universal coefficient theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Cohomology
+-- {: .hide}
[[!include cohomology - contents]]
=--
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _universal coefficient theorem_ states how [[ordinary homology]]/[[ordinary cohomology]] determines [[homology]]/[[cohomology]] with arbitrary [[coefficients]].

For $C_\bullet$ a [[chain complex]] (of [[abelian groups]]) and $F$ a [[field]] (the _coefficient field_), the [[homology group]] $H_p(C,F)$ and the [[cohomology group]] $H^p(C,F)$ are indeed related by [[dualization]]: $ H^p(C,F) \simeq Hom_F(H_p(C,F), F)$.
If the coefficients are not a [[field]] but an arbitrary [[abelian group]], then this relationship receives a correction by an [[Ext]]-group. This is discussed below in _[For ordinary cohomology](#InCohomology)_.

Dually, again if $F$ is a field then there is an isomorphism $H_n(C) \otimes F \simeq H_n(C \otimes F)$ and for more general $F$ this is corrected by a [[Tor]]-group. This is discussed below in _[For ordinary homology](#InHomology)_.

More generally, under suitable conditions there are universal coefficient theorems that relate [[generalized (Eilenberg-Steenrod) cohomology]] to the dual of [[generalized homology]]. This is discussed below in _[For generalized cohomology](#ForGeneralizedCohomology)_.

There is also a version of the theorem for Kasparov's [[KK-theory]], see the [references](#ReferencesForKK).

## Statement

### For ordinary cohomology
 {#InCohomology}

#### In terms of homology

Let $C_\bullet$ be a [[chain complex]] of  [[free abelian groups]]. Let $A$ be an arbitrary [[abelian group]].

Write

* $C^\bullet \coloneqq Hom_{Ab}(C_\bullet, A)$ for the dual [[cochain complex]] with respect to $A$;

* $H_n(C)$ for the [[chain homology]] of $C_\bullet$

* $H^n(C,A)$ for the [[cochain cohomology]] of $C^\bullet$ hence for the cochain cohomology of $C_\bullet$ with [[coefficients]] in $A$.

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

There is a canonical morphism of [[abelian groups]]

$$
  \int_{(-)}(-) : H^n(C,A) \to Hom_{Ab}(H_n(C,\mathbb{Z}), A)
$$

given by sending a [[cocycle]] to [[evaluation]] of that cocycle on a [[chain]]:

$$
  [\omega] \mapsto
   \left( [\sigma] \mapsto \int_{\sigma} \omega \coloneqq \omega(\sigma) \right)
$$

=--

+-- {: .num_theorem #OrdinaryStatementInCohomology}
###### Theorem
**(universal coefficient theorem in ordinary cohomology)**

The morphism $h$ is surjective and its [[kernel]] is the [[Ext group]] $Ext^1(H_{n-1}(C, \mathbb{Z}), A)$. In other words, there is
a [[short exact sequence]]

$$
  0 \to
   Ext^1(H_{n-1}(C), A)
   \to
   H^n(C, A)
   \to
   Hom_{Ab}(H_n(C), A)
   \to
   0
$$

hence

$$
  0 \to
   Ext^1(H_{n-1}(C), A)
   \to
   H^n(Hom_{Ab}(C_\bullet,A))
   \to
   Hom_{Ab}(H_n(C), A)
   \to
   0
$$

Moreover, this sequence [[split exact sequence|splits]] (non-canonically).

=--

We reproduce the direct proof given for instance in ([Boardman](#Boardman)).

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

Given a homomorphism $A_1 \stackrel{f}{\to} A_2 \stackrel{g}{\to} A_3$ of [[abelian groups]] together with a [[retract]] $s : A_3 \to A_2$ of $g$, there is a [[short exact sequence]] of [[cokernels]]

$$
  0 \to coker f \stackrel{g'}{\to} coker(g \circ f) \to coker(g) \to 0
  \,.
$$

=--

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

Since we work in [[Ab]], all the [[cokernels]] appearing here (as discussed there) may be expressed as [[quotients]], e.g $coker(f) \simeq A_2/im(f)$.

The sequence of inclusions $im(g \circ f) \hookrightarrow im(g) \hookrightarrow A_3$ induces the canonical [[short exact sequence]]

$$
  0
   \to
  \frac{im(g)}{im(g \circ f)}
    \to
  \frac{A_3}{im(g \circ f)}
    \to
  \frac{A_3}{im(g)}
   \to
  0
$$

and we claim that this is already isomorphic to the one stated in the lemma. This is manifestly true for the two terms on the right. For the term on the left observe that $g$ induces a morphism $g' : A_2 / im(f) \to A_3 / im(g \circ f)$. By the existence of the retract $s$ this has itself a retract. Moreover it factors as

$$
  g' : A_2/im(f) \to im(g)/im(g \circ f) \hookrightarrow A_3/ im(g \circ f)
  \,.
$$

Therefore the first morphism here on the left has to be an isomorphism, too.

=--

+-- {: .proof}
###### Proof (of theorem \ref{OrdinaryStatementInCohomology})

Write

$$
  0 \to B_n \to Z_n \to H_n \to 0
$$

for the [[short exact sequence]] of [[boundaries]], [[cycles]], and [[homology groups]] of $C_\bullet$ in degree $n$. Since $C_n$ is assumed to be a [[free abelian group]] and since $B_n$ and $Z_n$ are [[subgroups]], it follows that these are also free abelian, by the abelian [[Nielsen-Schreier theorem]]. Therefore this sequence exhibits a [[projective resolution]] of the group $H_n$. It follows that the [[Ext]]-group $Ext^1(H_n,A)$ is characterized by the short exact sequence

\[
  \label{ExtES}
  Hom(Z_n, A) \to Hom(B_n,A) \to Ext^1(H_n,A) \to 0
  \,.
\]

Notice also that the short exact sequence

\[
  \label{ChainCycleBoundarySES}
  0 \to Z_n \to C_n \stackrel{\partial}{\to} B_{n-1} \to 0
\]

is [[split exact sequence|split]] because, as before, $B_{n-1}$ is free abelian. Using these two exact sequences on the left and right of the short exact sequence

$$
  0 \to Z_n/B_n \to C_n/B_n \to C_n/Z_n \to 0
$$

shows that this is equivalent to

\[
  \label{HomologyBoundariesSES}
  0 \to H_n \to C_n/B_n \stackrel{\partial}{\to} B_{n-1}
  \,.
\]

Again this splits as $B_{n-1}$ is free abelian.

In addition to these exact sequence consider the decomposition

$$
  \partial
  :
  C_n
  \to
  C_n/B_n
  \to
  C_n/Z_n
  \stackrel{\simeq}{\to}
  B_{n-1}
  \hookrightarrow
  Z_{n-1}
  \hookrightarrow
  C_{n-1}
$$

and apply $Hom(-,A)$ to obtain the diagram

$$
  \array{
    && && 0
    \\
    && && \uparrow
    \\
    && && Hom(H_n,A)
    \\
    && && \uparrow
    \\
    Hom(B_n,A) &\leftarrow& Hom(C_n,A)
    &\stackrel{i}{\leftarrow}&
    Hom(C_n/B_n,A)
    &\leftarrow&
    0
    &&
    0
    \\
    && && \uparrow^{\mathrlap{Hom(\bar \partial,A)}}
    && && \uparrow
    \\
    0 &\leftarrow& Ext^1(H_n,A) &\leftarrow& Hom(B_{n-1},A)
    && \leftarrow &&
    Hom(Z_{n-1},A)
    \\
    && && \uparrow && \nwarrow && \uparrow
    \\
    && && 0 && && Hom(C_{n-1},A)
  }
$$

Here the right vertical sequence is exact, because (eq:ChainCycleBoundarySES) splits, and the left vertical sequence is exact because (eq:HomologyBoundariesSES) splits.  The upper horizontal sequence is exact because the [[hom functor]] takes [[cokernels]] to [[kernels]] and finally the lower horizontal sequence is the exact sequence (eq:ExtES).

Since therefore $i$ and $Hom(\bar \partial,A)$ are monomorphisms, it follows that the degree $n$-[[cocycles]] are

$$
  Z^{n-1} \coloneqq ker( Hom(C_{n-1},A) \to Hom(C_n,a) )
  \simeq
  ker( Hom(C_{n-1},A) \to Hom(B_{n-1}, A) )
  \,.
$$

Using this for $n-1$ replaced by $n$ shows by the upper horizontal exact sequence that

$$
  Z^n = Hom( C_n/B_n, A)
  \,.
$$

Similarly the [[coboundaries]] are seen to be

$$
  B^n \coloneqq im Hom(\partial,A) \simeq
  im ( Hom(Z_{n-1}, A) \to Hom(C_n/B_n), A )
  \,.
$$

Together this gives the cochain cohomology as

$$
  H^n(C,A) \coloneqq Z^n / B^n
  \simeq
  coker ( Hom(Z_n, A) \to Hom( C_n/B_n, A ) )
  \,.
$$

Now the universal coefficient theorem follows by
going into lemma \ref{LemmaForUCTInOrdinaryCohomology}
with the identifications
$A_1 = Hom(Z_{n-1}, A)$, $A_2 = Hom(B_{n-1}, A)$, $A_3 = Hom(C_n/B_n,A)$.

=--

#### In terms of cohomology
 {#UCTForOrdinaryCohomologyInTermsOfCohomology}

There is also a UCT relating cohomology to cohomology:

Let $A$ and $B$ be chain complexes of [[free modules]] over a [[ring]] $R$ which is a [[principal ideal domain]]. Let $N_1$ and $N_2$ be $R$-modules. Assume that

1. $Tor_R(N_1,N_2) = 0$ (the [[Tor]] group of $N_1$ with $N_2$ vanishes);

1. at least one of $H^\bullet(A,N_1)$ and $H^\bullet(B,N_2)$ is of [[finite type]]

then there are [[short exact sequence]] of the form

$$
  0
    \to
  \underset{p+q = n}{\oplus} H^p(A,N_1) \otimes H^q(B,N_2)
    \longrightarrow
  H^n(A \otimes B, N_1 \otimes N_2)
    \longrightarrow
  Tor(H^\bullet(A,N_1), H^\bullet(B,N_2))
   \to
  0
$$

([Spanier 66, section 5.5, theorem 11](#Spanier66))

### For ordinary homology
 {#InHomology}

Let $C_\bullet \in Ch_\bullet(Ab)$ be a [[chain complex]] of [[free abelian groups]], and let $A \in $ [[Ab]] be any abelian group. Write $C_\bullet \otimes A$ etc. for the degreewise [[tensor product of abelian groups]].

More generally, let $R$ be a [[ring]] which is a [[principal ideal domain]] (in the above $R = \mathbb{Z}$ is the ring of [[integers]]), let $C_\bullet \in Ch_\bullet(R Mod)$ be a chain complex of [[free modules]] over $R$, let $A \in R Mod$ be any $R$-[[module]] and write $C_k \otimes_R A$ for the [[tensor product of modules]] over $R$.

+-- {: .num_theorem #TheoremInOrdinaryHomology}
###### Theorem

For each $n \in \mathbb{N}$ there is a [[short exact sequence]]

$$
  0
   \to
  H_n(C_\bullet) \otimes_R A
   \to
  H_n(C_\bullet \otimes_R A)
   \to
  Tor_1^{R Mod}(H_{n-1}(C_\bullet), A)
   \to
  0
  \,
$$

where on the right we have the first [[Tor]]-module of the [[chain homology]] $H_{n-1}(C_\bullet)$ with $A$.

=--

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

A fairly direct generalization of this statement and its proof is the [[Künneth theorem]] in ordinary homology, see at _[Künneth theorem - In ordinary homology](http://ncatlab.org/nlab/show/K%C3%BCnneth+theorem#InOrdinaryHomology)_.

=--

We spell out a proof along the lines for instance given in ([Hatcher, 3.A](#Hatcher)) or ([Chen, section 3](#Chen)).

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

For $C_\bullet$ a [[chain complex]] of [[free abelian groups]] and $A \in$ [[Ab]] any [[abelian group]], there is a [[long exact sequence]] of the form

$$
  \cdots \to B_n \otimes A \stackrel{i_n \otimes A}{\to} Z_n \otimes A \to H_n(C_\bullet \otimes A) \to B_{n-1} \otimes A \stackrel{i_{n-1}\otimes A}{\to} Z_{n-1} \otimes A \to \cdots
  \,,
$$

where $B_n$ are the [[boundaries]] and $Z_n$ the [[cycles]] of $C_\bullet$ in degree $n$ and where $i_n \colon B_n \hookrightarrow Z_n$ is the canonical inclusion.

=--

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

Since, by the [[Dedekind-Nielsen-Schreier theorem]], every [[subgroup]] of a [[free abelian group]] is itself free abelian, such as the subgroup of [[cycles]] $Z_n \hookrightarrow C_n$, it follows that for each $n \in \mathbb{N}$ we have a [[split exact sequence|splitting]] of the [[short exact sequence]] $0 \to Z_n \to C_n \to B_{n-1}$ and hence (as discussed at _[[split exact sequence]]_) a [[direct sum]] decomposition

$$
  C_n \simeq  Z_n \oplus B_{n-1}
  \,.
$$

Here the second direct summand on the right identifies under the differential $\partial^C$ with the [[boundaries]] in one degree lower, since by construction $\partial^C$ is injective on $C_n/Z_n$.

Accordingly, if we regard the graded abelian groups $B_\bullet$ and $Z_\bullet$ as chain complexes with vanishing [[differential]], then we have a sequence of [[chain maps]]

$$
  0 \to Z_\bullet \hookrightarrow C_\bullet \to B_{\bullet-1} \to 0
$$

which is degreewise a [[short exact sequence]], hence is a short exact sequence of chain complexes. Now since the [[tensor product of abelian groups]] distributes over [[direct sum]], the image of this sequence under $(-)\otimes A$

$$
  0 \to Z_\bullet \otimes A \hookrightarrow C_\bullet \otimes A \to B_{\bullet-1} \otimes A \to 0
$$

is still a short exact sequence. The induced [[homology long exact sequence]], as discussed there, is the long exact sequence to be shown.

=--

+-- {: .proof}
###### Proof
**of theorem \ref{TheoremInOrdinaryHomology}**

By lemma \ref{LongSequenceForHomologyWithCoefficients} we have [[short exact sequences]]

$$
  0
    \to
  coker(i_n \otimes A)
    \to
  H_n(C_\bullet \otimes A)
    \to
  ker(i_n \otimes A)
    \to
  0
$$

Since the [[tensor product of abelian groups]] is a [[right exact functor]] it preserves [[cokernels]] and hence

$$
  coker(i_n \otimes A) \simeq coker(i_n) \otimes A = H_n(C) \otimes A
  \,.
$$

The dual statement were true if $(-)\otimes A$ were also a [[left exact functor]]. In general it is not, and the failure is measure by the [[Tor]]-group:

Notice that by assumption and by the [[Dedekind-Nielsen-Schreier theorem]] the defining [[short exact sequence]]

$$
  0 \to B_n \stackrel{i_n}{\to} Z_n \to H_n(C_\bullet) \to  0
$$

exhibits $[\cdots \to 0 \to B_n \to Z_n] \stackrel{\simeq_{qi}}{\to} H_n(C)$ as a [[projective resolution]] of $H_n(C_\bullet)$. Therefore by definition of [[Tor]] the group $Tor_1(H_n(C_\bullet), A)$ is the chain homology in degree 1 of

$$
  [\cdots \to 0 \to B_n \otimes G \stackrel{i_n \otimes A}{\to} Z_n \otimes A]
  \,,
$$

which is

$$
  Tor_1(H_n(C_\bullet), A)
  \simeq
  ker(i_n \otimes A)
  \,.
$$

=--

+-- {: .num_section #sectiona }
=--

### For generalised cohomology theories
 {#ForGeneralizedCohomology}

The situation for [[generalised cohomology theories]] is much more complicated than that for ordinary cohomology due to the fact that it is harder (or impossible!) to use the tools of [[chain complexes]]. Nonetheless, it is possible to say something. The general case was studied by Adams in [Ada69](#Ada69) (for use in the [[Adams spectral sequence]], see there for more) and the initial version of the rest of this section is heavily based on that treatment.  This was also considered in the slightly later work, [Ada74, III.13](#Ada74). Adams' opening paragraph in [Ada69](#Ada69) is worth quoting in its entirety as motivation for this study.

> It is an established practice to take old theorems about ordinary homology, and generalise them so as to obtain theorems about generalised homology theories. For example, this works very well for duality theorems about manifolds. We may ask the following question. Take all those theorems about ordinary homology which are standard results in every day use. Which are the ones which still lack a fully satisfactory generalisation to generalised homology theories? I want to devote this lecture to such problems.

> *J. F. Adams*{: style="text-align: right"}

The lecture concentrates on the Universal Coefficient Theorem and, as a by-product, the [[Künneth theorem]].

Let $E^*$ and $F^*$ be two [[generalized cohomology theories]] and $E_*$ and $F_*$ two [[generalized homology theories]], such that $E$ is [[multiplicative cohomology theory|multiplicative]] and $F$ is a [[module]] over $E$.
Then the general problems that a Universal Coefficient Theorem should apply to are the following:

1.

   +-- {: .num_enuma #enumiAa }
   =--
   Given $E_{*}(X)$, calculate $F_{*}(X)$.

2.

   +-- {: .num_enuma #enumiAb }
   =--
   Given $E_{*}(X)$, calculate $F^{*}(X)$.

3.

   +-- {: .num_enuma #enumiAc }
   =--
   Given $E^{*}(X)$, calculate $F_{*}(X)$.

4.

   +-- {: .num_enuma #enumiAd }
   =--
   Given $E^{*}(X)$, calculate $F^{*}(X)$.

In [Ada69](#Ada69), Adams works in a very general setting. On this page, we shall work in a more restricted situation (as spelled out in Note 2 in Adams' lectures). We assume that $E^{*}(-)$ is the [[generalised cohomology theory]] associated to a [[commutative ring spectrum]], $E$. The cohomology theory $F^{*}(-)$ is assumed to come from a left [[module-spectrum]] over $E$, which we shall denote by $F$. We do not assume that $F$ is itself a ring spectrum. Following Adams, we shall also assume that all our cohomology and homology theories are [[reduced cohomology|reduced]].

There are two statements that one would like to hold. These are not themselves theorems, rather the theorem would say "Under certain conditions, these statements hold". The statements are the following.

\begin{statement} _(UCT1)_ \label{ucta}

There is a [[spectral sequence]]
$$
\Tor_{p,*}^{E_{*}} (E_{*}(X), F_{*}) \xRightarrow[p]{}    F_{*}(X)
$$
with edge homomorphism
$$
E_{*}(X) \otimes _{E_{*}} F_{*} \to F_{*}(X).
$$

\end{statement}

\begin{statement} _(UCT2)_ \label{uctb}

There is a [[spectral sequence]]
$$
\Ext_{E_{*}}^{p,*} (E_{*}(X), F^{*}) \xRightarrow[p]{}    F^{*}(X)
$$
with edge homomorphism
$$
F^{*}(X) \to \Hom_{E_{*}} (E_{*}(X), F^{*}).
$$

\end{statement}

For finite [[CW-complexes]] then we can derive two further statements from the above by [[S-duality]]. We use the notation $D X$ for the [[Spanier-Whitehead dual]] of $X$.

For a finite CW-complex $X$, we can apply [UCT1](#ucta) and [UCT2](#uctb) to $D X$ in place of $X$ and then use the various isomorphisms relating the cohomologies of $X$ and $D X$ to reformulate them in terms of $X$. We thus get the following statements.

\begin{statement} _(UCT3)_ \label{uctc}

For $X$ a finite CW-complex, there is a spectral sequence
$$
\Tor_{p,*}^{E^{*}}(E^{*}(X), F^{*}) \xRightarrow[p]{}    F^{*}(X)
$$
with edge homomorphism
$$
E^{*}(X) \otimes _{E^{*}} F^{*} \to F^{*}(X).
$$

\end{statement}

\begin{statement} _(UCT4)_ \label{utcd}

For $X$ a finite CW-complex, there is a spectral sequence
$$
\Ext^{p,*}_{E^{*}} (E^{*}(X), F_{*}) \xRightarrow[p]{}    F_{*}(X)
$$
with edge homomorphism
$$
F_{*}(X) \to \Hom^{*}_{E^{*}}(E^{*}(X), F_{*}).
$$

\end{statement}

(This is a generalization of the [[Kronecker pairing]], see also e.g. [Schwede 12, prop. 6.20](#Schwede12)).

A particularly important special case of these statements is when we have a [[topological space]], say $Y$, and a [[cohomology theory]], $E^{*}(-)$. Then we define a new homology theory $F_{*}(-)$ by $F_{*}(X) = E_{*}(X \wedge Y)$ and a new cohomology theory $G^{*}(-)$ by $G^{*}(X) = E^{*}(X \wedge Y)$. These are representable, the homology theory by $Y \wedge E$ and the cohomology theory by the function spectrum $F(Y,E)$. Putting these into the statements of the universal coefficient theorem, we obtain similar statements for the [[Künneth theorem]].

\begin{statement} _(KT1)_ \label{kta}

There is a spectral sequence
$$
\Tor_{p,*}^{E_{*}} (E_{*}(X), E_{*}(Y)) \xRightarrow[p]{}    E_{*}(X \wedge Y)
$$
with edge homomorphism
$$
E_{*}(X) \otimes _{E_{*}} E_{*}(Y) \to E_{*}(X \wedge Y).
$$

\end{statement}

\begin{statement} _(KT2)_ \label{ktb}

There is a spectral sequence
$$
\Ext_{E_{*}}^{p,*}(E_{*}(X), E^{*}(Y)) \xRightarrow[p]{}    E^{*}(X \wedge Y)
$$
with edge homomorphism
$$
E^{*}(X \wedge Y) \to \Hom_{E_{*}}^{*} (E_{*}(X), E^{*}(Y)).
$$

\end{statement}

\begin{statement} _(KT3)_ \label{ktc}

For $X$ a finite CW-complex, there is a spectral sequence
$$
\Tor_{p,*}^{E^{*}} (E^{*}(X), E^{*}(Y)) \xRightarrow[p]{}    E^{*}(X \wedge Y)
$$
with edge homomorphism
$$
E^{*}(X) \otimes _{E^{*}} E^{*}(Y) \to E^{*}(X \wedge Y).
$$

\end{statement}

\begin{statement} _(KT4)_ \label{ktd}

For $X$ a finite CW-complex, there is a spectral sequence
$$
\Ext^{p,*}_{E^{*}} (E^{*}(X), E_{*}(Y)) \xRightarrow[p]{}    E_{*}(X \wedge Y)
$$
with edge homomorphism
$$
E_{*}(X \wedge Y) \to \Hom_{E^{*}}^{*} (E^{*}(X), E_{*}(Y)).
$$

\end{statement}

The key question is, thus: when do these statements hold? Adams gives some answers in [Ada69](#Ada69).

* If $F_{*}$ is [[flat module|flat]] over $E_{*}$ then [UCT1](#ucta) holds, whence [KT1](#kta) holds if either $E_{*}(X)$ or $E_{*}(Y)$ is flat.

* If $E = S$, the [[sphere spectrum]], then all the results are true.

* If $E$ is a strict [[ring spectrum]] then [KT1](#kta) holds, if also $F$ is a strict [[module spectrum]] over $E$ then [UCT1](#ucta) holds.

* [[Atiyah]] gives a method in [Ati62](#Ati62) for [KT3](#ktc) with $E =$ [[KU]] being complex [[K-theory]] and $X$, $Y$ finite complexes.

#### A Special Case ####

In both [Ada69](#Ada69) and [Ada74](#Ada74), there is a particular focus on the universal coefficient theorem coming from its applications to the [[Adams spectral sequence]].  With that aim in mind, he studies the universal coefficient theorems with considerably strong assumptions.  These assumptions are designed to allow Atiyah's method (from [Ati62](#Ati62)) to work.

\begin{assum} _(Condition 13.3 in [Ada74](#Ada74), see also Assumption 20 in [Ada69](#Ada69))_ \label{adams_assumption}

   The spectrum $E$ is the [[direct limit]] of finite spectra $E_\alpha$ for which $E_*(D E_\alpha)$ is [[projective object|projective]] over $E_*$ and

   $$
   F^*(D E_\alpha) \to \Hom^*_{E_*}(E_*(D E_\alpha), F_*)
   $$

   is an isomorphism for all [[module-spectra]] $F$ over $E$.  Here, $D E_\alpha$ is the [[S-dual]] of $E_\alpha$.

\end{assum}

The main difference between the two treatments is that in [Ada69](#Ada69), the condition involving $F$ is stated for a *single* module-spectrum, not for *all* module-spectra, and there are alternatives for homology ($F_*(-)$) and cohomology ($F^*(-)$).

In the comments following Assumption 20 in [Ada69](#Ada69), Adams remarks that this is implied by a stronger condition (Proposition 17) on $E$ which makes no reference to $F$.  As $E$ is a ring spectrum, this reduces to:

1. The spectral sequence $H^*(E_\alpha; E^*) \Rightarrow E^*(E_\alpha)$ is trivial, and
2. For each $p$, $H^p(E_\alpha; E^*)$ is projective as an $E^*$-module.

With this assumption, Adams shows the following result:

+-- {: .num_theorem #uctholds}
###### Theorem ######
Let $E$ be a ring spectrum satisfying Assumption \ref{adams_assumption}. Let $F$ be a module-spectrum over $E$. Then \ref{uctb} holds, and the spectral sequence is convergent.
=--

What "convergent" means here is spelled out in [Ada74, Theorem 8.2](#Ada74).

\begin{corollary}
\label{uctproj}
Let $E$ be a ring spectrum satisfying Assumption \ref{adams_assumption}.  Suppose that $E_*(X)$ is projective over $E_*$.  Then the spectral sequence from \ref{uctb} collapses at the $E^2$ term.  That is,
$$
F^*(X) \to \Hom^*_{E_*}(E_*(X),F_*)
$$
is an isomorphism.
\end{corollary}

In [Ada74](#Ada74), Adams lists several cohomology theories (for $E^*(-)$) where the assumption holds.  These are: $S$, $H\mathbb{Z}_p$, $MO$, $MU$, $MSp$, $K$, $KO$.

#### Freeness and Flatness

In [BJW95](#BJW95) and [Boa95](#Boa95) there are various versions of the universal coefficient theorems and Künneth theorems which are stated and proved (or indications given on how to prove) under assumptions of either [[free module|freeness]] or [[flat module|flatness]].

Here, we shall gather together all the statements made.  In all the following, $E^*(-)$ is a multiplicative generalised cohomology theory with representing ring spectrum $E$.  We use $E_*(-)$ for the associated homology theory.  Following [Boa95](#Boa95) and [BJW95](#BJW95), cohomology and homology are *not* reduced in this section.

+-- {: .num_theorem #homkun}
###### Theorem
([Boa95, 4.2](#Boa95))
Assume that $E_*(X)$ or $E_*(Y)$ is a free or flat $E^*$-module.  Then the pairing:
$$
\times \colon E_*(X) \otimes E_*(Y) \to E_*(X \times Y),
$$
induces the Künneth isomorphism $E_*(X \times Y) \cong E_*(X) \otimes E_*(Y)$ in homology.
=--

The next result relates homology and cohomology.

+-- {: .num_theorem #strdual}
###### Theorem
([Boa95, 4.14](#Boa95))
Assume that $E_*(X)$ is a free $E^*$-module.  Then $X$ has strong duality, i.e. the duality map $d \colon E^*(X) \to D E_*(X)$ is a homeomorphism between the profinite topology on $E^*(X)$ and the dual-finite topology on $D E_*(X)$.  In particular, $E^*(X)$ is complete Hausdorff.
=--

Combining these two gives the Künneth theorem for cohomology.

+-- {: .num_theorem #cokun}
###### Theorem
([Boa95, 4.19](#Boa95))
Assume that $E_*(X)$ and $E_*(Y)$ are free $E^*$-modules.  Then we have the Künneth homeomorphism $E^*(X \times Y) \cong E^*(X) \widehat{\otimes} E^*(Y)$ in cohomology.
=--

There are similar results for spectra.  Boardman, Johnson, and Wilson write reduced homology and cohomology as $E_*(X,o)$ and $E^*(X,o)$, even when $X$ is a spectrum (and so the reduced theories are all that there are).

+-- {: .num_theorem #homkunsp}
###### Theorem
([Boa95, 9.20](#Boa95))
Assume that $E_*(X,o)$ or $E_*(Y,o)$ is a free or flat $E^*$-module.  Then the pairing $\times \colon E_*(X,o) \otimes E_*(Y,o) \to E_*(X \wedge Y,o)$ is an isomorphism in homology.
=--

+-- {: .num_theorem #strdusp}
###### Theorem
([Boa95, 9.25](#Boa95))
Assume that $E_*(X,o)$ is a free $E^*$-module.  Then $X$ has strong duality, i.e. $d \colon E^*(X,o) \to D E_*(X,o)$ is a homeomorphism between the profinite topology on $E^*(X,o)$ and the dual-finite topology on $D E_*(X,o)$.  In particular, $E^*(X,o)$ is complete Hausdorff.
=--

+-- {: .num_theorem #cokunsp}
###### Theorem
([Boa95, 9.31](#Boa95))
Assume that $E_*(X,o)$ and $E_*(Y,o)$ are free $E^*$-modules.  Then the pairing
$$
\times \colon E^*(X,o) \widehat{\otimes} E^*(Y,o) \to E^*(X \wedge Y,o)
$$
induces the cohomology Künneth homeomorphism.
=--

## Examples

### For singular cohomology
 {#InTopology}

For $X$ a [[topological space]], write $Sing X$ for its [[singular simplicial complex]] and
$$
  C_\bullet(X)
    \coloneqq
  N \mathbb{Z}[Sing X]
$$
for the [[normalized chain complex]] of the [[simplicial abelian group]] obtained by forming degreewise the [[free abelian group]].

The _[[singular homology]]_ $H_\bullet(X)$ of $X$ is the [[chain homology]] of $C_\bullet(X)$, and for $A$ some [[coefficient]] [[abelian group]], the [[singular cohomology]] $H^\bullet(X,A)$ is the [[cochain cohomology]], of $C_\bullet(X)$ with coefficients in $A$.

Comparison with the ordinary universal coefficient theorem \ref{OrdinaryStatementInCohomology} shows that:

+-- {: .num_theorem #OrdinaryStatementInTopology}
###### Theorem
**(universal coefficient theorem in topology)**

For $X$ a [[topological space]], $A$ an [[abelian group]] and $n \geq 1 \in \mathbb{N}$, the [[singular homology]] and [[singular cohomology]] of $X$ fit into a [[split exact sequence|split]] [[short exact sequence]] of the form

$$
  0
    \to
  Ext^1(H_{n-1}(X), A)
    \longrightarrow
  H^n(X,A)
    \longrightarrow
  Hom_{Ab}(H_n(X), A)
    \to
  0
  \,.
$$

=--

+-- {: .num_example #UniversalCoefficientTheoremInRationalCohomology}
###### Example
**([[universal coefficient theorem]] in [[rational cohomology]])**

For [[rational numbers]]-[[coefficients]] $\mathbb{Q}$, the
[[Ext groups]] $Ext^1(-;\mathbb{Q})$ vanish, and hence the [[universal coefficient theorem]] identifies [[rational cohomology]] [[cohomology groups|groups]] with the [[dual vector space]] of the [[rational vector space]] of rational [[homology groups]]:

$$
  H^\bullet
  \big(
    -; \, \mathbb{Q}
  \big)
  \;\;
  \simeq
  \;\;
  Hom_{\mathbb{Z}}
  \Big(
    H_\bullet\big(-;\,\mathbb{Q} \big);
    \,
    \mathbb{Q}
  \Big)
  \,.
$$

=--

(e.g. [Moerman 15, Cor. 1.2.1](rational+homotopy+theory#Moerman15))

## Related concepts

* [[Künneth theorem]]

* [[Eilenberg-Zilber theorem]]

* [[bootstrap category]]

## References

### For ordinary (co)homology

* {#Spanier66}[[Edwin Spanier]], section 5.5 of _Algebraic topology_, 1966

An exposition of the universal coefficient theorem for ordinary cohomology and homology is in section 3.1 of

* {#Hatcher} [[Allen Hatcher]], _Algebraic topology_ ([pdf](http://www.math.cornell.edu/~hatcher/AT/AT.pdf)); also [section 3.A](http://www.math.cornell.edu/~hatcher/AT/ATch3.4.pdf).

The note

* Adam Clay, _The universal coefficient theorems and Künneth formulas_ ([pdf](http://www.math.ubc.ca/~aclay/Algtopology.pdf))

surveys and spells out statement and proof of the theorem. A detailed proof of the theorem in cohomology is also in

* {#Boardman} [[Michael Boardman]], _The universal coefficient theorem_ ([pdf](http://www.math.jhu.edu/~jmb/note/uctcoh.pdf))

and a detailed proof of the statement in homology is in section 3 of

* {#Chen} Chen, _Universal coefficient theorem for homology_ ([pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/ChenJ.pdf))

### For generalized (co)homology
 {#ForGeneralizedCoHomology}

The universal coefficient theorem in [[symmetric smash product of spectra|symmetric monoidal]] [[model categories of spectra]] is discussed in

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 8 of _[[Modern foundations for stable homotopy theory]]_ ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

Universal coefficient theorems for [[generalized homology]] are discussed in:

* [[Friedrich Bauer]], _Remarks on universal coefficient theorems for generalized homology theories_ Quaestiones Mathematicae **9** 1, 4 (1986) 29-54

More on the universal coefficient theorem in generalized homology is in:

* {: #Ada69 }  **Ada69** [[John Adams|J. F. Adams]]. Lectures on generalised cohomology. pages 1--138, Berlin, 1969. Springer.

* {: #Ada74 } **Ada74** [[John Adams|J. F. Adams]],  (1974). [[Stable homotopy and generalised homology]]. Chicago, Ill.: University of Chicago Press.

* {: #Ati62 }  **Ati62** [[Michael Atiyah|M. F. Atiyah]]. Vector bundles and the Künneth formula. Topology, 1:245--248, 1962.

* {: #Boa95} **Boa95** [[J. M. Boardman]],  _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

* {: BJW95} **BJW95** [[J. M. Boardman]], and Johnson, David Copeland and Wilson, W. Stephen. (1995). Unstable operations in generalized cohomology. (pp. 687–828). Amsterdam: North-Holland.

See also

* {#Schwede12} [[Stefan Schwede]], prop. 6.20 of _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

Further discussion along these lines includes

* [[Andrew Baker]], Andrey Lazarev, _On the Adams Spectral Sequence for $R$-modules_, Algebr. Geom. Topol. 1 (2001) 173-199 ([arXiv.0105079](http://arxiv.org/abs/math/0105079))

### For KK-theory
 {#ReferencesForKK}

Discussion for [[KK-theory]] is in

* [[Jonathan Rosenberg]], Claude Schochet, _The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor_,  Duke Math. J. Volume 55, Number 2 (1987), 431-474. ([EUCLID](http://projecteuclid.org/euclid.dmj/1077306030))

[[!redirects universal coefficient theorem]]
[[!redirects universal coefficient theorems]]


universal complex line bundle

#Contents#
* table of contents
{:toc}

## Idea

The _universal complex line bundle_ is the [[complex vector bundle|complex]] [[universal vector bundle]] of [[rank]] 1, hence the [[complex line bundle]] which is [[associated bundle|associated]] to the [[circle group]]-[[universal principal bundle]] $E U(1)$ over the [[classifying space]] $B U(1)$ via the canonical [[action]] of $U(1)$ on $\mathbb{C}$.

Under the identification $B \mathrm{U}(1) \,\simeq\, \mathbb{C}P^\infty$ of the $\mathrm{U}(1)$-[[classifying space]] with the infinite [[complex projective spaces]], this is the [[dual vector bundle|dual]] [[tautological line bundle]] on the latter.

Its [[pullback bundle]] along the canonical inclusion $S^2 \longrightarrow B U(1)$ (the map which represents $1 \in \pi_2(B U(1)) \simeq \mathbb{Z}$) is the _[[basic complex line bundle on the 2-sphere]]_.

## Properties

### Zero-section into Thom space is weak equivalence

See at _[[zero-section into Thom space of universal line bundle is weak equivalence]]_.

## Related concepts

* [[Möbius strip]]

* [[tautological line bundle]], [[canonical line bundle]]

* [[complex oriented cohomology theory]]

[[!redirects universal complex line bundles]]


universal complex orientation on MU

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

For $E$ a [[homotopy commutative ring spectrum]], there is a [[bijection]] between [[complex oriented cohomology theory|complex orientations]] on $E$ and homotopy ring spectrum homomorphism $MU \longrightarrow E$ from [[MU]].

Hence $MU$ is the universal [[complex oriented cohomology theory]].

(e.g [Lurie 10, lect. 6, theorem 8](#Lurie10), [Ravenel, chapter 4, lemma 4.1.13](#Ravenel))

## Details

##### Conner-Floyd $E$-Chern classes are $E$-Thom classes
  {#ConnerFloydChernClassesAreThomClasses}

We discuss that for $E$ a [[complex oriented cohomology theory]], the $n$th universal [[Conner-Floyd-Chern class]] $c^E_n$ is in fact a universal [[Thom class]] for rank $n$ [[complex vector bundles]]. On the one hand this says that the choice of a [[complex oriented cohomology theory|complex orientation]] on $E$ indeed universally [[orientation in generalized cohomology|orients]] all [[complex vector bundles]]. On the other hand, we interpret this fact [below](#ComplexOrientationAsRingSpectrumMaps) as the [[unitality]] condition on a [[homomorphism]] of [[homotopy commutative ring spectra]] $M U \to E$ which represent that universal orienation.

+-- {: .num_lemma #SphereBundleBunminus1}
###### Lemma

For $n \in \mathbb{N}$, the [[fiber sequence]] ([prop.](classifying+space#SphereFibrationOverInclusionOfClassifyingSpaces))

$$
  \array{
    S^{2n-1} &\longrightarrow& B U(n-1)
    \\
    && \downarrow
    \\
    && B U(n)
  }
$$

exhibits $B U(n-1)$ as the [[sphere bundle]] of the [[universal vector bundle|universal complex vector bundle]] over $B U(n)$.

=--

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

When exhibited by a fibration, here the vertical morphism is equivalently the quotient map

$$
 (E U(n))/U(n-1) \longrightarrow (E U(n))/U(n)
$$

(by the proof of [this prop.](classifying+space#SphereFibrationOverInclusionOfClassifyingSpaces)).

Now the [[universal principal bundle]] $E U(n)$ is ([def.](classifying+space#EOn)) equivalently the colimit

$$
  E U(n) \simeq \underset{\longrightarrow}{\lim}_k  U(k)/U(k-n)
  \,.
$$

Here each [[Stiefel manifold]]/[[coset spaces]] $U(k)/U(k-n)$ is equivalently the space of (complex) $n$-dimensional subspaces of $\mathbb{C}^k$ that are equipped with an orthonormal (hermitian) [[linear basis]]. The [[universal vector bundle]]

$$
  E U(n) \underset{U(n)}{\times} \mathbb{C}^n
    \simeq
  \underset{\longrightarrow}{\lim}_k  U(k)/U(k-n) \underset{U(n)}{\times} \mathbb{C}^n
$$

has as fiber precisely the linear span of any such choice of basis.

While the quotient $U(k)/(U(n-k)\times U(n))$ (the [[Grassmannian]]) divides out the entire choice of basis, the quotient $U(k)/(U(n-k) \times U(n-1))$ leaves the choice of precisly one unit vector. This is parameterized by the sphere $S^{2n-1}$ which is thereby identified as the [[unit sphere]] in the respective fiber of $E U(n) \underset{U(n)}{\times} \mathbb{C}^n$.

=--

In particular:

+-- {: .num_lemma #UniversalComplexLineBundleThomSpace}
###### Lemma
**([[zero-section into Thom space of universal line bundle is weak equivalence]])

The [[zero-section]] map from the [[classifying space]] $B U(1) \simeq \mathbb{C}P^\infty$ (the infinite [[complex projective space]]) to the [[Thom space]] of the [[universal vector bundle|universal]] [[complex line bundle]] (the [[tautological line bundle]] on infinite [[complex projective space]]) is a [[weak homotopy equivalence]]

$$
  B U(1)
    \overset{\in W_{cl}}{\longrightarrow}
  M U(1)
   \coloneqq
  Th( E U(1) \underset{U(1)}{\times} \mathbb{C})
  \,.
$$

=--

(e.g. [Adams 74, Part I, Example 2.1](#Adams74))

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

Observe that the [[circle group]] $U(1)$ is naturally identified with the [[unit sphere]] in $\mathbb{C}$: $U (1) \simeq S(\mathbb{S})$. Therefore the [[sphere bundle]] of the universal complex line bundle is equivalently the $U(1)$-[[universal principal bundle]]

$$
  \begin{aligned}
    E U(1) \underset{U(1)}{\times} S(\mathbb{C})
    &
    \;\simeq\;
    E U(1) \underset{U(1)}{\times} U(1)
    \\
    &
    \;\simeq\;
    E U(1)
  \end{aligned}
  \,.
$$

But the [[universal principal bundle]] is [[contractible topological space|contractible]]

$$
  E U(1) \overset{\in W_{cl}}{\longrightarrow} \ast
  \,.
$$

(Alternatively this is the special case of lemma \ref{SphereBundleBunminus1} for $n = 1$.)

Therefore the [[Thom space]] of the universal complex line bundle is:

$$
  \begin{aligned}
    Th( E U(1) \underset{U(1)}{\times} \mathbb{C} )
    &
    \;\coloneqq\;
    D( E U(1) \underset{U(1)}{\times} \mathbb{C} )
    /
    S( E U(1) \underset{U(1)}{\times} \mathbb{C} )
    \\
    &
    \;\overset{\in W_{cl}}{\longrightarrow}\;
    D( E U(1) \underset{U(1)}{\times} \mathbb{C} )
    \\
    &
    \;\overset{\in W_{cl}}{\longrightarrow}\;
    B U(1)
  \end{aligned}
  \,.
$$

=--

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

For $E$ a [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]] theory, the $E$-[[reduced cohomology]] of the [[Thom space]] of the complex [[universal vector bundle]] is equivalently the $E$-[[relative cohomology]] of $B U(n)$ relative $B U(n-1)$:

$$
  \tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) )
    \;\simeq\;
  E^\bullet( B U(n), B U(n-1))
  \,.
$$

If $E$ is equipped with the structure of a [[complex oriented cohomology theory]] then

$$
  \tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) )
   \simeq
  c^E_n \cdot (\pi_\bullet(E))[ [ c^E_1, \cdots, c^E_n ] ]
  \,,
$$

where the $c_i$ are the universal $E$-[[Conner-Floyd-Chern classes]].

=--

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

Regarding the first statement:

In view of lemma \ref{SphereBundleBunminus1} and using that the disk bundle is homotopy equivalent to the base space we have

$$
  \begin{aligned}
    \tilde E^\bullet( Th(E U(n) \underset{U(n)}{\times} \mathbb{C}^n ) )
    &
    \;=\;
    E^\bullet( D(E U(n) \underset{U(n)}{\times} \mathbb{C}^n), S(E U(n)
\underset{U(n)}{\times} \mathbb{C}^n) )
    \\
    &
    \;\simeq\;
    E^\bullet( B U(n), B U(n-1))
  \end{aligned}
  \,.
$$

Regarding the second statement: the [[Conner-Floyd-Chern classes]] freely generate the $E$-cohomology of $B U(n)$ for all $n$ ([prop.](Conner-Floyd+Chern+class#ConnerFloyedClasses)):

$$
  E^\bullet(B U(n))
  \simeq
  \pi_\bullet(E)[ [ c^E_1, \cdots, c^E_n ] ]
  \,.
$$

and the restriction morphism

$$
  E^\bullet(B U(n))
   \longrightarrow
  E^{\bullet}(B U(n-1))
$$

projects out $c_n^E$. Since this is in particular a surjective map, the [[relative cohomology]] $E^\bullet( B U(n), B U(n-1) )$ is just the [[kernel]] of this map.

=--

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

Let $E$ be a [[complex oriented cohomology theory]]. Then the $n$th $E$-[[Conner-Floyd-Chern class]]

$$
  c^E_n \in \tilde E(Th( E U(n) \underset{U(n)}{\times} \mathbb{C}^n ))
$$

(using the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}) is a [[Thom class]] in that its restriction to the Thom space of any fiber is a suspension of a unit in $\pi_0(E)$.

=--

(e.g. [Tamaki-Kono 06, p. 61](#TamakiKono06), [Lurie 10, lecture 5, prop. 6](#Lurie10))

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

Since $B U(n)$ is [[connected topological space|connected]], it is sufficient to check the statement over the base point. Since that fixed fiber is canonically isomorphic to the direct sum of $n$ complex lines, we may equivalently check that the restriction of $c^E_n$ to the pullback of the universal rank $n$ bundle along

$$
  i \colon B U(1)^n \longrightarrow B U(n)
$$

satisfies the required condition. By the [[splitting principle]], that restriction is the product of the $n$-copies of the first $E$-Conner-Floyd-Chern class

$$
  i^\ast c_n \simeq ( (c_1^E)_1 \cdots (c_1^E)_n )
  \,.
$$

Hence it is now sufficient to see that each factor restricts to a unit on the fiber, but that is precisely the condition that $c_1^E$ is a complex orientation of $E$. In fact by def. \ref{StrictComplexOrientation} the restriction is even equal to $1 \in \pi_0(E)$.

=--

##### Complex orientation as ring spectrum maps
 {#ComplexOrientationAsRingSpectrumMaps}

For the present purpose:

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

For $E$ a [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]] theory, a _[[complex oriented cohomology theory|complex orientation]]_ on $E$ is a choice of element

$$
 c_1^E \in E^2(B U(1))
$$

in the cohomology of the [[classifying space]] $B U(1)$ (given by the infinite [[complex projective space]]) such that its image under the restriction map

$$
  \phi
  \;\colon\;
  \tilde E^2( B U(1) )
    \longrightarrow
  \tilde E^2 (S^2)
    \simeq
  \pi_0(E)
$$

is the unit

$$
  \phi(c_1^E) = 1
  \,.
$$

=--

([Lurie 10, lecture 4, def. 2](#Lurie10))

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

Often one just requires that $\phi(c_1^E)$ is _a_ [[unit]], i.e. an invertible element. However we are after identifying $c_1^E$ with the degree-2 component $M U(1) \to E_2$ of homtopy ring spectrum morphisms $M U \to E$, and by unitality these necessarily send $S^2 \to M U(1)$ to the unit $\iota_2 \;\colon\; S^2 \to E$ (up to homotopy).

=--

+-- {: .num_lemma #S2SpectrumMapFromComplexOrientation}
###### Lemma

Let $E$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)) equipped with a [[complex oriented cohomology theory|complex orientation]] (def. \ref{StrictComplexOrientation}) represented by a map

$$
  c_1^E
  \;\colon\;
  B U(1)
    \longrightarrow
  E_2
  \,.
$$

Write $\{c^E_k\}_{k \in \mathbb{N}}$ for the induced [[Conner-Floyd-Chern classes]]. Then there exists a morphism of $S^2$-[[sequential spectra]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#SequentialTSpectra))

$$
  M U \longrightarrow E
$$

whose component map $M U_{2n} \longrightarrow E_{2n}$ represents $c_n^E$ (under the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}), for all $n \in \mathbb{N}$.

=--

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

Consider the standard model of [[MU]] as a sequential $S^2$-spectrum with component spaces the [[Thom spaces]] of the complex [[universal vector bundle]]

$$
  M U_{2n}
    \coloneqq
  Th( E U(n) \underset{U(n)}{\otimes} \mathbb{C}^n)
  \,.
$$

Notice that this is a [[CW-spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-1#CWSpectrum), [lemma](Thom+space#ThomSpaceCWStructure)).

In order to get a homomorphism of $S^2$-[[sequential spectra]], we need to find representatives $f _{2n} \;\colon\; M U_{2n} \longrightarrow E_{2n}$ of $c^E_n$ (under the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}) such that all the squares

$$
  \array{
    S^2 \wedge M U_{2n}
      &\overset{id \wedge f_{2n}}{\longrightarrow}&
    S^2 \wedge E_{2n}
    \\
    \downarrow
      &&
    \downarrow
    \\
    M U_{2(n+1)}
      &\underset{f_{2(n+1)}}{\longrightarrow}&
    E_{2n+1}
  }
$$

commute strictly (not just up to homotopy).

To begin with, pick a map

$$
  f_0 \;\colon\; M U_0 \simeq S^0 \longrightarrow E_0
$$

that represents $c_0 = 1$.

Assume then by [[induction]] that maps $f_{2k}$ have been found for $k \leq n$. Observe that we have a homotopy-commuting diagram of the form

$$
  \array{
    S^2 \wedge M U_{2n}
      &\overset{id \wedge f_{2n}}{\longrightarrow}&
    S^2 \wedge E_{2n}
    \\
    \downarrow
      &\swArrow&
    \downarrow
    \\
    M U_{2} \wedge M U_{2 n}
      &\overset{c_1 \wedge c_{n}}{\longrightarrow}&
    E_2 \wedge E_{2n}
    \\
    \downarrow
      &\swArrow&
    \downarrow^{\mathrlap{\mu_{2,2n}}}
    \\
    M U_{2(n+1)}
      &\underset{c_{n+1}}{\longrightarrow}&
    E_{2(n+1)}
  }
  \,,
$$

where the maps denoted $c_k$ are any representatives of the Chern classes of the same name, under the identification of lemma \ref{UniversalComplexVectorBundleThomSpace}. Here the homotopy in the top square exhibits the fact that $c_1^E$ is a complex orientation, while the homotopy in the bottom square exhibits the Whitney sum formula for Chern classes ([prop.](Chern+class#WhitneySumChernClasses)).

Now since $M U$ is a [[CW-spectrum]], the total left vertical morphism here is a [[classical model structure on topological spaces|Serre-cofibration]], hence a [[Hurewicz cofibration]], hence satisfies the [[homotopy extension property]]. This means precisely that we may find a map $f_{2n+1} \colon M U_{2(n+1)} \longrightarrow E_{2(n+1)}$ homotopic to the given representative $c_{n+1}$ such that the required square commutes strictly.

=--

+-- {: .num_lemma #HRingSpectrumS2SpectrumMapFromComplexOrientation}
###### Lemma

For $E$ a [[complex oriented cohomology theory|complex oriented]] [[homotopy commutative ring spectrum]], the morphism of spectra

$$
  c \;\colon\; M U \longrightarrow E
$$

that represents the complex orientation by lemma \ref{S2SpectrumMapFromComplexOrientation} is a [[homomorphism]] of [[homotopy commutative ring spectra]].

=--

([Lurie 10, lecture 6, prop. 6](#Lurie10))

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

The unitality condition demands that the diagram

$$
  \array{
    \mathbb{S}
      &\overset{}{\longrightarrow}&
    M U
    \\
    & \searrow & \downarrow^{\mathrlap{c}}
    \\
    && E
  }
$$

commutes in the [[stable homotopy category]] $Ho(Spectra)$. In components this means that

$$
  \array{
    S^{2n}
      &\overset{}{\longrightarrow}&
    M U_{2n}
    \\
    & \searrow & \downarrow^{\mathrlap{c_n}}
    \\
    && E_{2n}
  }
$$

commutes up to homotopy, hence that the restriction of $c_n$ to a fiber is the $2n$-fold suspension of the unit of $E_{2n}$. But this is the statement of prop. \ref{ThomClassesCFClass}: the Chern classes are universal Thom classes.

Hence componentwise all these triangles commute up to some homotopy. Now we invoke the [[Milnor sequence]] for generalized cohomology of spectra ([prop.](lim^1+and+Milnor+sequences#CohomologyOfSpectraMilnorSequence)). Observe that the [[tower]] of abelian groups $n \mapsto E^{n_1}(S^n)$ is actually constant ([[suspension isomorphism]]) hence trivially satisfies the [[Mittag-Leffler condition]] and so a homotopy of morphisms of spectra $\mathbb{S} \to E$ exists as soon as there are componentwise homotopies ([cor.](lim^1+and+Milnor+sequences#WithSomeLefflerTheHomsOfSpectraAreHomotopicIfComponentsAre)).

Next, the respect for the product demands that the square

$$
  \array{
    M U \wedge M U
      &\overset{c \wedge c}{\longrightarrow}&
    E \wedge E
    \\
    \downarrow
      &&
    \downarrow
    \\
    M U
      &\underset{c}{\longrightarrow}&
    E
  }
$$

commutes in the [[stable homotopy category]] $Ho(Spectra)$. In order to rephrase this as a condition on the components of the ring spectra, regard this as happening in the [[homotopy category of a model category|homotopy category]] $Ho(OrthSpec(Top_{cg}))_{stable}$ of the [[model structure on orthogonal spectra]], which is [[equivalence of categories|equivalent]] to the [[stable homotopy category]]  ([thm.](Introduction+to+Stable+homotopy+theory+--+1-2#SequentialSpectraQuillenEquivalence)).

Here the derived [[symmetric monoidal smash product of spectra]] is given by [[Day convolution]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#SsymModuleSymmetricSpectra)) and maps out of such a product are equivalently as in the above diagram is equivalent ([cor.](Introduction+to+Stable+homotopy+theory+--+1-2#DayConvolutionViaNaturalIsosInvolvingExternalTensorAndTensor)) to a suitably equivariant collection diagrams of the form

$$
  \array{
    M U_{2 n_1} \wedge M U_{2 n_2}
      &\overset{c_{n_1} \wedge c_{n_2}}{\longrightarrow}&
    E_{2 n_1} \wedge E_{2 n_2}
    \\
    \downarrow
      &&
    \downarrow
    \\
    M U_{2(n_1 + n_2)}
      &\underset{c_{(n_1 + n_2)}}{\longrightarrow}&
   E_{2 (n_1 + n_2)}
  }
  \,,
$$

where on the left we have the standard pairing operations for $M U$ ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#OrthogonalComplexThomSpectrum)) and on the right we have the given pairing on $E$.

That this indeed commutes up to homotopy is the Whitney sum formula for Chern classes ([prop.](Chern+class#WhitneySumChernClasses)).

Hence again we have componentwise homotopies. And again the relevant [[Mittag-Leffler condition]] on $n \mapsto E^{n-1}((MU \wedge MU)_n)$-holds, by the nature of the universal [[Conner-Floyd classes]] ([prop.](Conner-Floyd+Chern+class#ConnerFloyedClasses)). Therefore these componentwise homotopies imply the required homotopy of morphisms of spectra ([cor.](lim^1+and+Milnor+sequences#WithSomeLefflerTheHomsOfSpectraAreHomotopicIfComponentsAre)).

=--

+-- {: .num_theorem}
###### Theorem

Let $E$ be a [[homotopy commutative ring spectrum]] ([def.](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyCommutativeRingSpectrum)). Then the map

$$
  (M U \overset{c}{\longrightarrow} E)
  \;\mapsto\;
  (B U(1) \simeq M U_{2} \overset{c_1}{\longrightarrow} E_2)
$$

which sends a homomorphism $c$ of [[homotopy commutative ring spectra]] to its component map in degree 2, interpreted as a class on $B U(1)$ via lemma \ref{UniversalComplexLineBundleThomSpace}, constitutes a [[bijection]] from homotopy classes of homomorphisms of homotopy commutative ring spectra to complex orientations (def. \ref{StrictComplexOrientation}) on $E$.

=--

([Lurie 10, lecture 6, theorem 8](#Lurie10))

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

By lemma \ref{S2SpectrumMapFromComplexOrientation} and lemma \ref{HRingSpectrumS2SpectrumMapFromComplexOrientation} the map is surjective, hence it only remains to show that it is injective.

So let $c, c' \colon M U \to E$ be two morphisms of homotopy commutative ring spectra that have the same restriction, up to homotopy, to $c_1 \simeq c_1'\colon M U_2 \simeq B U(1)$. Since both are homotopy ring spectrum homomophisms, the restriction of their components $c_n, c'_n \colon M U_{2n} \to E_{2 n}$ to $B U(1)^{\wedge^n}$ is a product of $c_1 \simeq c'_1$, hence $c_n$ becomes homotopic to $c_n'$ after this restriction. But by the [[splitting principle]] this restriction is injective on cohomology classes, hence $c_n$ itself ist already homotopic to $c'_n$.

It remains to see that these homotopies may be chosen compatibly such as to form a single homotopy of maps of spectra

$$
  f \;\colon\; M U \wedge I_+ \longrightarrow E
  \,,
$$

This follows due to the existence of the [[Milnor exact sequence|Milnor]] [[short exact sequence]] of the form

$$
  0
    \to
  \underset{\longleftarrow}{\lim}^1_n E^{-1}( \Sigma^{-2n} M U_{2n} )
    \longrightarrow
  E^0(M U)
    \longrightarrow
  \underset{\longleftarrow}{\lim}_n E^0( \Sigma^{-2n} M U_{2n} )
    \to
  0
$$

([prop.](lim^1+and+Milnor+sequences#CohomologyOfSpectraMilnorSequence)).

Here the [[Mittag-Leffler condition]] is clearly satisfied (by lemma \ref{UniversalComplexVectorBundleThomSpace} all relevant maps are epimorphisms). Hence the [[lim^1]]-term vanishes, and so by exactness the canonical morphism

$$
  E^0(M U)
    \overset{\simeq}{\longrightarrow}
  \underset{\longleftarrow}{\lim}_n E^0( \Sigma^{-2n} M U_{2n} )
$$

is an [[isomorphism]]. This says that two homotopy classes of morphisms $M U \to E$ are equal precisely already if all their component morphisms are homotopic (represent the same cohomology class).

=--

##### Universal finite-rank orientation on MΩΩSU(n)

The analogous universal finite-rank complex orientation on [[MΩΩSU(n)]]: [Hopkins 84,  Prop. 1.2.1](MOmegaSUn#Hopkins84).

## Related concepts

* [[zero-section into Thom space of universal line bundle is weak equivalence]]

* [[Lazard's theorem]]

* [[Landweber exact functor theorem]]

* [[Quillen's theorem on MU]]

* [[Landweber-Novikov theorem]]

## References

* {#Adams74} [[Frank Adams]], part II, lemma 4.6, example 4.7 in: _[[Stable homotopy and generalised homology]]_, 1974

* {#Ravenel} [[Doug Ravenel]], chapter 4 of: _[[Complex cobordism and stable homotopy groups of spheres]]_, 1986

* {#Kochmann96} [[Stanley Kochman]], section 4.4 of: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Section 3.7 of: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* {#Lurie10} [[Jacob Lurie]], _MU and complex orientations_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture6.pdf)), Lecture 6 in: _[[Chromatic Homotopy Theory]]_, Lecture series 2010 ([web](http://www.math.harvard.edu/~lurie/252x.html)),

[[!redirects MU and complex orientation]]

[[!redirects complex orientation and MU]]
[[!redirects complex orientations and MU]]

[[!redirects universal complex orientation of MU]]
[[!redirects MU and complex orientations]]


universal connection

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[compact space|compact]] [[Lie group]] there is a way to equip the topological [[classifying space]] $B G$ with [[smooth structure]] such that the corresponding smooth [[universal principal bundle]] $E G \to B G$ carries a [[connection on a bundle|smooth connection]] $\nabla_{univ}$ with the property that for every $G$-[[principal bundle]] $P \to X$ with [[connection on a bundle|connection]] $\nabla$ there is a smooth representative $f : X \to B G$ of the classifying map, such that $(P, \nabla) \simeq (P, f^* \nabla_{univ})$. This $\nabla_{univ}$ is called the **universal $G$-connection**.

## Related concepts

* [[connection on a bundle]]

* [[universal principal ∞-bundle]] , [[groupal model for universal principal ∞-bundles]]

* [[universal connection]]

## References

### In bounded dimension

Universal connections for manifolds of some bounded dimension $\leq n$ are appealed to in

* {#Chern50} [[Shiing-shen Chern]], _Differential geometry of fiber bundles_, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411,  Amer. Math. Soc., Providence, R. I. (1952) ([[Chern-DifferentialGeometryOfFiberBundles.pdf:file]], [full proceedings vol 2 pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1950.2/ICM1950.2.ocr.pdf))

  (in the context of [[Chern-Weil theory]])

and discussed in detail in

* [[Mudumbai Narasimhan]] and [[Sundararaman Ramanan]], _Existence of Universal Connections_, American Journal of Mathematics Vol. 83, No. 3 (Jul., 1961), pp. 563-572 ([jstor:2372896 ](https://www.jstor.org/stable/2372896 ))

* [[Mudumbai Narasimhan]] and [[Sundararaman Ramanan]], _Existence of Universal Connections II_, American Journal of Mathematics Vol. 85, No. 2 (Apr., 1963), pp. 223-231 ([jstor:2373211](https://www.jstor.org/stable/2373211))

* [[Roger Schlafly]], _Universal connections_, Invent Math 59, 59–65 (1980) ([doi:10.1007/BF01390314](https://doi.org/10.1007/BF01390314))

* [[Roger Schlafly]], _Universal connections: the local problem_, Pacific J. Math. Volume 98, Number 1 (1982), 157-171 ([euclid:pjm/1102734394](https://projecteuclid.org/euclid.pjm/1102734394))

See also

* Shrawan Kumar, _A Remark on Universal Connections_, Mathematische Annalen 260 (1982): 453-462 ([dml:163680](http://eudml.org/doc/163680))

### In unbounded dimension

Discussion of universal connections on some smooth incarnation of the full [[classifying space]]:

Using [[diffeological spaces]]:

* [[Mark Mostow]], _The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations_, J. Differential Geom. Volume 14, Number 2 (1979), 255-293 ([euclid:jdg/1214434974](https://projecteuclid.org/euclid.jdg/1214434974))

[[!redirects universal connections]]


universal construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
Universal properties are commonly used in mathematics, often without mentioning the term "universal property".

For example, if one were asked to give a map $\mathbb{R} \to \mathbb{R} \times \mathbb{C}$, they might write down something like $x \mapsto (x^2, x + i x)$. In effect, what is done is that a pair of maps $\mathbb{R} \to \mathbb{R}$ and $\mathbb{R} \to \mathbb{C}$ is given, namely $(x \mapsto x^2, x \mapsto x + i x)$. The **universal property** of the [[product]] says that giving a map to $A \times B$ is the same as giving a map to $A$ and a map to $B$, and moreover this correspondence is natural in some precise sense.

Similarly, given [[rings]] $R$ and $S$, if we want to extend a [[ring homomorphism]] $R \to S$ to a homomorphism from the [[polynomial ring]] $R[x] \to S$, all we have to do is to specify an element of $S$ that we send $x$ to. In other words, a homomorphism $R[x] \to S$ is the same as a homomorphism $R \to S$ and an element of $S$.

For it to be a universal property, just the existence of such a bijection is not sufficient. We will need some conditions to make sure the bijection is "natural". Abstractly, this says that the bijection is given by a [[natural isomorphism]] of certain [[functors]]. More concretely, by the [[Yoneda lemma]], this is equivalent to saying the bijection is "mediated" by some "universal maps", which is how universal properties are usually formulated. See [Concrete examples](#ConcreteExample) for more details.

Recall that by the [[Yoneda lemma]], specifying how we can map in or out of an object uniquely determines the object up to [[isomorphism]]. So we can use these universal properties as *definitions* of the constructions! These are known as **universal constructions**. Of course, these definitions are not actually "constructions". We still have to do the concrete constructions the good, old way to show that there are objects satisfying the universal property (or apply general theorems such as the [[adjoint functor theorem]]).

## Example

### Concrete examples {#ConcreteExample}
We first look at a few concrete examples of universal properties. These are all special cases of the ones described below.

+-- {: .num_example}
###### Example (products)
The [[product]] of two objects (eg. sets, groups, rings etc.) is specified by the property that maps $f: X \to A \times B$ biject naturally with pairs of maps $(f_1: X \to A, f_2: X \to B)$.

The naturality condition is that if $f: X \to A \times B$ corresponds to $f_1: X \to A, f_2: X \to B$, and $g: Y \to X$ is a map, then $f \circ g$ corresponds to $f_1 \circ g$ and $f_2 \circ g$, so that the bijection respects composition.

In particular, the [[identity map]] $id: A \times B \to A \times B$ corresponds to a pair of [[projection maps]] $\pi_1: A \times B \to A$ and $\pi_2: A \times B \to B$. Then if $f: X \to A \times B$ is a map, then by naturality, it corresponds to $\pi_1 \circ f: X \to A$ and $\pi_2 \circ f: X \to B$.

Suppose we are not given a bijection, but just an object $P$ with maps $\pi_1: P \to A$ and $\pi_2: P \to B$. Then as above, we obtain a function from maps $X \to P$ to pairs of maps $X \to A, X \to B$ by composition. This makes $P$ into the product of $A$ and $B$ exactly when this function is a bijection, ie. for any pair of maps $f_1 : X \to A, f_2: X \to B$, there is a unique map $f: X \to P$ whose compositions with $\pi_1, \pi_2$ are $f_1, f_2$ respectively (naturality is easy to check).

(The experienced reader will notice that this is just a special case of the [[Yoneda lemma]])

Thus, the universal property can be stated as follows: $C$ is a product of $A$ and $B$ if there exists maps $\pi_1: C \to A$ and $\pi_2: C \to B$ such that given any pair of maps $f_1: X \to A$ and $f_2: X \to B$, there is a unique map $f: X \to C$ such that the following diagram commutes:
$$
\array{
& & X & & \\
& ^\mathllap{f_1}\swarrow & \downarrow^\mathrlap{f} & \searrow^\mathrlap{f_2}\\
A & \underset{\pi_1}{\leftarrow} & C & \underset{\pi_2}{\rightarrow} & B
}
$$
In this case, we tend to write $A \times B$ for $C$.
=--

Note that if we are talking about sets, then an element of a set $X$ is equivalent to a map $1 \to X$ from the singleton set $1$. Thus in particular, the above definition says an element of $A \times B$ is the same as a pair of elements $(a, b)$, where $a \in A$ and $b \in B$.

+-- {: .num_example}
###### Example (free groups)
The [[free group|free]] [[group]] on $n$ generators is a group $F_n$ such that [[group homomorphisms]] $F_n \to G$ bijects (naturally) with $n$ elements of $G$ (not necessarily distinct).

Similar to the above, the naturality condition says if $f: F_n \to G$ corresponds to $g_1, ..., g_n \in G$, and $h: G \to H$ is a map, then $h \circ f$ corresponds to the elements $h(g_1), ..., h(g_n)$. In particular, suppose the identity map $id: F_n \to F_n$ corresponds to $n$ elements $x_1, ..., x_n\in F_n$. Then any homomorphism $f: F_n \to G$ corresponds to the elements $f(x_1), ..., f(x_n)$ of $G$.

Thus, given the specified elements $x_1, ..., x_n \in F_n$, the universal property says given any $n$ elements of $G$, we can find a unique homomorphism $f: F_n \to G$ that sends $x_1, ..., x_n$ to the $n$ elements.

Diagrammatically, picking $n$ elements out of a set $X$ is the same as a [[function]] (of sets) $n \to X$. If we write $U(G)$ for the underlying set of the group $G$ (ie. $U$ is the [[forgetful functor]] to $Set$), the universal property of the free group says that there is a specified function $\phi: n \to U(F_n)$, such that for every function $f: n \to U(G)$, we can find a unique group homomorphism $\tilde{f}: F_n \to G$ such that the following diagram commutes:
$$
\array{
  U(F_n) & \overset{U(\tilde{f})}{\to} & U(G)\\
  ^\mathllap{\phi}\uparrow & \nearrow_{\mathrlap{f}}\\
  n
}
$$
In other words, every map $f: n \to U(G)$ factors through the universal map $\phi: n \to U(F_n)$ uniquely.
=--

+-- {: .num_example}
###### Example (tensor products)
The [[tensor product]] of [[vector spaces]] has the universal property that a [[bilinear map]] $V \times W \to U$ bijects naturally with linear maps $V \otimes W \to U$. The naturality condition is given by the existence of a universal bilinear map $\phi: V \times W \to V \otimes W$ such that every bilinear map $V \times W \to U$ factors through $\phi$ uniquely.
=--

We have more degenerate examples such as the terminal object:
+-- {: .num_example}
###### Example (terminal and initial objects)
In the [[category of sets]], the singleton $1$ satisfies the property that there is always a unique map from any object to $1$. So we can say that the maps $A \to 1$ biject (necessarily naturally) with the set $1$. More generally, in any category, if an object $X$ is such that there is always a unique map from any object to $X$, then $X$ is called the [[terminal object]].

Dually, an initial object is an object $0$ such that there is a unique map from $0$ to any object $X$.
=--

### Classes of examples
In general, the __universal constructions__ in [[category theory]] include

* [[representable functor]]

* [[adjoint functor]]

* [[limit]]/[[colimit]]

* [[end]]/[[coend]]

* [[Kan extension]]

* [[dependent sum]]/[[dependent product]]

Each of these may be defined by requiring it to satisfy a __universal property__. A universal property is a property of some construction which boils down to (is manifestly equivalent to) the property that an associated object is a universal initial object of some (auxiliary) category.

In good cases, every single one of these is a special case of every other, so somehow one single concept here comes to us with many different faces.

Some or all of these have analogs in [[higher category theory]], notably in [[2-category]] theory and [[(∞,1)-category theory]]:

* [[2-limit]]

* [[2-adjunction]]

* [[limit in a quasi-category]]

* [[adjoint (∞,1)-functor]]

## References

An introductory but thorough treatment of universal properties and their relationship with the [[Yoneda lemma]] is given in

* [[Emily Riehl]], _Category Theory in context_, Chapter 2. ([pdf](http://www.math.jhu.edu/~eriehl/context.pdf))

A more informal and intuitive account is given in

* [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_, Chapter 2. ([arXiv](http://arxiv.org/abs/1912.10642))

[[!redirects universal construction]]
[[!redirects universal constructions]]
[[!redirects universal mapping]]
[[!redirects universal mappings]]

[[!redirects universal property]]
[[!redirects universal properties]]
[[!redirects universal mapping property]]
[[!redirects universal mapping properties]]
[[!redirects universal]]


universal constructions of topological spaces -- table

**examples of [universal constructions of topological spaces](Top#UniversalConstructions):**

| $\phantom{AAAA}$[[limits]] | $\phantom{AAAA}$[[colimits]] |
|------------|--------------|
|  $\,$ [[point space]]$\,$ | $\,$ [[empty space]] $\,$ |
| $\,$ [[product topological space]]  $\,$| $\,$ [[disjoint union topological space]] $\,$ |
| $\,$ [[topological subspace]] $\,$ | $\,$ [[quotient topological space]] $\,$ |
| $\,$ fiber space $\,$ | $\,$ [[space attachment]] $\,$ |
| $\,$ [[mapping cocylinder]], [[mapping cocone]] $\,$ | $\,$ [[mapping cylinder]], [[mapping cone]], [[mapping telescope]] $\,$ |
|  |  $\,$ [[cell complex]], [[CW-complex]] $\,$  |


universal covering space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### In point-set topology
 {#Definition}

The following is the classical discussion of universal covering spaces in [[point-set topology]].

+-- {: .num_prop #SimplyConnectedCoveringUniqueness}
###### Proposition
**([[essentially unique|essential uniqueness]] of simply connected covering spaces)**

Let $X$ be a [[topological space]] which is

1. [[path-connected topological space|path-connected]],

1. [[locally path-connected topological space|locally path connected]].

Then if $E_i \overset{p_i}{\to} X$ are two [[covering spaces]]
over $X$, $i \in \{1,2\}$, which are both [[path-connected topological space|path-connected]] and [[simply connected topological space|simply connected]], then they are [[isomorphism|isomorphic]] as covering spaces.

=--

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

Since both $E_1$ and $E_2$ are simply connected, the assumption of the lifting theorem for covering spaces is satisfied ([this prop.](covering+space#TheTheoremLifting)). This says that there are horizontal continuous function making the following diagrams commute:

$$
  \array{
    E_1 && \overset{f}{\longrightarrow} && E_2
    \\
    & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}}
    \\
    && X
  }
  \phantom{AAAAAA}
  \array{
    E_2 && \overset{g}{\longrightarrow} && E_1
    \\
    & {}_{\mathllap{p_2}}\searrow && \swarrow_{\mathrlap{p_1}}
    \\
    && X
  }
$$

$$
  \array{
    E_i && \overset{id}{\longrightarrow} && E_i
    \\
    & {}_{\mathllap{p_i}}\searrow && \swarrow_{\mathrlap{p_i}}
    \\
    && X
  }
$$

and that these are _unique_ once we specify the image of a single point, which we may freely do (in the given fiber).

So if we pick any point $x \in X$ and $\hat x_1 \in E_1$ with $p(\hat x) = x$ and $\hat x_2 \in E_2$ with $p(\hat x_2) = x$ and specify that $f(\hat x_1) = \hat x_2$ and $g(\hat x_2) = \hat x_1$ then uniqueness applied to the composites implies $f \circ g = id_{E_{2}}$ and $g \circ f = id_{E_1}$.

=--

+-- {: .num_defn #CoveringUniversal}
###### Definition
**(universal covering space)**

Let $X$ be a [[topological space]] which is

1. [[path-connected topological space|path-connected]],

1. [[locally path-connected topological space|locally path connected]].

Then a [[path-connected topological space|path-connected]] and [[simply connected topological space|simply connected]] [[covering space]], is called [[generalized the|the]] _universal covering space_ of $X$. This is well-defined, if it exists, up to [[isomorphism]], by prop. \ref{SimplyConnectedCoveringUniqueness}.

=--

+-- {: .num_prop #ReconstructCoveringForFreeAndTransitiveMonodromyRepresentation}
###### Proposition
**([[universal covering space]] reconstructed from [[free action|free]] and [[transitive action|transitive]] [[fundamental group]] [[representation]])**

Let $X$ be a topological space which is _[[well-connected space|well-connected]]_ in that it is

* [[path-connected space|path-connected]],

* [[locally path-connected space|locally path-connected]]

* and [[semi-locally simply-connected space|semi-locally simply-connected]]

Then a universal covering space of $X$ (def. \ref{CoveringUniversal}) exists.

=--

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

By [this prop.](covering+space#CoveringConnectivityViaMonodromy) the covering space is connected and simply connected precisely if its [[monodromy]] representation is free and transitive. By the [[fundamental theorem of covering spaces]] every [[permutation representation]] of the [[fundamental groupoid]] $\Pi_1(X)$ arises as the monodromy of some covering space. Hence it remains to see that a free and transitive representation of $\Pi_1(X)$ exists. Let $x \in X$ be any point, then $Hom_{\Pi_1(X)}(x,-)$ is such a representation.

=--

This is a  functorial construction of a universal covering spaces

$$
  Top_*^{wc} \longrightarrow Cov_*
$$

where $Top_*^{wc}$ denotes the [[full subcategory]] of [[pointed topological spaces]] $Top_*$ on the well-connected spaces and $Cov$ is the subcategory of $Top_*^2$ of pointed maps of spaces with objects the covering space maps.

Specifically, if $X$ is a space with basepoint $x_0$, we define $X^{(1)}$ to be the space whose points are homotopy classes of paths in $X$ starting at $x_0$, with the projection $X^{(1)}\to X$ projecting to the endpoint of a path.  We can equip this set $X^{(1)}\to X$ with a topology coming from $X$ so that it becomes a universal covering space as above.  As described at [[covering space]], under the correspondence between covering spaces and $\Pi_1(X)$-actions, the space $X^{(1)}$ corresponds to the "regular representation" of $\Pi_1(X)$.

+-- {: .un_remark}
###### Remark
This [[regular representation]] can be seen to arise by taking a [[category of elements]] in the same way that the regular representation of a group is gotten by taking its action on itself: we can see that the  _universal covering groupoid_ $\Pi(X)^{(1)}$ in the [[slice category]] $\mathbf{Grpd}/\Pi(X)$ (see the universal covering $\infty$-groupoid below) is just the category of elements of the action of $\Pi(X)$ on itself, and can be topologized in a natural way by lifting the topology on $X$ along the canonical projection $\Pi(X)^{(1)} \to \Pi(X)$; decategorifying this yields $X^{(1)}$.
=--

### In cohesive homotopy theory
 {#InCohesiveHomotopyTheory}

We describe now how the universal cover construction may be understood from the [[nPOV]], using a [[fragment]] of [[cohesive homotopy theory]].

 The basic idea is that the universal cover of a space $X$ is the [[homotopy fiber]] of the canonical morphism

$$
  X \longrightarrow \Pi_1(X)
$$

from $X$ to its [[fundamental groupoid]], which exists if both objects are regarded as "[[topological ∞-groupoids]]".  We may think of this as a precise way of the intuitive idea of "[[forcing]] $\Pi_1(X)$ to become trivial in a universal way".

+-- {: .num_example #ExamplesOfCohesiveToposesOfTopologicalGroupoids}
###### Example
**(cohesive higher toposes of topological groupoids)**

Let $\mathcal{S}$ be some [[∞-cohesive site]] of [[spaces]] and consider

$$
  \mathbf{H} \coloneqq Sh_\infty(\mathcal{S})
$$

the corresponding [[cohesive (∞,1)-topos]] over this site. This is the [[(∞,1)-category]] of "[[topological ∞-groupoids]]" modeled on $\mathcal{S}$.

A canonical choice relating to the traditional discussion would be $\mathcal{S} = Top^\kappa_{lcont}$ a [[small category|small]] [[full subcategory]] of [[Top]] on [[locally contractible topological spaces]], in which case the objects of $\mathbf{H}$ might be called "[[locally contractible topological ∞-groupoids]]".

A more restrictive choice would be $\mathcal{S} = $ [[CartSp]], in which case the objects of $\mathbf{H}$ might be called [[Euclidean-topological ∞-groupoids]].

In fact for the discussion of just universal [[covering spaces]] as opposed to the higher stages in the [[Whitehead tower]] it would be sufficient and more natural to take $\mathcal{S}$ the full subcategory of [[locally simply connected topological spaces]] and consider just the [[(2,1)-category]] $\mathbf{H}$ of [[stacks]] over this site.

=--

But the following discussion is completely formal and applies globally to all such realizations. Namely all we need is that

+-- {: .num_defn #AssumptionOnH}
###### Assumption

Assume that

1. $\mathbf{H}$ is a [[cohesive (∞,1)-topos|cohesive (n,1)-topos]]

   $$
     \mathbf{H}
       \array{
         \overset{\Pi}{\longrightarrow}
         \\
         \overset{\Delta}{\longleftarrow}
         \\
         \overset{\Gamma}{\longrightarrow}
         \\
         \overset{\nabla}{\longleftarrow}
       }
     \mathbf{B}
   $$

   over the given [[base (∞,1)-topos|base (n,1)-topos]] of [[n-groupoids]].

1. such that its [[shape modality]] preserves [[homotopy fiber products]] over [[discrete objects]] (objects in the essential image of $\Delta$).

We write

$$
  ʃ \coloneqq \Delta \Pi
$$

for the induced [[shape modality]] and

$$
  X \overset{\eta_X}{\longrightarrow} ʃ X
$$

for its [[unit of an adjunction|unit]] morphism ona given object $X$.

=--

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

Assumption \ref{AssumptionOnH} is satisfied for the case of $(\infty,1)$-toposes over an [[∞-cohesive site]] and for $(n,1)$-toposes by an $n$-cohesive site as in example \ref{ExamplesOfCohesiveToposesOfTopologicalGroupoids}: by [this prop.](cohesive+infinity-topos#PiPreservesPullbacksOverDiscretes).

=--

Now consider $X \in \mathbf{H}$ any object, for instance a [[topological space]] regarded as a [[0-truncated object|0-truncated]] [[topological infinity-groupoid]].

Assume, just for ease of discussion, that $X$ is _geometrically connected_ in that the [[0-truncated object|0-truncation]] of its [[shape]] is [[contractible]]:

$$
  \tau_0(ʃ X) \simeq \ast
  \,.
$$

Using the [[axiom of choice]] in the base topos $\mathbf{H}$ we choose a point

$$
  \ast \to \tau_1(ʃ X)
  \,.
$$

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

The _universal cover_ of $X$ is the [[homotopy pullback]] in

$$
  \array{
    \hat X &\longrightarrow& \ast
    \\
    \downarrow &(pb)& \downarrow
    \\
    X &\underset{L_{\tau_1} \circ \eta_X}{\longrightarrow}& \tau_1(ʃ X)
  }
  \,.
$$

=--

The [[universal property]] of $\hat X$ is immediate from the abstract setup:

1. $\hat X$ is simply connected, in that $\tau_1 ʃ \hat X \simeq \ast$

   This is because $ʃ X$ is discrete by construction, and hence so is $\tau_1(ʃ X)$. So by assumption \ref{AssumptionOnH} applying $ʃ$ to the above square yields another homotopy pullback of the form

   $$
     \array{
       ʃ \hat X &\longrightarrow& \ast
       \\
       \downarrow &(pb)& \downarrow
       \\
       ʃ X &\underset{ʃ \eta_X}{\longrightarrow}& \tau_1(ʃ X)
     }
     \,.
   $$

   Now the [[long exact sequence of homotopy groups]] (in $\mathbf{B}$) applied to this [[homotopy fiber sequence]] is of the form

   $$
     0 \to \pi_1(ʃ \hat X) \longrightarrow \pi_1(ʃ X) \overset{=}{\longrightarrow} \pi_1(\tau_1 ʃ X) \to \cdots
   $$

   which implies that $\pi_1( ʃ \hat X )$ and hence $\tau_1 ʃ \hat X$ is indeed trivial.

1. Let $E \longrightarrow X$ be any other object of $\mathbf{H}_{/X}$ such that $\tau_1(ʃ E) \simeq \ast$, then there is a morphism

   $$
     \array{
       E && \longrightarrow && \hat X
       \\
       & \searrow && \swarrow
       \\
       && X
     }
   $$

   This is because the naturality of the shape unit and of the truncation unit gives a homotopy commuting square of the form

   $$
     \array{
       E &\overset{\eta_E}{\longrightarrow}& ʃ E &\overset{}{\longrightarrow}& \tau_1(ʃ E ) \simeq \ast
       \\
       \downarrow && && \downarrow
       \\
       X &\underset{\eta_X}{\longrightarrow}& ʃ X &\longrightarrow& \tau_1 ʃ X
     }
   $$

   and thus a [[cone]] over the diagram which defines $\hat X$ via its [[universal property]].

This shows that $\hat X$ is the universal cover on abstract grounds.

We may also check explicitly that $\hat X$ is given as the space of homotopy classes of [[paths]] in $X$ from the given basepoint. To that end we use that, at least over the site [[CartSp]], the [[shape modality|shape]] of $X$ is represented by the topological [[path ∞-groupoid]]. See at _[[shape via cohesive path ∞-groupoid]]_.

> The following is old material that deserves to be harmonized a bit more with the above stuff.

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

Let $X$ be a suitably well behaved pointed space. The universal cover $X^{(1)}$ of $X$ is (equivalent to) the [[homotopy fiber]] of $X \to \Pi(X)$ in the [[(∞,1)-category]] $\mathbf{H} = Sh_{(\infty,1)}(Top_{cg})$ of [[topological ∞-groupoid]]s.

In other words, the [[principal ∞-bundle]] classified by the [[cocycle]] $X \to \Pi_1(X)$ is the universal cover $X^{(1)}$: we have a [[homotopy pullback]] square

$$
  \array{
    X^{(1)} &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    X &\to& \Pi(X)
  }
  \,.
$$

=--

+-- {: .query}

[[Urs Schreiber]]: may need polishing.

=--

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

We place ourselves in the context of [[topological ∞-groupoid]]s and regard both the space $X$ as well as its [[fundamental ∞-groupoid]] $\Pi(X)$ and its truncation to the [[fundamental groupoid]] $\Pi_1(X)$ as objects in there.

The canonical morphism $X \to \Pi(X)$ hence $X \to \Pi_1(X)$ given by the inclusion of constant paths may be regarded as a [[cocycle]] for a $\Pi(X)$-[[principal ∞-bundle]], respectively for a $\Pi_1(X)$-principal bundle.

Let $\pi_0(X)$ be the set of connected components of $X$, regarded as a topological $\infty$-groupoid, and choose any [[section]]  $\pi_0(X) \to \Pi(X)$ of the projection $\Pi(X) \to \pi_0(X)$.

Then the $\Pi(X)$-principal $\infty$-bundle classified by the [[cocycle]] $X \to \Pi(X)$ is its [[homotopy fiber]], i.e. the [[homotopy pullback]]

$$
  \array{
    UCov(X) &\to& \pi_0(X)
    \\
    \downarrow && \downarrow
    \\
    X &\to& \Pi(X)
  }
  \,.
$$

We think of this topological $\infty$-groupoid $UCov(X)$ as the **universal covering $\infty$-groupoid** of $X$. To break this down, we check that its [[decategorification]] gives the ordinary universal covering space:

for this we compute the [[homotopy pullback]]

$$
  \array{
    UCov_1(X) &\to& {*}
    \\
    \downarrow && \downarrow^{\mathrlap{x}}
    \\
    X &\to& \Pi_1(X)
  }
  \,,
$$

where we assume $X$ to be connected with chosen baspoint $x$ just to shorten the exposition a little. By the laws of [[homotopy pullback]]s in general and [[homotopy fiber]]s in particular, we may compute this as the ordinary [[pullback]] of a weakly equivalent diagram, where the point $*$ is resolved to the [[generalized universal bundle|universal]] $\Pi_1(X)$-principal bundle

$$
  \mathbf{E}_x \Pi_1(X) = T_x \Pi_1(X)
  \,.
$$

> (More in detail, what we do behind the scenes is this: we regard the diagram as a diagram in the _global_ [[model structure on simplicial presheaves]] on [[Top]]. In there all our topological groupoids are fibrant, hence all we have to ensure is that one of the morphisms of the diagram becomes a fibration, which is what the passage to $\mathbf{E}_x \Pi_1(X)$ achieves. Then the ordinary pullback in the category of simplicial presheaves is the homotopy pullback in $\infty$-prestacks. Then by left exactness of $\infty$-stackification, the image of that in $\infty$-stacks is still a homotopy pullback. )

The topological groupoid $\mathbf{E}_x \Pi_1(X)$ has as objects homotopy classes rel endpoints of paths in $X$  starting at $x$ and as morphisms $\kappa : \gamma \to \gamma'$ it has commuting triangles

$$
  \array{
    && x
    \\
    &{}^{\mathllap{\gamma}}\swarrow && \searrow^{\mathrlap{\gamma'}}
    \\
    y &&\stackrel{\kappa}{\to}&& y'
  }
$$

in $\Pi_1(X)$. The topology on this can be deduced from thinking of this as the [[pullback]]

$$
  \array{
   \mathbf{E}_x \Pi_1(X) &\to& {*}
   \\
   \downarrow && \downarrow^{\mathrlap{x}}
   \\
   \Pi_1(X)^I &\stackrel{d_0}{\to}& \Pi_1(X)
  }
$$

in simplicial presheaves on [[Top]]. Unwinding what this means we find that the open sets in $Mor(\mathbf{E}_x \Pi_1(X))$ are those where the endpoint evaluation produces an open set in $X$.

Now it is immediate to read off the homotopy pullback as the ordinary pullback

$$
  \array{
    UCov_1(X) &\to& \mathbf{E}_x \Pi_1(X)
    \\
    \downarrow && \downarrow
    \\
    X &\to& \Pi_1(X)
    \,.
  }
$$

Since $X$ is categorically discrete, this simply produces the space of objects of $\mathbf{E}_x \Pi_1(X)$ over the points of $X$, which is just the space of all paths in $X$ starting at $x$ with the projection $UCov_1(X) \to X$ being endpoint evaluation.

This indeed is then the usual construction of the universal covering space in terms of paths, as described for instance in

* _[Covering spaces, Building a universal cover](http://www.mathreference.com/at-cov,build.html)_ , [Math Reference Project](http://www.mathreference.com/)

=--

### In the petit $\infty$-toposes over the space

On the other hand, we can view a space $X$ as the little $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$ of $(\infty,1)$-sheaves on $X$.  If $X$ is locally connected and locally simply connected in the "coverings" sense, then $Sh(X)$ is locally 1-connected.

In fact, for the construction of the universal cover we require only the (2,1)-topos $Sh_{(2,1)}(X)$ of sheaves (stacks) of [[groupoids]] on $X$, so we will work in that context because it is simpler.  The construction can be adapted, however, to produce a "universal cover" of any locally 1-connected $(\infty,1)$-topos.

Let $E$ be any (2,1)-topos which is locally 1-connected.  This means that in the unique [[global sections]] [[geometric morphism]] $(E^*,E_*)\colon E\to Gpd$, the functor $E^*$ has a left adjoint $E_!\colon E \to Gpd$, which is automatically $Gpd$-[[indexed functor|indexed]].  The [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|fundamental groupoid]] of $E$ is defined to be $\Pi_1(E)\coloneqq E_!(*)$, where $*$ is the [[terminal object]] of $E$.

As discussed [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|here]] in the $(\infty,1)$-case, the construction of $\Pi_1(E)$ is a left adjoint to the inclusion of groupoids into locally 1-connected (2,1)-toposes (which sends $G\mapsto Gpd/G \simeq Gpd^G$).  Thus we have a geometric morphism $E\to \Pi_1(E)$ (where we regard $\Pi_1(E)$ as the (2,1)-topos $Gpd^{\Pi_1(E)}$).

Suppose, for simplicity, that $E$ is connected.  Then $\Pi_1(E)$ is also connected, and so we have an essentially unique functor $*\to \Pi_1(E)$.  We define the **universal cover** of $E$ to be the pullback (2,1)-topos:
$$\array{ E^{(1)} & \to & E \\ \downarrow & & \downarrow \\ * & \to & \Pi_1(E)}$$
(where of course $*$ denotes the terminal (2,1)-topos $Gpd$).

Now observe that $*\to \Pi_1(E)$ is a [[local homeomorphism of toposes]], since we have $Gpd \simeq Gpd/\Pi_1(E)/(*\to \Pi_1(E))$.  Since local homeomorphisms of toposes are stable under pullback, $E^{(1)}\to E$ is also a local homeomorphism, i.e. there exists an object $\widetilde{E}\in E$ and an equivalence $E^{(1)} \simeq E/\widetilde{E}$ over $E$.  Moreover, it is not hard to see that $\widetilde{E}$ can be identified with the pullback
$$\array{ \widetilde{E} & \to & * \\ \downarrow & & \downarrow \mathrlap{\eta} \\ * & \to & E^*(\Pi_1(E)) \mathrlap{= E^*(E_!(*))}}$$
in $E$.  Note that the bottom map $*\to E^*(E_!(*))$ is $E^*$ applied to the unique map $*\to E_!(*)$, while the right-hand map is the unit of the adjunction $E_!\dashv E^*$.

In order to see that this is a sensible definition, we first observe that $E^{(1)}$ is itself locally 1-connected (since it is etale over $E$).  Moreover, it is actually 1-connected, which is equivalent to saying that $E_!(\widetilde{E}) = *$.  This is because the "Frobenius reciprocity" condition for the adjunction $E_!\dashv E^*$ (which is equivalent to saying that $E_!$ is $Gpd$-indexed) applied to the defining pullback of $\widetilde{E}$ implies that we also have a pullback
$$\array{ E_!(\widetilde{E}) & \to & E_!(*) \\ \downarrow & & \downarrow \mathrlap{id} \\ * & \to & E_!(*)}$$
which clearly implies that $E_!(\widetilde{E}) = *$.

Thus, $E^{(1)}$ is a connected and simply connected space with a local homeomorphism to $E$, but is it a covering space?  In other words, is it locally trivial?  Since we have supposed that $E$ is locally 1-connected, as a (2,1)-category it can be generated by 1-connected objects, i.e. objects $U$ such that $E_!(U)\simeq *$.  In particular, we have a 1-connected object $U$ and a [[regular 1-epic]] $U\to *$.

We claim that if $U$ is any 1-connected object of $E$, then $\widetilde{E}$ is trivialized (or split) over $U$, in that $U\times \widetilde{E}$ is equivalent, over $U$, to $U\times E^*S$ for some $S\in Gpd$.  For pulling back the defining pullback to $U$, we obtain
$$\array{ U\times \widetilde{E} & \to & U \\ \downarrow & & \downarrow \mathrlap{U\times \eta} \\ U & \to & U\times E^*(\Pi_1(E)).}$$
But $U\times E^*(\Pi_1(E)) \cong (E/U)^* (\Pi_1(E))$, so to give a map $U \to (E/U)^* (\Pi_1(E))$ over $U$ is the same as to give a map $(E/U)_!(*) \to \Pi_1(E)$ in $Gpd$.  But $(E/U)_!(*)\simeq *$, since $U$ is 1-connected, and $\Pi_1(E)$ is connected, so there is only one such morphism.  Therefore, the two maps $U\to U\times E^*(\Pi_1(E))$ in the pullback above are in fact the same, and in particular both are the pullback to $E/U$ of the map $*\to \Pi_1(E)$.  Thus, $U\times \widetilde{E}$ is equivalent to $(E/U)^*(S) \cong U\times E^*(S)$, where $S= \Omega(\Pi_1(E))$ is the loop object of $\Pi_1(E)$, i.e. what we might call the fundamental *group* of the connected (2,1)-topos $E$.

Therefore, since $\widetilde{E}$ is trivialized over any 1-connected object, and $E$ is generated by 1-connected objects, $\widetilde{E}$ is locally trivial.  Moreover, since $*$ is a [[discrete object]] of $E$, so is $\widetilde{E}$.  Thus, if we specialize all this to the case $E=Sh_{(2,1)}(X)$ of (2,1)-sheaves on a topological space, then we conclude that $\widetilde{E}$ is an honest 1-sheaf on $X$ which, when regarded as a local homeomorphism over $X$, is locally trivial (hence a covering space), connected, and 1-connected---i.e. a universal cover of $X$.

## Examples
  {#Example}

+-- {: .num_example #UniversalCoveringOfCircleRealLine}
###### Example
**([[real line]] is universal covering of [[circle]])**

Let

1. $\mathbb{R}^1$ be the [[real line]] with its [[Euclidean space|Euclidean]] [[metric topology]];

1. $S^1 \coloneqq \left\{ x\in \mathbb{R}^2 \;\vert\; {\Vert x\Vert} = 1 \right\} \subset \mathbb{R}^2$ be the [[circle]] with its [[subspace topology]] induced from the [[Euclidean plane]].

Consider the [[continuous function]]

$$
  \array{
     \mathbb{R}^1
       &\overset{p}{\longrightarrow}&
     S^1 \subset \mathbb{R}^2
     \\
     t &\mapsto& (\cos(2\pi t), \sin(2\pi t))
  }
  \,.
$$

This exhibits the universal covering space (def. \ref{CoveringUniversal}) of the circle.

=--

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

Let $p \in S^1$ be any point. It is clear that we have a [[homeomorphism]] of the form

$$
  \array{
    S^1 \setminus p
      &\overset{\simeq}{\longrightarrow}&
    (0, 1)
  }
  \,.
$$

and hence a [[homeomorphism]] of the form

$$
  \array{
    S^1 \times Disc(\mathbb{Z}) &\simeq& (0,1) \times Disc(\mathbb{Z})
    &\overset{\simeq}{\longrightarrow}&
    p^{-1}(S^1 \setminus \{p\})
    \\
    && (t,n) &\mapsto& (cos(2\pi n t), \sin(2\pi n t))
  }
  \,.
$$

Now for $p_1 \neq p_2$ two distinct point in $S^1$, their [[complements]] constitute an [[open cover]]

$$
  \left\{ S^1 \setminus p_i \subset S^1  \right\}_{i \in \{1,2\}}
$$

and so this exhibits $p \colon \mathbb{R}^1 \to S^1$ as being covering spaces.

Now

1. $S^1$ is [[path-connected topological space|path-connected]] and [[locally path-connected topological space|locally path connected]] ([this example](LocallyPathConnectedCircle));

1. $\mathbb{R}^1$ is [[simply connected topological space|simply connected]] ([this example](fundamental+group#EuclideanSpaceFundamentalGroup)).

Therefore $p$ exhibits $\mathbb{R}^1$ as a universal covering space of $S^1$, by def. \ref{CoveringUniversal}.

=--

## Related entries

* [[fundamental theorem of covering spaces]]

* [homotopy of smooth paths relative to their endpoints](smooth+homotopy#HomotopyOfSmoothPathsRelativeToTheirEndpoints)

## References

See also the references at _[[covering space]]_.

A detailed treatment is available in

* [[Nicolas Bourbaki]], _Topologie Algébrique_, Chapitres 1 à 4, Springer (1998, 2016)  [doi:10.1007/978-3-662-49361-8](https://doi.org/10.1007/978-3-662-49361-8), ISBN 978-3-662-49361-8.

Original discussion of universal covering spaces in terms of equivalence classes of based paths:

* [[William S. Massey]], Chapter 5, Section 10 of: *Algebraic Topology: An Introduction*, Harcourt Brace & World 1967, reprinted in: Graduate Texts in Mathematics, Springer 1977 ([ISBN:978-0-387-90271-5](https://link.springer.com/book/9780387902715))

Formulation in [[cohesive homotopy theory]]

* [[Urs Schreiber]], [Def. 3.8.10](https://arxiv.org/pdf/1310.7930v1.pdf#page=356) in *[[schreiber:differential cohomology in a cohesive topos|Differential cohomology in a cohesive $\infty$-topos]]* (2013)

and explicitly so in [[cohesive homotopy type theory]]:

* {#CherubiniRijke20} [[Felix Cherubini]], [[Egbert Rijke]], Def. 8.5: *Modal Descent*, Mathematical Structures in Computer Science (2020)  ([arXiv:2003.09713](https://arxiv.org/abs/2003.09713), [doi:10.1017/S0960129520000201](https://doi.org/10.1017/S0960129520000201))

Discussion of [[equivariant universal covers]] in the generality of [[equivariant homotopy theory]]:

* [[Tammo tom Dieck]], Section I (10.13) of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

[[!redirects universal cover]]
[[!redirects universal covers]]

[[!redirects universal covering]]
[[!redirects universal coverings]]

[[!redirects universal covering spaces]]
[[!redirects universal cover of a topos]]
[[!redirects universal cover of a (2,1)-topos]]
[[!redirects universal cover of an (∞,1)-topos]]
[[!redirects universal cover of an (infinity,1)-topos]]
[[!redirects universal covering topos]]
[[!redirects universal covering (2,1)-topos]]
[[!redirects universal covering (∞,1)-topos]]
[[!redirects universal covering (infinity,1)-topos]]
[[!redirects universal covering groupoid]]


universal differential envelope

##Motivation##

$k$-linear [[derivations]] of a (not necessarily commutative) associative unital $k$-[[associative unital algebra|algebra]] $A$ with values in $A$-[[bimodules]] are often considered: these are $k$-linear maps $d: A\to M$ satisfying the Leibniz identity: $d(a b)=d(a) b+ a d(b)$. One would like to construct a universal construction of that kind.

##Definitions##

The triple $(A,M,d)$ is then called a __differential calculus__ over $A$, provided $M$ is in the $k$-linear span of all expressions of the form $a d(b_1) d(b_2)\cdots d(b_n)$ where $a,b_1,\ldots,b_n\in A$. Morphisms of differential calculi over $A$ are straightforward to define. A universal object in that category is the __universal differential calculus__ and $\Omega A = M$ in that case is called the __universal differential envelope__ of $A$. It is constructed as the quotient of the tensor algebra $T A$ modulo the ideal generated by the ("augmentation") kernel $I\subset A\otimes A$ of the multiplication map $A\otimes A\to A$; then $d a = 1\otimes a - a\otimes 1$.

+--{.query}
This is a bit confusing. The subbimodule $I$ should rather be the degree-1 component of $\Omega A$ (see Bourbaki, Algèbre III, paragraph Problème universel pour les dérivations: cas non commutatif).
—[[Benoit Jubin]]

[[Eric]]: I'm probably confused, but $A$ is degree-0 and $A\otimes A$ is degree-1.

[[Benoit Jubin]]: I agree. To me, we should have $\Omega A = A \oplus I \oplus \ldots$, whereas the article currently proposes $\Omega A = TA / \langle I \rangle$.
=--

The bimodule $\Omega A$ is in fact a $k$-algebra, generated by degree-$1$ elements (in $A$); differential $d:A\to \Omega A$ can be naturally extended along the inclusion $A\hookrightarrow \Omega A$ to a differential $d:\Omega A\to \Omega A$ satisfying the Leibniz rule, by setting $d 1 = 0$, $d (w y) =  d(w) y + (-1)^p w d(y)$, where $w,y$ are forms and $w\in\Omega^p A$ is homogeneous.

##References

* A. Connes, [Noncommutative Geometry](http://www.alainconnes.org/en/downloads.php)
* R. Coquereaux and D. Kastler, [Remarks on the differential envelopes of associative algebras](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102650385)
* F. Bonechi, R. Giachetti, R. Maciocco, E. Sorace, M.Tarlini, [Cohomological Properties of Differential Calculi on Hopf Algebras](http://arxiv.org/abs/q-alg/9612019)
* H. C. Baehr, A. Dimakis, F. Müller-Hoissen, [Differential Calculi on Commutative Algebras](http://arxiv.org/abs/hep-th/9412069)
* [[Folkert Müller-Hoissen]], [Introduction to Noncommutative Geometry of Commutative Algebras and Applications in Physics](http://wwwuser.gwdg.de/%7Efmuelle/download/mexico.ps)

##Discussion

[[Eric]]: I have some notes on [[ericforgy:differential envelope|my personal web]], but they need work before transferring them here.

[[!redirects universal differential envelopes]]


universal element

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Definition

A **universal element** of a [[functor]] $F: C \to Set$ is an element $\theta \in F(x)$, where $x$ is some [[object]] of $C$, which exhibits [[representable functor|representability]] of $F$ via the [[Yoneda lemma]]. That is, any element $\theta \in F(x)$ induces, in natural bijective fashion, a [[natural transformation]]

$$\hat{\theta}: \hom(x, -) \to F$$

$$\hat{\theta}_y(f: x \to y) \stackrel{def}{=} F(f)(\theta)$$

and $\theta$ is _universal_ if $\hat{\theta}$ is an isomorphism.

Thus, universal elements are part and parcel of any discussion involving representability. Well-known examples include [[adjoint functor]]s, where one has representability

$$\hom(F(c), -) \cong \hom(c, G-),$$

the [[Brown representability theorem]], and there are many others. A few more examples are discussed below.

## Examples

### Internal logic in toposes

Quite often, logical constructions that work for arbitrary [[topos|toposes]] can be deduced by arguing from universal elements. Some simple examples follow.

Consider first the construction of internal conjunction $\wedge: \Omega \times \Omega \to \Omega$.

### Colimits of nerves

A [question](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=1284) was brought to the nForum on [[colimit]]s of [[nerve]]s, conjecturing the following which we state as a proposition (proof below):

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

If $C$ is a [[small category]], then the [[colimit]] of the composite

$$C^{op} \stackrel{(- \downarrow C)}{\to} Cat \stackrel{Nerve}{\to} Set^{\Delta^{op}}$$

is equivalent to the [[nerve]] of $C$.

=--

As a warm-up:

+-- {: .un_lemma}
###### Lemma

The colimit of

$$C^{op} \stackrel{(- \downarrow C)}{\to} Cat$$

is [[isomorphic]] to $C$.

=--

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

The objects of $colim_{c: C^{op}} (c \downarrow C)$  are equivalence classes of arrows or [[generalized element|generalized elements]] $c \to d$ where the equivalence $\sim$ is generated by

$$(c \stackrel{f}{\to} d) \sim (c' \stackrel{g}{\to} c \stackrel{f}{\to} d)$$

and it is immediate that every generalized element $g: c \to d$ is equivalent to the universal generalized element $1_d: d \to d$; in spirit, this is the [[Yoneda lemma]] in disguise.

=--

Now we prove the proposition on colimits of nerves.

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

[[limits and colimits by example|Since]] colimits in $Set^{\Delta^{op}}$ ([[sSet]]) are computed pointwise, we just have to show the colimit of

$$C^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{nerve}{\to} Set^{\Delta^{op}} \stackrel{ev_n}{\to} Set,$$

where $ev_n$ is evaluation at an object $n$, agrees with $nerve(C)_n$. This is

$$C^{op} \stackrel{(c \downarrow C)}{\to} Cat \stackrel{\hom([n], -)}{\to} Set$$

Now an $n$-[[simplex]] in the [[comma category]] $(c \downarrow C)$, which is an element of this composite, is the same as an $(n+1)$-simplex beginning with the vertex $c$, and the colimit (in $Set$) consists of equivalence classes of $(n+1)$-simplices where a simplex beginning with $c$ is deemed equivalent to a simplex beginning with $c'$ obtained by pulling back along any $g: c' \to c$. And again, it is a triviality that each $(n+1)$-simplex

$$c \to (d_0 \to \ldots \to d_n)$$

is equivalent to

$$d_0 \stackrel{1_{d_0}}{\to} (d_0 \to \ldots \to d_n)$$

but the collection of such $d_0 \to \ldots \to d_n$ is the same as $nerve(C)_n$. This completes the verification.

=--

[[!redirects universal elements]]