequence is also a long exact sequence.
To identify the morphism $\partial$, notice that it comes from pulling back $E$-cohomology classes along the inclusions $A \cap B \to A$ and $A\cap B \to B$. Comonentwise these are the inclusions of each $X_n$ into the left and the right end of its cylinder inside the mapping telescope, respectively. By the construction of the mapping telescope, one of these ends is embedded via $i_n \colon X_n \hookrightarrow X_{n+1}$ into the cylinder over $X_{n+1}$. In conclusion, $\partial$ acts by
(The relative sign is the one in $(\iota_{A_x})^\ast - (\iota_{B_x})^\ast$ originating in the definition of the Mayer-Vietoris sequence and properly propagated to the bottom sequence while ensuring that $\tilde E^\bullet(X)\to \prod_n \tilde E^\bullet(X_n)$ is really $(i_n^\ast)_n$ and not $(-1)^n(i_n^\ast)_n$, as needed for the statement to be proven.)
This is the morphism from def. 1 for the sequence
Hence truncating the above long exact sequence by forming kernel and cokernel of $\partial$, the result follows via remark 1 and definition 2.
In contrast:
Let $X$ be a pointed CW-complex, $X = \underset{\longleftarrow}{\lim}_n X_n$.
For $\tilde E_\bullet$ an additive reduced generalized homology theory, then
is an isomorphism.
For $X, E \in Ho(Spectra)$ two spectra, then the $E$-generalized cohomology of $X$ is the graded group of homs in the stable homotopy category (def., exmpl.)
The stable homotopy category is, in particular, the homotopy category of the stable model structure on orthogonal spectra, in that its localization at the stable weak homotopy equivalences is of the form
In the following when considering an orthogonal spectrum $X \in OrthSpec(Top_{cg})$, we use, for brevity, the same symbol for its image under $\gamma$.
For $X, E \in OrthSpec(Top_{cg})$ two orthogonal spectra (or two symmetric spectra such that $X$ is a semistable symmetric spectrum) then there is a short exact sequence of the form
where $\underset{\longleftarrow}{\lim}^1$ denotes the lim^1, and where this and the limit on the right are taken over the following structure morphisms
(Schwede 12, chapter II prop. 6.5 (ii)) (using that symmetric spectra underlying orthogonal spectra are semistable (Schwede 12, p. 40))
For $X,E \in Ho(Spectra)$ two spectra such that the tower $n \mapsto E^{n -1}(X_{n})$ satisfies the Mittag-Leffler condition (def. 3), then two morphisms of spectra $X \longrightarrow E$ are homotopic already if all their morphisms of component spaces $X_n \to E_n$ are.
By prop. 4 the assumption implies that the $lim^1$-term in prop. 8 vanishes, hence by exactness it follows that in this case there is an isomorphism
John Milnor, On axiomatic homology theory, Pacific J. Math. Volume 12, Number 1 (1962), 337-341 (Euclid)
Z. Z. Yeh, Higher Inverse Limits and Homology Theories, Thesis, Princeton, 1959.
Aldridge Bousfield, Daniel Kan, section IX.2 of Homotopy limits, completions and localization, Springer 1972
Robert Switzer, section 7 from def. 7.57 on in Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Charles Weibel, section 3.5 of An Introduction to Homological Algebra, Cambridge University Press (1994)
Stanley Kochmann, section 4.2 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Paul Goerss, Rick Jardine, section VI.2 of Simplicial homotopy theory, Modern Birkhäuser Classics (1999)
(that’s section VII.6 of the 1996 Progress of Mathematics edition )
For generalized cohomology of spectra
Discussion in the context of categories of fibrant objects is in
Discussion in the context of conditional convergence of spectral sequences is in