lift

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Factorization systems
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[[!include factorization systems - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _lifting property_ is a property of a pair of [[morphism|morphisms]] in a [[category]]. It is used in [[homotopy theory]] within [[algebraic topology]] to define properties of morphisms starting from an explicitly given class of morphisms. It appears in a prominent way in the theory of [[model categories]], an axiomatic framework for [[homotopy theory]] introduced by [[Daniel Quillen]]. It is also used in the definition of a [[factorization system]], and of a [[weak factorization system]], notions related to but less restrictive than the notion of a model category.
A number of elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.

{#AsQualitativeNegation} Often it is useful to think of lifting properties as a expressing a kind of qualitative negation ("[[Quillen negation]]"): The morphisms with the left/right lifting property against those in a class $P$ tend to be characterized by properties opposite of those in $P$.
For example, a morphism in [[Sets]] is [[surjective]] iff it has the right lifting property against the archetypical non-surjective map $\varnothing \to \{*\}$, and [[injective]] iff it has either left or right lifting property against the archetypical non-injective map  $\{x_1,x_2\}\to \{*\}$. (For more such examples see at *[[separation axioms in terms of lifting properties]]*.)

## Definition

\begin{definition}\label{LiftingPropertiesOfMorphisms}
**(lifting properties of morphisms)**
\linebreak
A morphism $i$ in a category has the *left lifting property* with respect to a morphism $p$, and $p$ also has the *right lifting property* with respect to $i$, sometimes denoted $i\,\,⧄\,\, p$ or $i\downarrow p$, iff the following implication holds for each morphism $f$ and $g$ in the category:

* if the outer square of the following diagram commutes, then there exists $h$ completing the diagram, i.e. for each $f:A\to X$ and $g:B\to Y$ such that $p\circ f = g \circ i$ there exists $h:B\to X$ such that $h\circ i = f$ and $p\circ h = g$.

\begin{tikzcd}
  A
  \ar[dd, "i"]
  \ar[rr, "f"]
  && X\ar[dd,"p"]
  \\
  \\
  B
  \ar[
    rr,"g"
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"{description}
  ]
  &&Y
\end{tikzcd}
\end{definition}

This is sometimes also known as the morphism $i$ being ''weakly orthogonal to'' the morphism $p$; however, ''orthogonal to'' will refer to the stronger property that whenever $f$ and $g$ are as above, the diagonal morphism $h$ exists and is also required to be unique.

\begin{remark} \label{LiftingForObjects}
**(lifting properties of objects)**
\linebreak
One also speaks of *[[objects]]* having left or right lifting properties (for instance in the definition of [[projective objects]] and [[injective objects]], respectively, or in the characterization of [[cofibrant objects]] and [[fibrant objects]], respectively), by which one then means, respectively:
\end{remark}

* that the corresponding unique morphisms $\varnothing \longrightarrow B$ (from the [[initial object]]) has the left lifting property in the sense of Def. \ref{LiftingPropertiesOfMorphisms} (against the given morphism $p$):

\begin{tikzcd}
  \color{lightgray}
  \varnothing
  \ar[dd, "\exists !", lightgray]
  \ar[rr, "\exists !", lightgray]
  &&
  X \ar[dd,"p"]
  \\
  \\
  B
  \ar[
    rr,"g"
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"{description}
  ]
  &&
  Y
  \mathrlap{\,,}
\end{tikzcd}

* that the unique morphism $X \longrightarrow \ast$ (to the [[terminal object]]) has the right lifting property (against the given morphism $i$):

\begin{tikzcd}
  A
  \ar[dd, "i"]
  \ar[rr, "f"]
  &&
  X\ar[dd,"\exists !", lightgray]
  \\
  \\
  B
  \ar[
    rr, "\exists !", lightgray
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"{description}
  ]
  &&
  \color{lightgray}
  \ast
  \mathrlap{\,,}
\end{tikzcd}

\begin{definition}
\label{QuillenNegation}
**([[orthogonal class]]/[[Quillen negation]])**
\linebreak
Given a [[class]] $M \;\subset\; Mor(\mathcal{C})$ of [[morphisms]] in a [[category]] $\mathcal{C}$, its

* *left weak orthogonal class* or *left [[Quillen negation]]*
 $\multiscripts{^⧄}{M}{}$

or

* *right weak orthogonal class* or *right [[Quillen negation]]*  $\multiscripts{}{M}{^⧄}$

is the class of all morphisms which have the left, respectively right, lifting property (in the sense of Def. \ref{LiftingPropertiesOfMorphisms}) with respect to each morphism in the class $M$:

$$
  M^{⧄}
  \;\coloneqq\;
  \Big\{
    p \,\in\, Mor(\mathcal{C})
    \;\big\vert\;
   \underset{i \in M}{\forall}
   \; i \,⧄\, p
  \Big\},
  \;\;\;\;\;
  {}^{⧄}M
  \;\coloneqq\;
  \Big\{
    i \,\in\, Mor(\mathcal{C})
    \;\big\vert\;
   \underset{p \in M}{\forall}
   \; i \,⧄\, p
  \Big\}
  \,.
$$

\end{definition}

## Examples of lifting properties
 {#ExamplesOfLiftingProperties}

Decyphering notation in most of the examples below leads to standard definitions or reformulations. The intuition behind most examples below
is that the class of morphisms consists of simple or archetypal examples
related to the property defined.

We use the notation of Def. \ref{QuillenNegation}.

### Elementary examples

#### Sets

In _[[Set]]_,

* $
    \big\{
      \varnothing \to \{*\}\big
    \}^{⧄}
    \;\;\;
    =
    \;\;\;
    Srjctv
  $

  is the class of [[surjective functions]],

* $
    \big( \{a,b\}\to \{*\} \big)^{⧄}
    \;\;\;
    =
    \;\;\;
    Injctv
  $

  is the class of [[injective functions]].

#### Modules
 {#ExamplesModules}

In the category *[[RMod]]* of [[module|modules]] over a [[commutative ring]] $R$ (recalling that we use thenotation of Def. \ref{QuillenNegation}):

The [[surjection|surjective]] [[homomorphisms]] are those with the [[right lifting property]] against the [[initial object|initial]] [[homomorphism]] from the [[zero]] [[module]] into the [[ground ring]]:

\[
  \label{SurjectiveModuleMapsAsRightOrthClass}
  \{ 0 \to R \}^{⧄}
  \;\;\;=\;\;\;
  Srjctv
  \,.
\]

The [[injection|injective]] [[homomorphisms]] are those with the [[right lifting property]] against the [[terminal object|terminal]] [[homomorphism]] from the [[ground ring]]
into the [[zero]] [[module]]:

\[
  \label{InjectiveModuleMapsAsRightOrthClass}
  \{ R \to 0 \}^{⧄}
  \;\;\;=\;\;\;
  Injctv
  \,.
\]

An $R$-[[module]] $M$ is [[projective module|projective]] iff (by direct unwinding of the definitions of [[projective objects]] and lifts) the [[initial object|initial morphism]] $0 \to R$ (out of the [[zero]] [[module]] into the [[ground ring]]) has the [[left lifting property]] against all surjective homomorphisms.

With the notation of Def. \ref{QuillenNegation} this reads as follows:

  $$
    M\;\text{projective}
    \;\;\;\;\;
      \Leftrightarrow
    \;\;\;\;\;
    \{ 0 \to M \}
    \;⧄\;
    Srjctv
    \;\;\;\;\;
      \overset{
        \text{(eq:SurjectiveModuleMapsAsRightOrthClass)}
      }{
      \Leftrightarrow
      }
    \;\;\;\;\;
    \{ 0 \to M \}
    \;⧄\;
    \Big(
      \{ 0 \to R \}^{⧄}
    \Big)
    \;\;\;\;\;
      \Leftrightarrow
    \;\;\;\;\;
    \{ 0 \to M \}
    \;\in\;
    \multiscripts{^{⧄}}
    {
    \Big(
      \{ 0 \to R \}^{^⧄}
    \Big)
    }{}
  $$

#### Groups
 {#ExamplesGroups}

In the category _[[Grp]]_ of [[Group|groups]],

* $\{\mathbb{Z} \to 0\}^{⧄  r}$, resp. $\{0\to \mathbb{Z}\}^{⧄  r}$, is the class of injections, resp. surjections (where $\mathbb{Z}$ denotes the infinite [[cyclic group]]),

* A group $F$ is a [[free group]] iff $0\to F$ is in $\{0\to \mathbb{Z} \}^{⧄  r\ell},$

* A group $A$ is [[torsion|torsion-free]] iff $0\to A$ is in $\{ n \mathbb{Z} \to \mathbb{Z} : n\ge0 \}^{⧄  r},$

* A [[subgroup]] $A$ of $B$ is [[pure subgroup|pure]] iff $A \to B$ is in $\{ n\mathbb{Z}\to \mathbb{Z} : n\ge0 \}^{⧄  r}.$

* $(*\to 1)^{⧄ l}$ is the class of retracts
*  $(1\to *)^{⧄ r}$ is the class of split homomorphisms

* $(0\longrightarrow \mathbb{Z})^{⧄ r}$ is the class of surjections
* $(\mathbb{Z}\to 1)^{⧄ r}$ is the class of injections
*     a group $F$ is free iff
 $1\to F$ is in  $(0\longrightarrow \mathbb{Z})^{⧄rl}$

* a group $A$ is Abelian iff $A\to 1$ is in $( \mathbb{F}_2 \to \mathbb{Z}\times\mathbb{Z})^{⧄ r}$

* group $G$ can be obtained from $H$ by adding commutation relations, i.e.~the kernel of $H\to G$ is generated by commutators $[h_1,h_2]$, $h_1,h_2\in H$,
iff
 $H\to G$ is in $( \mathbb{F}_2 \to \mathbb{Z}\times\mathbb{Z})^{⧄rl}$

* subgroup $H$ of $G$ is the normal span of substitutions in words $w_1,..,w_i$ of the free group $\mathbb{F}_n$ iff
$G \to G/H$ is in $( \mathbb{F}_n \to \mathbb{F}_n/\le\!w_1,...,w_i\!\ge)^{⧄rl}$

* $\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell l}$ is the class
of homomorphisms whose kernel is perfect

For a [[finite group]] $G$, in the category of finite groups,

* $\{0\to {\mathbb{Z}}/p{\mathbb{Z}}\} \,\,⧄\,\, G\to 1$ iff the [[order]] of $G$ is prime to $p$,

* $G\to 1 \in (0\to {\mathbb{Z}}/p{\mathbb{Z}})^{⧄  rr}$ iff $G$ is a [[p-group|$p$-group]],

* $H$ is nilpotent iff the diagonal map $H\to H\times H$ is in $(1\to *)^{⧄ \ell r}$ where $(1\to *)$ denotes the class of maps $\{ 1\to G : G \text{ arbitrary}\},$

* a finite group $H$ is [[soluble]] iff $1\to H$ is in $\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell r}=\{[G,G]\to G : G\,\,\text{ arbitrary } \}^{⧄ \ell r}.$

Moreover,

* $\{0\to G : G\,\,\text{ arbitrary}\}^{⧄ \ell r}$ is the class of subnormal subgroups

*  $\{0\to A : A\,\,\text{ abelian}\}^{⧄ \ell r}=\{[G,G]\to G : G\,\,\text{ arbitrary } \}^{⧄ \ell r}$,
 and is the class of subgroups $H\leq G$ such that there is a chain of subnormal subgroups
$H=G_0 \vartriangleleft G_1  \vartriangleleft \ldots  \vartriangleleft G_n =G$ such that $G_{i+1}/G_{i}$ is Abelian, for $i=0,...,n-1$.

* $\{1 \to S\}^{⧄ \ell r}$ is the class of subgroups $H\leq G$ such that there is a chain of subnormal subgroups
$H=G_0 \vartriangleleft G_1  \vartriangleleft \ldots  \vartriangleleft G_n =G$ such that $G_{i+1}/G_{i}$ embeds into $S$, for $i=0,...,n-1$.

*  $(\mathbb{Z}/p\mathbb{Z}\longrightarrow 0)^{⧄r}$ is the class of homomorphisms whose kernel has no elements of order $p$
*  $(\mathbb{Z}/p\mathbb{Z}\longrightarrow 0)^{⧄rr}$ is the class of surjective homomorphisms whose kernel is a $p$-group

### In algebraic topology and in model categories

Lifting properties are paramount in [[homotopy theory]] and [[algebraic topology]]. In "abstract homotopy theory" lifting properties are encoded in the structures of [[model categories]], whose defintion revolves all around compatible classes of [[weak factorization systems]]. In particular:

* the [[cofibrations]] in a model category are precisely the class with the left lifting property against the [[acyclic fibrations]],

* the [[fibrations]] in a model category are precisely the class with the right lifting property against the [[acyclic cofibrations]],

#### Serre fibrations of topological spaces

The [[classical model structure on topological spaces]] $Top_{Qu}$ is controlled by the following lifting properties:

consider let $C_0$ be the class of maps $S^n\to D^{n+1}$, [[embeddings]] of the boundary $S^n=\partial D^{n+1}$ of a ball into the ball $D^{n+1}$. Let $WC_0$ be the class of maps embedding the upper semi-sphere into the disk. $WC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r}$ are the classes of [[Serre fibrations]], acyclic cofibrations, acyclic fibrations, and cofibrations. [Hovey, Model Categories, Def. 2.4.3, Th.2.4.9](https://archive.org/details/arxiv-math9803002)

#### Hurewicz fibrations of topological spaces

A map $f:U\to B$ has the ''path lifting property'' iff $\{0\}\to [0,1] \,\,⧄\,\, f$ where $\{0\} \to [0,1]$ is the inclusion of one end point of the closed interval into the interval $[0,1]$.

A map $f:U\to B$ has the [[homotopy lifting property]] iff $X \to X\times [0,1] \,\,⧄\,\, f$ where $X\to X\times [0,1]$ is the map $x \mapsto (x,0)$.

#### Kan fibrations of simplicial sets

The [[classical model structure on simplicial sets]] $sSet_{Qu}$ is controlled by the following lifting properties:

Let $C_0$ be the class of boundary inclusions $\partial \Delta[n] \to \Delta[n]$, and let $WC_0$ be the class of horn inclusions $\Lambda^i[n] \to \Delta[n]$. Then the classes of [[Kan fibrations]], acyclic cofibrations, acyclic fibrations, and cofibrations are, respectively, $WC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r}$.
[(Model Categories, Def. 3.2.1, Th.3.6.5)](https://archive.org/details/arxiv-math9803002)

#### Degreewise surjections of chain complexes

A [[model structure on chain complexes]] is controlled by the following lifting properties:

* Let _Ch_($R$) be the category of [[chain complexes]] over a [[commutative ring]] $R$. Let $C_0$ be the class of maps of form
$\cdots\to 0\to R \to 0 \to 0 \to \cdots \to \cdots \to R \xrightarrow{\operatorname{id}} R \to 0 \to 0 \to \cdots,$
and $WC_0$ be
 $\cdots \to 0\to 0 \to 0 \to 0 \to \cdots \to \cdots \to R \xrightarrow{\operatorname{id}} R \to 0 \to 0 \to \cdots.$
Then $WC_0^{⧄ \ell}, WC_0^{⧄ \ell r}, C_0^{⧄ \ell}, C_0^{⧄ \ell r}$ are the classes of fibrations, acyclic cofibrations, acyclic fibrations, and cofibrations.
 [(Model Categories, Def. 2.3.3, Th.2.3.11)](https://archive.org/details/arxiv-math9803002)

### Topology

Many elementary properties in [[general topology]], such as compactness, being dense or open,
can be expressed as iterated Quillen negation of morphisms of finite topological spaces
in the category _Top_ of topological spaces. This leads to a concise, if useless, notation
for a number of properties. Items below use notation for morphisms of finite topological spaces
defined in the page on [[separation axioms in terms of lifting properties]],
and some examples are explained there in detail.

#### Uniform spaces

In the category of [[uniform  spaces]] or [[metric spaces]] with [[uniformly continuous]] maps.

* A space $X$ is [[complete]] iff $\{1/n\}_{n \in \mathbb{N}} \to \{0\}\cup \{1/n\}_{n \in \mathbb{N}} \,\,⧄\,\, X\to \{0\}$ where $\{1/n\}_{n \in \mathbb{N}} \to \{0\}\cup \{1/n\}_{n \in \mathbb{N}}$ is the obvious inclusion between the two subspaces of the real line with induced metric, and $\{0\}$ is the metric space consisting of a single point,

* A subspace $i:A\to X$ is closed iff $\{1/n\}_{n \in \mathbb{N}} \to \{0\}\cup \{1/n\}_{n \in \mathbb{N}} \,\,⧄\,\, A\to X.$

#### In topological spaces

The following lifting properties are calculated in the category of (all) topological spaces. Below
we use notation defined in
[[separation axioms in terms of lifting properties#NotationForFiniteTopologicalSpaces|the page on lifting properties]]

#### Iterated lifting properties

*         $(\emptyset\longrightarrow \{o\})^{⧄r}$   is the class of surjections
*         $(\emptyset\longrightarrow \{o\})^{⧄r}$   is the class of maps $A\longrightarrow B$ where $A\neq \emptyset$ or $A=B$
*         $(\emptyset\longrightarrow \{o\})^{⧄rr}=\{\{x\leftrightarrow y\rightarrow c\}\longrightarrow\{x=y=c\}\}^{⧄l}=\{\{x\leftrightarrow y\leftarrow c\}\longrightarrow\{x=y=c\}\}^{⧄l}$ is the class of subsets, i.e. injective maps $A\hookrightarrow B$ where the topology on $A$ is induced from $B$

*  $(\emptyset\longrightarrow \{o\})^{⧄lr}$ is the class of maps $\emptyset\longrightarrow B$, $B$ arbitrary
*         $(\emptyset\longrightarrow \{o\})^{⧄lrr}$ is the class of maps $A\longrightarrow B$ which admit a section

*   $(\emptyset\longrightarrow \{o\})^{⧄l}$ consists of maps $f:A\longrightarrow B$ such that either  $A\neq \emptyset$ or  $A=B=\emptyset$

*  $(\emptyset\longrightarrow \{o\})^{⧄rl}$ is the class of maps of form $A\longrightarrow A\sqcup D$ where $D$ is discrete

*  $(\emptyset\longrightarrow \{o\})^{⧄rll}$ is the class of maps
$A\to B$ such that each connected subset of $B$ intersects the image of $A$; for "nice" spaces it means that the map $\pi_0(A)\to \pi_0(B)$ is surjective, where "nice" means that connected componets are both open and closed.

*  $(\emptyset\longrightarrow \{o\})^{⧄rllr}$ is the class of maps
of form $A\to A\sqcup B$ where $A\sqcup B$ denotes the disjoint union of $A$ and $B$.

*  $\{ \{z\leftrightarrow  x\leftrightarrow  y\rightarrow c\}\longrightarrow\{z=x\leftrightarrow  y=c\} \}^{⧄l}  = \{\{c\}\longrightarrow \{o\rightarrow c\}\}^{⧄lr}$ is the class of closed inclusions $A\subset B$ where $A$ is closed

*  $\{ \{z\leftrightarrow  x\leftrightarrow  y\leftarrow c\}\longrightarrow\{z=x\leftrightarrow  y=c\} \}^{⧄l}$ is the class of open inclusions
$A\subset B$ where $A$ is open

*  $\{ \{x\leftrightarrow  y\rightarrow c\}\longrightarrow\{x\leftrightarrow  y=c\} \}^{⧄l} $ is the class of closed maps $A\longrightarrow B$
where the topology on $A$ is pulled back from $B$
*  $\{ \{x\leftrightarrow  y\leftarrow c\}\longrightarrow\{x\leftrightarrow  y=c\} \}^{⧄l}$ is the class of open maps $A\longrightarrow B$
where the topology on $A$ is pulled back from $B$

*         $(\{b\}\longrightarrow \{a{ \searrow}b\})^{⧄l}$ is the class of maps with dense image
*         $(\{b\}\longrightarrow \{a{ \searrow}b\})^{⧄lr}=\{ \{z\leftrightarrow x \leftrightarrow y\rightarrow c\}\longleftarrow\{z=x\leftrightarrow y=c\} \}^{⧄l}$ is the class of closed subsets $A \subset  X$, $A$ a closed subset of $X$

*         $\{ \{z\leftrightarrow x \leftrightarrow y\leftarrow c\}\longleftarrow\{z=x\leftrightarrow y=c\} \}^{⧄l}$ is the class of open subsets $A \subset  X$, $A$ a open subset of $X$

*         $(\{a\}\longrightarrow \{a{ \searrow}b\})^{⧄lr}$ is the class of subsets $A \subset  X$ such that $A$ is the intersection of open subsets containing $A$

*         $((\{a\}\longrightarrow \{a \searrow b\})^{⧄r}_{\le 4})^{⧄lr}$ is roughly the class of proper maps

#### Separation axioms

Here follows a list of examples of well-known properties defined by
iterated Quillen negation starting from maps between [[finite topological spaces]], often with less than 5 elements.  See at *[[separation axioms in terms of lifting properties]]*  for more on the following.

*   a space $K$ is non-empty iff $K\longrightarrow \{o\}$ is in $   (\emptyset\longrightarrow \{o\})^{⧄l}$
*   a space $K$ is empty iff $K \longrightarrow \{o\}$ is in $   (\emptyset\longrightarrow \{o\})^{⧄ll}$
*   a space $K$ is $T_0$ iff $K  \longrightarrow \{o\}$ is in $   (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^{⧄r}$
*    a space $K$ is $T_1$ iff $K  \longrightarrow \{o\}$ is in $   (\{a{ \searrow}b\}\longrightarrow \{a=b\})^{⧄r}$
*   a space $X$ is Hausdorff iff for each injective map $\{x,y\} \hookrightarrow  X$
it holds $\{x,y\} \hookrightarrow   {X} \,⧄\,  \{  {x} { \searrow}  {o} { \swarrow}  {y} \} \longrightarrow  \{ x=o=y \}$

*   a non-empty space $X$ is regular (T3) iff for each arrow $    \{x\} \longrightarrow  X$ it holds
    $    \{x\} \longrightarrow   {X} \,⧄\,  \{x{ \searrow}X{ \swarrow}U{ \searrow}F\} \longrightarrow  \{x=X=U{ \searrow}F\}$
*   a space $X$ is normal (T4) iff $\emptyset \longrightarrow  {X} \,⧄\,   \{a{ \swarrow}U{ \searrow}x{ \swarrow}V{ \searrow}b\}\longrightarrow \{a{ \swarrow}U=x=V{ \searrow}b\}$

*   a space $X$ is completely normal iff $\emptyset\longrightarrow  {X} \,⧄\,  [0,1]\longrightarrow \{0{ \swarrow}x{ \searrow}1\}$
 where the map $[0,1]\longrightarrow \{0{ \swarrow}x{ \searrow}1\}$ sends $0$ to $0$, $1$ to $1$, and the rest $(0,1)$ to $x$

*  a space $X$ is hereditary normal iff
$ \emptyset \to X ⧄
\{ x \leftarrow au \leftrightarrow u' \leftarrow u \leftarrow uv \rightarrow v \rightarrow v'\leftrightarrow bv \rightarrow x \}
\longrightarrow
\{ x \leftarrow au \leftrightarrow u' = u \leftarrow uv \rightarrow v = v'\leftrightarrow bv \rightarrow x \} $

*   a space $X$ is path-connected iff $\{0,1\} \longrightarrow  [0,1] \,⧄\,   {X} \longrightarrow  \{o\}$
*   a space $X$ is path-connected iff for each Hausdorff compact space $K$ and each injective map $\{x,y\} \hookrightarrow  K$ it holds
   $\{x,y\} \hookrightarrow   {K} \,⧄\,   {X} \longrightarrow  \{o\}$

* A map $X\longrightarrow Y$ is a quotient iff $X\to Y \,\,⧄\,\, \{o \rightarrow c\}\longrightarrow \{o\leftrightarrow c\}$
* For every pair of disjoint closed subsets of $X$, the closures of their images  of $Y$ do not intersect, if
$X\to Y \,\,⧄\,\,  \{x\leftarrow o\rightarrow y\}\longrightarrow \{x=o=y\}$

* A topological space $X$ is extremally disconnected iff
$\emptyset\to X \,\,⧄\,\, \{u\rightarrow a,b\leftarrow v\}\longrightarrow \{u\rightarrow a=b\leftarrow v\}$

* A topological space $X$ is zero-dimensional iff
$\emptyset\to X \,\,⧄\,\, \{a\leftarrow u,v\rightarrow b\}\longrightarrow \{a\leftarrow u=v\rightarrow b\}$

* A topological space $X$ is ultranormal iff
$\emptyset\to X \,\,⧄\,\, \{u\rightarrow a,b\leftarrow v\}\longrightarrow \{a\leftarrow u=v\rightarrow v\}$

*  $\{\bullet\}\longrightarrow A$ is in $(\emptyset\longrightarrow \{o\})^{⧄rll}$ iff $A$ is connected

*  $Y$ is totally disconnected iff $\{\bullet\}\xrightarrow y Y$ is in $(\emptyset\longrightarrow \{o\})^{⧄rllr}$ for each map  $\{\bullet\}\xrightarrow y Y$ (or,
in other words, each point $y\in Y$).

*   a Hausdorff space $K$ is compact iff $K\longrightarrow \{o\}$ is in  $((\{o\}\longrightarrow \{o{ \searrow}c\})^{⧄r}_{\le5})^{⧄lr}$
*   a  Hausdorff space $K$ is compact iff $K\longrightarrow \{o\}$ is in  $$
     \{\, \{a\leftrightarrow b\}\longrightarrow \{a=b\},\, \{o{ \searrow}c\}\longrightarrow \{o=c\},\,
     \{c\}\longrightarrow \{o{ \searrow}c\},\,\{a{ \swarrow}o{ \searrow}b\}\longrightarrow \{a=o=b\}\,\,\}^{⧄lr}$$

* a topological space $X$ is compactly generated iff
$\varnothing\longrightarrow X$ is in  $\big(\{\{0 \leftrightarrow 1\}\to\{0=1\}\}\cup\{\varnothing \to K \,\,:\,\, K\,\, \text{ compact}\}\big)^{⧄rl}$

*   a space $D$ is discrete iff $ \emptyset \longrightarrow  D$ is in $   (\emptyset\longrightarrow \{o\})^{⧄rl}      $

*   a space $D$ is codiscrete iff $  {D} \longrightarrow  \{o\} $ is in
$(\{a,b\}\longrightarrow \{a=b\})^{⧄rr}= (\{a\leftrightarrow b\}\longrightarrow \{a=b\})^{⧄lr} $

*   a space $K$ is connected or empty iff $K\longrightarrow \{o\}$ is in  $(\{a,b\}\longrightarrow \{a=b\})^{⧄l} $
*   a space $K$ is totally disconnected and non-empty iff $K\longrightarrow \{o\}$ is in  $(\{a,b\}\longrightarrow \{a=b\})^{⧄lr} $

*   a space $K$ is connected and non-empty
 iff
 for some arrow $\{o\}\longrightarrow K$
it holds that $\{o\}\longrightarrow K$ is in
            $   (\emptyset\longrightarrow \{o\})^{⧄rll} = (\{a\}\longrightarrow \{a,b\})^{⧄l}$

* A topological space $X$ has Lebesgue dimension at most $n$  iff
for each finite set $I$
$\emptyset\to X \,\,⧄\,\,
\{ (F,J): 1\leq |F|\leq n+1, F\subset J\subset I\}\longrightarrow
\{ J: 1\leq |J|, J\subset I\}
$
where the order on the domain
$\{ (F,J): 1\leq |F|\leq n+1, F\subset J\subset I\}$ is given by
$(F,J)\to (F',J')$ iff $F\subset F'$ and $J\subset J'$.

* A topological space $X$ has Lebesgue dimension at most $n$  iff
for each closed subset $A$ of $X$ $
A\to X \,\,⧄\,\, \mathbb{S}^n\to \{o\}$
where $\mathbb{S}^n$ denotes the $n$-sphere.

#### Homotopy theory

A finite CW complex $X$ is contractible iff  $X  \longrightarrow  {\{\bullet\}} \in \{   \{a{ \swarrow}U{ \searrow}x{ \swarrow}V{ \searrow}b\}\longrightarrow \{a{ \swarrow}U=x=V{ \searrow}b\}\}^{⧄rl}$

The map defining Separation Axiom $T_4$ above
is a trivial Serre fibration, hence their ${}^{⧄rl}$-orthogonals are classes of trivial fibrations.

\begin{conjecture} If $f$ is a "nice" map, then
$f$ is a trivial fibration iff
$$f\in\{  \{a{ \swarrow}U{ \searrow}x{ \swarrow}V{ \searrow}b\}\longrightarrow \{a{ \swarrow}U=x=V{ \searrow}b\}
\}^{⧄rl}$$
\end{conjecture}

One can make the same conjecture for the map
defining Separation Axiom $T_6$ (hereditary normal) since it is also a trivial Serre fibration.

### Model theory

In model theory,
a number of the Shelah's divining lines,
namely  $NOP, NSOP, NSOP_i, NTP, NTP_i$, and $NATP$
are [[situs|expressed as Quillen lifting properties of form]]
$$A_\bullet \to B_\bullet \rightthreetimes M_\bullet\to\top$$
where $\top$ is the terminal object, and
$M$ is a situs associated with a model and a formula,
and $A$ and $B$ are objects of combinatorial nature,
in the category of simplicial objects in the [[category of filters]].

## Related concepts

* [[factorization system]], [[weak factorization system]]

* [[homotopy lifting property]]

* [[Kan lift]]

* [[Joyal-Tierney calculus]]

* [[lifts and extensions]]

* [Tychonoff theorem -- Proof via lifting properties](Tychonoff+theorem#ProofViaTaimanovTheoremAndLiftingProperties)

* [covering space lifting properties](covering+space#LiftingProperties)

[[!redirects lifts]]

[[!redirects lifting]]
[[!redirects liftings]]

[[!redirects lifting property]]
[[!redirects lifting properties]]

[[!redirects right lifting property]]
[[!redirects left lifting property]]
[[!redirects right lifting properties]]
[[!redirects left lifting properties]]

[[!redirects lifting problem]]
[[!redirects lifting problems]]


lifted limit

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## Definition

A [[functor]] $F\colon \mathcal{C}\to \mathcal{D}$ is said to _lift_ [[limits]] of a particular shape $I$ if for any diagram $J:I\to C$, any limiting [[cone]] for $F \circ J$ in $\mathcal{D}$ is the image of a limiting cone for $J$ in $\mathcal{C}$.

The above definition is not invariant under [[equivalences of categories]].
It can be made invariant if we demand instead
that any limiting cone for $F\circ J$ is isomorphic
to the image of a limiting cone for $J$. Alternatively, this says that, if $F \circ J$ has a limit, then $J$ also has a limit and that limit is [[preserved limit|preserved]] by $F$.

## Terminological remarks

Lifting limits is closely related to [[created limits|creating them]].  The relationships between these notions were the subject of a post by Aleks Kissinger at the categories mailing list, [here](https://raw.githubusercontent.com/punkdit/categories/26b751bbcbe6765fe447e805629ff6416f4c38b1/gmane/science/mathematics/categories/6644), but there is [some dispute](https://nforum.ncatlab.org/discussion/7024/lifted-limit/?Focus=56765#Comment_56765) about its correctness.

## Related concepts

* [[preserved limit]]

* [[reflected limit]]

* [[created limit]]

* [[topological concrete category]]

* **lifted limit**

[[!redirects lifted limits]]

## References

See Definition 13.17 in [[Adamek]], [[Herrlich]], [[Strecker]]: [Abstract and Concrete Categories](http://katmat.math.uni-bremen.de/acc/acc.pdf).  Remark 13.38 provides a useful diagram of relations between reflected/created/lifted limits.


lifting gerbe

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## Idea

A lifting gerbe is a [[cohomology|cocycle]]/[[gerbe]] -- or equivalently the [[principal infinity-bundle]] that it classifies -- arising as the obstruction to the lift of a $G$-[[principal bundle]] or more generally of any $G$-[[principal infinity-bundle]], through a [[group extension|shifted central extension]]

$$
  \mathbf{B}^{n-1} A \to \hat G \to G
$$

of $G$, for some abelian group $A$.

## Definition

Such an extension is classified (or rather its [[delooping]] is) by a cocycle

$$
  \mathbf{B}G \to \mathbf{B}^{n+1} A
$$

in [[group cohomology]].

For $X \to \mathbf{B}G$ the cocycle in [[nonabelian cohomology]] classifying a $G$-[[principal bundle]] or [[principal infinity-bundle]], the corresponding lifting $A$-gerbe is the thing classified by the composite cocycle

$$
  X \to \mathbf{B}G \to \mathbf{B}^n A
  \,.
$$

## Examples

### Lifting 1-gerbes

In the literature the following simplest case of the general situation is usually considered exclusively:

For $G$ an ordinary topological or Lie [[group]] and

$$
  U(1) \to \hat G \to G
$$

a $U(1)$-central [[group extension]], classified by a 2-cocycle
$\mathbf{B}G \to \mathbf{B}^2 U(1)$ in ordinary [[group cohomology]], and for $X \to \mathbf{B}G$ the cocycle of a $G$-[[principal bundle]], the corresponding lifting gerbe is given by the cocycle

$$
  X \to \mathbf{B}G \to \mathbf{B}^2 U(1)
  \,.
$$

In the literature this is often discussed in terms of the model for gerbes given by $U(1)$-principal [[bundle gerbe]]s: let $P \to X$ be the total space of the [[bundle]] classified by $X \to \mathbf{B}G$. Regard this as the [[surjective submersion]] that is part of the data of the [[bundle gerbe]] to be constructed. By principality of $P$, there is a canonical morphism

$$
  P \times_X P \to G
$$

that sends two elements in the fiber of the bundle to the unique group element that takes the first to the second. Along this morphism, form the [[pullback]] of the extension $\hat G \to G$ to get $L$ in

$$
  \array{
    L &\to& \hat G
    \\
    \downarrow && \downarrow
    \\
    P \times_X P &\to& G
  }
  \,.
$$

This $L$ is the line bundle ingredient in the lifting bundle gerbe. There is canocically a multiplication map that completes the definition.

+-- {: .num_prop}
###### Proposition

Let $\mathcal{H}$ be a separable infinite-dimensiona [[Hilbert space]], $U(\mathcal{H})$ its [[unitary group]] and $P U(\mathcal{H})$ its [[projective unitary group]]. Then the extension

$$
  0 \to U(1) \to U(\mathcal{H}) \to P U(\mathcal{H}) \to 0
$$

is universal with the above property: every $U(1)$-[[bundle gerbe]]/[[circle n-bundle with connection|circle 2-bundle]] is the lifting gerbe of some $P U(\mathcal{H})$-[[principal bundle]] through this extension.

=--

See [[projective unitary group]] for details.

To contrast, in ([Roberts 2021](#Roberts21)), _finite dimensional_ lifting gerbes are very often those gerbes with torsion [[Dixmier-Douady class]]. This is in particular true if the [[fundamental group]] of the base is finite, or the Lie group $G$ is compact.

### Lifting 2-gerbes

The archetypical example of a lifting 2-gerbe is a [[Chern-Simons gerbe]].

## References

A review is for instance in section 2.2 of

* {#CBMMS}  [[Alan Carey]], [[Peter Bouwknegt]], [[Varghese Mathai]], [[Michael Murray]] and [[Danny Stevenson]], _K-theory of bundle gerbes and twisted K-theory_ ,  Commun Math Phys, 228 (2002) 17-49 ([arXiv](http://arxiv.org/abs/hep-th/0106194))

* {#Roberts21} [[David Roberts]], _Many finite-dimensional lifting bundle gerbes are torsion_ (2021), arXiv:[2104.07936](https://arxiv.org/abs/2104.07936).

[[!redirects lifting gerbes]]
[[!redirects lifting bundle gerbe]]
[[!redirects lifting bundle gerbes]]


lifting to locators

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{: toc}

## Definition ##

Let $\mathbb{R}_D$ be the [[Dedekind real numbers]], and let $\mathbb{R}_\mathcal{L}$ be the subset of all Dedekind real numbers with a [[locator]], with canonical [[embedding]] $\eta:\mathbb{R}_\mathcal{L} \to \mathbb{R}_D$. A [[function]] $f:\mathbb{R}_D \to \mathbb{R}_D$ __lift to [[locators]]__ if it comes with a function $f_\mathcal{L}:\mathbb{R}_\mathcal{L} \to \mathbb{R}_\mathcal{L}$ such that the following [[square]] in [[Set]] [[commutative square|commutes]]:

$$
  \array{& \mathbb{R}_\mathcal{L} & \overset{f_\mathcal{L}}\rightarrow & \mathbb{R}_\mathcal{L} & \\
          \eta & \downarrow &&\downarrow & \eta\\
          &\mathbb{R}_D & \underset{f}\rightarrow& \mathbb{R}_D & \\
}$$

## See also ##

* [[locator]]

* [[Dedekind real numbers]]

* [[intermediate value theorem]]

* [[computable analysis]]

## References ##

* Auke Booij, *Extensional constructive real analysis via locators*, ([abs:1805.06781](https://arxiv.org/abs/1805.06781))

[[!redirects lift to locators]]
[[!redirects lifts to locators]]


lifts and extensions

[[!redirects lifts and sections]]

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## Idea

Given any [[morphism]] in some [[category]] a basic question is to ask for [[lifts]]/[[extensions]] along it (which are [[formal duality|dual]] notions), and in particular for [[retracts]]/[[sections]].

We survey how these concepts relate to each other. See the respective entries for more details and pointers.

## Definitions

+-- {: .num_example}
###### Extension problem

Given morphisms $i:A\to X$, $f:A\to Y$ find an [[extension]] of $f$ to $X$, i.e. a morphism $\tilde{f}:X\to Y$ such that $i\circ\tilde{f}=f$. Notice that if $i:A\hookrightarrow X$ is a [[subobject]], then $i\circ\tilde{f}$ is the [[restriction]] $\tilde{f}{|_A}$, and the condition is $\tilde{f}{|_A} = f$.

=--

+-- {: .num_example}
###### Retraction problem

Let $i:A\to X$ be a morphism. Find a [[retraction]] of $i$, that is a morphism $r:X\to A$ such that $r\circ i = id_A$.

=--

The retraction problem is a special case of the extension problem for $Y=A$ and $f=id_A$. Conversely, the general extension problem may (in [[Top]] and many other categories) be reduced to a retraction problem:

+-- {: .num_prop}
###### Proposition (Reducing an extension to a retraction)

If the [[pushout]] $Y\coprod_A X$ exists (for $i$, $f$ as above) then the extensions $\tilde{f}$ of $f$ along $i$ are in 1--1 correspondence with the retractions of $i_*(f) : Y\to Y\coprod_A X$.

=--

+-- {: .num_example}
###### Lifting problem

Given morphisms $p:E\to B$ and $g:Z\to B$, find a [[lifting]] of $g$ to $E$, i.e. a morphism $\tilde{g}:Z\to E$ such that $p\circ\tilde{g}=g$.

=--

+-- {: .num_example}
###### Section problem

For any $p:E\to B$ find a [[section]] $s: B\to E$, i.e. a morphism $s$ such that $p\circ s = id_B$.

=--

The section problem is a special case of a lifting problem where $g = id_B : B\to B$. Then the lifting is the section: $\tilde{g} = s$.  A converse is true in the sense

+-- {: .num_prop}
###### Proposition (Reducing a lifting to a section)

If the [[pullback]] $Z\times_B E$ exists then the general liftings for of $G$ along $p$ as above are in a bijection with the section of $g^*(p)=Z\times_B p : Z\times_B E\to Z$.

=--

## Related concepts

* [[lift]]/[[extension]]

* [[retract]]/[[section]]

category: disambiguation


light

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## Idea

The [[waves]] of the [[electromagnetic field]] in [[electromagnetism]] are called _light_ (sometimes the term is limited to the [[frequency]] range of _visible light_ and a bit below that ([[infrared]]) and above that ([[ultraviolet]])): light is appearing as _[[electromagnetic radiation]]_).

## Properties

The [[quanta]] of light are [[photons]]. The [[speed of light]] in [[vacuum]] plays a special role in the laws of physics, see at _[[special relativity]]_.

## Terminology

In physics, the term "light" is usually applied to cover the frequency range that includes humanly visible light and a little beyond in either direction, where photons have the requisite energy to slightly alter molecular configurations within the retina to produce sight sensations in animals, (approximately 390 to 740 nm in wavelength). Other terms (e.g. radio waves, X-rays) are then applied to cover other regions in the electromagnetic spectrum; the general term is just "electromagnetic radiation".

Some physicists however use the term "light" to cover any form of electromagnetic radiation. From Feynman's _QED: The Strange Theory of Light and Matter_, we have (p. 10)

* Another possibility, especially if the lecturer is a physicist, is that he uses ordinary words in a funny way. Physicists often use ordinary words such as "work" or "action" or"energy" or even, as you shall see, "light" for some technical purpose. Thus, when I talk about "work" in physics, I don't mean the same thing as when I talk about "work" on the street. During this lecture I might use one of those words without noticing that it is being used in this unusual way.I'll try my best to catch myself -- that's my job -- but it is an error that is easy to make.

and (p. 13)

* When I say "light" in these lectures, I don't mean simply the light we can see, from red to blue. It turns out that visible light is just a part of a long scale that's analogous to a musical scale in which there are notes higher than you can hear and other notes lower than you can hear. The scale of light can be described by numbers—called the frequency—and as the numbers get higher, the light goes from red to blue to violet to ultraviolet. We can't see ultraviolet light, but it can affect photographic plates. Its still light— only the number is different. (We shouldn't be so provincial: what we can detect directly with our own instrument, the eye, isn't the only thing in the world!) If we continue simply to change the number, we go out into X-rays, gamma rays, and so on. If we change the number in the other direction, we go from blue to red to infrared (heat) waves, then television waves and radio waves. For me, all of that is "light" I'm going to use just red light for most of my examples, but the theory of quantum electrodynamics extends over the entire range that I have described.

## Related concepts

* [[photon]], [[laser]]

* [[optics]]

* [[topological photonics]]

* [[gravitational wave]]

* [[pp-wave spacetime]]

* [[radiation]]

* [[photonic crystal]]

## References

* {#Orfanidis99} [[Sophocles J. Orfanidis]], *Electromagnetic Waves and Antennas* (1999-2016) [[web](http://eceweb1.rutgers.edu/~orfanidi/ewa), [pdf](http://eceweb1.rutgers.edu/~orfanidi/ewa/ewa-1up.pdf)]

See also:

* Wikipedia, _[Light](http://en.wikipedia.org/wiki/Light)_

[[!redirects light]]

[[!redirects electromagnetic wave]]
[[!redirects electromagnetic waves]]
[[!redirects EM wave]]
[[!redirects EM waves]]
[[!redirects em wave]]
[[!redirects em waves]]

[[!redirects light wave]]
[[!redirects light waves]]


light cone

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#### Gravity
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=--
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#Contents#
* table of contents
{:toc}

## Idea

Given a point in [[Minkowski spacetime]], its _light cone_ is the subspace of all [[lightlike]] vectors at that point.

More generally for a point $x$ in a [[Lorentzian manifold]] $(X,g)$, its lightcone is the lightcone in the [[tangent space]] $T_x X$.

Sometimes one also calls the collection of all [[time-like]] vectors the light cone, then denoted $V_x \subset T_x X$. On a [[time orientation|time oriented]] spacetime this decomposes into the _[[future cone]]_ $V_x^+$ and the _[[past cone]]_ $V_x^-$.

All these are _[[conical sets]]_.

## Related concepts

* [[causal cone]]

* [[celestial sphere]]

[[!redirects light cones]]


light mill

Think you know everything under the sun?

Then ponder the puzzle of [Light Mills](http://golem.ph.utexas.edu/category/2008/07/light_mills.html), and check out Greg Egan's magnificent [Light Mill Simulation Applet](http://www.gregegan.net/SCIENCE/LightMill/LightMill.html).

[[!redirects Light Mills]]
[[!redirects Light mill]]


Light mill > history

< [[Light mill]]
[[!redirects Light mill -- history]]


Light Mills > history

< [[Light Mills]]
[[!redirects Light Mills -- history]]


light-cone quantization

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## Idea

### General
 {#General}

In [[Hamiltonian]] formulation of [[relativistic quantum field theory]] on [[Minkowski spacetime]] there are three essential choices of forms of [[hypersurfaces]] to [[foliation|foliate]] [[spacetime]] by [[Dirac (1949)](#Dirac49)]:

> (Fig. 1 from [Brodsky, Pauli & Pinsky (1998)](#BrodskyPauliPinsky98))

The "instant form" on the left [[foliation|foliates]] [[spacetime]] by [[spacelike]] [[hypersurfaces]] (cf. [[Cauchy surface]]). This is the form considered for [[Hamiltonian mechanics]] in most of the literature (certainly in the case of [[quantum mechanics]], cf. [[Dirac (1949) §3](#Dirac49)]).

In contrast, *light-cone quantization* corresponds to the choice of [[foliation]] by [[lightlike]] [[wavefronts]] [[Dirac (1949) §5](#Dirac49)].

[Dirac (1949) §8](#Dirac49):

> The instant form has the advantage of being the one people are most familiar with, but I do not believe it is intrinsically any better for this reason. [...]

> The front form has the advantage that it requires only three Hamiltonians, instead of the four of the other forms. This makes it mathematically the most interesting form, and makes any problem of finding Hamiltonians substantially easier.

> The front form has the further advantage that there is no square root in the Hamiltonians, which means that one can avoid negative energies for particles by suitably choosing the values of the dynamical variables in the front, without having to make a special convention about the sign
of a square root.

(...)

### Discretized light-cone quantization
 {#DiscreteLightConeQuantization}

For [[nonperturbative quantum field theory|non-perturbative]] computations it turns out at least convenient (if not necessary) to in addition assume/impose periodic [[boundary conditions]] along the lightlike front, hence, in flat lightcone coordinates $x^{\pm}$, so that all fields $\phi$ satisfy

$$
  \phi\big(
    x^- + L,\, x^+
  \big)
  \;=\;
  \phi\big(
    x^-,\, x^+
  \big)
$$

for some $L \in \mathbb{R}_{\gt 0}$ giving the circumference of a lightlike [[circle]] (which some authors take to be literally [[lightlike]], of the kind shown eg. in [Bär 2004, Exp. 2.21 (2)](Lorentizan+geometry#Bär04), while others, like [Seiberg 1997](#Seiberg97), consider to be the [[limit of a sequence|limit]] of [[boosts]] of a [[spacelike]] [[circle]] [[fiber]]).

Since, [[Fourier transform|dually]], this means that the corresponding lightcone momentum $p^+$ becomes quantized in units of the inverse [[radius]] $R$ of this circle

\[
  \label{LightConeMomentumQuantization}
  p^+ \;\propto\; N/R
  \phantom{AAA}
  N \in \mathbb{Z}
  \,,
\]

hence one speaks of *discretized light-cone quantization*  or _DLCQ_, for short. This goes back to [Maskawa & Yamakawi 1976](#MaskawaYamakawi76); [Casher 1976](#Casher76); [Thorn 1977](#Thorn77), [1978](#Thorn78); [Pauli & Brodsky (1985b)](#PauliBrodsky85b), [(1985a)](#PauliBrodsky85a); [Tang, Brodsky & Pauli 1991 (2.6)](#TangBrodskyPauli91); see also [McCartor & Robertson 1994, §3](#McCartorRobertson94); for review see [Burkardt 1996, §5.1](#Burkardt96); [Heinzl 2001, §3.4](#Heinzl01).

Often it turns out that [[negative number|negative]] values of $N$ in (eq:LightConeMomentumQuantization) can be neglected or integrated out, so that

$$
  N \in \mathbb{N}
$$

becomes a [[natural number]]-parameter akin to that considered in the [['t Hooft double line construction]] of [[gauge theories]], and then the [[large N limit]] of the discrete light cone quantization becomes of interest.

### For sigma-models

In [[quantization]] of [[sigma models]] whose [[target space]] has a [[lightlike]] [[Killing vector]], the strategy of light-cone gauge quantization is to [[gauge fixing|gauge fix]] the [[metric]] and [[diffeomorphisms]] of the [[worldvolume]] such that also the worldvolume has a [[lightlike]] [[Killing vector field]] and such that this is mapped to the given one on target space.

This typically fixes most of the available gauge freedom, and the strategy is then to apply [[quantization]] to what remains. For more on this general idea see at _[[quantization commutes with reduction]]_.

Often this is considered for [[target space]] being [[Minkowski spacetime]] $\mathbb{R}^{d-1,1}$ and with $X^+ \coloneqq X^0 - X^1$ denoting one of its canonical lightlike [[coordinates]]. If $\tau$ denotes similarly a lightlike coordinate function on the [[worldvolume]], then the condition of light-cone gauge reads

$$
  X^+ = p^+ \tau
$$

for some proportionality constant $p^+$, the *light cone momentum*.  This is how light cone gauge appears in much of the [[string theory]] literature.

## Applications

### Quantization of Green-Schwarz super $p$-brane sigma models

Light cone gauge quantization is the only method by which [[Green-Schwarz super p-brane sigma models]] have been quantized, to date.

### BFSS matrix model

Specifically, applying light-cone gauge quantization to the [[Green-Schwarz sigma model]] for the  [[M2-brane]] on 11d [[Minkowski spacetime]], combined with a certain regularization of the remaining light-cone [[Hamiltonian]] yields the [[BFSS matrix model]].

### BMN matrix model

Alternatively, applying the light cone gauge quantization of the [[Green-Schwarz sigma-model]] of the [[M2-brane]] not on [[Minkowski spacetime]] but, more generally, on 11d [[pp-wave spacetimes]] (which are [[Penrose limits]] of the [[near horizon geometry]] of the [[black brane|black]] [[M2-branes]]/[[M5-branes]]) yields the [[BMN matrix model]].

## Related concepts

* [[large N limit]]

* [[holographic light front QCD]]

## References

### General

The concept originates with

* {#Dirac49} [[Paul A. M. Dirac]], §5 in "The Front Form", in: *Forms of Relativistic Dynamics*, Rev. Mod. Phys. **21** (1949) 392-399 [[doi:10.1103/RevModPhys.21.392](https://doi.org/10.1103/RevModPhys.21.392)]

reviewed in

* K. K. Wan, J. J. Powis, *Quantum mechanics in Dirac's front form*,  Int J Theor Phys **33** (1994) 553–574 [[doi:10.1007/BF00670516](https://doi.org/10.1007/BF00670516)]

and was rediscovered several times, such as in the guise of "infinite momentum frame" field theory:

* [[Steven Weinberg]], *Dynamics at Infinite Momentum*, Phys. Rev. **150** (1966) 1313,  Erratum Phys. Rev. 158, 1638 (1967) [[doi:10.1103/PhysRev.150.1313](https://doi.org/10.1103/PhysRev.150.1313)]

see also [Chang & Ma 1969](#ChangMa69) and [Kogut & Soper 1970](#KogutSoper70)

Early review:

* Shau-Jin Chang, Robert G. Root, and Tung-Mow Yan, *Quantum Field Theories in the Infinite-Momentum Frame. I. Quantization of Scalar and Dirac Fields*, Phys. Rev. D **7**,  (1973) 1133 [[doi:10.1103/PhysRevD.7.1133](https://doi.org/10.1103/PhysRevD.7.1133)]

* [[Stanley J. Brodsky]], [[Gary McCartor]], [[Hans-Christian Pauli]], [[Stephen S. Pinsky]], *The challenge of light-cone quantization of gauge field theory*, Particle World **3** 3 (1993) 109-124  [[cds:240388](http://cds.cern.ch/record/240388), [inspire:335247](https://inspirehep.net/literature/335247)]

* Wei-Min Zhang, *Light-Front Dynamics and Light-Front QCD*, Chin. J. Phys. **32** (1994) 717-808 [[arXiv:hep-ph/9412244](https://arxiv.org/abs/hep-ph/9412244)]

* {#Burkardt96} [[Matthias Burkardt]], *Light Front Quantization*, Adv. Nucl. Phys. **23** (1996) 1-74 [[arXiv:hep-ph/9505259](https://arxiv.org/abs/hep-ph/9505259), [doi:10.1007/0-306-47067-5_1](https://doi.org/10.1007/0-306-47067-5_1)]

Further review:

* Guillaume Beuf, Meijian Li, Tolga Altinoluk, Fabio Dominguez, *Course on Light-Cone Techniques applied to QCD*  (2022) [[indico:1203236](https://indico.cern.ch/event/1203236)]

* [[Stanley Brodsky]], [[Thomas Heinzl]] et al. *Light-Front Quantum Chromodynamics: A framework for the analysis of hadron physics*, White Paper of International Light Cone Advisory Committee, Nuclear Physics B - Proceedings Supplements **251**, **252** (2014) 165-174 [[doi:10.1016/j.nuclphysbps.2014.05.004](https://doi.org/10.1016/j.nuclphysbps.2014.05.004), [arXiv:1309.6333](https://arxiv.org/abs/1309.6333)]

* J. Vary et al. *Basis Light-Front Quantization: Recent Progress and Future Prospects*, Few-Body Syst **57** (2016) 695–702 [[doi:10.1007/s00601-016-1117-x](https://doi.org/10.1007/s00601-016-1117-x)]

* Wikipedia, *[Light front quantization](https://en.wikipedia.org/wiki/Light_front_quantization)*

* Wikipedia, [Light-front computational methods](https://en.wikipedia.org/wiki/Light-front_computational_methods)

See also:

* A. Harindranath, *[Light Front Dynamics](https://www.saha.ac.in/theory/a.harindranath/light/light.html)* (web resources)

The notion of lightlike circle compactification ("discretized light-cone quantization"):

* {#MaskawaYamakawi76} [[Toshihide Maskawa]], [[Koichi Yamakawi]], *The Problem of $P^+ = 0$ Mode in the Null-Plane Field Theory and Dirac's Method of Quantization*, Progress of Theoretical Physics **56** 1 (1976) 270–283 [[doi:10.1143/PTP.56.270](https://doi.org/10.1143/PTP.56.270)]

* {#Casher76} [[Aharon Casher]], *Gauge fields on the null plane*, Phys. Rev. D **14** (1976) 452 [[doi:10.1103/PhysRevD.14.452](https://doi.org/10.1103/PhysRevD.14.452)]

* {#Thorn77} [[Charles B. Thorn]], *On the derivation of dual models from field theory*, Physics Letters B **70** 1  (1977) 85-87 [doi:10.1016/0370-2693(77)90351-3]

* {#Thorn78} [[Charles B. Thorn]], *Derivation of dual models from field theory. II*, Phys. Rev. D **17** (1978) 1073 [[doi:10.1103/PhysRevD.17.1073](https://doi.org/10.1103/PhysRevD.17.1073)]

* {#PauliBrodsky85a} [[Hans-Christian Pauli]], [[Stanley J. Brodsky]], *Solving field theory in one space and one time dimension*, Phys. Rev. D **32** (1985) 1993 [[doi:10.1103/PhysRevD.32.1993](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.32.1993)]

* {#Pauli99} [[Hans-Christian Pauli]], *Discretized light-cone quantization and the effective interaction in hadrons*, AIP Conf. Proc. **494** (1999) 80-139 [[arXiv:hep-ph/9910203](https://arxiv.org/abs/hep-ph/9910203), [doi:10.1063/1.1301662](https://doi.org/10.1063/1.1301662)]

and on the issue of excluding $p^+ = 0$:

* [[Joel S. Rozowsky]], [[Charles B. Thorn]], *Spontaneous Symmetry Breaking at Infinite Momentum without $P^+$ Zero-Modes*, Phys. Rev. Lett. **85** (2000) 1614-1617 [[arXiv:hep-th/0003301](https://arxiv.org/abs/hep-th/0003301), [doi:10.1103/PhysRevLett.85.1614](https://doi.org/10.1103/PhysRevLett.85.1614)]

See also:

* [[Glenn Barnich]], Sucheta Majumdar, [[Simone Speziale]], Wen-Di Tan, *Lessons from discrete light-cone quantization for physics at null infinity: Bosons in two dimensions* [[arXiv:2401.14873](https://arxiv.org/abs/2401.14873)]

Lecture notes:

* {#Heinzl01} [[Thomas Heinzl]], *Light-Cone Quantization: Foundations and Applications*, in *Methods of Quantization*, Lecture Notes in Physics **572**, Springer (2001) [[arXiv:hep-th/0008096](https://arxiv.org/abs/hep-th/0008096), [doi:10.1007/3-540-45114-5_2](https://doi.org/10.1007/3-540-45114-5_2)]

More on the case of [[gauge fields]]:

* {#McCartorRobertson94} [[Gary McCartor]], David G. Robertson, *Light-cone quantization of gauge fields*, Zeitschrift für Physik C Particles and Fields **62** (1994) 349–355 [[doi:10.1007/BF01560250](https://doi.org/10.1007/BF01560250)]

In relation to [[spontaneous symmetry breaking]]:

* [[Stephen S. Pinsky]], *The Light-Cone Field Theory Paradigm for Spontaneous Symmetry Breaking*, in: *Proceedings, Hadron Structure 93 : Banská Štiavnica, September 5-10, 1993* [[arXiv:hep-th/9405091](https://arxiv.org/abs/hep-th/9405091), [inspire:364380](https://inspirehep.net/literature/364380)]

In relation to instant-time quantization:

* Philip D. Mannheim, _Light-front quantization is instant-time quantization_ [[arXiv:1909.03548](https://arxiv.org/abs/1909.03548)]

* Philip D. Mannheim, Peter Lowdon, [[Stanley Brodsky]],
*Comparing light-front quantization with instant-time quantization*, Physics Reports **891** (2021) 1-65 [[arXiv:2005.00109](https://arxiv.org/abs/2005.00109), [doi:10.1016/j.physrep.2020.09.001](https://doi.org/10.1016/j.physrep.2020.09.001)]

Discussion of ([[reduced phase space|reduced]]) [[phase spaces]] in light-cone coordinates:

* Prem P. Srivastava, *Constraints and Hamiltonian in light-front quantized field theory*, Nuovo Cim. A **107** (1994) 549-558 [[arXiv:hep-th/9308046](https://arxiv.org/abs/hep-th/9308046), [doi:10.1007/BF02768789](https://doi.org/10.1007/BF02768789)]

* Kianoosh Kargar, Ahmad Shirzad, Majid Monemzadeh, *Dynamical structure of fields in light cone coordinates*, Phys. Rev. D **99** (2019) 045019 [[arXiv:1608.03255](https://arxiv.org/abs/1608.03255), [doi:10.1103/PhysRevD.99.045019](https://doi.org/10.1103/PhysRevD.99.045019)]

### Application to quantum electrodynamics

Application to [[quantum electrodynamics]]:

* {#ChangMa69} Shau-Jin Chang, Shang-Keng Ma, *Feynman Rules and Quantum Electrodynamics at Infinite Momentum*, Phys. Rev. **180** (1969) 1506 [[doi:10.1103/PhysRev.180.1506](https://doi.org/10.1103/PhysRev.180.1506)]

* {#KogutSoper70} John B. Kogut and Davison E. Soper, *Quantum Electrodynamics in the Infinite-Momentum Frame*, Phys. Rev. D **1** (1970) 2901 [[doi:10.1103/PhysRevD.1.2901](https://doi.org/10.1103/PhysRevD.1.2901)]

* [[Andrew C. Tang]], *Discretized light cone quantization: Application to quantum electrodynamics*, PhD thesis, Stanford (1990) [[spire:296776](https://inspirehep.net/literature/296776), [pdf](https://inspirehep.net/files/5ef318e56c87e7561d36559f54895bd1), [[Tang-DLCQ.pdf:file]]]

* {#TangBrodskyPauli91} [[Andrew C. Tang]], [[Stanley J. Brodsky]], [[Hans-Christian Pauli]], *Discretized light-cone quantization: Formalism for quantum electrodynamics*, Phys. Rev. D **44** (1991) 1842 [[doi:10.1103/PhysRevD.44.1842](https://doi.org/10.1103/PhysRevD.44.1842)]

and to pure [[electromagnetism]]:

* Martina M. Brisudova, *Electromagnetic duality and light-front coordinates*, Phys. Rev. D **59** (1999) 087702 [[arXiv:hep-th/9806196](https://arxiv.org/abs/hep-th/9806196), [doi:10.1103/PhysRevD.59.087702](https://doi.org/10.1103/PhysRevD.59.087702)]

* Sucheta Majumdar, *Residual gauge symmetry in light-cone electromagnetism*,  J. High Energ. Phys. **2023** 215 (2023) [[arXiv:2212.10637](https://arxiv.org/abs/2212.10637), doi:10.1007/JHEP02(2023)215]

### Application to quantum chromodynamics
 {#ReferencesApplicationToQCD}

Application of (discretized) light cone quantization to to [[QCD]]:

* [[G. Peter Lepage]], [[Stanley J. Brodsky]], *Exclusive processes in perturbative quantum chromodynamics*, Phys. Rev. D **22** (1980) 2157 [[doi:10.1103/PhysRevD.22.2157](https://doi.org/10.1103/PhysRevD.22.2157)]

* {#PauliBrodsky85b} [[Hans-Christian Pauli]], [[Stanley J. Brodsky]], *Discretized light-cone quantization: Solution to a field theory in one space and one time dimension*, Phys. Rev. D **32** (1985) 2001 [[doi:10.1103/PhysRevD.32.2001](https://doi.org/10.1103/PhysRevD.32.2001)]

* {#BrodskyPauliPinsky98} [[Stanley Brodsky]], [[Hans-Christian Pauli]], [[Stephen S. Pinsky]], *Quantum Chromodynamics and Other Field Theories on the Light Cone*, Phys. Rept. **301** (1998) 299-486 [[arXiv:hep-ph/9705477](https://arxiv.org/abs/hep-ph/9705477), doi:10.1016/S0370-1573(97)00089-6]

Review in the broader context of [[non-perturbative quantum field theory]]:

* [[Yitzhak Frishman]], [[Jacob Sonnenschein]], §12 in: *Non-Perturbative Field Theory -- From Two Dimensional Conformal Field Theory to QCD in Four Dimensions*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511770838](https://doi.org/10.1017/CBO9780511770838), summary: [arXiv:1004.4859](https://arxiv.org/abs/1004.4859)] open access (2023) [[doi:10.1017/9781009401654](https://doi.org/10.1017/9781009401654)]

A light-cone QCD-[[Lagrangian density]] adapted to [[MHV amplitudes]]:

* Hiren Kakkad, Piotr Kotko, Anna Stasto, *Quantum correction to a new Wilson line-based action for Gluodynamics* [[arXiv:2311.04101](https://arxiv.org/abs/2311.04101)]

### Application in string theory

Light cone quantization of the [[string]] [[sigma-model]] originates with

* [[Peter Goddard]], [[Jeffrey Goldstone]], [[Claudio Rebbi]], [[Charles B. Thorn]],  §2.2 in: *Quantum dynamics of a massless relativistic string*, Nuclear Physics B **56** 1 (1973) 109-135 [doi:10.1016/0550-3213(73)90223-X]

see also

* [[Charles B. Thorn]], *Quantum Newtonian Dynamics on a Light Front*, Phys.Rev. D **59** (1999) 025005 [[arXiv:hep-th/9807151](https://arxiv.org/abs/hep-th/9807151), [doi:10.1103/PhysRevD.59.025005](https://doi.org/10.1103/PhysRevD.59.025005)]

All the standard introductory texts on [[string theory]] have sections devoted to light-cone quantization. For instance:

* {#GreenSchwarzWitten88} [[Michael Green]], [[John Schwarz]], [[Edward Witten]], section 5.2.1 of: _Superstring theory_, 3 vols. Cambridge Monographs on Mathematical Physics 1988 (Vol 1: [spire:250488](http://inspirehep.net/record/250488), [ISBN:9781107029118](https://www.cambridge.org/us/academic/subjects/physics/theoretical-physics-and-mathematical-physics/superstring-theory-25th-anniversary-edition-volume-1?format=HB&isbn=9781107029118))

In the context of the [[BFSS matrix model]] as a discrete light-cone formulation of [[M-theory]]:

* {#Seiberg97} [[Nathan Seiberg]], *Why is the Matrix Model Correct?*, Phys. Rev. Lett. **79** (1997) 3577-3580 [[arXiv:hep-th/9710009](https://arxiv.org/abs/hep-th/9710009), [doi:10.1103/PhysRevLett.79.3577](https://doi.org/10.1103/PhysRevLett.79.3577)]

* [[Adel Bilal]], *DLCQ of M-Theory as the Light-Like Limit*, Phys. Lett. B **435** (1998) 312-318 [[arXiv:hep-th/9805070](https://arxiv.org/abs/hep-th/9805070), doi:10.1016/S0370-2693(98)00811-9]

Review:

* {#Ydri18} [[Badis Ydri]], Section 3.9 of: _Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory_ [[arXiv:1708.00734](https://arxiv.org/abs/1708.00734)], published as: _Matrix Models of String Theory_, IOP (2018) [[ISBN:978-0-7503-1726-9](https://iopscience.iop.org/book/978-0-7503-1726-9)]

[[!include quantization of M2-brane on Minkowski spacetime to BFSS matrix model -- references]]

[[!redirects light cone quantization]]

[[!redirects light-cone gauge quantization]]
[[!redirects light cone gauge quantization]]

[[!redirects discretized light-cone quantization]]
[[!redirects discretized light cone quantization]]

[[!redirects light-cone gauge]]
[[!redirects light cone gauge]]
[[!redirects lightcone gauge]]

[[!redirects light front quantization]]

[[!redirects discrete light front quantization]]

[[!redirects discrete light cone quantization]]
[[!redirects discrete light-cone quantization]]
[[!redirects discrete lightcone quantization]]
[[!redirects discrete light cone quantizations]]
[[!redirects discrete light-cone quantizations]]
[[!redirects discrete lightcone quantizations]]

[[!redirects DLCQ]]

[[!redirects light-front quantization]]

[[!redirects light cone momentum]]
[[!redirects light-cone momentum]]
[[!redirects lightcone momentum]]

[[!redirects light cone longitudal]]
[[!redirects light cone transversal]]
[[!redirects light-cone longitudal]]
[[!redirects light-cone transversal]]
[[!redirects lightcone longitudal]]
[[!redirects lightcone transversal]]

[[!redirects light-front formalism]]


LIGO-Virgo

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Experiments
+-- {: .hide}
[[!include experiments -- contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[experiment]] designed to directly detect [[gravitational waves]].

Reported detections include: [LIGO 16](#LIGO16), [LIGO-Virgo 17](#LIGOVirgo17).

## Related entries

* [[LHC]], [[Super-Kamiokande]], [[EDGES]], [[RHIC]]

## References

* [LIGO website](https://www.ligo.caltech.edu)

* [Virgo website](http://www.virgo-gw.eu/)

* Wikipedia, _[LIGO](https://en.wikipedia.org/wiki/LIGO)_, _[Virgo](https://en.wikipedia.org/wiki/Virgo_interferometer)_ ,

Signal from two [[black holes]] merging:

* {#LIGO16} B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), _Observation of Gravitational Waves from a Binary Black Hole Merger_, Phys. Rev. Lett. 116, 061102, 11 February 2016, doi:[10.1103/PhysRevLett.116.061102](http://dx.doi.org/10.1103/PhysRevLett.116.061102)

Signal from two [[neutron stars]] merging:

* {#LIGOVirgo17} B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), _GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral_ Phys. Rev. Lett. 119, 161101, 2017 ([doi:10.1103/PhysRevLett.119.161101](https://doi.org/10.1103/PhysRevLett.119.161101))

[[!redirects LIGO]]
[[!redirects Virgo]]


Lijie Chen

* [personal page](https://www.mit.edu/~lijieche/)

## Selected writins

On [[complexity theory|complexity theoretic]] aspects of [[quantum supremacy]]:

* [[Scott Aaronson]], [[Lijie Chen]], *Complexity-Theoretic Foundations of Quantum Supremacy Experiments*, CCC '17: Proceedings of the 32nd Computational Complexity Conference (2017) **22** 1–67 [[arXiv:1612.05903](https://arxiv.org/abs/1612.05903), [doi;10.5555/3135595.3135617](https://dl.acm.org/doi/10.5555/3135595.3135617)]

category: people


lim^1 and Milnor sequences

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## 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]] _Milnor sequence_ ([below](#MilnorSequences)).

## Definition
 {#Definition}

+-- {: .num_defn #TheBoundaryMapDefiningLim1}
###### Definition

Given a [[tower]] $A_\bullet$ of [[abelian groups]]

$$
  \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0
$$

write

$$
  \partial
    \;\colon\;
  \underset{n}{\prod} A_n
    \longrightarrow
  \underset{n}{\prod} A_n
$$

for the homomorphism given by

$$
  \partial \;\colon\;
    (a_n)_{n \in \mathbb{N}}
  \mapsto
    (a_n - f_n(a_{n+1}))_{n \in \mathbb{N}}.
$$
=--

+-- {: .num_remark #LimitAsKernelAnalogousToLim1}
###### Remark

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

$$
  \array{
     && \underset{\longleftarrow}{\lim}_n A_n
     \\
     & {}^{\mathllap{p_{n+1}}}\swarrow && \searrow^{\mathrlap{p_n}}
     \\
     A_{n+1} && \overset{f_n}{\longrightarrow} && A_n
  }
$$

[[commuting diagram|commute]] -- is equivalently the [[kernel]] of the morphism $\partial$ in def. \ref{TheBoundaryMapDefiningLim1}.

=--

+-- {: .num_defn #Lim1ViaCokernel}
###### Definition

Given a [[tower]] $A_\bullet$ of [[abelian groups]]

$$
  \cdots \to A_3 \stackrel{f_2}{\to} A_2 \stackrel{f_1}{\to} A_1 \stackrel{f_0}{\to} A_0
$$

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

$$
  0 \to
  \underset{\longleftarrow}{\lim}_n A_n
  \longrightarrow
  \underset{n}{\prod} A_n
   \stackrel{\partial}{\longrightarrow}
  \underset{n}{\prod} A_n
    \longrightarrow
  \underset{\longleftarrow}{\lim}^1_n A_n
  \to 0
  \,,
$$

=--

There is a generalization to groups (not necessarily abelian). ([Bousfield-Kan 72, IX.2.1](#BousfieldKan72))

## Properties

### Abstract characterizations

+-- {: .num_prop #PropertiesOfLim1}
###### Proposition

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

1. 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

   $$
    0
    \to
    \underset{\longleftarrow}{\lim}_n A_n
    \to
    \underset{\longleftarrow}{\lim}_n B_n
    \to
    \underset{\longleftarrow}{\lim}_n C_n
    \to
    \underset{\longleftarrow}{\lim}_n^1  A_n
    \to
    \underset{\longleftarrow}{\lim}_n^1 B_n
    \to
    \underset{\longleftarrow}{\lim}_n^1 C_n
    \to 0
   $$

   is a [[long exact sequence]] of abelian groups;

1. 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](#BousfieldKan72), [Switzer 75, prop. 7.63](#Switzer75), [Goerss-Jardine 96, section VI. lemma 2.11](#GoerssJardine96))

+-- {: .proof}
###### Proof

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

$$
  L^\bullet(A_\bullet)
  \coloneqq
  \left[
    0
   \to
  \underset{deg \, 0}{\underbrace{\underset{n}{\prod} A_n}}
    \overset{\partial}{\longrightarrow}
  \underset{deg\, 1}{\underbrace{\underset{n}{\prod} A_n}}
   \to
   0
  \right]
$$

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

$$
  0
    \to
  L^\bullet(A_\bullet)
    \longrightarrow
  L^\bullet(B_\bullet)
    \longrightarrow
  L^\bullet(C_\bullet)
    \to
  0
  \,.
$$

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.

=--

+-- {: .num_lemma #TowersOfAbelianGroupsHasEnoughInjectives}
###### Lemma

The category $Ab^{(\mathbb{N}, \geq)}$ of [[towers]] of [[abelian groups]] has [[enough injectives]].

=--

+-- {: .proof}
###### 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

$$
  r_n
  \coloneqq
  \left[
    \cdots \overset{id}{\to}
    A \overset{id}{\to}
    \underset{= (r_n)_{n+1}}{\underbrace{A}} \overset{id}{\to}
    \underset{= (r_n)_n}{\underbrace{A}}
       \overset{}{\to}
    \underset{= (r_n)_{n-1}}{\underbrace{0}}
    \to
    0
    \to
    \cdots
    \to
    0
    \to
    0
  \right]
  \,.
$$

Since $(-)_n$ itself is evidently an [[exact functor]], its right adjoint preserves injective objects ([prop.](injective+object#RightAdjointsOfExactFunctorsPreserveInjectives)).

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

$$
  A_\bullet
    \overset{(\eta_n)_{n \in \mathbb{N}}}{\longrightarrow}
  \underset{n \in \mathbb{R}}{\prod}
  r_n A_n
  \hookrightarrow
  \underset{n \in \mathbb{N}}{\prod} r_n \tilde A_n
$$

is an injective resolution for $A_\bullet$.

=--

+-- {: .num_prop #Lim1IsDerivedLimit}
###### Proposition

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 $\underset{\longleftarrow}{\lim} \colon Ab^{(\mathbb{N},\geq)} \longrightarrow Ab$.

=--

([Bousfield-Kan 72, chapter IX.2, remark 2.6](#BousfieldKan72))

+-- {: .proof}
###### 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

$$
  0
    \to
  A_\bullet
    \overset{j^0}{\longrightarrow}
  J^0_\bullet
    \overset{j^1}{\longrightarrow}
  J^1_\bullet
    \overset{j^2}{\longrightarrow}
  J^2_\bullet
    \overset{}{\longrightarrow}
  \cdots
$$

be an [[injective resolution]]. We need to show that

$$
  \underset{\longleftarrow}{\lim}^1 A_\bullet
    \simeq
  ker(\underset{\longleftarrow}{\lim}(j^2))/im(\underset{\longleftarrow}{\lim}(j^1))
  \,.
$$

Since limits preserve [[kernels]], this is equivalently

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

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

$$
  \array{
      \cdots &\to & 0 &\to& 0 &\longrightarrow& 0 &\longrightarrow& \mathbb{Z} &\overset{id}{\longrightarrow}& \cdots &\overset{id}{\longrightarrow}& \mathbb{Z} &\overset{id}{\longrightarrow}& \mathbb{Z}
     \\
     \cdots
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \downarrow^{\mathrlap{incl}}
     &&
     \downarrow^{\mathrlap{id}}
     &&
     &&
     \downarrow^{\mathrlap{id}}
     &&
     \downarrow^{\mathrlap{id}}
     \\
     \cdots &\to& 0 &\to& 0 &\to & \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \cdots &\underset{id}{\longrightarrow}& \mathbb{Z} &\underset{id}{\longrightarrow}& \mathbb{Z}
  }
$$

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]]

$$
  0
   \to
  A_\bullet
   \overset{j^0}{\longrightarrow}
  J^0_\bullet
   \overset{j^1}{\longrightarrow}
  ker(j^2)_\bullet
   \to
  0
$$

becomes

$$
   0
     \to
   \underset{\longleftarrow}{\lim} A_\bullet
    \overset{\underset{\longleftarrow}{\lim} j^0}{\longrightarrow}
  \underset{\longleftarrow}{\lim} J^0_\bullet
    \overset{\underset{\longleftarrow}{\lim}j^1}{\longrightarrow}
  \underset{\longleftarrow}{\lim}(ker(j^2)_\bullet)
    \longrightarrow
  \underset{\longleftarrow}{\lim}^1 A_\bullet
    \longrightarrow
  0
  \,.
$$

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))
 \,.
$

=--

+-- {: .num_prop #AbstractCharacterizationOfLim1}
###### Proposition

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}.

=--

+-- {: .proof}
###### 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.

=--

### Vanishing of $\lim^1$

+-- {: .num_defn #MittagLefflerCondition}
###### Definition

A tower $A_\bullet$ of [[abelian groups]]

$$
  \cdots \to A_3 \to A_2 \to A_1 \to A_0
$$

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 $A_j \to A_k$

$$
  im(A_i \to A_k) \simeq im(A_j \to A_k)
  \,.
$$

=--

(e.g. [Switzer 75, def. 7.74](#Switzer75))

+-- {: .num_example #MittagLefflerSatisfiedInParticularForTowerOfSurjections}
###### Example

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]]).

=--

+-- {: .num_prop #Lim1VanihesUnderMittagLeffler}
###### Proposition

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

$$
  \underset{\longleftarrow}{\lim}^1 A_\bullet = 0
  \,.
$$

=--

e.g. ([Switzer 75, theorem 7.75](#Switzer75), [Kochmann 96, prop. 4.2.3](#Kochmann96), [Weibel 94, prop. 3.5.7](#Weibel94))

+-- {: .proof}
###### Proof idea

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 $n$.

=--

### Relation to $Ext$-groups

+-- {: .num_lemma #lim1AndExt1}
###### Lemma

Given a [[cotower]]

$$
  A_\bullet = (A_0 \overset{f_0}{\to} A _1 \overset{f_1}{\to} A_2 \to \cdots)
$$

of [[abelian groups]], then for every abelian group $B \in Ab$ there is a [[short exact sequence]] of the form

$$
  0
    \to
  \underset{\longleftarrow}{\lim}^1_n Hom(A_n, B)
    \longrightarrow
  Ext^1( \underset{\longrightarrow}{\lim}_n A_n, B )
    \longrightarrow
  \underset{\longleftarrow}{\lim}_n Ext^1( A_n, B)
    \to
  0
  \,,
$$

where $Hom(-,-)$ denotes the [[hom-object|hom-group]],  $Ext^1(-,-)$ denotes the first [[Ext]]-group (and so $Hom(-,-) = Ext^0(-,-)$).

=--

+-- {: .proof}
###### Proof

Consider the homomorphism

$$
  \tilde \partial
  \;\colon\;
  \underset{n}{\oplus} A_n
    \longrightarrow
  \underset{n}{\oplus} A_n
$$

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

$$
  0
    \to
  \underset{n}{\oplus} A_n
    \overset{\tilde \partial}{\longrightarrow}
  \underset{n}{\oplus} A_n
    \overset{}{\longrightarrow}
  \underset{\longrightarrow}{lim}_n A_n
   \to
  0
  \,.
$$

Every short exact sequence gives rise to a [[long exact sequence]] of [[derived functors]] ([prop.](derived+functor+in+homological+algebra#LongExactSequenceOfRightDerivedFunctorsFromShortExactSequence)) which in the present case starts out as

$$
  0
    \to
  Hom(\underset{\longrightarrow}{\lim}_n A_n,B)
    \longrightarrow
  \underset{n}{\prod} Hom( A_n, B )
    \overset{\partial}{\longrightarrow}
  \underset{n}{\prod} Hom( A_n, B )
    \longrightarrow
  Ext^1(\underset{\longrightarrow}{\lim}_n A_n,B)
    \longrightarrow
  \underset{n}{\prod} Ext^1(  A_n, B )
    \overset{\partial}{\longrightarrow}
  \underset{n}{\prod} Ext^1(  A_n, B )
    \longrightarrow
  \cdots
$$

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]]

$$
  Hom(A_\bullet,B)
  =
  (  \cdots \to Hom(A_2,B) \to Hom(A_1,B) \to  Hom(A_0,B) )
  \,.
$$

Hence truncating this long sequence by forming kernel and cokernel of $\partial$, respectively, it becomes the short exact sequence in question.

=--

## Milnor exact sequences
 {#MilnorSequences}

### For homotopy groups

+-- {: .num_prop #MilnorExactSequence}
###### Proposition
**(Milnor exact sequence for homotopy groups)**

Let

$$
  \cdots \to X_3 \overset{p_2}{\longrightarrow} X_2 \overset{p_1}{\longrightarrow} X_1 \overset{p_0}{\longrightarrow} X_0
$$

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]]

$$
  0
  \to
  \underset{\longleftarrow}{\lim}^1_i \pi_{q+1}(X_i)
  \longrightarrow
  \pi_q(\underset{\longleftarrow}{\lim}_i X_i)
  \longrightarrow
  \underset{\longleftarrow}{\lim}_i \pi_q(X_i)
  \to
  0
  \,,
$$

for $\pi_\bullet$ the [[homotopy group]]-functor (exact as [[pointed sets]] for $i = 0$, as [[groups]] for $i \geq 1$) which says that

1. 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;

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

=--

e.g. ([Bousfield-Kan 72, chapter IX, theorem 3.1](#BousfieldKan72), [Goerss-Jardine 96, section VI. prop. 2.15](#GoerssJardine96))

+-- {: .proof}
###### 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]] $[(\mathbb{N},\geq), sSet]_{inj}$ ($[(\mathbb{N},\geq), Top]_{inj}$) ([prop](projectively+cofibrant+diagram#CofibrantCotowerDiagram)). 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

$$
  \array{
    holim X_\bullet &\longrightarrow& \underset{n}{\prod} Path(X_n)
    \\
    \downarrow &(pb)& \downarrow
    \\
    \underset{n}{\prod} X_n &\underset{(id,p_n)_n}{\longrightarrow}&
    \underset{n}{\prod} X_ n \times X_n
  }
  \,,
$$

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]]

$$
  \underset{n}{\prod} \Omega X_n
    \longrightarrow
  holim X_\bullet
    \longrightarrow
  \underset{n}{\prod} X_n
  \,.
$$

The [[long exact sequence of homotopy groups]] of this fiber sequence goes

$$
  \cdots
   \to
  \underset{n}{\prod} \pi_{q+1}(X_n)
   \longrightarrow
  \underset{n}{\prod} \pi_{q+1}(X_n)
    \longrightarrow
  \pi_q (\underset{\longleftarrow}{\lim} X_\bullet)
    \longrightarrow
  \underset{n}{\prod} \pi_q(X_n)
    \longrightarrow
  \underset{n}{\prod} \pi_q(X_n)
   \to
  \cdots
  \,.
$$

Chopping that off by forming kernel and cokernel yields the claim for positive $q$. For $q = 0$ it follows by inspection.

=--

### For chain homology

+-- {: .num_example}
###### Example

Let

$$
  \cdots \to C_3 \to C_2 \to C_1 \to C_0
$$

be a tower of [[chain complexes]] (of [[abelian groups]]) such that it satisfies degree-wise the [[Mittag-Leffler condition]], def. \ref{MittagLefflerCondition}, and write

$$
  C \coloneqq \underset{\longleftarrow}{\lim}_n C_n
$$

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:

$$
  0 \to \underset{\longleftarrow}{\lim}^1_i H_{q+1}(C_i)
  \longrightarrow H_q(C)
  \longrightarrow \underset{\longleftarrow}{\lim}_i H_q(C_i)
  \to 0
  \,.
$$

=--

(e.g. [Weibel 94, prop. 3.5.8](#Weibel94))

### For generalized cohomology groups
 {#MilnorSequenceForGeneralizedCohomology}

#### Of spaces

+-- {: .num_prop #MilorSequenceForReducedCohomologyOnCWComplex}
###### Proposition
**(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 $\tilde E^\bullet$ be an additive [[reduced cohomology theory]].

Then the canonical morphisms make a [[short exact sequence]]

   $$
     0
      \to
     \underset{\longleftarrow}{\lim}^1_n \tilde E^{\bullet-1}(X_n)
      \longrightarrow
     \tilde E^{\bullet}(X)
      \longrightarrow
     \underset{\longleftarrow}{\lim}_n \tilde E^{\bullet}(X_n)
     \to
     0
     \,,
   $$

saying that

1. 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]];

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

=--

e.g. ([Switzer 75, prop. 7.66](#Switzer75), [Kochmann 96, prop. 4.2.2](#Kochmann96))

+-- {: .proof}
###### Proof

For

$$
  X_\bullet
  =
  \left(
    X_0
      \overset{i_0}{\hookrightarrow}
    X_1
      \overset{i_1}{\hookrightarrow}
    X_2
      \overset{i_1}{\hookrightarrow}
    \cdots
  \right)
$$

the sequence of stages of the ([[pointed topological space|pointed]]) [[CW-complex]] $X = \underset{\longleftarrow}{\lim}_n X_n$, write

$$
  \begin{aligned}
    A_X
    &\coloneqq
    \underset{n \in \mathbb{N}}{\sqcup} X_{2n} \times [2n,{2n}+1];
  \\
    B_X
    &\coloneqq
    \underset{n \in \mathbb{N}}{\sqcup} X_{(2n+1)} \times [2n+1,{2n}+2].
  \end{aligned}
$$

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.](mapping+telescope#MappingTelescope)), which we denote by

$$
  \array{
    A_X && && B_X
    \\
    & {}_{\mathllap{\iota_{A_x}}}\searrow && \swarrow_{\mathrlap{\iota_{B_x}}}
    \\
    && Tel(X_\bullet)
  }
  \,.
$$

Observe that

1. $A_X \cup B_X \simeq Tel(X_\bullet)$;

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

and that there are [[homotopy equivalences]]

1. $A_X \simeq \underset{n \in \mathbb{N}}{\sqcup} X_{2n+1}$

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

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

The first two are obvious, the third is [this proposition](mapping+telescope#TelescopeOfCWComplexEquivalentToTheOriginal).

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

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

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

$$
  \partial
    \;\colon\;
  (a_n)_{n \in \mathbb{N}}
    \mapsto
  ( a_n - i_n^\ast(a_{n+1}) )
  \,.
$$

(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

$$
   \cdots
    \to
  \tilde E^\bullet(X_{n+1})
    \overset{i_n^\ast}{\longrightarrow}
  \tilde E^\bullet(X_n)
    \overset{i_n^\ast}{\longrightarrow}
  \tilde E^{\bullet}(X_{n-1})
   \to
   \cdots
 \,.
$$

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}.

=--

In contrast:

+-- {: .num_prop}
###### Proposition

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

   $$
     \underset{\longrightarrow}{\lim}_n \tilde E_\bullet(X_n)
     \overset{\simeq}{\longrightarrow}
     \tilde E_\bullet(X)
   $$

   is an [[isomorphism]].

=--

([Switzer 75, prop. 7.53](#Switzer75))

#### Of spectra
 {#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]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#GradedAbelianGroupStructureOnHomsInTheHomotopyCategory), [exmpl.](Introduction+to+Stable+homotopy+theory+--+1-1#ForASpectrumXGeneralizedECohomology))

$$
  \begin{aligned}
    E^\bullet(X)
      & \coloneqq
    [X,E]_{-\bullet}
    \\
      & \coloneqq
    [\Sigma^\bullet X, E]
  \end{aligned}
  \,.
$$

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

$$
  \gamma
    \;\colon\;
  OrthSpec(Top_{cg})_{stable}
    \longrightarrow
  Ho(Spectra)
  \,.
$$

In the following when considering an [[orthogonal spectrum]] $X \in OrthSpec(Top_{cg})$, we use, for brevity, the same symbol for its image under $\gamma$.

+-- {: .num_prop #CohomologyOfSpectraMilnorSequence}
###### Proposition

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

$$
  0
    \to
  \underset{\longleftarrow}{\lim}^1_n E^{\bullet + n -1}(X_{n})
    \longrightarrow
  E^\bullet(X)
    \longrightarrow
  \underset{\longleftarrow}{\lim}_n E^{\bullet + n}(X_n)
    \to
  0
$$

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

$$
  E^{\bullet + n + 1}(X_{n+1})
    \overset{E^{\bullet+1n+1}(\Sigma^X_n)}{\longrightarrow}
  E^{\bullet+n+1}(X_n \wedge S^1)
    \overset{\simeq}{\longrightarrow}
  E^{\bullet + n}(X_n)
  \,.
$$

=--

([Schwede 12, chapter II prop. 6.5 (ii)](#Schwede12)) (using that symmetric spectra underlying orthogonal spectra are semistable ([Schwede 12, p. 40](#Schwede12)))

+-- {: .num_cor #WithSomeLefflerTheHomsOfSpectraAreHomotopicIfComponentsAre}
###### Corollary

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.

=--

+-- {: .proof}
###### 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]]

$$
  [X,E]
    \simeq
   E^0(X)
     \overset{\simeq}{\longrightarrow}
   \underset{\longleftarrow}{\lim}_n [X_n, E_n]
  \,.
$$

=--

## Related concepts

* [[homotopy limit]]

* [[spectral sequence of a tower of fibrations]]

* [[conditional convergence of spectral sequences]])

## References

* {#Milnor62} [[John Milnor]], _On axiomatic homology theory_, Pacific J. Math. Volume 12, Number 1 (1962), 337-341 ([Euclid](http://projecteuclid.org/euclid.pjm/1103036730))

* Z. Z. Yeh, _Higher Inverse Limits and Homology Theories_, Thesis, Princeton, 1959.

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], section IX.2 of _Homotopy limits, completions and localization_, Springer 1972

* {#Switzer75} [[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.

* {#Weibel94} [[Charles Weibel]], section 3.5 of _[[An Introduction to Homological Algebra]]_, Cambridge University Press (1994)

* {#Kochmann96} [[Stanley Kochmann]], section 4.2 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#GoerssJardine99} [[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

* {#Schwede12} [[Stefan Schwede]], around prop.6.5 of _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

Discussion in the context of [[categories of fibrant objects]] is in

* [[Kenneth Brown]], section 5 of _[[Abstract Homotopy Theory and Generalized Sheaf Cohomology]]_, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458

Discussion in the context of [[conditional convergence of spectral sequences]] is in

* {#Boardman99} [[Michael Boardman]], section I.1 of _Conditionally convergent spectral sequences_, 1999 ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/boardman-conditionally-1999.pdf))

[[!redirects lim1 and Milnor sequences]]
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[[!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]]