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\begin{document}

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\section*{lim^1 and Milnor sequences}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{abstract_characterizations}{Abstract characterizations}\dotfill \pageref*{abstract_characterizations} \linebreak
\noindent\hyperlink{vanishing_of_}{Vanishing of $\lim^1$}\dotfill \pageref*{vanishing_of_} \linebreak
\noindent\hyperlink{relation_to_groups}{Relation to $Ext$-groups}\dotfill \pageref*{relation_to_groups} \linebreak
\noindent\hyperlink{MilnorSequences}{Milnor exact sequences}\dotfill \pageref*{MilnorSequences} \linebreak
\noindent\hyperlink{for_homotopy_groups}{For homotopy groups}\dotfill \pageref*{for_homotopy_groups} \linebreak
\noindent\hyperlink{for_chain_homology}{For chain homology}\dotfill \pageref*{for_chain_homology} \linebreak
\noindent\hyperlink{for_generalized_cohomology_groups}{For generalized cohomology groups}\dotfill \pageref*{for_generalized_cohomology_groups} \linebreak
\noindent\hyperlink{of_spaces}{Of spaces}\dotfill \pageref*{of_spaces} \linebreak
\noindent\hyperlink{ForGeneralizedCohomologyOfSpectra}{Of spectra}\dotfill \pageref*{ForGeneralizedCohomologyOfSpectra} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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. \ref{Lim1IsDerivedLimit} 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 sequence|short exact]] \emph{Milnor sequence} (\hyperlink{MilnorSequences}{below}).

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\begin{defn}
\label{TheBoundaryMapDefiningLim1}\hypertarget{TheBoundaryMapDefiningLim1}{}
Given a [[tower]] $A_\bullet$ of [[abelian groups]]

\begin{displaymath}
\cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0
\end{displaymath}
write

\begin{displaymath}
\partial
    \;\colon\;
  \underset{n}{\prod} A_n
    \longrightarrow
  \underset{n}{\prod} A_n
\end{displaymath}
for the homomorphism given by

\begin{displaymath}
\partial \;\colon\; 
    (a_n)_{n \in \mathbb{N}} 
  \mapsto  
    (a_n - f_n(a_{n+1}))_{n \in \mathbb{N}}.
\end{displaymath}
\end{defn}
\begin{remark}
\label{LimitAsKernelAnalogousToLim1}\hypertarget{LimitAsKernelAnalogousToLim1}{}
The [[limit]] of a sequence as in def. \ref{TheBoundaryMapDefiningLim1} -- 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

\begin{displaymath}
\itexarray{
     && \underset{\longleftarrow}{\lim}_n A_n
     \\
     & {}^{\mathllap{p_{n+1}}}\swarrow && \searrow^{\mathrlap{p_n}}
     \\
     A_{n+1} && \overset{f_n}{\longrightarrow} && A_n
  }
\end{displaymath}
[[commuting diagram|commute]] -- is equivalently the [[kernel]] of the morphism $\partial$ in def. \ref{TheBoundaryMapDefiningLim1}.

\end{remark}
\begin{defn}
\label{Lim1ViaCokernel}\hypertarget{Lim1ViaCokernel}{}
Given a [[tower]] $A_\bullet$ of [[abelian groups]]

\begin{displaymath}
\cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0
\end{displaymath}
then $\underset{\longleftarrow}{\lim}^1 A_\bullet$ is the [[cokernel]] of the map $\partial$ in def. \ref{TheBoundaryMapDefiningLim1}, hence the group that makes a [[long exact sequence]] of the form

\begin{displaymath}
0 \to 
  \underset{\longleftarrow}{\lim}_n A_n
  \longrightarrow
  \underset{n}{\prod} A_n
   \stackrel{\partial}{\longrightarrow}
  \underset{n}{\prod} A_n
    \longrightarrow
  \underset{\longleftarrow}{\lim}^1_n A_n
  \to 0
  \,,
\end{displaymath}
\end{defn}
There is a generalization to groups (not necessarily abelian). (\hyperlink{BousfieldKan72}{Bousfield-Kan 72, IX.2.1})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{abstract_characterizations}{}\subsubsection*{{Abstract characterizations}}\label{abstract_characterizations}

\begin{prop}
\label{PropertiesOfLim1}\hypertarget{PropertiesOfLim1}{}
The [[functor]] $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. \ref{Lim1ViaCokernel}) satisfies

\begin{enumerate}%
\item 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

\begin{displaymath}
0 
 \to
 \underset{\longleftarrow}{\lim}_n A_n
 \to
 \underset{\longleftarrow}{\lim}_n B_n
 \to
 \underset{\longleftarrow}{\lim}_n C_n
 \to
 \underset{\longleftarrow}{\lim}_n^1  A_n
 \to
 \underset{\longleftarrow}{\lim}_n^1 B_n
 \to
 \underset{\longleftarrow}{\lim}_n^1 C_n
 \to 0
\end{displaymath}
is a [[long exact sequence]] of abelian groups;


\item if $A_\bullet$ is a tower such that all maps are [[surjections]], then $\underset{\longleftarrow}{\lim}^1_n A_n \simeq 0$.



\end{enumerate}
\end{prop}
(e.g. \hyperlink{BousfieldKan72}{Bousfield-Kan 72, ch IX, prop. 2.3 and 2.4}, \hyperlink{Switzer75}{Switzer 75, prop. 7.63}, \hyperlink{GoerssJardine96}{Goerss-Jardine 96, section VI. lemma 2.11})

\begin{proof}
For the first property: Given $A_\bullet$ a tower of abelian groups, write

\begin{displaymath}
L^\bullet(A_\bullet)
  \coloneqq
  \left[
    0
   \to 
  \underset{deg \, 0}{\underbrace{\underset{n}{\prod} A_n}}
    \overset{\partial}{\longrightarrow}
  \underset{deg\, 1}{\underbrace{\underset{n}{\prod} A_n}}
   \to 
   0
  \right]
\end{displaymath}
for the homomorphism from def. \ref{TheBoundaryMapDefiningLim1} regarded as the single non-trivial differential in a [[cochain complex]] of abelian groups. Then by remark \ref{LimitAsKernelAnalogousToLim1} and def. \ref{Lim1ViaCokernel} 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

\begin{displaymath}
0 
    \to 
  L^\bullet(A_\bullet) 
    \longrightarrow 
  L^\bullet(B_\bullet)
    \longrightarrow
  L^\bullet(C_\bullet)
    \to
  0
  \,.
\end{displaymath}
For the second statement: If all the $f_k$ are surjective, then inspection shows that the homomorphism $\partial$ in def. \ref{TheBoundaryMapDefiningLim1} is surjective. Hence its [[cokernel]] vanishes.

\end{proof}
\begin{lemma}
\label{TowersOfAbelianGroupsHasEnoughInjectives}\hypertarget{TowersOfAbelianGroupsHasEnoughInjectives}{}
The category $Ab^{(\mathbb{N}, \geq)}$ of [[towers]] of [[abelian groups]] has [[enough injectives]].

\end{lemma}
\begin{proof}
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

\begin{displaymath}
r_n
  \coloneqq
  \left[
    \cdots \overset{id}{\to}
    A \overset{id}{\to}
    \underset{= (r_n)_{n+1}}{\underbrace{A}} \overset{id}{\to}
    \underset{= (r_n)_n}{\underbrace{A}} 
       \overset{id}{\to}
    \underset{= (r_n)_{n-1}}{\underbrace{0}}
    \to 
    0
    \to
    \cdots
    \to
    0
    \to 
    0
  \right]
  \,.
\end{displaymath}
Since $(-)_n$ itself is evidently an [[exact functor]], its right adjoint preserves injective objects (\href{injective+object#RightAdjointsOfExactFunctorsPreserveInjectives}{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

\begin{displaymath}
A_\bullet 
    \overset{(\eta_n)_{n \in \mathbb{N}}}{\longrightarrow}
  \underset{n \in \mathbb{R}}{\prod}
  r_n A_n
  \hookrightarrow
  \underset{n \in \mathbb{N}}{\prod} r_n \tilde A_n
\end{displaymath}
is an injective resolution for $A_\bullet$.

\end{proof}
\begin{prop}
\label{Lim1IsDerivedLimit}\hypertarget{Lim1IsDerivedLimit}{}
The [[functor]] $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. \ref{Lim1ViaCokernel}) is the [[derived functor in homological algebra|first right derived functor]] of the [[limit]] functor $\underset{\longleftarrow}{\lim} \colon Ab^{(\mathbb{N},\geq)} \longrightarrow Ab$.

\end{prop}
(\hyperlink{BousfieldKan72}{Bousfield-Kan 72, chapter IX.2, remark 2.6})

\begin{proof}
By lemma \ref{TowersOfAbelianGroupsHasEnoughInjectives} 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

\begin{displaymath}
0 
    \to 
  A_\bullet 
    \overset{j^0}{\longrightarrow} 
  J^0_\bullet 
    \overset{j^1}{\longrightarrow}
  J^1_\bullet
    \overset{j^2}{\longrightarrow}
  J^2_\bullet
    \overset{}{\longrightarrow}
  \cdots
\end{displaymath}
be an [[injective resolution]]. We need to show that

\begin{displaymath}
\underset{\longleftarrow}{\lim}^1 A_\bullet
    \simeq
  ker(\underset{\longleftarrow}{\lim}(j^2))/im(\underset{\longleftarrow}{\lim}(j^1))
  \,.
\end{displaymath}
Since limits preserve [[kernels]], this is equivalently

\begin{displaymath}
\underset{\longleftarrow}{\lim}^1 A_\bullet
    \simeq
  (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1))
\end{displaymath}
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

\begin{displaymath}
\itexarray{
      \cdots &\to & 0 &\to& 0 &\longrightarrow& 0 &\longrightarrow& \mathbb{Z} &\overset{id}{\longrightarrow}& \cdots &\overset{id}{\longrightarrow}& \mathbb{Z} &\overset{id}{\longrightarrow}& \mathbb{Z}
     \\
     \cdots
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \downarrow^{\mathrlap{incl}}
     &&
     \downarrow^{\mathrlap{id}}
     &&
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \downarrow^{\mathrlap{id}}
     \\
     \cdots &\to& 0 &\to& 0 &\to & \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \cdots &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z}
  }
\end{displaymath}
Therefore by the second item of prop. \ref{PropertiesOfLim1} the long exact sequence from the first item of prop. \ref{PropertiesOfLim1} applied to the [[short exact sequence]]

\begin{displaymath}
0
   \to 
  A_\bullet
   \overset{j^0}{\longrightarrow}
  J^0_\bullet
   \overset{j^1}{\longrightarrow}
  ker(j^2)_\bullet
   \to
  0
\end{displaymath}
becomes

\begin{displaymath}
0
     \to
   \underset{\longleftarrow}{\lim} A_\bullet
    \overset{\underset{\longleftarrow}{\lim} j^0}{\longrightarrow}
  \underset{\longleftarrow}{\lim} J^0_\bullet
    \overset{\underset{\longleftarrow}{\lim}j^1}{\longrightarrow}
  \underset{\longleftarrow}{\lim}(ker(j^2)_\bullet)
    \longrightarrow
  \underset{\longleftarrow}{\lim}^1 A_\bullet
    \longrightarrow 
  0 
  \,.
\end{displaymath}
Exactness of this sequence gives the desired identification $\underset{\longleftarrow}{\lim}^1 A_\bullet
    \simeq
  (\underset{\longleftarrow}{\lim}(ker(j^2)_\bullet))/im(\underset{\longleftarrow}{\lim}(j^1))
 \,.$

\end{proof}
\begin{prop}
\label{AbstractCharacterizationOfLim1}\hypertarget{AbstractCharacterizationOfLim1}{}
The [[functor]] $\underset{\longleftarrow}{\lim}^1 \colon Ab^{(\mathbb{N}, \geq)} \longrightarrow Ab$ (def. \ref{Lim1ViaCokernel}) is in fact the unique functor, up to [[natural isomorphism]], satisfying the conditions in prop. \ref{AbstractCharacterizationOfLim1}.

\end{prop}
\begin{proof}
The proof of prop. \ref{Lim1IsDerivedLimit} only used the conditions from prop. \ref{PropertiesOfLim1}, hence any functor satisfying these conditions is the first right derived functor of $\underset{\longleftarrow}{\lim}$, up to natural isomorphism.

\end{proof}
\hypertarget{vanishing_of_}{}\subsubsection*{{Vanishing of $\lim^1$}}\label{vanishing_of_}

\begin{defn}
\label{MittagLefflerCondition}\hypertarget{MittagLefflerCondition}{}
A tower $A_\bullet$ of [[abelian groups]]

\begin{displaymath}
\cdots \to A_3 \to A_2 \to A_1 \to A_0
\end{displaymath}
is said to satisfy the \textbf{[[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 $A_j \to A_k$

\begin{displaymath}
im(A_i \to A_k) \simeq im(A_j \to A_k)
  \,.
\end{displaymath}
\end{defn}
(e.g. \hyperlink{Switzer75}{Switzer 75, def. 7.74})

\begin{example}
\label{MittagLefflerSatisfiedInParticularForTowerOfSurjections}\hypertarget{MittagLefflerSatisfiedInParticularForTowerOfSurjections}{}
The Mittag-Leffler condition, def. \ref{MittagLefflerCondition}, is satisfied in particular when all morphisms $A_{i+1}\to A_i$ are [[epimorphisms]] (hence [[surjections]] of the underlying [[sets]]).

\end{example}
\begin{prop}
\label{Lim1VanihesUnderMittagLeffler}\hypertarget{Lim1VanihesUnderMittagLeffler}{}
If a tower $A_\bullet$ satisfies the [[Mittag-Leffler condition]], def. \ref{MittagLefflerCondition}, then its $\underset{\leftarrow}{\lim}^1$ vanishes:

\begin{displaymath}
\underset{\longleftarrow}{\lim}^1 A_\bullet = 0
  \,.
\end{displaymath}
\end{prop}
e.g. (\hyperlink{Switzer75}{Switzer 75, theorem 7.75}, \hyperlink{Kochmann96}{Kochmann 96, prop. 4.2.3}, \hyperlink{Weibel94}{Weibel 94, prop. 3.5.7})

\begin{proof}
One needs to show that with the Mittag-Leffler condition, then the [[cokernel]] of $\partial$ in def. \ref{TheBoundaryMapDefiningLim1} 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 $n$.

\end{proof}
\hypertarget{relation_to_groups}{}\subsubsection*{{Relation to $Ext$-groups}}\label{relation_to_groups}

\begin{lemma}
\label{lim1AndExt1}\hypertarget{lim1AndExt1}{}
Given a [[cotower]]

\begin{displaymath}
A_\bullet = (A_0 \overset{f_0}{\to} A _1 \overset{f_1}{\to} A_2 \to \cdots)
\end{displaymath}
of [[abelian groups]], then for every abelian group $B \in Ab$ there is a [[short exact sequence]] of the form

\begin{displaymath}
0 
    \to 
  \underset{\longleftarrow}{\lim}^1_n Hom(A_n, B)
    \longrightarrow
  Ext^1( \underset{\longrightarrow}{\lim}_n A_n, B )
    \longrightarrow
  \underset{\longleftarrow}{\lim}_n Ext^1( A_n, B)
    \to
  0
  \,,
\end{displaymath}
where $Hom(-,-)$ denotes the [[hom-object|hom-group]], $Ext^1(-,-)$ denotes the first [[Ext]]-group (and so $Hom(-,-) = Ext^0(-,-)$).

\end{lemma}
\begin{proof}
Consider the homomorphism

\begin{displaymath}
\tilde \partial
  \;\colon\;
  \underset{n}{\oplus} A_n
    \longrightarrow
  \underset{n}{\oplus} A_n
\end{displaymath}
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 [[formal dual|formally dual]] situation in remark \ref{LimitAsKernelAnalogousToLim1}). Hence (as opposed to the long exact sequence in def. \ref{Lim1ViaCokernel}) there is a [[short exact sequence]] of the form

\begin{displaymath}
0 
    \to
  \underset{n}{\oplus} A_n
    \overset{\tilde \partial}{\longrightarrow}
  \underset{n}{\oplus} A_n
    \overset{}{\longrightarrow}
  \underset{\longrightarrow}{lim}_n A_n
   \to
  0
  \,.
\end{displaymath}
Every short exact sequence gives rise to a [[long exact sequence]] of [[derived functors]] (\href{derived+functor+in+homological+algebra#LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence}{prop.}) which in the present case starts out as

\begin{displaymath}
0
    \to
  Hom(\underset{\longrightarrow}{\lim}_n A_n,B)
    \longrightarrow
  \underset{n}{\prod} Hom( A_n, B )
    \overset{\partial}{\longrightarrow}
  \underset{n}{\prod} Hom( A_n, B )
    \longrightarrow
  Ext^1(\underset{\longrightarrow}{\lim}_n A_n,B)
    \longrightarrow
  \underset{n}{\prod} Ext^1(  A_n, B )
    \overset{\partial}{\longrightarrow}
  \underset{n}{\prod} Ext^1(  A_n, B )
    \longrightarrow
  \cdots
\end{displaymath}
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. \ref{TheBoundaryMapDefiningLim1} corresponding to the [[tower]]

\begin{displaymath}
Hom(A_\bullet,B)
  = 
  (  \cdots \to Hom(A_2,B) \to Hom(A_1,B) \to  Hom(A_0,B) )
  \,.
\end{displaymath}
Hence truncating this long sequence by forming kernel and cokernel of $\partial$, respectively, it becomes the short exact sequence in question.

\end{proof}
\hypertarget{MilnorSequences}{}\subsection*{{Milnor exact sequences}}\label{MilnorSequences}

\hypertarget{for_homotopy_groups}{}\subsubsection*{{For homotopy groups}}\label{for_homotopy_groups}

\begin{prop}
\label{MilnorExactSequence}\hypertarget{MilnorExactSequence}{}
\textbf{(Milnor exact sequence for homotopy groups)}

Let

\begin{displaymath}
\cdots \to X_3 \overset{p_2}{\longrightarrow} X_2 \overset{p_1}{\longrightarrow} X_1 \overset{p_0}{\longrightarrow} X_0
\end{displaymath}
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]]

\begin{displaymath}
0 
  \to
  \underset{\longleftarrow}{\lim}^1_i \pi_{q+1}(X_i)
  \longrightarrow
  \pi_q(\underset{\longleftarrow}{\lim}_i X_i)
  \longrightarrow
  \underset{\longleftarrow}{\lim}_i \pi_q(X_i)
  \to 
  0
  \,,
\end{displaymath}
for $\pi_\bullet$ the [[homotopy group]]-functor (exact as [[pointed sets]] for $i = 0$, as [[groups]] for $i \geq 1$) which says that

\begin{enumerate}%
\item the failure of the [[limit]] over the homotopy groups of the stages of the tower to equal the homotopy groups of the [[limit]] of the tower is at most in the [[kernel]] of the canonical comparison map;


\item that kernel is the $\underset{\longleftarrow}{\lim}^1$ (def. \ref{Lim1ViaCokernel}) of the homotopy groups of the stages.



\end{enumerate}
\end{prop}
e.g. (\hyperlink{BousfieldKan72}{Bousfield-Kan 72, chapter IX, theorem 3.1}, \hyperlink{GoerssJardine96}{Goerss-Jardine 96, section VI. prop. 2.15})

\begin{proof}
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]] $[(\mathbb{N},\geq), sSet]_{inj}$ ($[(\mathbb{N},\geq), Top]_{inj}$) (\href{projectively+cofibrant+diagram#CofibrantCotowerDiagram}{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

\begin{displaymath}
\itexarray{
    holim X_\bullet &\longrightarrow& \underset{n}{\prod} Path(X_n)
    \\
    \downarrow &(pb)& \downarrow
    \\
    \underset{n}{\prod} X_n &\underset{(id,p_n)_n}{\longrightarrow}&
    \underset{n}{\prod} X_ n \times X_n
  }
  \,,
\end{displaymath}
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]]

\begin{displaymath}
\underset{n}{\prod} \Omega X_n
    \longrightarrow
  holim X_\bullet
    \longrightarrow
  \underset{n}{\prod} X_n
  \,.
\end{displaymath}
The [[long exact sequence of homotopy groups]] of this fiber sequence goes

\begin{displaymath}
\cdots
   \to
  \underset{n}{\prod} \pi_{q+1}(X_n)  
   \longrightarrow
  \underset{n}{\prod} \pi_{q+1}(X_n)
    \longrightarrow
  \pi_q (\underset{\longleftarrow}{\lim} X_\bullet)
    \longrightarrow
  \underset{n}{\prod} \pi_q(X_n)
    \longrightarrow
  \underset{n}{\prod} \pi_q(X_n)
   \to
  \cdots
  \,.
\end{displaymath}
Chopping that off by forming kernel and cokernel yields the claim for positive $q$. For $q = 0$ it follows by inspection.

\end{proof}
\hypertarget{for_chain_homology}{}\subsubsection*{{For chain homology}}\label{for_chain_homology}

\begin{example}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
\cdots \to C_3 \to C_2 \to C_1 \to C_0
\end{displaymath}
be a tower of [[chain complexes]] (of [[abelian groups]]) such that it satisfies degree-wise the [[Mittag-Leffler condition]], def. \ref{MittagLefflerCondition}, and write

\begin{displaymath}
C \coloneqq \underset{\longleftarrow}{\lim}_n C_n
\end{displaymath}
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:

\begin{displaymath}
0 \to \underset{\longleftarrow}{\lim}^1_i H_{q+1}(C_i)
  \longrightarrow H_q(C)
  \longrightarrow \underset{\longleftarrow}{\lim}_i H_q(C_i)
  \to 0
  \,.
\end{displaymath}
\end{example}
(e.g. \hyperlink{Weibel94}{Weibel 94, prop. 3.5.8})

\hypertarget{for_generalized_cohomology_groups}{}\subsubsection*{{For generalized cohomology groups}}\label{for_generalized_cohomology_groups}

\hypertarget{of_spaces}{}\paragraph*{{Of spaces}}\label{of_spaces}

\begin{prop}
\label{MilorSequenceForReducedCohomologyOnCWComplex}\hypertarget{MilorSequenceForReducedCohomologyOnCWComplex}{}
\textbf{(Milnor exact sequence for generalized cohomology)}

Let $X$ be a [[pointed topological space|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]]

\begin{displaymath}
0 
      \to 
     \underset{\longleftarrow}{\lim}^1_n \tilde E^{\bullet-1}(X_n)
      \longrightarrow
     \tilde E^{\bullet}(X)
      \longrightarrow
     \underset{\longleftarrow}{\lim}_n \tilde E^{\bullet}(X_n)
     \to 
     0
     \,,
\end{displaymath}
saying that

\begin{enumerate}%
\item 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]];


\item this kernel is precisely the $\lim^1$ (def. \ref{Lim1ViaCokernel}) of the cohomology groups at the finite stages in one degree lower.



\end{enumerate}
\end{prop}
e.g. (\hyperlink{Switzer75}{Switzer 75, prop. 7.66}, \hyperlink{Kochmann96}{Kochmann 96, prop. 4.2.2})

\begin{proof}
For

\begin{displaymath}
X_\bullet
  = 
  \left(
    X_0 
      \overset{i_0}{\hookrightarrow}
    X_1 
      \overset{i_1}{\hookrightarrow}
    X_2 
      \overset{i_1}{\hookrightarrow}
    \cdots
  \right)
\end{displaymath}
the sequence of stages of the ([[pointed topological space|pointed]]) [[CW-complex]] $X = \underset{\longleftarrow}{\lim}_n X_n$, write

\begin{displaymath}
\begin{aligned}
    A_X 
    &\coloneqq 
    \underset{n \in \mathbb{N}}{\sqcup} X_{2n} \times [2n,{2n}+1];
  \\
    B_X 
    &\coloneqq 
    \underset{n \in \mathbb{N}}{\sqcup} X_{(2n+1)} \times [2n+1,{2n}+2].
  \end{aligned}
\end{displaymath}
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]] $Tel(X_\bullet)$ (\href{mapping+telescope#MappingTelescope}{def.}), which we denote by

\begin{displaymath}
\itexarray{
    A_X && && B_X
    \\
    & {}_{\mathllap{\iota_{A_x}}}\searrow && \swarrow_{\mathrlap{\iota_{B_x}}}
    \\
    && Tel(X_\bullet)
  }
  \,.
\end{displaymath}
Observe that

\begin{enumerate}%
\item $A_X \cup B_X \simeq Tel(X_\bullet)$;


\item $A_X \cap B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_n$;



\end{enumerate}
and that there are [[homotopy equivalences]]

\begin{enumerate}%
\item $A_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n+1}$


\item $B_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n}$


\item $Tel(X_\bullet) \simeq X$.



\end{enumerate}
The first two are obvious, the third is \href{mapping+telescope#TelescopeOfCWComplexEquivalentToTheOriginal}{this proposition}.

This implies that the [[Mayer-Vietoris sequence]] (\href{generalized+cohomology#MayerVietorisSequenceInGeneralizedCohomology}{prop.}) for $\tilde E^\bullet$ on the cover $A \sqcup B \to X$ is isomorphic to the bottom horizontal sequence in the following diagram:

\begin{displaymath}
\itexarray{
     \tilde E^{\bullet-1}(A_X)\oplus \tilde E^{\bullet-1}(B_X)
      &\longrightarrow&
    \tilde E^{\bullet-1}(A_X \cap B_X)
      &\longrightarrow& 
    \tilde E^\bullet(X)
      &\overset{(\iota_{A_x})^\ast - (\iota_{B_x})^\ast}{\longrightarrow}&
    \tilde E^\bullet(A_X)\oplus \tilde E^\bullet(B_X)
      &\overset{}{\longrightarrow}&
    \tilde E^\bullet(A_X \cap B_X)
    \\
    \downarrow^{\mathrlap{\simeq}}
     &&
    \downarrow^{\mathrlap{\simeq}} 
      && 
    \downarrow^{\mathrlap{=}}
      &&
    {}^{\mathllap{(id, -id)}}\downarrow^{\mathrlap{\simeq}}
      &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \underset{n}{\prod}\tilde E^{\bullet-1}(X_n)
      &\underset{\partial}{\longrightarrow}&
    \underset{n}{\prod}\tilde E^{\bullet-1}(X_n) 
      &\longrightarrow& 
    \tilde E^\bullet(X)
      &\overset{(i_n^\ast)_{n}}{\longrightarrow}&
    \underset{n}{\prod}\tilde E^\bullet(X_n)
      &\underset{\partial}{\longrightarrow}&
    \underset{n}{\prod}\tilde E^\bullet(X_n)
  }
  \,,
\end{displaymath}
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

\begin{displaymath}
\partial
    \;\colon\;
  (a_n)_{n \in \mathbb{N}}
    \mapsto
  ( a_n - i_n^\ast(a_{n+1}) )
  \,.
\end{displaymath}
(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. \ref{TheBoundaryMapDefiningLim1} for the sequence

\begin{displaymath}
\cdots
    \to 
  \tilde E^\bullet(X_{n+1}) 
    \overset{i_n^\ast}{\longrightarrow}
  \tilde E^\bullet(X_n)
    \overset{i_n^\ast}{\longrightarrow}
  \tilde E^{\bullet}(X_{n-1})
   \to
   \cdots
 \,.
\end{displaymath}
Hence truncating the above long exact sequence by forming kernel and cokernel of $\partial$, the result follows via remark \ref{LimitAsKernelAnalogousToLim1} and definition \ref{Lim1ViaCokernel}.

\end{proof}
In contrast:

\begin{prop}
\label{}\hypertarget{}{}
Let $X$ be a [[pointed topological space|pointed]] [[CW-complex]], $X = \underset{\longrightarrow}{\lim}_n X_n$.

For $\tilde E_\bullet$ an additive reduced [[generalized homology theory]], then

\begin{displaymath}
\underset{\longrightarrow}{\lim}_n \tilde E_\bullet(X_n)
     \overset{\simeq}{\longrightarrow}
     \tilde E_\bullet(X)
\end{displaymath}
is an [[isomorphism]].

\end{prop}
(\hyperlink{Switzer75}{Switzer 75, prop. 7.53})

\hypertarget{ForGeneralizedCohomologyOfSpectra}{}\paragraph*{{Of spectra}}\label{ForGeneralizedCohomologyOfSpectra}

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]] (\href{Introduction+to+Stable+homotopy+theory+--+1-1#GradedAbelianGroupStructureOnHomsInTheHomotopyCategory}{def.}, \href{Introduction+to+Stable+homotopy+theory+--+1-1#ForASpectrumXGeneralizedECohomology}{exmpl.})

\begin{displaymath}
\begin{aligned}
    E^\bullet(X)
      & \coloneqq
    [X,E]_{-\bullet}
    \\
      & \coloneqq
    [\Sigma^\bullet X, E]
  \end{aligned}
  \,.
\end{displaymath}
The [[stable homotopy category]] is, in particular, the [[homotopy category of a model category|homotopy category]] of the stable [[model structure on orthogonal spectra]], in that its [[localization]] at the [[stable weak homotopy equivalences]] is of the form

\begin{displaymath}
\gamma
    \;\colon\;
  OrthSpec(Top_{cg})_{stable}
    \longrightarrow
  Ho(Spectra)
  \,.
\end{displaymath}
In the following when considering an [[orthogonal spectrum]] $X \in OrthSpec(Top_{cg})$, we use, for brevity, the same symbol for its image under $\gamma$.

\begin{prop}
\label{CohomologyOfSpectraMilnorSequence}\hypertarget{CohomologyOfSpectraMilnorSequence}{}
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

\begin{displaymath}
0
    \to
  \underset{\longleftarrow}{\lim}^1_n E^{\bullet + n -1}(X_{n})
    \longrightarrow
  E^\bullet(X)
    \longrightarrow
  \underset{\longleftarrow}{\lim}_n E^{\bullet + n}(X_n)
    \to
  0
\end{displaymath}
where $\underset{\longleftarrow}{\lim}^1$ denotes the [[lim{\tt \symbol{94}}1]], and where this and the limit on the right are taken over the following structure morphisms

\begin{displaymath}
E^{\bullet + n + 1}(X_{n+1})
    \overset{E^{\bullet+1n+1}(\Sigma^X_n)}{\longrightarrow}
  E^{\bullet+n+1}(X_n \wedge S^1)
    \overset{\simeq}{\longrightarrow}
  E^{\bullet + n}(X_n)
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Schwede12}{Schwede 12, chapter II prop. 6.5 (ii)}) (using that symmetric spectra underlying orthogonal spectra are semistable (\hyperlink{Schwede12}{Schwede 12, p. 40}))

\begin{cor}
\label{WithSomeLefflerTheHomsOfSpectraAreHomotopicIfComponentsAre}\hypertarget{WithSomeLefflerTheHomsOfSpectraAreHomotopicIfComponentsAre}{}
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. \ref{MittagLefflerCondition}), then two morphisms of spectra $X \longrightarrow E$ are homotopic already if all their morphisms of component spaces $X_n \to E_n$ are.

\end{cor}
\begin{proof}
By prop. \ref{Lim1VanihesUnderMittagLeffler} the assumption implies that the $lim^1$-term in prop. \ref{CohomologyOfSpectraMilnorSequence} vanishes, hence by exactness it follows that in this case there is an [[isomorphism]]

\begin{displaymath}
[X,E] 
    \simeq
   E^0(X)
     \overset{\simeq}{\longrightarrow}
   \underset{\longleftarrow}{\lim}_n [X_n, E_n]
  \,.
\end{displaymath}
\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[homotopy limit]]


\item [[spectral sequence of a tower of fibrations]]


\item [[conditional convergence]] (of [[spectral sequences]])



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[John Milnor]], \emph{On axiomatic homology theory}, Pacific J. Math. Volume 12, Number 1 (1962), 337-341 (\href{http://projecteuclid.org/euclid.pjm/1103036730}{Euclid})


\item Z. Z. Yeh, \emph{Higher Inverse Limits and Homology Theories}, Thesis, Princeton, 1959.


\item [[Aldridge Bousfield]], [[Daniel Kan]], section IX.2 of \emph{Homotopy limits, completions and localization}, Springer 1972


\item [[Robert Switzer]], section 7 from def. 7.57 on in \emph{Algebraic Topology - Homotopy and Homology}, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.


\item [[Charles Weibel]], section 3.5 of \emph{[[An Introduction to Homological Algebra]]}, Cambridge University Press (1994)


\item [[Stanley Kochmann]], section 4.2 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Paul Goerss]], [[Rick Jardine]], section VI.2 of \emph{[[Simplicial homotopy theory]]}, Modern Birkh\"a{}user Classics (1999)

(that's section VII.6 of the 1996 \emph{Progress of Mathematics} edition )



\end{itemize}
For generalized cohomology of spectra

\begin{itemize}%
\item [[Stefan Schwede]], around prop.6.5 of \emph{[[Symmetric spectra]]}, 2012 (\href{http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf}{pdf})

\end{itemize}
Discussion in the context of [[categories of fibrant objects]] is in

\begin{itemize}%
\item [[Kenneth Brown]], section 5 of \emph{[[Abstract Homotopy Theory and Generalized Sheaf Cohomology]]}, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458

\end{itemize}
Discussion in the context of [[conditional convergence]] of [[spectral sequences]] is in

\begin{itemize}%
\item [[Michael Boardman]], section I.1 of \emph{Conditionally convergent spectral sequences}, 1999 (\href{http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf}{pdf})

\end{itemize}
[[!redirects lim1]] [[!redirects lim{\tt \symbol{94}}1]] [[!redirects Milnor sequence]] [[!redirects Milnor sequences]] [[!redirects Milnor exact sequence]] [[!redirects Milnor exact sequences]] [[!redirects Milnor short exact sequence]] [[!redirects Milnor short exact sequences]] [[!redirects Milnor long exact sequence]] [[!redirects Milnor long exact sequences]]



\end{document}