Ehrenfeucht-Fraïssé games

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### Context
#### Model theory
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[[!include model theory - contents]]
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=--
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#Contents#
* table of contents
{:toc}

## Idea

**Ehrenfeucht-Fraïssé games** are a technique in [[model theory]] for
studying whether two [[structures in model theory|structures]] are [[elementary equivalence|elementary equivalent]]. Variations
can be used to study equivalence with respect to logics other than
[[first-order logic]].

The games are played by two players: the _spoiler_, whose objective is
to establish a difference between the two structures, and the
_duplicator_, whose objective is to show that the two structures are
similar.

A game is played in turns for a number of rounds (in general an
ordinal, but usually a finite number or $\omega$). In each round the
spoiler picks an element of one of the structures and then the
duplicator responds by picking an element of the other structure. At
the end of the game they have together selected two sequences of equal
length of elements of the two structures, and the duplicator wins if
the sequences generate isomorphic substructures of the given
structures. (See below for variations in the winning conditions.)

## Definition

We first define the **Ehrenfeucht-Fraïssé game of length $\gamma$ on
$\mathfrak{A}$ and $\mathfrak{B}$**,
$EF_\gamma(\mathfrak{A},\mathfrak{B})$. It is a game of perfect
information played as follows. The ordinal $\gamma$ is the length of
the game, and at step $i\lt\gamma$, the spoiler chooses an element of
either $\mathfrak{A}$ or $\mathfrak{B}$; then the duplicator chooses
an element of the other structure. Between them they choose $a_i\in A$
and $b_i\in B$. At the end of the game we have a **play** which is a
pair $(\bar a,\bar b)$ of sequences of length $\gamma$. We count this
play as a **win for the duplicator** if there is an isomorphism $f :
\langle\bar a\rangle_{\mathfrak{A}} \to \langle\bar
b\rangle_{\mathfrak{B}}$ such that $f(a_i)=b_i$ for $i\lt\gamma$. (The
notation $\langle\bar a\rangle_{\mathfrak{A}}$ denotes the
substructure of $\mathfrak{A}$ generated by the sequence $\bar a$.)

An alternative way of phrasing the winning condition is to say that
the play $(\bar a,\bar b)$ is a win for the duplicator if and only if
$(\mathfrak{A},\bar a) \equiv_0 (\mathfrak{B},\bar b)$, i.e., the
expansions of the two structures by the given sequences satisfy the
same atomic (or the same quantifier-free) sentences.

+-- {: .num_example #DefEF}
###### Definition
We write $\mathfrak{A} \sim_\gamma \mathfrak{B}$ to mean that the
duplicator has a winning strategy in the game
$EF_\gamma(\mathfrak{A},\mathfrak{B})$.
=--

It is easy to see that this is an equivalence relation on structures,
and $\mathfrak{A} \sim_\gamma \mathfrak{B}$ always holds when
$\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic.
Note the $EF_\gamma(\mathfrak{A},\mathfrak{B})$ is an infinite game if
$\gamma\ge\omega$.

## Back-and-forth systems

For the game $EF_\omega(\mathfrak{A},\mathfrak{B})$ we have the
following alternative characterization of $\mathfrak{A} \sim_\omega
\mathfrak{B}$.

A **back-and-forth system** from $\mathfrak{A}$ to $\mathfrak{B}$ is
an inhabited set $I$ of pairs $(\bar a,\bar b)$ of equal length
sequences from $\mathfrak{A}$ and $\mathfrak{B}$ such that

* for every $(\bar a,\bar b)\in I$ we have $(\mathfrak{A},\bar a)
  \equiv_0 (\mathfrak{B},\bar b)$, and

* for every $(\bar a,\bar b)\in I$ and every $a\in A$, there is a
  $b\in B$ such that $(\bar a a, \bar b b)\in I$, and

* for every $(\bar a,\bar b)\in I$ and every $b\in B$, there is an $a
  \in A$ such that $(\bar a a, \bar b b)\in I$.

+-- {: .num_example #PropBF}
###### Proposition
There is a back-and-forth system from $\mathfrak{A}$ to $\mathfrak{B}$
if and only if $\mathfrak{A} \sim_\omega \mathfrak{B}$.
=--

For this reason we say that $\mathfrak{A}$ and $\mathfrak{B}$ are
**back-and-forth equivalent** if and only if $\mathfrak{A} \sim_\omega
\mathfrak{B}$. (Other names for back-and-forth equivalence are partial isomorphism or potential isomorphism.)

+-- {: .num_example #ACFBF}
###### Example
Any two algebraically closed fields of equal characteristic and
infinite transcendence degree over their respective prime subfields
are back-and-forth equivalent. A back-and-forth system is in this case
given by the pairs of generating sequences for isomorphic finitely generated
subfields.
=--

+-- {: .num_example #ThmCountableBF}
###### Theorem
Suppose $\mathfrak{A}$ and $\mathfrak{B}$ are at most countable
back-and-forth equivalent structures. Then $\mathfrak{A}$ and
$\mathfrak{B}$ are isomorphic, and if the play $(\bar a,\bar b)$ is a
win for the duplicator in $EF_\omega(\mathfrak{A},\mathfrak{B})$, then
there is an isomorphism $f : \mathfrak{A} \to \mathfrak{B}$ with
$f(a_i)=b_i$ for $i\lt\omega$.
=--

+-- {: .num_example #CantorThm}
###### Corollary
**(Cantor's Theorem)**
Any two countable [[DLO|dense linear orders
without endpoints]] are isomorphic.  A back-and-forth system is given
by the pairs of finite sequences of isomorphic finite suborders.
=--

## Unnested Ehrenfeucht-Fraïssé games

One major application of Ehrenfeucht-Fraïssé games is to study
elementary equivalence. For languages without (non-constant) function
symbols, however, we need a slight modification to the above
definitions.

The **unnested Ehrenfeucht-Fraïssé game of length $\gamma$ on
$\mathfrak{A}$ and $\mathfrak{B}$**,
$EF_\gamma[\mathfrak{A},\mathfrak{B}]$ is played exactly like
$EF_\gamma(\mathfrak{A},\mathfrak{B})$, but a play $(\bar a,\bar b)$
counts as a **win for the duplicator** if and only $(\mathfrak{A},\bar
a)$ and $(\mathfrak{B},\bar b)$ satisfy the same **unnested** atomic
formulas.

Recall that an **unnested atomic formula** for a first-order language
of one of the following forms:

* $x = y$ for variables $x$ and $y$,

* $x = c$ for variable $x$ and constant symbol $c$,

* $x = f(\bar y)$ for variables $x$ and $\bar y$ and function symbol $f$,

* $R(\bar x)$ for variables $\bar x$ and relation symbol $R$.

Any atomic formula is equivalent to a $\Delta$-formula built from
unnested atomic formulas (but the quantifier rank grows with the size
of the involved terms).

+-- {: .num_example #DefEFunnested}
###### Definition
We write $\mathfrak{A} \approx_\gamma \mathfrak{B}$ to mean that the
duplicator has a winning strategy in the game
$EF_\gamma[\mathfrak{A},\mathfrak{B}]$.
=--

Note that $\mathfrak{A} \sim_\gamma \mathfrak{B}$ implies
$\mathfrak{A} \approx_\gamma \mathfrak{B}$ and the converse holds
whenever the language has no constant and function symbols (as in that
case every atomic formula is automatically unnested).

The key reason to introduce the unnested formulas and games is the
following theorem:

+-- {: .num_example #FraisseHintikka}
###### Theorem
**(Fraïssé-Hintikka theorem)**
Suppose the language $L$ has finitely many symbols. Then we can
effectively find for each $k,n\lt\omega$ a finite set $\Theta_{n,k}$
of unnested formulas in $n$ free variables of quantifier rank at most
$k$, such that

* for every $L$-structure $\mathfrak{A}$, all $k,n\lt\omega$, and all
  $n$-tuples $\bar a$ from $\mathfrak{A}$, there is exactly one
  formula $\theta$ in $\Theta_{n,k}$ with $(\mathfrak{A},\bar a)
  \vDash \theta$,

* for all $k,n\lt\omega$ and all $L$-structures $\mathfrak{A}$ and
  $\mathfrak{B}$ and corresponding $n$-tuples $\bar a$ and $\bar b$ we
  have $(\mathfrak{A},\bar a) \approx_k (\mathfrak{B},\bar b)$ if and
  only if there is $\theta\in\Theta_{n,k}$ satisfied by both
  $(\mathfrak{A},\bar a)$ and $(\mathfrak{B},\bar b)$,

* for all $k,n\lt\omega$ it holds that every unnested formula in $n$
  free variables of quantifier rank at most $k$ is equivalent to a
  disjunction of formulas in $\Theta_{n,k}$.
=--

This appears as Theorem 3.3.2 in [Hodges 93](#Hodges93).

+-- {: .num_example #FraisseTheorem}
###### Corollary
**(Fraïssé's theorem)** Two structures $\mathfrak{A}$ and
$\mathfrak{B}$ in a finite language are elementarily equivalent,
$\mathfrak{A} \equiv \mathfrak{B}$, if and only if for every
$k\lt\omega$, $\mathfrak{A} \approx_k \mathfrak{B}$.
=--

## Dynamic Ehrenfeucht-Fraïssé games

The condition occurring in Fraïssé's theorem, that for every
$k\lt\omega$, $\mathfrak{A} \approx_k \mathfrak{B}$, is nicely
captured by another variation of either the ordinary or the unnested
Ehrenfeucht-Fraïssé game. Namely, instead of taking an ordinal
$\gamma$ to be the length of the game, we can let an ordinal $\alpha$
be part of the playing field, so to speak, indicating in a sense how
many times the spoiler can change his mind about the length of the
game, yet keeping the game finite.

More precisely, we define the **dynamic Ehrenfeucht-Fraïssé game
measured by $\alpha$ on $\mathfrak{A}$ and $\mathfrak{B}$**,
$EFD_\alpha(\mathfrak{A},\mathfrak{B})$, to be a finite game of
perfect information, defined by induction on $\alpha$ as follows:

* for $\alpha=0$ there is a unique play (i.e., the players do nothing)
  that is a win for the duplicator if and only if $\mathfrak{A}
  \equiv_0 \mathfrak{B}$,

* for $\alpha=\beta+1$, a play consists of the spoiler picking an
  element of one structure, followed by the duplicator picking an
  element of the other structure, followed by a play of
  $EFD_\beta((\mathfrak{A},a),(\mathfrak{B},b))$, where $a\in A$ and
  $b\in B$ are the two elements picked, and this play in
  $EFD_\alpha(\mathfrak{A},\mathfrak{B})$ is a win for duplicator if
  and only if the play in
  $EFD_\beta((\mathfrak{A},a),(\mathfrak{B},b))$ is a win for
  duplicator,

* for $\alpha$ a limit ordinal, a play consists of the spoiler picking
  an ordinal $\beta\lt\alpha$, and then a continued play in
  $EFD_\beta(\mathfrak{A},\mathfrak{B})$, and this play in
  $EFD_\alpha(\mathfrak{A},\mathfrak{B})$ is a win for duplicator if
  and only if the play in
  $EFD_\beta(\mathfrak{A},\mathfrak{B})$ is a win for
  duplicator.

The game $EFD_\alpha[\mathfrak{A},\mathfrak{B}]$ is defined similarly,
using unnested atomic formulas in the base case.

+-- {: .num_example #DefEFD}
###### Definition
We write $\mathfrak{A} \sim^\alpha_\omega \mathfrak{B}$ to mean that
the duplicator has a winning strategy in the game
$EFD_\alpha(\mathfrak{A},\mathfrak{B})$ and $\mathfrak{A}
\approx^\alpha_\omega \mathfrak{B}$ to mean that the duplicator has a
winning strategy in the game $EFD_\alpha[\mathfrak{A},\mathfrak{B}]$.
=--

We use a subscript $\omega$ to indicate that the relations
${\sim^\alpha_\omega}$ and ${\approx^\alpha_\omega}$ form approximations
to ${\sim_\omega}$ and ${\approx_\omega}$, respectively.

## Back-and-forth sequences

There is an alternative characterization of the relations
${\sim^\alpha_\omega}$ in terms of **back-and-forth sequences**.

We define a **back-and-forth sequence of length $\alpha$ from
$\mathfrak{A}$ to $\mathfrak{B}$** to be a sequence
$(I_\beta)_{\beta\lt\alpha}$ of inhabited sets of pairs $(\bar a,\bar
b)$ of equal length sequences from $\mathfrak{A}$ and $\mathfrak{B}$
such that

* $I_\beta \subseteq I_\gamma$ for $\gamma\lt\beta\lt\alpha$,

* for every $\beta\lt\alpha$ and every $(\bar a,\bar b)\in I_\beta$ we
  have $(\mathfrak{A},\bar a) \equiv_0 (\mathfrak{B},\bar b)$, and

* for every $\beta+1\lt\alpha$ and every $(\bar a,\bar b)\in
  I_{\beta+1}$ and every $a\in A$, there is a $b\in B$ such that
  $(\bar a a, \bar b b)\in I_\beta$, and

* for every $\beta+1\lt\alpha$ and every $(\bar a,\bar b)\in
  I_{\beta+1}$ and every $b\in B$, there is an $a \in A$ such that
  $(\bar a a, \bar b b)\in I_\beta$.

+-- {: .num_example #PropBFD}
###### Proposition
There is a back-and-forth sequence of length $\alpha$ from
$\mathfrak{A}$ to $\mathfrak{B}$ if and only if $\mathfrak{A}
\sim^\alpha_\omega \mathfrak{B}$.
=--

Clearly, if $\mathfrak{A}$ and $\mathfrak{B}$ are back-and-forth
equivalent, then we can make a back-and-forth sequences of any length by
taking the constant sequence at a back-and-forth system.

## Use in infinitary logic

Recall that the formulas in the logic $L_{\infty\omega}$ can be built
using infinitary disjunctions and conjunctions. We write $\mathfrak{A}
\equiv_{\infty\omega} \mathfrak{B}$ (resp. $\mathfrak{A}
\equiv_{\infty\omega}^\alpha \mathfrak{B}$) to mean that
$\mathfrak{A}$ and $\mathfrak{B}$ satisfy the same sentences of
$L_{\infty\omega}$ (resp. of quantifier rank at most $\alpha$).

There is a close relationship between the expressivity of this
infinitary logic and the relations ${\sim_\omega}$ and
${\sim_\omega^\alpha}$, as the following theorem shows:

+-- {: .num_example #KarpTheorem}
###### Theorem
We have $\mathfrak{A} \equiv_{\infty\omega}^\alpha \mathfrak{B}$ if
and only if $\mathfrak{A} \sim_\omega^\alpha \mathfrak{B}$ and
$\mathfrak{A} \equiv_{\infty\omega} \mathfrak{B}$ if and only if
$\mathfrak{A} \sim_\omega \mathfrak{B}$.
=--

This is Theorem 4.47 and Proposition 7.48 in [Väänänen 2011](#Vaananen2011).

## Related concepts

* [[type in model theory]]
* [[Ehrenfeucht-Fraïssé comonad]]

## References

* {#Hodges93} [[Wilfrid Hodges]], _Model Theory_, Cambridge University Press 1993

* {#Vaananen2011} Jouko Väänänen, _Models and Games_, Cambridge University Press 2011

* Wikipedia, _[Ehrenfeucht–Fraïssé game](https://en.wikipedia.org/wiki/Ehrenfeucht%E2%80%93Fra%C3%AFss%C3%A9_game)_

[[!redirects Ehrenfeucht-Fraïssé game]]
[[!redirects Ehrenfeucht-Fraisse games]]
[[!redirects Ehrenfeucht-Fraisse game]]
[[!redirects back-and-forth games]]
[[!redirects back-and-forth game]]
[[!redirects back-and-forth argument]]


Ehresmann > history

< [[Ehresmann]]
[[!redirects Ehresmann -- history]]


Ehresmann connection

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###Context###
#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
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#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
=--
=--
=--

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

## Idea

The notion of _Ehresmann connection_ is one of the various equivalent definitions of [[connection on a bundle]].

## Definition

### In terms of differential forms {#InTermsDiffForms}

Let $G$ be a [[Lie group]] with [[Lie algebra]] $\mathfrak{g}$ and $P \to X$ a $G$-[[principal bundle]]. Let

$$
  \rho : P \times G \to P
$$

be the [[action]] of $G$ on $P$ and

$$
  \rho_* : \mathfrak{g} \to \Gamma(T P)
$$

its derivative, sending each element $x \in \mathfrak{g}$ to the [[vector field]] on $P$ that at $p \in P$ is the push-forward $\rho(p,-)_*(x)$.

For $v \in \Gamma(T P)$ and $\omega$ a differential form on $P$ write $\iota_v \omega$ for the contraction.

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

Given a $G$-[[principal bundle]] $P$ over $X$,  a **Cartan-Ehresmann connection** on $P$ is a [[Lie algebra-valued 1-form]]

$$
  A\in \Omega^1(P, \mathfrak{g})
$$

on the total space $P$ satisfying two conditions:

1. **first Ehresmann condition**

   for every $x \in \mathfrak{g}$ we have

   $$
     \iota_{\rho_*(x)} A = x
     \,.
   $$

2. **second Ehresmann condition**

   for every $x \in \mathfrak{g}$ we have

   $$
     \mathcal{L}_{\rho_*(x)} A = ad_x A
     \,,
   $$

   where $\mathcal{L}_{\rho_*(x)}$ is the [[Lie derivative]] along $\rho_*(x)$
   and where $ad_x : \mathfrak{g} \to \mathfrak{g}$ is the [[adjoint action]]
   of $\mathfrak{g}$ on itself.

=--

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

This is equivalent to

1. **first Ehresmann condition**

   for every $x \in \mathfrak{g}$ we have

   $$
     \iota_{\rho_*(x)} A = x
     \,.
   $$

2. **second Ehresmann condition**

   for every $x \in \mathfrak{g}$ we have

   $$
     \iota_{\rho_*(x)} F_A = 0
     \,,
   $$

   where $F_A \in \Omega^2(P, \mathfrak{g})$ is the [[curvature]] 2-form of
   $A$.

=--

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

Using $\iota_{\rho_*(x)} A = x$ we have by [[Cartan calculus]]

$$
  \begin{aligned}
     \iota_{\rho_*(x)} F_A &=
     \iota_{\rho_*(x)} ( d_{dR} A + \frac{1}{2} [A\wedge A] )
     \\
     & =
     \mathcal{L}_{\rho_*(x)} A - d_{dR} \iota_{\rho_*(x)} A +
     [x,A]
     \\
     & = \mathcal{L}_{\rho_*(x)} A + [x,A]
     \,.
  \end{aligned}
$$

=--

### In terms of distributions

Given a smooth bundle $\pi: E\to X$ with typical fiber $F$ (e.g. a smooth vector bundle or a smooth principal $G$-bundle), there is a well defined vector subbundle $V E\subset T E$ over $E$ such that $V_p$ consists of the  tangent vectors $v_p$ such that $(T_p \pi)(v_p) = 0$. A __smooth distribution (field) of horizontal subspaces__ is a choice of a vector subspace $H_p E\subset T_p E$ for every $p$ such that

E1. (complementarity) $T_u E \: = \: H_u E \oplus V_u E$

E2. $p\mapsto H_p E$ is smooth. That means that in the unique decomposition of any smooth vector field $X$ on $E$ into vector fields $X^H \in \Gamma(H_u E)$ and $X^V \in \Gamma(V_u E)$ such that $X = X^H + X^V$ the vector field $X^H$ is smooth (or equivalently $X^V$ is smooth, or equivalently both) as a section of $T E$ (there exist yet several other equivalent formulations of the smooothness criterion).

An Ehresmann connection describes a [[connection on a bundle|connection on]] a $G$-[[principal bundle]] $\pi : P \to X$ (for $G$ some [[Lie group]]) in terms of a distribution of horizontal subspaces $H \subset T P$ which is a subbundle of the [[tangent bundle]] of $P$ complementary at each point to the vertical tangent bundle to the fiber. More precisely, an __Ehresmann connection__ on a principal $G$-bundle $\pi:P\to X$ is a smooth distribution of horizontal subspaces $p\mapsto H_p P$ which is __equivariant__:

E3. $H_{p g}P = (T_p R_g) H_p P$ for every $p \in P$ and $g \in G$.

This subbundle $H = \cup_p H_p\subset T X$ over $X$ can be expressed as a field of subspaces $H_x = Ker A_x = Ann A_x\subset T P$ ($x\in P$) which are pointwise annihilators of a smooth [[Lie algebra]]-valued $1$-[[differential form|form]] $A \in \Omega^1(P,Lie(G))$ on $P$ that satisfies two Ehresmann conditions from the previous subsection.

The Ehresmann connections on a principal $G$-bundle are in 1-1 correspondence with an appropriate notion of a connection on the associated
bundle. Namely, if $T^H P\subset T P$ is the smooth horizontal distrubution of subspaces defining the principal connection on a principal $G$-bundle $P$ over $X$, where $G$ is a Lie group and $F$ a smooth left $G$-space, then consider the total space $P\times_G F$ of the associated bundle with typical fiber $F$. Then, for a fixed $f\in F$ one defines a map $\rho_f : P\to P\times_G F$ assigning the class $[p,f]$ to $p\in P$. If $(T_p \rho_f)(T^H_p P) =: T_{[p,f]}^H P\times_G F$ defines the horizontal subspace $T_{[p,f]}^H P\times_G F\subset T_{[p,f]} P\times_G F$, the collection of such subspaces does not depend on the choice of $(p,f)$ in the class $[p,f]$, and the correspondence $p\mapsto T_{[p,f]}^H P\times_G F$ is a connection on the associated bundle $P\times_G F\to X$.

### In terms of cohesive homotopy type theory

One may also describe(flat) Ehresmann connections in
[[cohesive homotopy type theory]].

The general abstract discussion is [here](cohesive+%28infinity%2C1%29-topos+--+structures#FlatEhresmannConnections). The discussion of how in [[smooth infinity-groupoids]] this reduces to the traditional notion is [here](smooth+infinity-groupoid+--+structures#FlatEhresmannConnections).

## Properties

### General

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

The two definitions in terms of 1-forms and in terms of horizontal distributions are equivalent.

=--

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

At each $p \in P$ take the horizontal subspace $H_p P$ to be the [[kernel]] of $A(p) : T_p P \to \mathfrak{g}$

$$
  H_p P := ker A(p)
$$

=--

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

This means we may think of $A$ as measuring how [[infinitesimal object|infinitesimal]] paths in $P$ fail to be _horizontal_ or _parallel_ to $X$ in the sense of [[parallel transport]].

=--

### Curvature characteristic forms

Let $\langle - \rangle \in W(\mathfrak{g})$  be an [[invariant polynomial]] on the Lie algebra. For $A \in \Omega^1(P,\mathfrak{g})$ an Ehresmann connection, write

$$
  \langle F_A \rangle = \langle F_A \wedge F_A \wedge \cdots F_A\rangle
$$

for the [[curvature characteristic form]] obtained by evaluating this on wedge powers of the [[curvature]] 2-form.

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

The forms $\langle F_A \rangle \in \Omega^{2k}(P)$ are closed,  descend along $p : P \to X$, in that they are pullbacks of forms along $p$, and their class in [[de Rham cohomology]] $H^{2k}(X)$ are independent of the choice of $A$ on $P$.

=--

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

That the forms are closed follows from the [[Bianchi identity]]

$$
  d F_A = [A\wedge F_A]
$$

satisfied by the curvature 2-form and the defining as-invariance of $\langle-\rangle$.  More abstractly, the 1-form $A$ itself may be identified with a morphism of [[dg-algebra]]s out of the [[Weil algebra]] $W(\mathfrak{g})$ (see there)

$$
  \Omega^\bullet(P) \leftarrow W(\mathfrak{g}) : A
$$

and the evaluation of the curvature in the invariant polynomials corresponds to the precomposition with the morphism

$$
  W(\mathfrak{g}) \leftarrow CE(b^{2k-1}\mathbb{R} ) : \langle - \rangle
$$

described at [[∞-Lie algebra cohomology]].

to show that these forms descend, it is sufficient  to show that for all $x \in \mathfrak{g}$ we have

1. $\iota_{\rho_*(X)} \langle F_A \rangle = 0$

1. $\mathcal{L}_{\rho_*(x)} \langle F_A \rangle = 0$

The first follows from $\iota_{\rho_*(x)} F_A = 0$. The second from this, the $d_{dR}$-closure just discussed and [[Cartan's magic formula]] for the [[Lie derivative]].

=--

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

The form $\langle F_A \rangle$ is called the [[curvature characteristic form]] of the connection $A$. The map

$$
  \inv(\mathfrak{g}) \to \Omega^\bullet(X)
$$

induced by $(P,A)$ as above is the [[Chern-Weil homomorphism]].

=--

## Note on terminology

The terminology for the various incarnations of the single notion of [[connection on a bundle]] varies throughout the literature. What we here call an Ehresmann connection is sometimes, but not always, called **principal connection** (as it is defined for [[principal bundle]]s).

## References

The notion originates with:

* [[Charles Ehresmann]], *Les connexions infinitésimales dans un espace fibré différentiable*,

   Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris (1951) 29–55

  [[Séminaire Bourbaki]], **1** 24 (1952) 153–168 [[numdam:SB_1948-1951__1__153_0](http://www.numdam.org/item/?id=SB_1948-1951__1__153_0), [pdf](http://www.numdam.org/article/SB_1948-1951__1__153_0.pdf)]

See also:

* [[Mikio Nakahara]], _[[Geometry, Topology and Physics]]_, IOP (2003) [[doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf]

A formulation and discussion of Ehresmann connections using language and tools from [[synthetic differential geometry]]:

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], section 6 of: _[[Models for Smooth Infinitesimal Analysis]]_

Generalization to [[principal 2-bundles]]:

* {#Waldorf16} [[Konrad Waldorf]], _A global perspective to connections on principal 2-bundles_ ([arXiv:1608.00401](https://arxiv.org/abs/1608.00401))

Generalization to [[connections on principal ∞-bundles]]:

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes]] -- An $\infty$-Lie theoretic construction_, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735), [Euclid](http://projecteuclid.org/euclid.atmp/1358950853))

A more comprehensive account is in sections 3.9.6, 3.9.7 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

[[!redirects Ehresmann-connection]]

[[!redirects Ehresmann connections]]

[[!redirects Cartan-Ehresmann connection]]
[[!redirects Cartan-Ehresmann connections]]


Ehresmann coring

Given a coalgebra-Galois extension $U\hookrightarrow E$ of a $k$-algebra $U$, which is the appropriate generalization of a [[Hopf-Galois extension]], where $E$ is faithfully flat over the base $U$ as a left $E$-module, one constructs a coring, __Ehresmann coring__, out of these data. Its role is somewhat analogous to the gauge groupoid (see [[Atiyah Lie groupoid]]), and in Hopf-Galois case it is an intermediate stage in constructing another analogue, (Ehresmann-)[[Schauenburg bialgebroid]], see there.

* [[T. Brzeziński]], R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series __309__, Cambridge 2003.


Ehresmann's theorem

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### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Geometry
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[[!include higher geometry - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

A theorem of [Ehresmann 1951](#Ehresmann51) states that a [[proper morphism|proper]] [[submersion]] of [[smooth manifolds]] $f : X \to Y$ is a [[locally trivial bundle|locally trivial]] [[fibration]] (cf. [Kolář, Slovák & Michor 1993, Lem. 92](#KolářSlovákMichor93); [Voisin 2002](#Voisin02), [Thm. 9.3](/nlab/files/VoisinOnEhresmannTheorem.jpg)).

## Relation to Gauss-Manin connections

The Ehresmann therem implies that the [[higher direct image]] $R^i f_\ast \underline{\mathbb{C}}$ of the [[constant sheaf]] $\underline{\mathbb{C}}$ on such $X$ are ($\mathbb{C}$-)[[local systems]] on $Y$.

> (In the [[algebraic geometry|algebraic category]], instead of the constant sheaf $\underline{\mathbb{C}}$ use the [[de Rham cohomology|de Rham complex]] $\Omega_X^\bullet$ and instead of the higher direct images take the [[hypercohomology|hyper]]-higher direct images.)

The [[Riemann-Hilbert correspondence|corresponding]] [[vector bundle]] then has a canonical [[flat connection|flat]] [[connection]], known as the *[[Gauss-Manin connection]]* (e.g. [Voisin 2002, Def. 9.13](#Voisin02)).

This is the typical setup one considers when studying variations of [[Hodge theory|Hodge]] structure.

## Related concepts

* [[Gauss-Manin connection]]

## References

The theorem is named after:

* {#Ehresmann51} [[Charles Ehresmann]], *Les connexions infinitésimales dans un espace fibré différentiable*,

   Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris (1951) 29–55

  [[Séminaire Bourbaki]], **1** 24 (1952) 153–168 [[numdam:SB_1948-1951__1__153_0](http://www.numdam.org/item/?id=SB_1948-1951__1__153_0), [pdf](http://www.numdam.org/article/SB_1948-1951__1__153_0.pdf)]

Discussion in the context of [[natural operations]] on [[natural bundles]]:

* {#KolářSlovákMichor93} [[Ivan Kolář]], [[Jan Slovák]], [[Peter Michor]], Lemma 9.2 in: *[[Natural operations in differential geometry]]*, Springer (1993) [[book webpage](https://www.emis.de/monographs/KSM/), [doi:10.1007/978-3-662-02950-3](https://link.springer.com/book/10.1007/978-3-662-02950-3), [pdf](http://www.emis.de/monographs/KSM/kmsbookh.pdf)]

Discussion in the context of [[Hodge theory]]:

* {#Voisin02} [[Claire Voisin]] (translated by [[Leila Schneps]]), [Thm. 9.3](/nlab/files/VoisinOnEhresmannTheorem.jpg) in _[[Hodge theory and Complex algebraic geometry]] I_, Cambridge Stud. in Adv. Math. **76** (2002) [[doi:10.1017/CBO9780511615344](https://doi.org/10.1017/CBO9780511615344), [[Gauss-Manin connection]] def: [[Voisin-DefinitionOfGaussManinConn.pdf:file]]]

See also

* Wikipedia, *[Ehresmann's lemma](https://en.wikipedia.org/wiki/Ehresmann%27s_lemma)*

[[!redirects Ehresmann's theorem]]
[[!redirects Ehresmann's theorem]]
[[!redirects Ehresmann theorem]]

[[!redirects Ehresmann's lemma]]
[[!redirects Ehresmann's lemma]]
[[!redirects Ehresmann lemma]]


Ehsan Hatefi

## Selected writings

* [[Luis Alvarez-Gaumé]], [[Ehsan Hatefi]], _Critical Collapse in the Axion-Dilaton System in Diverse Dimensions_, Class. Quant. Grav. 29 (2012) 025006 ([arXiv:1108.0078](https://arxiv.org/abs/1108.0078))

* S. de Alwis, Rajesh Gupta, E. Hatefi , F. Quevedo, _Stability, Tunneling and Flux Changing de Sitter Transitions in the Large Volume String Scenario_, JHEP 1311 (2013) 179 ([arXiv:1308.1222](https://arxiv.org/abs/1308.1222))

* [[Ehsan Hatefi]], [[Alexei Nurmagambetov]], I.Y. Park, _ADM reduction of IIB on $H^{p,q}$ to dS braneworld_, JHEP 1304 (2013) 170 ([arXiv:1210.3825](https://arxiv.org/abs/1210.3825))

category: people


Ehud Hrushovski

_Ehud Hrushovski_ is an Israeli mathematician, working mainly in logics and [[model theory]].

Cf. [[motivic integration]].

category: people


EI-category

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## Definition

An _EI-category_ is a [[category]] in which every [[endomorphism]] is an [[isomorphism]], hence an [[automorphism]].

Similarly an _EI $(\infty,1)$-category_ is an [[(∞,1)-category]] in which every [[endomorphism]] is an [[equivalence in an (∞,1)-category|equivalence]].

## Examples

* A [[poset]]

* The [[path category]] of an acyclic [[quiver]].

* A group, $G$, understood as [delooped](group#delooping) to a pointed connected [[groupoid]].

* A [[one-way category]], which is a category in which every endomorphism is an identity, is trivially an EI-category.

Let $\mathcal{S}$ be a set of [[subgroups]] of a [[group]] $G$. The following are all EI-categories ([Webb08, p. 4078](#Webb08)):

\begin{example}
The *transporter category* $\mathcal{T}_{\mathcal{S}}$ has as its objects the members of $\mathcal{S}$, and morphisms $Hom(H,K) = N_G(H,K) = \{g \in G|{}^{g}H \subseteq K\}$.
\end{example}

\begin{example}\label{OrbitCategory}
The [[orbit category]] $\mathcal{O}_{\mathcal{S}}$ associated to $\mathcal{S}$ in which the objects are the [[coset spaces]] $G/H$ where $H \in S$ and the morphisms are the $G$-[[equivariant functions]].
\end{example}

\begin{example}\label{FundamentalCategoryOfAGSpace}
 More generally: the [fundamental category of a $G$-space](orbit category#FundamentalCategoryOfAGSpace) is an EI-category.
\end{example}

\begin{example}
The *Frobenius category*  $\mathcal{F}_{\mathcal{S}}$ has the elements of $\mathcal{S}$ as its objects, and $Hom_{\mathcal{F}_{\mathcal{S}}} = N_G(H,K)/C_G(H)$. The morphisms may be identified with the set of group homomorphisms $H \to K$ which are of the form 'conjugation by $g$' for some $g \in G$.
\end{example}

## Properties

### General

EI-categories may be seen as those categories satisfying a  kind of [[Schröder–Bernstein theorem]].

\begin{proposition}
A category $C$ is EI if and only if every antiparallel pair $X \rightleftarrows Y$ exhibits a pair of isomorphisms.
\end{proposition}

\begin{proof}
Assume that $C$ is EI, and let $f \colon X \rightleftarrows Y : g$ be an antiparallel pair. Consider $X \xrightarrow{f} Y \xrightarrow{g} X \xrightarrow{f} Y$. Since isomorphisms have the [[2-out-of-6 property]], and $gf$ and $fg$ are isomorphisms, $f$ and $g$ are also isomorphisms. Conversely, suppose that $C$ satisfies the assumption of the proposition. Let $i \colon X \to X$ be an endomorphism. Then $i \colon X \rightleftarrows X : i$ exhibits an antiparallel pair, so in particular $i$ is an isomorphism.
\end{proof}

In particular, assuming [[excluded middle]], the [[Schröder–Bernstein theorem]] states that [[Inj]], the wide subcategory of [[Set]] spanned by [[monomorphisms]], is an EI-category.

### Partial ordering

Given an EI-category, $C$, the [[set]] of [[isomorphism classes]] $[x]$ of [[objects]] $x \in C$ forms a [[partially ordered set]] under the [[relation]]

$$
  [x] \leq [y]
  \phantom{AA} \text{if and only if}
  \phantom{AA}
  \text{there is a morphism } x \to y
$$

### Representation theory

A finite EI-category contains finitely many morphisms.

The [[category algebra]] $k C$ of a finite EI-category, $C$, for a fixed [[base ring]] $k$  and has as [[linear basis|basis]] the set of morphisms in $C$ with multiplication induced by composition of morphisms. It is thus a generalization of the [[group algebra]] of a [[finite group]], the [[path algebra]] of a finite [[quiver]] without oriented cycles or the [[incidence algebra]] of a finite poset. There is a stratification of $k C$ of depth equal to the number of isomorphism classes in the category.

The [[category of modules]] over the [[category algebra]] $k C$ is equivalent to the category of $k$-[[linear representations]] of $C$, i.e., the [[functor category]] $Fun(C, Mod k)$.

## Related concepts

* [[inverse EI (∞,1)-category]]

## References

Maybe the earliest explicit observation that in an [[orbit category]], and its relatives, *endomorphisms are automorphisms* is in:

* {#tomDieck1987} [[Tammo tom Dieck]], p. 73 of: *[[Transformation Groups]]*, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

Discussion in the context of [[algebraic K-theory]]:

* {#Lueck89} [[Wolfgang Lück]], _Transformation Groups and Algebraic K-Theory_, Lecture Notes in Mathematics **1408** (Springer 1989) ([doi:10.1007/BFb0083681](https://doi.org/10.1007/BFb0083681))

On the [[representation theory]] of EI-categories:

* {#Webb08} [[Peter Webb]], _Standard stratifications of EI categories and Alperin's weight conjecture_, Journal of Algebra
**320** 12 (2008) 4073-4091 ([doi:10.1016/j.jalgebra.2006.03.052](https://doi.org/10.1016/j.jalgebra.2006.03.052))

* Karsten Dietrich, _Representation Theory of EI-categories_, 2010 ([urn:nbn:de:hbz:466-20100701014](https://digital.ub.uni-paderborn.de/hsmig/content/titleinfo/1708), [pdf](https://d-nb.info/1007029781/34), [[Dietrich_EICategories.pdf:file]])

* Liping Li, *A generalized Koszul theory and its application*, Transactions of the American Mathematical Society **366** 2 (2014) 931-977 ([arXiv:1109.5760](https://arxiv.org/abs/1109.5760), [jstor:23812975](https://www.jstor.org/stable/23812975))

* Ergün Yalçın, *Projective resolutions over EI-categories*, 2012 ([hdl:11693/15472](http://hdl.handle.net/11693/15472))

[[!redirects EI-categories]]
[[!redirects EI category]]
[[!redirects EI categories]]

[[!redirects EI (∞,1)-categories]]
[[!redirects EI (∞,1)-category]]


eigenspace

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For $E : V \to V$ a [[linear operator]] on a finite dimensional [[vector space]] (which can be identified with a [[matrix]]), an _eigenspace_ of $E$ is a subspace of $V$ on which $E$ acts via multiplication by some fixed element in the ground [[field]], the corresponding _[[eigenvalue]]_.

[[!redirects eigenspaces]]


eigenstate thermalization hypothesis

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## Idea

A hypothesis about the emergence of [[thermodynamics]] in [[quantum many-body systems]].

## References

* [[Mark Srednicki]], *Chaos and Quantum Thermalization*, Phys Rev E 50 (1994) 888  ([arXiv:cond-mat/9403051v2](https://arxiv.org/abs/cond-mat/9403051v2))

See also

* Wikipedia, *[Eigenstate thermalization hypothesis](https://en.wikipedia.org/wiki/Eigenstate_thermalization_hypothesis)*

Relation to [[quantum error correcting codes]] obtained from [[holographic tensor networks]]:

* Ning Bao, Newton Cheng, *Eigenstate Thermalization Hypothesis and Approximate Quantum Error Correction*, JHEP 08 (2019) 152 ([arXiv:1906.03669](https://arxiv.org/abs/1906.03669))


eigenvalue

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## Definition

Given a [[linear map|linear]] [[endomorphism]], an _[[eigenvector]]_ of it is a [[vector]] which is taken by the linear map to a multiple of itself. This multiple is called the _eigenvalue_ of the eigenvector.

## Properties

* [[Gershgorin circle theorem]]

* [[normal eigenvalue]]

## References

See also:

* Wikipedia, *[Eigenvalues and eigenvectors](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors)*

[[!redirects eigenvalues]]


eigenvariety

# Contents
* table of contents
{: toc}

## Idea

An _eigenvariety_ is a [[rigid analytic space]] that parametrizes p-adic families of systems of Hecke eigenvalues. Historically the first example is the _eigencurve_ as constructed by Coleman and Mazur.

## Construction of the Eigencurve

Let $T_{\ell}$ and $S_{\ell}$ be the Hecke operators corresponding to the double cosets $\GL_{\mathbb{Z}_{\ell}}\begin{pmatrix}\ell & 0 \\ 0 & 1\end{pmatrix}\GL_{\mathbb{Z}_{\ell}}$ and $\GL_{\mathbb{Z}_{\ell}}\begin{pmatrix}\ell & 0 \\ 0 & \ell\end{pmatrix}\GL_{\mathbb{Z}_{\ell}}$ respectively. Let $\mathcal{M}_{k}(N)$ be the space of modular forms of weight $k$ and level $\Gamma_{1}(N)$. Let $\mathbb{T}_{\leq k}^{(p)}$ be the ring of endomorphisms of $\bigoplus_{i=1}^{k}\mathcal{M}_{i}(N)$ generated by the Hecke operators $T_{\ell}$ and $S_{\ell}$ as $\ell$ ranges over all primes not dividing $N p$.

\begin{definition}([Emerton09](#Emerton09), Definition 2.5)

The _p-adic Hecke algebra_ is the inverse limit

$$\mathbb{T}:=\underset
    {\underset{k}{\leftarrow}}
    {\lim}
  \;\mathbb{Z}_{p}\otimes_{\mathbb{Z}}\mathbb{T}_{\leq k}^{(p)}
$$

\end{definition}

\begin{definition}([Emerton09](#Emerton09), Definition 2.24)

Let $\mathcal{X}$ be the set of $\overline{\mathbb{Q}}_{p}$-valued points in $\Spec\mathbb{T}\times\mathbb{G}_{m}$ consisting of pairs $(\xi,\alpha)$ where $\xi:\mathbb{T}\to \overline{\mathbb{Z}}_{p}$ is classical, corresponding to some system of Hecke eigenvalues $\mathbb{T}_{k}\to \overline{\mathbb{Z}}_{p}$ and $\alpha$ is a root of the $p$-th Hecke polynomial. The _eigencurve_ is the rigid analytic Zariski closure of $\mathcal{X}$ in $(\Spec\mathbb{T}\times\mathbb{G}_{m})^{\an}$.

\end{definition}

## Related concepts

* [[p-adic modular form]]

* [[Hida theory]]

## References

* {#Emerton09} Matthew Emerton, _p-adic Families of Modular Forms_, 2009 ([pdf](http://www.math.uchicago.edu/~emerton/pdffiles/padic.pdf))


eigenvector

## Definition

Given a [[linear map|linear]] [[endomorphism]], an _eigenvector_ of it is a [[vector]] which is taken by the linear map to a multiple of itself. This multiple is called the _[[eigenvalue]]_ of the eigenvector.

It is sometimes required that an eigenvector is not the zero vector.

If the linear map acts as a [[differential operator]] on a space of functions, its eigenvectors are sometimes called **eigenfunctions**.

[[!redirects eigenvectors]]
[[!redirects eigenfunction]]
[[!redirects eigenfunctions]]

[[!redirects eigenstate]]
[[!redirects eigenstates]]


Eigil F. Rischel

* [personal page](https://erischel.com/)

* [GoogleScholar page](https://scholar.google.com/citations?user=qx0Sp-oAAAAJ&hl=en)

* [underline.io page](https://underline.io/speakers/119563-eigil-f-rischel)

## Selected writings

On [[categorical systems theory]] ([[cybernetics]]):

* [[Matteo Capucci]], [[Bruno Gavranović]], [[Jules Hedges]], [[Eigil F. Rischel]], *Towards Foundations of Categorical Cybernetics*, EPTCS **372** (2022) 235-248 [[arXiv:2105.06332](https://arxiv.org/abs/2105.06332), [doi:10.4204/EPTCS.372.17](https://doi.org/10.4204/EPTCS.372.17)]

category: people

[[!redirects Eigil Rischel]]


Eilenberg subcomplex

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## Definition

For $X$ a [[simplicial set]], for $x : \Delta[0] \to X$ a point in $X$, and for $n \in \mathbb{N}$, the **$n$th Eilenberg subcomplex** $E_n(X,x)$ of $X$ at $x$ is the [[fiber]] of the $(n-1)$-[[coskeleton]]-projection over $x$, hence the [[pullback]]

$$
  \array{
    E_n(X,x) &\to& X
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{x}{\to}& cosk_{n-1}X
  }
  \,.
$$

By the [[skeleton]]/[[coskeleton]] [[adjunction]] $(sk_{n-1} \dashv cosk_{n-1})$ the $n$th Eilenberg subcomplex is the [[subobject]] of $X$ consisting of those simplices whose $(n-1)$-skeleton is constant on the point $x$.

## Properties

### Restriction to Kan complexes

If $X$ is a [[Kan complex]] , then so is $E_n(X,x)$ for all $n \in \mathbb{N}$ and $x \in X_0$.

### Relation to $n$-connected objects

If $X$ is a Kan complex and [[n-connected|(n-1)-connected]], then the canonical morphism $E_n(X,x) \to X$ is a [[homotopy equivalence]].

See ([May, theorem 8.4](#May)).

### Relation to pointed $n$-connected objects

The inclusion $sSet_{(n-1)} \hookrightarrow sSet^{*/}$ of $n$-fold [[reduced simplicial set]]  (those with a single $k$-simplex for all $k \leq n-1$) into all [[pointed object|pointed]] simplicial sets is a [[coreflective subcategory]] with coreflector being forming of the $n$th Eilenberg subcomplex

$$
  sSet^{*/}
    \underoverset
      {\underset{E_n(-,*)}{\longrightarrow}}
      {\overset{}{\hookleftarrow}}
      {\bot}
  sSet_{n-1}
  \,.
$$

the [[counit of an adjunction|counit]] of this adjunction is the defining inclusion $E_n(X,*) \to X$.

So if $(* \to X) \in sSet^{*/}$ such that $X \in sSet$ is a [[Kan complex]] and [[n-connected|(n-1)-connected]], then the [[counit of an adjunction|counit]] $E_n(X,*) \to X$   is a [[homotopy equivalence]].

Accordingly, the coreflection presents the inclusion of [[n-connected|(n-1)-connected]] [[pointed object|pointed]] [[infinity-groupoids]] into all [[pointed object|pointed]] [[infinity-groupoids]]

$$
  \infty Grpd_{\geq (n-1)}^{*/} \hookrightarrow \infty Grpd^{*/}
  \,.
$$

## Related concepts

* [[Whitehead tower]], [[Postnikov tower]]

## References

Around def. 8.3 in

* [[Peter May]], _Simplicial objects in algebraic topology_ ([djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu))
 {#May}

[[!redirects Eilenberg subcomplexes]]


Eilenberg swindle

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## Idea

In [[homological algebra|homological]] [[algebra]], what came to be known [[Bass (1963)](#Bass63)] as the *Eilenberg swindle* (in honor of [[Samuel Eilenberg]]) is an argument (originally due to [Mazur (1961)](#Mazur61)) proving that the [[Grothendieck group]] (the 0th [[algebraic K-theory]] group $K_0$) of many [[abelian categories]] is [[trivial group|trivial]].

The content of the "swindle" is essentially the following [Hilbert hotel](https://en.m.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel)-type argument:

In an [[abelian category]] $A$ with [[countable set|countable]] [[direct sums]], we have for any [[object]]  $X \in A$ an [[isomorphism]]

$$
  X
    \oplus   \bigoplus_{i=1}^{\infty}
  X
  \,\simeq\,  \bigoplus_{i=1}^{\infty}
  X
  \,,
$$

which implies $[X]=0$ in $K_0(A)$.

This is the reason that, for instance, one has to restrict oneself to categories of *[[finitely generated module|finitely generated]]* ([[projective modules|projective]]) [[modules]] (which lack infinite direct sums) in defining (nontrivial) [[algebraic K-theory]] groups $K_0$ of a [[ring]].

## References

The earliest known appearance of Eilenberg's swindle in writing is in Lemma 3 of

* {#Mazur61} [[Barry C. Mazur]], _On embeddings of spheres_,  Acta Mathematica **105** 1–2 (1961) 1–17   [[doi:10.1007/BF02559532](https://doi.org/10.1007%2FBF02559532)]

The earliest attribution to [[Samuel Eilenberg]] as well as the use of the term “swindle” is in §1 of

* {#Bass63} [[Hyman Bass]], _Big projective modules are free_, Illinois Journal of Mathematics **7** (1963) 24–31  [[doi:10.1215/ijm/1255637479](https://doi.org/10.1215%2Fijm%2F1255637479)]


Eilenberg-Mac Lane object

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## Idea

The notion of _Eilenberg-Mac Lane object_ in an [[(∞,1)-topos]] or [[stable (∞,1)-category]] generalizes the notion of [[Eilenberg-Mac Lane space]] from the [[(∞,1)-topos]] [[Top]] of [[topological space]]s or the [[stable (∞,1)-category]] of [[spectrum|spectra]]:

it is an object $\mathbf{B}^n A$ obtained from an abelian [[group object]] $A$ by [[delooping]] that $n$ times.

An object that is both $n$-[[truncated]] as well as $n$-connected.

## Definition

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

Let $\mathbf{H}$ be an [[(∞,1)-topos]].

For $n \in \mathbb{N}$ an **Eilenberg-MacLane object** $X$ of degree $n$

* a [[pointed object]] $* \to X \in \mathbf{H}$

* which is both $n$-[[n-connective|connective]] as well as $n$-[[truncated]].

=--

This appears as [[Higher Topos Theory|HTT, def. 7.2.2.1]]

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

If one drops the condition that $X$ has a global point, then this is the definition of [[∞-gerbe]]s.

=--

## Properties

The next proposition asserts that Eilenberg-MacLane objects defined this way are shifted [[group object in an (∞,1)-category|(∞,1)-categorical group objects]]:

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

For $\mathbf{H}$ an [[(∞,1)-topos]], $\mathbf{H}_*$ its [[(∞,1)-category]] of [[pointed object]]s, $Disc(\mathbf{H})$ the full sub-[[(∞,1)-category]] on discrete objects (0-[[truncated]] objects) and $n \in \mathbb{N}$, write

$$
  \pi_n : \mathbf{H}_* \to Disc(\mathbf{H}_*)
$$

for the [[(∞,1)-functor]] that assigns the $n$-th [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]].

* For $n = 0$ this establishes an equivalence between the full subcategory on degree 0 Eilenberg-MacLane objects and [[pointed object]]s of $Disc(\mathbf{H})$; moreover, the restriction $\pi_0:Disc(\mathbf{H}_*)\to Disc(\mathbf{H}_*)$ is equivalent to the identity.

* For $n = 1$ this establishes an equivalence between the full subcategory on degree 1 Eilenberg-MacLane objects and the category of [[group object in an (∞,1)-category|group objects]] in $Disc(\mathbf{H})$.

* For $n \geq 2$ this establishes an equivalence between the full subcategory on degree $n$ Eilenberg-MacLane objects and the category of commutative [[group object in an (∞,1)-category|group objects]] in $Disc(\mathbf{H})$.

=--

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

This is [[Higher Topos Theory|HTT, prop. 7.2.2.12]].

=--

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

For $\mathbf{H}$ an [[(∞,1)-topos]] and $n \in \mathbb{N}$ write $K(-,n)$ for [[generalized the|the]] homotopy inverse to the equivalence induced by $\pi_n$ by the above proposition. For $A \in Disc(\mathbf{H})$ an (abelian) group object we say that

$$
  K(A,n) \in \mathbf{H}
$$

is the degree $n$-Eilenberg-MacLane object of $A$.

=--

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

We have that

$$
  K(A,n) \simeq \mathbf{B}^n A
$$

is [[generalized the|the]] $n$-fold [[delooping]] of the discrete group object $A$.

=--

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

> check

The [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]] are defined in terms of the canonical [[power]]ing of $\mathbf{H}$ over [[∞Grpd]]

$$
  (-)^{(-)} : \infty Grpd \times \mathbf{H} \to \mathbf{H}
  \,.
$$

For fixed [[∞-groupoid]] $K$ this

$$
  (-)^{K} : \mathbf{H} \to \mathbf{H}
  \,.
$$

preserves $(\infty,1)$-[[limit]]s and hence [[pullback]]s. It follows that the categorical homotopy groups of the [[loop space object]] $\Omega K(A,n)$ are those of $K(A,n)$, shifted down by one degree.

By the above proposition on the equivalence between Eilenberg-MacLane objects and group objects, this identifies $\Omega K(A,n) \simeq K(A,n-1)$.

=--

## Examples

### In $Top$

In the archetypical [[(∞,1)-topos]] [[Top]]$\simeq$ [[∞Grpd]] the notion of Eilenberg-MacLane object reduces to the traditional notion of [[Eilenberg-MacLane space]].

### In $(\infty,1)$-sheaf $(\infty,1)$-toposes

Recall that an [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaf]]/[[∞-stack]] [[(∞,1)-topos]] $\mathbf{H} = Sh_{(\infty,1)}(C)$ may be [[presentable (∞,1)-category|presented]] by the [[model structure on simplicial sheaves]] on $C$.

In terms of this model the Eilenberg-Mac Lane objects $K(A,n) \in \mathbf{H}$ (for abelian $A$) are the **Eilenberg-MacLane [[sheaf|sheaves]]** of [[abelian sheaf cohomology]] theory.

Under the [[Dold?Kan correspondence]]

$$
  N : sAb \stackrel{\leftarrow}{\to} Ch_+ : \Gamma
$$

[[chain complex]]es $A[n]$ of abelian groups concentrated in degree $n$ map into [[simplicial set]]s

$$
  K(A,n) := \Gamma(A[n])
$$

and these to the corresponding constant simplicial sheaves on the [[site]] $C$, that we denote by the same symbol, for convenience.

Under the equivalence

$$
  \mathbf{H} = Sh_{(\infty,1)}(C) \simeq (sSh(C)_{loc})^\circ
$$

of $\mathbf{H}$ with the [[Kan complex]]-enriched full subcategory of $sSh(C)$ on fibrant cofibrant objects, this identifies the fibrant reeplacement -- the [[∞-stackification]] -- of $\Gamma(A[n])$ with the Eilenberg-MacLane object in $\mathbf{H}$.

## Cohomology

The notion of [[cohomology]] in the [[(∞,1)-topos]] $\mathbf{H}$ with coefficients in an object $\mathcal{A} \in \mathbf{H}$ is often taken to be restricted to the case where $\mathcal{A}$ is an Eilenberg-MacLane object.

For $A \in Disc(\mathbf{A})$ an abelian group object, and $n \in \mathbb{N}$, the degree $n$-cohomology of an object $ X \in \mathbf{H}$ is the cohomology with coefficients in $K(A,n)$:

$$
  H^n(X, A) := H(X, K(A,n)) := \pi_0 \mathbf{H}(X, K(A,n))
  \,.
$$

## References

The general discussion of Eilenberg-MacLane objects is in section 7.2.2 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

For a discussion of Eilenberg-MacLane objects in the context of the [[model structure on simplicial presheaves]] see top of page 4 of

* [[Rick Jardine]], _Fields Lectures: Simplicial Presheaves_ ([pdf](http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf))

Discussion in [[equivariant homotopy theory]] (see also at _[[Bredon cohomology]]_) is in

* {Lewis92} L. G. Lewis, _Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen suspension theorems_, Topology Appl., 48 (1992), no. 1, pp. 25–61.

Formalization in [[homotopy type theory]] is in

* {#LicataFinster14} [[Dan Licata]], [[Eric Finster]], _Eilenberg-MacLane spaces in homotopy type theory_, LICS 2014 ([pdf text](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf), [Agda HoTT code](https://github.com/dlicata335/hott-agda/blob/master/homotopy/KGn.agda), [web discussion](http://homotopytypetheory.org/2014/04/15/eilenberg-maclane-spaces-in-hott/))

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Eilenberg-Mac Lane space

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

#Contents#
* table of contents
{:toc}

## Idea

An _Eilenberg--Mac Lane space_ is a connected [[topological space]] with nontrivial [[homotopy groups]] only in a single degree.

## Definition

### Characterization

For $n \in \mathbb{N}$ and $G$ a group, and an [[abelian group]] if $n \geq 2$, then an _Eilenberg-MacLane space_ $K(G,n)$ is a [[topological space]] with the property that all its [[homotopy groups]] are trivial, except that in degree $n$, which is $G$.

### Constructions

#### Via geometric realization of higher groupoids

For $G$ a [[group]], the _Eilenberg--Mac Lane space_ $K(G,1)$ is the image under the [[homotopy hypothesis]] [[Quillen equivalence]] $|-| : \infty Grpd \to Top$ of the one-object groupoid $\mathbf{B}G$ whose [[hom-set]] is $G$:

$$
  K(G,1) = | \mathbf{B} G |
  \,.
$$

The construction of EM-spaces $K(A,n)$ for an [[abelian group]] $A$ may be given by the [[Dold-Kan correspondence]] between [[chain complexes]] and [[simplicial abelian groups]]: let $A[-n]$ be the chain complex which is $A$ in dimension $n$ and zero elsewhere; the geometric realisation of the corresponding simplicial abelian group is then a $K(A,n)$.

We can include the case $n=1$ when $G$ may be nonabelian, by regarding $C(G,n)$ as a [[crossed complex]]. Its [[classifying space]] $B(C(G,n))$ is then a $K(G,n)$. (This also includes the case $n=0$ when $G$ is just a set!) This method also allows for the construction of $K(M,n;G,1)$ where $G$ is a group, or groupoid,  and $M$ is a $G$-module. This gives a space with $\pi_1 =G$, $\pi_n=M$ all other homotopy trivial, and with the given operation of $\pi_1$ on $\pi_n$.

For $A$ an abelian group, the Eilenberg--Mac Lane space $K(A,n)$ is the image of the [[∞-groupoid]] $\mathbf{B}^n A$ that is the [[strict ∞-groupoid]] given by the [[crossed complex]] $[\mathbf{B}^n A]$ that is trivial everywhere except in degree $n$, where it is $A$:

$$
  \begin{aligned}
    [\mathbf{B}^n A] &=
     (  \cdots \to [\mathbf{B}^n A]_{n+1} \to [\mathbf{B}^n A]_{n} \to
     [\mathbf{B}^n A]_{n-1} \cdots \to [\mathbf{B}^n A]_{1} \stackrel{\to}{\to} [\mathbf{B}^n A]_{0})
      \\ &=
    (  \cdots \to {*} \to A \to
    {*} \cdots \to {*} \stackrel{\to}{\to} {*})
  \end{aligned}
  \,.
$$

So

$$
  K(A,n) = |\mathbf{B}^n A|
  \,.
$$

Therefore Eilenberg--Mac Lane spaces constitute a [[spectrum]]: the [[Eilenberg-Mac Lane spectrum]].

In general, if $A$ is an abelian [[topological group]], then there exist a model for the [[classifying space]] $\mathcal{B}A$ which is an abelian topological group. Iterating this construction, one has a notion of $\mathcal{B}^n A$ and a model for it which is an abelian topological group. If moreover $A$ is discrete, then $\mathcal{B}A=|\mathbf{B}A|=K(A,1)$, and one inductively sees that $\mathcal{B}^n A=|\mathbf{B}^n A|=K(A,n)$. Therefore one has a model for $K(A,n)$ which is an abelian topological group.

See for instance ([May, chapter 16, section 5](#May))

#### Via linearization of spheres

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

For $A$ an [[abelian group]] and $n \in \mathbb{N}$, the **reduced $A$-linearization** $A[S^n]_\ast$ of the [[n-sphere]] $S^n$ is the [[topological space]], whose underlying set is the [[quotient]]

$$
  \underset{k \in \mathbb{N}}{\sqcup}
   A^k \times (S^n)^k
  \longrightarrow
  A[S^n]_\ast
$$

of the [[tensor product]]  with $A$ of the [[free abelian group]] on the underlying set of $S^n$, by the relation that identifies every [[formal linear combination]] of the (any fixed) basepoint of $S^n$ with 0. The [[topological space|topology]] is the induced [[quotient topology]] (of the [[disjoint union]] of [[product topological spaces]], where $A$ is equipped with the [[discrete topology]]).

=--

([Aguilar-Gitler-Prieto 02, def. 6.4.20](#AguilarGitlerPrieto02))

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

For $A$ a [[countable set|countable]] [[abelian group]], then the reduced $A$-linearization $A[S^n]_\ast$ (def. \ref{ReducedALinearizationOfnSphere}) is an [[Eilenberg-MacLane space]], in that its [[homotopy groups]] are

$$
  \pi_q(A[S^n]_\ast)
   \simeq
   \left\{
     \array{
       A & if \; q = n
       \\
       \ast & otherwise
     }
   \right.
$$

(in particular for $n \geq 1$ then there is a unique connected component and hence we need not specify a basepoint for the homotopy group).

=--

([Aguilar-Gitler-Prieto 02, corollary 6.4.23](#AguilarGitlerPrieto02))

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

The topological space $A[S^n]_\ast$ in definition \ref{ReducedALinearizationOfnSphere} has a canonical continuous [[action]] of the [[orthogonal group]] $O(n)$, by regarding $S^n \simeq (\mathbb{R}^n)^\ast$ as the [[one-point compactification]] of [[Cartesian space]]. Hence, in view of prop. \ref{ReducedALinearizationOfnSphereIsEMSpace}, these models for EM-spaces lend themselves to the defintiion of [[Eilenberg-MacLane spectra]] as [[orthogonal spectra]]. See [there](Eilenberg-Mac+Lane+spectrum#AsSymmetricSpectra).

=--

## Cohomology

### With coefficients being EM-spaces

One common use of Eilenberg--Mac Lane spaces is as coefficient objects for "ordinary" [[cohomology]] (see e.g. [May, chapter 22](#May)).

The $n$th "ordinary" [[cohomology]] of a [[topological space]] $X$ with coefficients in $G$ (when $n=1$) or $A$ (generally) is the collection of [[homotopy]] classes of maps from $X$ into $K(G,1)$ or $K(A,n)$, respectively:

$$
  H^1(X,G) = Ho_{Top}(X, K(G,1)) = Ho_{\infty Grpd}(X, \mathbf{B} G)
$$

$$
  H^n(X,A) = Ho_{Top}(X, K(A,n)) = Ho_{\infty Grpd}(X, \mathbf{B}^n A)
  \,.
$$

Here on the right $Ho_{Top}$ and $Ho_{\infty Grpd}$ denotes the [[homotopy category]] of the [[(∞,1)-categories]] of [[topological spaces]] and of [[∞-groupoids]], respectively.

Not only the set $\pi_0\mathbf{Top}(X, K(A,n))=Ho_{Top}(X, K(A,n))$ is related to the cohomology of $X$ with coefficients in $A$, but also the higher homotopy groups $\pi_i\mathbf{Top}(X, K(A,n))$ are, and in the most obvious way: if $X$ is a connected CW-complex, then
$$
H^{n-i}(X,A)=\pi_i\mathbf{Top}(X, K(A,n))=\pi_i\mathbf{\infty Grpd}(X, \mathbf{B}^n A),
$$
for any choice of base point on the right hand sides. This fact, which appears to have first been remarked by Thom and Federer, is an immediate consequence of the natural homotopy equivalences
$$
\Omega\mathbf{H}(X,Y)\simeq \mathbf{H}(X,\Omega Y)
$$
and
$$
\Omega K(A,n)\simeq K(A,n-1)
$$
one has in every $(\infty,1)$-[[(infinity,1)-topos|topos]], see [[loop space object]]. For $G$ a nonabelian group, Gottlieb proves the following nonabelian analogue of the above result: let $X$ be a finite dimensional connected CW-complex; for a fixed map $f:X\to K(G,1)$, let $C_f$ be the centralizer in $G=\pi_1 K(G,1)$ of $f_*(\pi_1(X))$. Then the connected component of $f$ in $\mathbf{Top}(X,K(G,1))$ is a $K(C_f,1)$.

Notice that for $G$ a nonabelian group, $H^1(X,G)$ is a simple (and the most familiar) example of [[nonabelian cohomology]]. Nonabelian cohomology in higher degrees is obtained by replacing here the coefficient $\infty$-groupoids of the simple form $\mathbf{B}^n A$ with more general $\infty$-groupoids.

### Of EM spaces
 {#CohomologyOfEMSpaces}

On the other hand there is the [[cohomology]] of Eilenberg-MacLane spaces itself. This is in general rich. Classical results by [[Serre]] and [[Henri Cartan]] are reviewed in ([Clement 02, section  2](#Clement02)).

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

For all even $n \in \mathbb{N}$, the [[ordinary cohomology|ordinary]] [[cohomology ring]] of $K(\mathbb{Z},n)$ with [[coefficients]] in the [[rational numbers]] is the [[polynomial algebra]] on the generator $a \in H^n(K(\mathbb{Z},n),\mathbb{Q}) \simeq \mathbb{Q}$. For all odd $n$ it is the [[exterior algebra]] on this generator:

$$
  H^\bullet(K(\mathbb{Z},n),\mathbb{Q})
  \simeq
  \left\{
    \array{
      \mathbb{Q}[a] & n \; even
      \\
      \mathbb{Q}[a]/(a^2) & n \; odd
    }
  \right.
  \,.
$$

=--

This is reviewed for instance in ([Yin, section 4](#Yin)).

In p.130 of [Eilenberg & MacLane (1954)](#EilenbergMacLane54II), some low-degree cohomology groups of $K(A,n)$ are computed.

+-- {: .num_prop #RationalCohomologyOfKZn}
###### Proposition
There are stable isomorphisms

$$
H^n(K(A,n),G) \cong \text{Hom}(A,G)
$$
$$
H^{m+1}(K(A,m),G)\cong \text{Extabel}(A,G)\subset \text{Ext}(A,G)\cong H^2(K(A,1),G)
$$
for $n\geq 1$ and $m\geq 2$.

=--

+-- {: .num_prop #RationalCohomologyOfKZn}
###### Proposition
There exists an isomorphism
$$
H^4(K(A,2),G)\cong \text{Hom}(\Gamma_4(A),G)
$$
where $\Gamma_4(A)$ is the group of quadratic functions from $A$ valued in $G$.
=--

## Generalizations

The notion of [[Eilenberg?Mac Lane object]] makes sense in every $(\infty,1)$-[[(infinity,1)-topos|topos]], not just in $L_{whe}$[[Top]]. See at _[[Eilenberg-MacLane object]]_.

## Related concepts

* [[generalized Eilenberg-MacLane space]]

* [[Kan-Thurston theorem]]

* [[Moore space]]

* [[Peterson space]]

* [[Eilenberg-MacLane space type]]

* [[Eilenberg-MacLane spectrum]]

## References

### General

Eilenberg-MacLane spaces originate with:

* {#EilenbergMacLane53I} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _On the Groups $H(\Pi,n)$, I_, Annals of Mathematics Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 55-106 ([jstor:1969820](https://www.jstor.org/stable/1969820))

* {#EilenbergMacLane54II} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _On the Groups $H(\Pi,n)$, II: Methods of Computation_, Annals of Mathematics Second Series, Vol. 60, No. 1 (Jul., 1954), pp. 49-139 ([jstor:1969702](https://www.jstor.org/stable/1969702))

* {#EilenbergMacLane54III} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _On the Groups $H(\Pi,n)$, III: Operations and Obstructions_, Annals of Mathematics Second Series, Vol. 60, No. 3 (Nov., 1954), pp. 513-557 ([jstor:1969849](https://www.jstor.org/stable/1969849))

That Eilenberg-MacLane spaces represent [[ordinary cohomology]] is due to:

* [[Samuel Eilenberg]], p. 243 of: _Cohomology and Continuous Mappings_, Annals of Mathematics Second Series, Vol. 41, No. 1 (Jan., 1940), pp. 231-251  ([jstor:1968828](https://www.jstor.org/stable/1968828))

* [Eilenberg-MacLane 54 III, p. 520-521](#EilenbergMacLane54III)

Early review is in

* [[Norman Steenrod]], Section 19 of: _Cohomology operations, and obstructions to extending continuous functions_,  Advances in Math. 8, 371–416. (1972)  (doi:10.1016/0001-8708(72)90004-7, [pdf](http://www.maths.ed.ac.uk/%7Eaar/papers/steen6.pdf))

Textbook accounts:

* {#AguilarGitlerPrieto02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf), [doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586))

  > (EM-spaces are constructed in section 6, the cohomology theory they represent is discussed in section 7.1, and its equivalence to singular cohomology is Corollary 12.1.20)

* {#May} [[Peter May]], Chapter 22 of: _A concise course in algebraic topology_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/maybook.pdf))

Lecture notes:

* {#Kobin16} Andrew Kobin, Section 7.6 of: _Algebraic Topology_, 2016 ([[KobinAT2016.pdf:file]])

The construction via reduced linearization of spheres is considered in

* {#Schwede12} [[Stefan Schwede]], Example I.1.14 in _Symmetric spectra_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec.pdf))

Quick review includes

* {#Yin} Xi Yin, _On Eilenberg-MacLane spaces_ ([pdf](http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf))

Formalization of Eilenberg-MacLane spaces in [[homotopy type theory]] (cf. *[[Eilenberg-MacLane space type]]*):

* {#LicataFinster14} [[Dan Licata]], [[Eric Finster]], _Eilenberg-MacLane spaces in homotopy type theory_, CSL-LICS '14 **66** (2014) 1-9 [[pdf text](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf), [Agda HoTT code](https://github.com/dlicata335/hott-agda/blob/master/homotopy/KGn.agda), [web discussion](http://homotopytypetheory.org/2014/04/15/eilenberg-maclane-spaces-in-hott/), [doi:10.1145/2603088.2603153](https://doi.org/10.1145/2603088.2603153)]

* {#vanDoorn18} [[Floris van Doorn]], §42 in: *On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory* (2018) [[arXiv:1808.10690](https://arxiv.org/abs/1808.10690)]

* [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

* [[David Wärn]], *Eilenberg-MacLane spaces and stabilisation in homotopy type theory*, J. Homotopy Relat. Struct. **18**  (2023) 357–368 [[arXiv:2301.03685](https://arxiv.org/abs/2301.03685), [doi:10.1007/s40062-023-00330-5](https://doi.org/10.1007/s40062-023-00330-5)]

### Cohomology

The [[ordinary cohomology]] of Eilenberg-MacLane spaces is discussed in

* {#Clement02} Alain Clément, _Integral Cohomology of Finite Postnikov Towers_, 2002 ([pdf](http://doc.rero.ch/record/482/files/Clement_these.pdf))

The [[topological K-theory]] of EM-spaces is discussed in

* {#AndersonHodgkin68} D.W. Anderson, Luke Hodgkin _The K-theory of Eilenberg-Maclane complexes_, Topology, Volume 7, Issue 3, August 1968, Pages 317-329 ([doi:10.1016/0040-9383(68)90009-8](https://doi.org/10.1016/0040-9383(68)90009-8))

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Eilenberg-Mac Lane space > history

< [[Eilenberg-Mac Lane space]]


Eilenberg-Mac Lane spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

For $A$ an [[abelian group]], the _Eilenberg-Mac Lane spectrum_ $H A$ is the [[spectrum]] that [[Brown representability theorem|represents]] the [[ordinary cohomology|ordinary]] [[cohomology theory]]/[[ordinary homology]]  with [[coefficients]] in $A$.

By default $A$ is taken to be the [[integers]] and hence "the Eilenberg-MacLane spectrum" is $H \mathbb{Z}$, representing [[integral cohomology]].

## Definition

## Incarnations

### As symmetric/orthogonal spectra
 {#AsSymmetricSpectra}

We discuss the model of Eilenberg-MacLane spectra as [[symmetric spectra]] and [[orthogonal spectra]]. To that end, notice the following model for [[Eilenberg-MacLane spaces]].

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

For $A$ an [[abelian group]] and $n \in \mathbb{N}$, the **reduced $A$-linearization** $A[S^n]_\ast$ of the [[n-sphere]] $S^n$ is the [[topological space]], whose underlying set is the [[quotient]]

$$
  \underset{k \in \mathbb{N}}{\sqcup}
   A^k \times (S^n)^k
  \longrightarrow
  A[S^n]_\ast
$$

of the [[tensor product]]  with $A$ of the [[free abelian group]] on the underlying set of $S^n$, by the relation that identifies every [[formal linear combination]] of the (any fixed) basepoint of $S^n$ with 0. The [[topological space|topology]] is the induced [[quotient topology]] (of the [[disjoint union]] of [[product topological spaces]], where $A$ is equipped with the [[discrete topology]]).

=--

([Aguilar-Gitler-Prieto 02, def. 6.4.20](Eilenberg-MacLane+space#AguilarGitlerPrieto02))

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

For $A$ a [[countable set|countable]] [[abelian group]], then the reduced $A$-linearization $A[S^n]_\ast$ (def. \ref{ReducedALinearizationOfnSphere}) is an [[Eilenberg-MacLane space]], in that its [[homotopy groups]] are

$$
  \pi_q(A[S^n]_\ast)
   \simeq
   \left\{
     \array{
       A & if \; q = n
       \\
       \ast & otherwise
     }
   \right.
$$

(in particular for $n \geq 1$ then there is a unique connected component and hence we need not specify a basepoint for the homotopy group).

=--

([Aguilar-Gitler-Prieto 02, corollary 6.4.23](Eilenberg-MacLane+space#AguilarGitlerPrieto02))

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

For $A$ a [[countable set|countable]] [[abelian group]], then the [[orthogonal spectrum]] incarnation of the **[[Eilenberg-MacLane spectrum]]** $H A$ is the [[orthogonal spectrum]] with

* components spaces

  $$
    (H A)_V \coloneqq A[S^V]_\ast
  $$

  being the reduced $A$-linearization (def. \ref{ReducedALinearizationOfnSphere}) of the [[representation sphere]] $S^V$;

  hence for $V = \mathbb{R}^n$ then

  $$
    (H A)_n = A[S^n]_\ast
  $$

* $O(V)$-[[action]] on $A[S^V]_\ast$ induced from the canonical $O(V)$-action on $S^V$ ([[representation sphere]]);

* structure maps

  $$
    \sigma_{V,W}
      \;\colon\;
    (H A)_V \wedge S^W
      \longrightarrow
    (H A)_{V\oplus W}
  $$

  hence

  $$
    A[S^V] \wedge S^W
      \longrightarrow
    A[S^{V \oplus W}]
  $$

  given by

  $$
    \left(
    \left(
       \underset{i}{\sum} a_i x_i
    \right),
    y
    \right)
      \mapsto
    \underset{i}{\sum} a_i (x_i, y)
    \,.
  $$

The incarnation of $H A$ as a [[symmetric spectrum]] is the same, with the group action of $O(n)$ replaced by the [[subgroup]] action of the [[symmetric group]] $\Sigma(n) \hookrightarrow O(n)$.

If $R$ is a [[commutative ring]], then the Eilenberg-MacLane spectrum $H R$ becomes a commutative [[orthogonal ring spectrum]] (or [[symmetric ring spectrum]], respectively) by

1. taking the multiplication

   $$
     (H R)_{V_1} \wedge (H R)_{V_2}
       =
     R[S^{V_1}]_\ast \wedge R[S^{V_2}]_\ast
       \longrightarrow
     R[S^{V_1 \oplus V_2}]
      =
     (H R)_{V_1 \oplus V_2}
   $$

   to be given by

   $$
     \left(
     \left(
        \underset{i}{\sum} a_i x_i
     \right)
     ,
     \left(
       \underset{j}{\sum} b_j y_j
     \right)
     \right)
      \;\mapsto\;
     \underset{i,j}{\sum}
      (a_i \cdot b_j)(x_i, y_j)
   $$

1. taking the unit maps

   $$
     S^V
       \longrightarrow
     A[S^V]_\ast = (H R)_V
   $$

   to be given by the canonical inclusion of generators

   $$
     x \mapsto 1 x
     \,.
   $$

=--

([Schwede 12, example I.1.14](#Schwede12), [Schwede 15, V, costruction 3.21](#Schwede15))

### As symmetric monoidal $\infty$-groupoids

Under the identification of [[connective spectra]] with "[[abelian infinity-groups]]" the Eilenberg-MacLane spectrum $H A$ simply _is_ the group $A$.

Here the set $A$ is regarded as a [[discrete category|discrete groupoid]] (one object per integer, no nontrivial morphisms) whose symmetric monoidal structure is that given by the additive group structure on the integers.

Accordingly, the infinite tower of [[suspensions]] induced by this is the sequence of [[∞-groupoids]]

$$
  A, \mathbf{B} A, \mathbf{B}^2 A, \mathbf{B}^3 A, \cdots
$$

that in this case happen to be [[strict omega-groupoids]]. The [[strict omega-groupoid]] $\mathbf{B}^n A$ has only [[identity]] $k$-morphisms for all $k$, except for $k = n$, where $\mathrm{Mor}_n(\mathbf{B}^n A) = A$ are the endomorphisms of the unique identity $(n-1)$-morphism.

The [[strict ∞-groupoid]] $\mathbf{B}^n A$ is the one given under the [[Dold-Kan correspondence]] by the [[crossed complex]] of groupoids that is trivial everywhere and has the group $\mathbb{Z}$ in degree $n$.

$$
  \begin{aligned}
    [\mathbf{B}^n A] &=
     (  \cdots \to [\mathbf{B}^n A]_{n+1} \to [\mathbf{B}^n A]_{n} \to
     [\mathbf{B}^n A]_{n-1} \cdots \to [\mathbf{B}^n A]_{1} \stackrel{\to}{\to} [\mathbf{B}^n A]_{0})
      \\ &=
    (  \cdots \to {*} \to A \to
    {*} \cdots \to {*} \stackrel{\to}{\to} {*})
  \end{aligned}
  \,.
$$

Under the  [[Quillen equivalence]]

$$
  |-| : \infty Grpds \to Top
$$

between [[infinity-groupoids]] and [[topological spaces]] (see _[[homotopy hypothesis]]_) this sequence of suspensions of $A$ maps to the sequence of [[Eilenberg?Mac Lane spaces]]

$$
  |\mathbf{B}^n A|
  \simeq
  K(A, n)
  \,.
$$

## Properties

### Chromatic filtration

[[!include chromatic tower examples - table]]

### Ordinary homology spectra split

**[[ordinary homology spectra split]]**:
For $S$ any [[spectrum]] and $H A$ an Eilenberg-MacLane spectrum,
then the [[smash product]] $S\wedge H A$ (the $A$-[[ordinary homology]] spectrum)
is non-canonically equivalent to a product of EM-spectra (hence a [[wedge sum]] of EM-spectra in the finite case).

### Fibrant-cofibrant models

[MO comment](http://mathoverflow.net/a/218069/381)

## Related concepts

* [[generalized Eilenberg-MacLane spectrum]]

* [[Moore spectrum]]

* [[cohomology theory]], [[ordinary cohomology]]

## References

Textbook accounts include

* {#Adams74} [[Frank Adams]], part III, section 2 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Kochmann96} [[Stanley Kochmann]], section 3.5 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Lecture notes include

* {#May} [[Peter May]], chapter 22 of _A concise course in algebraic topology_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/maybook.pdf))

* {#Rognes12} [[John Rognes]], section 3.2 3.4 of _The Adams spectral sequence_, 2012 ([pdf](http://folk.uio.no/rognes/papers/notes.050612.pdf))

As [[symmetric spectra]]

* {#Schwede12} [[Stefan Schwede]], Example I.1.14 in _Symmetric spectra_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec.pdf))

* [[John Greenlees]], example 3.6, example 4.16 in _Spectra for commutative algebraists_ ([pdf](http://www.math.uic.edu/~bshipley/greenlees.SpectraMSRI.pdf))

as [[orthogonal spectra]]:

* {#Schwede15} [[Stefan Schwede]], construction V 3.21 in _[[Global homotopy theory]]_, 2015

[[!redirects Eilenberg-MacLane spectrum]]
[[!redirects Eilenberg--MacLane spectrum]]
[[!redirects Eilenberg--Mac Lane spectrum]]
[[!redirects Eilenberg?MacLane spectrum]]
[[!redirects Eilenberg?Mac Lane spectrum]]
[[!redirects Eilenberg-MacLane spectra]]
[[!redirects Eilenberg-Mac Lane spectra]]
[[!redirects Eilenberg--MacLane spectra]]
[[!redirects Eilenberg--Mac Lane spectra]]
[[!redirects Eilenberg?MacLane spectra]]
[[!redirects Eilenberg?Mac Lane spectra]]

[[!redirects HR]]

[[!redirects HA]]
[[!redirects HC]]
[[!redirects HZ]]


Eilenberg-MacLane space type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##
Let $G$ be an [[abelian group]], and $n$ a [[natural number]]. The [[Eilenberg-MacLane space]] $K(G,n)$ is the unique space that has $\pi_n(K(G,n))\cong G$ and its other [[homotopy groups]] trivial. These spaces are a basic tool in classical algebraic topology, they can be used to define [[cohomology]].

There is an analogous construction in [[homotopy type theory]].

## Definition ##
Let $G$ be a group. The **Eilenberg-MacLane space type** $K(G,1)$ is the [[higher inductive type|higher inductive 1-type]] with the following constructors:

 * $base : K(G,1)$
 * $loop : G \to (base = base)$
 * $id   : loop(e) = refl_{base}$
 * $comp : \prod_{(x,y:G)} loop(x \cdot y) = loop(y) \circ loop(x)$

$base$ is a point of $K(G,1)$, $loop$ is a function that constructs a [[path]] from $base$ to $base$ for each element of $G$. $id$ says the path constructed from the identity element is the [[identity type|trivial loop]]. Finally, $comp$ says that the path constructed from group multiplication of elements is the concatenation of paths.

It should be noted that this type is 1-[[truncated]] which could be added as another constructor:

 * $\displaystyle \prod_{x,y:K(G,1)} \prod_{p,q:x=y} \prod_{r,s:p=q} r = s$

### Recursion principle ###

To define a function $f : K(G,1) \to C$ for some type $C$, it suffices to give

 * a point $c : C$
 * a family of loops $l : G \to (c = c)$
 * a path $l(e)=id$
 * a path $l(x \cdot y) = l(y) \circ l(x)$
 * a proof that $C$ is a 1-type.

Then $f$ satisfies the equations

$$f(base) \equiv c$$
$$ap_{f} (loop(x))=l(x)$$

It should be noted that the above can be compressed, to specify a function $f : K(G,1) \to C$, it suffices to give:

 * a proof that $C$ is a 1-type
 * a point $c : C$
 * a group homomorphism from $G$ to $\Omega(C,c)$

## See also

* [[higher inductive type]]

* [[Eilenberg-MacLane space]]

* [[cohomology in homotopy type theory]]

## References

* {#LicataFinster14} [[Dan Licata]], [[Eric Finster]], _Eilenberg-MacLane spaces in homotopy type theory_, CSL-LICS '14 **66** (2014) 1-9 [[pdf text](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf), [Agda HoTT code](https://github.com/dlicata335/hott-agda/blob/master/homotopy/KGn.agda), [web discussion](http://homotopytypetheory.org/2014/04/15/eilenberg-maclane-spaces-in-hott/), [doi:10.1145/2603088.2603153](https://doi.org/10.1145/2603088.2603153)]

* {#vanDoorn18} [[Floris van Doorn]], §42 in: *On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory* (2018) [[arXiv:1808.10690](https://arxiv.org/abs/1808.10690)]

* [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

* [[David Wärn]], *Eilenberg-MacLane spaces and stabilisation in homotopy type theory*, J. Homotopy Relat. Struct. **18**  (2023) 357–368 [[arXiv:2301.03685](https://arxiv.org/abs/2301.03685), [doi:10.1007/s40062-023-00330-5](https://doi.org/10.1007/s40062-023-00330-5)]

Formal proof that the $K(G,-)$ form a [[spectrum]]:

* [github.com/HoTT/HoTT-Agda/blob/master/homotopy/EilenbergMacLane.agda](https://github.com/HoTT/HoTT-Agda/blob/master/homotopy/EilenbergMacLane.agda)

[[!redirects Eilenberg-MacLane space types]]

[[!redirects Eilenberg-MacLane type]]
[[!redirects Eilenberg-MacLane types]]


* [[Dan Licata]], [[Eric Finster]]:

  **Eilenberg-MacLane Spaces in Homotopy Type Theory**

  CSL-LICS '14 **66** (2014) 1-9

  [doi:10.1145/2603088.2603153](https://doi.org/10.1145/2603088.2603153)

  [Agda code](https://github.com/dlicata335/hott-agda/blob/master/homotopy/KGn.agda)

  [pdf from Dan's homepage](http://dlicata.web.wesleyan.edu/pubs/lf14em/lf14em.pdf),

  [pdf from Eric's homepage](http://ericfinster.github.io/files/emhott.pdf),

on [[Eilenberg-MacLane spaces]] in [[homotopy type theory]].

> **Abstract.** [[homotopy type theory|Homotopy type theory]] is an extension of [[Martin-Löf type theory]] with principles inspired by [[category theory]] and [[homotopy theory]]. With these extensions, type theory can be used to construct [[proofs]] of [[homotopy theory|homotopy-theoretic]] [[theorems]], in a way that is very amenable to computer-checked proofs in [[proof assistants]] such as [[Coq]] and [[Agda]]. In this paper, we give a computer-checked construction of [[Eilenberg-MacLane spaces]]. For an [[abelian group]] $G$, an Eilenberg-MacLane space $K(G,n)$ is a space ([[type]]) whose $n^{th}$ [[homotopy group]] is $G$, and whose homotopy groups are trivial otherwise. These spaces are a basic tool in [[algebraic topology]]; for example, they can be used to build spaces with specified homotopy groups, and to define the notion of [[ordinary cohomology|cohomology]] with [[coefficients]] in $G$. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far.

## See also ##

 * [[homotopy type theory - references]]

 * [[Eilenberg-Mac Lane space]]

 * [[synthetic homotopy theory]]

category: reference


Eilenberg-Moore category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The category of [[algebras over a monad]] (also: "modules over a monad") is traditionally called its _Eilenberg--Moore category_ (EM). Dually, the EM category of a [[comonad]] is its category of coalgebras (co-modules).

The [[subcategory]] of [[free functor|(co-)free]] (co-)algebras is traditionally called the _[[Kleisli category]]_ of the (co-)monad.

The EM and Kleisli categories have universal properties which make sense for (co-)monads in any [[2-category]] (not necessarily [[Cat]]).

## Definition
 {#Definition}

Let $(T,\eta,\mu)$ be a [[monad]] in [[Cat]], where $T \colon C\to C$ is an [[endofunctor]] with multiplication $\mu \colon T T\to T$ and unit $\eta \colon Id_C\to T$.

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

A (left) $T$-[[algebra for a monad|module]] (or $T$-algebra) in $C$ is a pair $(A,\nu)$ of an object $A$ in $C$ and a morphism $\nu\colon T(A)\to A$ which is a __$T$-[[action]]__, in that

$$
  \nu\circ T(\nu)=\nu\circ\mu_{A} \colon T(T(A))\to A
$$

and

$$
  \nu\circ\eta_A = id_A
  \,.
$$

A [[homomorphism]] of $T$-modules $f\colon (A,\nu^A)\to (B,\nu^B)$ is a morphism $f\colon A \to B$ in $C$ that commutes with the action, in that

$$
  f\circ\nu^A=\nu^B\circ T(f)\colon T(A)\to B
  \,.
$$

The composition of morphisms of $T$-modules is the composition of underlying morphisms in $C$. The resulting [[category]] $C^T$ of $T$-modules/algebras is called the __Eilenberg--Moore category__ of the monad $T$, also be written $Alg(T)$, or $T\,Alg$, etc.

By construction, there is a [[forgetful functor]]

$$
  U^T \colon C^T \to C
$$

(which may be thought of as the [[universal property|universal]] $T$-module) with a [[left adjoint]] [[free functor]] $F^T$ such that the monad $U^T F^T$ arising from the adjunction is isomorphic to $T$.

=--
### Eilenberg--Moore object

More generally, for $t \colon a \to a$ is a monad in any [[2-category]] $K$, then the __Eilenberg--Moore object__ $a^t$ of $t$ is, if it exists, the universal (left) $t$-module.  That is, there is a "forgetful" 1-cell $u^t \colon a^t \to a$ and a 2-cell $\beta \colon t u^t \Rightarrow u^t$ that mediate a natural isomorphism $K(x, a^t) \cong LMod(x,t)$ between morphisms $h \colon x \to a^t$ and $t$-modules $(m \colon x \to a, \lambda \colon t m \Rightarrow m)$.  Not every 2-category admits Eilenberg--Moore objects.

\begin{center}
\begin{tikzcd}
x
\arrow[rrr, "m", bend left=60, "" {name=A, xshift=6ex, below} ]
\arrow[rr, "m", bend left ]
\arrow[r, dashed, "h"']
&a^t
\arrow[rr, "u^t"', bend right=80, "" {name=B, above} ]
\arrow[r, "u^t"']
&a
\arrow[Rightarrow, "\lambda", shorten >= 0.8ex, shorten <=0.4ex, to=A]
\arrow[Rightarrow, "\beta"', shorten >= 0.4ex, shorten <=0.4ex,  to=B]
\arrow[r, "t"']
& a
\end{tikzcd}
\end{center}

### (Non)example

Let [[Rel|REL]] be the (locally posetal) 2-category of sets and relations. A monad on a set $X$ is just an endorelation $R$ satisfying $id_X\subseteq R$ (reflexivity) and $R\circ R \subseteq R$ (transitivity) i.e. a [[preorder]] on $X$. When $R$ arises from an adjunction it is necessarily of the form $R=F\circ F^{op}$ implying $R=R^{op}$ (symmetry) since adjunctions in $REL$ consist of functional relations as the left adjoint with their opposite relation as right adjoint. In other words, only [[equivalence relation|equivalence relations]] admit Eilenberg--Moore objects which then consist of the set $X^R$ of equivalence classes with "free algebra functor" $F^R$ relating $x\in X$ to its equivalence class $[x]\in X^R$.

## Properties

### Universal properties

Apart from being the universal left $T$-module, the EM category of a monad $T$ in $Cat$ has some other interesting properties.

There is a [[full subcategory]] $RAdj(C)$ of the [[slice category]] $Cat/C$ on the functors $X \to C$ that have [[left adjoints]].  For any monad $T$ on $C$ there is a full subcategory of this consisting of the adjoint pairs that compose to give $T$.  The functor $U^T \colon C^T \to C$ is the [[terminal object]] of this category.

### As a colimit completion of the Kleisli category
 {#AsColimitCompletionOfKleisliCategory}

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

Every $T$-algebra $(A,\nu)$ is the [[coequalizer]] of the first stage of its [[bar resolution]]:

$$
  (T^2 A, \mu_{T A})
  \stackrel{\overset{\mu_A}{\longrightarrow}}{\underset{T \nu}{\longrightarrow}}
  (T A, \mu_A)
  \stackrel{\nu}{\longrightarrow}
  (A,\nu)
  \,.
$$

This is a [[reflexive coequalizer]] of $T$-algebras. Moreover, the underlying [[fork]] in $C$ is a [[split coequalizer]], hence in particular an [[absolute coequalizer]] (sometimes called the _Beck coequalizer_, due to its role in the [[Beck monadicity theorem]]). A splitting is given by

$$
  T^2 A  \stackrel{\eta_{T A}}{\longleftarrow} T A \stackrel{\eta_A}{\longleftarrow} A
\,.
$$

=--

(e.g. [MacLane, bottom of p. 148 and exercise 4 on p. 151](#MacLane)) See also at [split coequalizer -- Beck coequalizer for algebras over a monad](split+coequalizer#BeckCoequalizerForAlgebrasOverAMonad).

In particular this says that every $T$-algebra is [[generators and relations|presented]] by free $T$-algebras. The nature of $T$-algebras as a kind of completion of free $T$-algebras under colimits is made more explicit as follows.

Write $C_T$ for the [[Kleisli category]] of $T$, the category of [[free construction|free]] $T$-algebras. Write $F_T \colon C \to C_T$ the [[free functor]].
Observe that via the inclusion $C_T \hookrightarrow C^T$ every $T$-algebra [[representable functor|represents]] a [[presheaf]] on $C_T$. Recall that the [[category of presheaves]] $[C_T^{op}, Set]$ is the [[free cocompletion]] of $C_T$.

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

The $T$-algebras in $C$ are equivalently those presheaves on the category of free $T$-algebras whose restriction along the free functor is [[representable functor|representable]] in $C$. In other words, the Eilenberg--Moore category $C^T$ is the (1-category theoretic) [[pullback]]

\begin{center}
 \begin{tikzcd}
 \mathcal C^T
 \arrow[d, ""']
 \arrow[r, ""]
\arrow[dr, phantom,  , very near start, "\lrcorner"]
 & {[\mathcal C_T^{op}, \mathbf{Set}]}
 \arrow[d, "{[F_T^{op}, \mathbf{Set}]}"]
 \\
 \mathcal C
 \arrow[r, "Y"']
 &
 {[\mathcal C^{op}, \mathbf{Set}]}
  \end{tikzcd}
\end{center}

of the [[category of presheaves]] on the [[Kleisli category]] along the [[Yoneda embedding]] $Y$ of $C$. (The top arrow is given by a functor isomorphic to the nerve of the inclusion of the Kleisli category into the Eilenberg--Moore category.)

=--

This statement appears as [(Linton 69, Observation 1.1)](#Linton69) (cf. [(Street 72, Theorem 14)](#Street72)). [(Street-Walters 78)](#StreetWalters78) show that it holds in any 2-category equipped with a [[Yoneda structure]].

+-- {: .proof}
###### Sketch of proof
It is easy to see that the square commutes. To see that it is a pullback, assume that $P:C_T^{op}\to Set$ is a presheaf on the Kleisli category and $A$ is an object of $C$ such that $YA=P\circ F_T^{op}$. Then a $T$-algebra structure $\alpha:TA\to A$ on $A$
is given by $\alpha=P(1_{TA})(1_A)$, where $1_{TA}$ is viewed as a [[Kleisli category#in_terms_of_kleisli_morphisms|Kleisli morphism]] from $TA$ to $A$ in $C_T$.
=--

### By lax 2-limits

Just as the [[Kleisli object]] of a monad $t$ in a 2-category $K$ can be defined as the [[lax colimit]] of the [[lax functor]] $\ast \to K$ [[monad|corresponding]] to $t$, the EM object of $t$ is its [[lax limit]].

[[Steve Lack]] has shown how Eilenberg--Moore objects $C^T$ can be obtained as combinations of certain simpler lax limits, when the 2-category $K$ in question is the 2-category of 2-algebras over a 2-monad $\mathbf{G}$ and lax, colax or pseudo morphisms of such:

* [[Steve Lack]], _Limits for lax morphisms_, Applied Categorical Structures __13__:3 (2005) , pp. 189--203(15)

This encompasses for example the theory of (op)monoidal monads and corresponding monoidal Eilenberg--Moore categories.

If $(T,\mu,\eta)$ is a monad in a small category $A$, and $B$ is another category, then consider the functor category $[B,A]$. There is a tautological monad $[B,T]$ on $[B,A]$ defined by $[B,T](F)(b) = T(F(b))$, $b\in Ob B$, $[B,T](F)(f) = T(F(f))$, $f\in Mor B$, $\mu^{[B,T]}_F : TTF\Rightarrow TF$, $(\mu^{[B,T]}_F)_b = \mu_{Fb}$
$(\eta^{[B,T]}_F)_b = \eta_{Fb} : Fb\to TFb$. Then there is a canonical isomorphism of EM categories

$$
[B,A^T] \cong [B,A]^{[B,T]}.
$$

Namely, write the object part of a functor $G : B\to A^T$ as $(G^A,G^\rho)$, where $G^A :B\to A$ and $G^\rho(b) : TG^A(b)\to G^A(b)$ is the $T$-action of $G^A(b)$ and the morphism part simply as $f\mapsto G(f)$. Then, $G^\rho : b\mapsto G^\rho(b) : TG^A\Rightarrow G^A$ is a natural transformation because for any morphism $f:b\to b'$, $G(f) : (G^A(b),G^\rho(b))\to (G^A(b'),G^\rho(b'))$ is by the definition of $G$, a morphism of $T$-algebras. $G^\rho$ is, by the same argument, an action $[B,T](G^A)\Rightarrow G^A$. Conversely, for any $[B,T]$-module $(G^A,G^\sigma)$ for any $b\in Ob B$, $G^\sigma(b)$ will evaluate to a $T$-action on $G^A(b)$, hence $b\mapsto (G^A(b), G^\sigma(b))$ is an object part of a functor in $[B,A^T]$ with morphism part again $f\mapsto G(g)$. The correspondence for the natural transformations, $g: (G^A,G^\sigma)\Rightarrow (H^A,H^\tau)$ is similar.

Dually, for a comonad $\Omega$ in $B$, there is a canonical comonad $[A, \Omega]$ on $[A, B]$ and an isomorphism of categories

$$
[A, B^\Omega] \cong [A, B]^{[A, \Omega]}
$$

### Limits and colimits in EM categories

* The Eilenberg--Moore category of a monad $T$ on a category $C$ has all [[limits]] which exist in $C$, and they are [[created limit|created]] by the forgetful functor.

* In contrast, the subject of [[colimits in categories of algebras]] is less easy, but a good deal can be said.

### Local presentability
 {#LocalPresentability}

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

An _[[accessible monad]]_ is a [[monad]] on an [[accessible category]] whose underlying [[functor]] is an [[accessible functor]].

=--

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

The Eilenberg--Moore category of a $\kappa$-accessible monad, def. \ref{AccessibleMonad}, is a  $\kappa$-[[accessible category]]. If in addition the category on which the monad acts is a $\kappa$-[[locally presentable category]] then so is the EM-category.

=--

([Adamek-Rosicky, 2.78](#AdamekRosicky))

Moreover, let $C$ be a [[topos]]. Then

* if a [[monad]] $T : C \to C$ has a [[right adjoint]] then $T Alg(C)= C^T$ is itself a topos;

* if a [[comonad]] $T : C \to C$ is [[exact functor|left exact]], then $T CoAlg(C) = C_T$ is itself a topos.

See at _[[topos of algebras over a monad]]_ for details.

## Examples

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

Given a [[reflective subcategory]] $\mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{\hookrightarrow}{i}} \mathcal{D}$ then the Eilenberg--Moore category of the induced [[idempotent monad]] $i\circ L$ on $\mathcal{D}$ recovers the subcategory $\mathcal{C}$.

=--

For instance ([Borceux, vol 2, cor. 4.2.4](#Borceux)).

## Related concepts

* [[(∞,1)-category of algebras over an (∞,1)-monad]]
* [[Eilenberg-Moore category]], [[Kleisli category]]
* [[Eilenberg-Moore object]], [[Kleisli object]]
* [[module over a monad]]

## References

Original reference:

* [[Samuel Eilenberg]], [[John Moore]], _Adjoint functors and triples_, Illinois J. Math. **9** 3 (1965) 381-398 [[doi:10.1215/ijm/1256068141](https://doi.org/10.1215/ijm/1256068141)]

General discussion:

* [[Fred Linton]], §7 in: *An outline of functorial semantics*, in *[[Seminar on Triples and Categorical Homology Theory]]*, Lecture Notes in Mathematics **80**, Springer (1969) 7-52 [[doi:10.1007/BFb0083080](https://doi.org/10.1007/BFb0083080)]

* {#Linton69} [[Fred Linton]], _Relative functorial semantics: adjointness results_, Lecture Notes in Mathematics, **99** (1969)

* {#Pumplün70} [[Dieter Pumplün]], *Eine Bemerkung über Monaden und adjungierte Funktoren*, Mathematische Annalen **185** (1970) 329-337 [[eudml:161964](https://eudml.org/doc/161964), [[Pumpluen-Monaden.pdf:file]]]

* [[Saunders MacLane]], §VI.2 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5**  Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Street72} [[Ross Street]], *The formal theory of monads*, Journal of Pure and Applied Algebra **2** 2 (1972) 149-168 [doi:10.1016/0022-4049(72)90019-9]

* [[Michael Barr]], [[Charles Wells]], §3.2 of: *[[Toposes, Triples, and Theories]]*, Grundlehren der math. Wissenschaften **278**, Springer (1983) [[TAC:12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)]

* {#StreetWalters78} [[Ross Street]], [[Bob Walters]], _Yoneda structures_, J. Algebra __50__, 1978

* [[Stephen Lack]], [[Ross Street]], *The formal theory of monads II*, Journal of Pure and Applied Algebra
**175** 1–3 (2002) 243-265 [doi:10.1016/S0022-4049(02)00137-8]

On [[locally presentable category|local presentability]] of EM-categories:

* {#AdamekRosicky} [[Jiří Adámek]], [[Jiří Rosický]], p. 123, 124 of: _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)

The following paper of Melliès compares the representability condition of ([Linton 69](#Linton69)) with the [[Segal condition]] that distinguishes those [[simplicial sets]] that are the [[nerves]] of categories.

* {#Mellies10} [[Paul-André Melliès]], _Segal condition meets computational effects_, LICS 2010 ([pdf](http://www.pps.jussieu.fr/~mellies/papers/segal-lics-2010.pdf))

The example of [[idempotent monads]] is discussed also in

* [[Francis Borceux]], _[[Handbook of Categorical Algebra]]_, vol.2, p. 196.
 {#Borceux}

Discussion of the [[universal property]] as the final adjoint decomposition of the monad:

* [[Anthony Voutas]], *The basic theory of monads and their connection to universal algebra*, (2012) [[pdf](https://voutasaur.us/monad-algebra.pdf), [[Voutas-Monads.pdf:file]]]

Discussion for [[(infinity,1)-monads]] realized in the context of [[quasi-categories]] is around def. 6.1.7 of

* [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_ ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279))
 {#RiehlVerity13}

An analogue of the pullback theorem for [[lax algebras]] (there confusingly called "pseudoagebras") of a [[2-monad]] is given in:

* [[Albert Burroni]]. _Structures pseudo-algébriques (1ère partie)_. Cahiers de topologie et géométrie différentielle catégoriques 16.4 (1975): 343-393. ([pdf](http://www.numdam.org/item/CTGDC_1975__16_4_343_0.pdf))

[[!redirects Eilenberg--Moore categories]]
[[!redirects Eilenberg–Moore category]]
[[!redirects Eilenberg--Moore category]]
[[!redirects Eilenberg-Moore categories]]
[[!redirects Eilenberg--Moore object]]
[[!redirects Eilenberg--Moore objects]]
[[!redirects Eilenberg-Moore object]]
[[!redirects Eilenberg-Moore objects]]
[[!redirects Eilenberg–Moore object]]
[[!redirects Eilenberg–Moore objects]]
[[!redirects Alg(T)]]
[[!redirects T-Alg]]
[[!redirects T-alg]]
[[!redirects T Alg]]
[[!redirects T alg]]

[[!redirects Beck coequalizer]]
[[!redirects Beck coequalizers]]
[[!redirects Beck coequaliser]]
[[!redirects Beck coequalisers]]

[[!redirects Beck equalizer]]
[[!redirects Beck equalizers]]
[[!redirects Beck equaliser]]
[[!redirects Beck equalisers]]

[[!redirects category of algebras over a monad]]
[[!redirects category of algebras for a monad]]