end

> This article is about ends (and coends) in [[category theory]].  For ends in [[topology]], see at *[[end compactification]]*.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Enriched category theory
+-- {: .hide}
[[!include enriched category theory contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

An _end_ is a special kind of [[limit]] over a [[functor]] of the form $F : C^{op} \times C \to D$ (sometimes called a _[[bifunctor]]_).

If we think of such a functor in the sense of [[profunctors]] as encoding a left and right [[action]] on the object

$$
  \prod_{c \in C} F(c,c)
$$

then the _end_ of the functor picks out the universal [[subobject]] on which the left and right action coincides.  Dually, the _coend_ of $F$ is the universal quotient of $\coprod_{c \in C} F(c,c)$ that forces the two actions of $F$ on that object to be equal.

A classical example of an _end_ is the $V$-object of [[natural transformations]] between $V$-[[enriched functors]] in [[enriched category theory]].  Perhaps the most common way in which ends and coends arise is through homs and tensor products of (generalized) modules, and their close cousins, weighted limits and weighted colimits. These concepts are fundamental in enriched category theory.

## Definition

### In ordinary category theory

#### Definition via extranatural transformations
In ordinary [[category theory]], given a [[functor]] $F: C^{op} \times C \to X$, an __end__ of $F$ in $X$ is an object $e$ of $X$ equipped with a [[universal construction|universal]] [[extranatural transformation]] from $e$ to $F$. This means that given any extranatural transformation from an object $x$ of $X$ to $F$, there exists a unique map $x \to e$ which respects the extranatural transformations.

In more detail: the end of $F$ is traditionally denoted $\int_{c: C} F(c, c)$, and the components of the universal extranatural transformation,
$$\pi_c: \int_{c: C} F(c, c) \to F(c, c)$$
are called the _projection maps_ of the end. Then, given any extranatural transformation with components
$$\theta_c: x \to F(c, c),$$
there exists a unique map $f: x \to e$ such that
$$\theta_c = \pi_c f$$
for every object $c$ of $C$.

The notion of __coend__ is dual to the notion of end.  The coend of $F$ is written $\int^{c: C} F(c, c)$, and comes equipped with a universal extranatural transformation with components
$$\iota_c \colon F(c,c) \to \int^{c: C} F(c,c)$$

#### Explicit definition
We unwrap the definition of an extranatural transformation to obtain a more explicit description of an end.

+-- {: .num_defn #wedge}
###### Definition
Let $F: C^{op} \times C \to X$ be a [[functor]]. A **wedge** $e: w \to F$ is an object $w$ and maps $e_c: w\to F(c, c)$ for each $c$, such that given any morphism $f: c \to c'$, the following diagram commutes:
$$
\array{
w & \overset{e_{c'}}{\to} & F(c', c')\\
^\mathllap{e_c}\downarrow & & \downarrow^\mathrlap{F(f, c')}\\
F(c, c) & \underset{F(c, f)}{\to} & F(c, c')
}
$$
=--

Given a wedge $e: w \to F$ and a map $f: v \to w$, we obtain a wedge $e f: v \to F$ by composition. Then we define the end as follows:

+-- {: .num_defn}
###### Definition
Let $F: C^{op} \times C \to X$ be a functor. An **end** of $F$ is a universal wedge, ie. a wedge $e: w \to F$ such that any other wedge $e': w' \to F$ factors through $e$ via a unique map $w' \to w$.
=--

Dually, a cowedge is given by maps $F(c, c) \to w$ satisfying similar commutativity conditions, and a coend is a universal cowedge.

#### Ends as right adjoint functors

In complete analogy to how [[limits]] are right adjoint functors
to the diagonal functor,
ends are right adjoint functors to the hom functor.

In more detail, suppose $C$ and $X$ are categories.

If any diagram $C^{op}\times C\to X$ admits an end,
then we have a functor
$$end\colon Fun(C^{op}\times C,X)\to X$$
whose left adjoint is the hom functor
$$hom\colon X\to Fun(C^{op}\times C,X)$$
that sends an object $x\in X$
to the functor $hom(x)\colon C^{op}\times C\to X$
that sends $(c,d)$ to $\coprod_{hom(c,d)}x=hom(c,d)\otimes x$.
(For coends one uses $x^{hom(c,d)}$ instead.)

This immediately implies a Fubini theorem for ends and coends.

### In enriched category theory

There is a definition of _end_ in [[enriched category theory]], as follows.

#### End of $V$-valued functors

Let $V$ be a [[symmetric monoidal category]], and let $C$ be a $V$-[[enriched category]]. Assuming $V$ is also [[closed monoidal category|closed monoidal]], $V$ may be considered as $V$-enriched; in that case, suppose $F: C^{op} \otimes C \to V$ is a $V$-[[enriched functor]].

Then in particular there is a covariant [[action]] of $C$ on $F$, with components
$$\lambda_{c, d, e}: F(c, d) \otimes C(d, e) \to F(c, e),$$
(where $C(d, e)$ is customary notation for the [[hom-object]] $\hom_C(d, e)$ of $C$ in $V$), and a contravariant action of $C$ on $F$, with components
$$\rho_{c, d, e}: F(d, e) \otimes C(c, d) \to F(c, e).$$

In detail, the covariant action is the [[adjunct]] of the morphism

$$
  (F(c,-)
    \colon
  C(d,e) \to [F(c,d), F(c,e)])
    \in
  Hom_V(C(d,e),[F(c,d), F(c,e)])
$$

under the [[closed monoidal category|Hom-adjunction]]

$$
  Hom_V(C(d,e),[F(c,d), F(c,e)])
    \stackrel{\simeq}{\longrightarrow}
  Hom_V(C(d,e)\otimes F(c,d),F(c,e))
$$

in $V$. Similarly for the contravariant action.

+-- {: .un_remark}
###### Remark
Even if $V$ is not closed monoidal, we can still define a **notion** of covariant $C$-action, sometimes called a "left" $C$-[[module]], as consisting of a function $F \colon Ob(C) \to Ob(V)$ together with an $Ob(V) \times Ob(V)$-indexed collection of morphisms

$$F(c) \times C(c, d) \to F(d)$$

satisfying some evident unit and associativity axioms, and regard this notion as a stand-in for the notion of $V$-functor $C \to V$. Similarly we have an evident notion of contravariant $C$-action as a stand-in for a $V$-functor $C^{op} \to V$; notice that we don't even require symmetry to make sense of this. Finally, we can combine these notions into one of $C$-bimodule, where we have a function $F \colon Ob(C) \times Ob(C) \to Ob(V)$ together with a collection of morphisms

$$C(a, b) \otimes F(b, c) \otimes C(c, d) \to F(a, d)$$

with evident axioms for a bimodule structure, as a stand-in for a $V$-functor of the form $C^{op} \otimes C \to V$.
=--

A $V$- **extranatural transformation**
$$\theta: v \stackrel{\bullet}{\to} F$$
from $v$ to $F$ consists of a family of arrows in $V$,
$$\theta_c: v \to F(c, c),$$
indexed over objects $c$ of $C$, such that for every pair of objects $(c, d)$ in $C$, the composites of (1) and (2) agree:
$$v \otimes C(c, d) \stackrel{\theta_c \otimes 1}{\to} F(c, c) \otimes C(c, d) \stackrel{\lambda_{c, c, d}}{\to} F(c, d) \qquad (1)$$
$$v \otimes C(c, d) \stackrel{\theta_d \otimes 1}{\to} F(d, d) \otimes C(c, d) \stackrel{\rho_{c, d, d}}{\to} F(c, d) \qquad (2)$$

A $V$-enriched **end** of $F$ is an object $\int_{c: C} F(c, c)$ of $V$ equipped with a $V$-extranatural transformation
$$\pi: \int_{c: C} F(c, c) \stackrel{\bullet}{\to} F$$
such any $V$-extranatural transformation $\theta$ from $v$ to $F$ is obtained by pulling back the components of $\pi$ along $f: v \to \int_{c: C} F(c, c)$, for some unique map $f$. That is,
$$\theta_c = \pi_c f$$
for all objects $c$ of $C$.

#### End of $C$-valued functors for $C \in V\Cat$

If $X$ is any $V$-enriched category and $F: C^{op} \otimes C \to X$ is a $V$-enriched functor, then the **end** of $F$ in $X$ is, if it exists, an object $\int_{c: C} F(c, c)$ of $X$ that [[representable functor|represents]] the functor

$$
\int_{c: C} X(-,F(c,c))\,.
$$

That means that the end $\int_{c: C} F(c,c)$ comes equipped with an $Ob(C)$-indexed family of arrows

$$
\pi_c: I \to X(\int_{c: C} F(c, c), F(c, c))
$$

in $V$, such that for every object $x$ of $X$, the family
of maps

$$
X(x, \pi_c): X(x, \int_{c: C} F(c, c)) \to X(x, F(c, c))
$$

are the projection maps realizing $X(x, \int_{c: C} F(c, c))$ as the corresponding end $\int_{c: C} X(x, F(c, c))$ in $V$.

#### End as an equalizer

##### Ordinary ends as equalizers

Now we motivate and define the _end_ in [[enriched category theory]] in terms of [[equalizers]].

Recall from the discussion at the end of [[limit]] that the [[limit]] over an (ordinary, i.e. not enriched) [[functor]]

$$
  F : C^{op} \to Set
$$

is given by the [[equalizer]] of

$$
  \prod_{c \in Obj(C)}
  F(c)
  \stackrel{\prod_{f \in Mor(c)} (F(f) \circ p_{t(f)}) }{\to}
  \prod_{f \in Mor(C)}
  F(s(f))
$$

and

$$
  \prod_{c \in Obj(C)}
  F(c)
  \stackrel{\prod_{f \in Mor(c)} (p_{s(f)}) }{\to}
  \prod_{f \in Mor(C)}
  F(s(f))
  \,.
$$

If we want to generalize an expression like this to [[enriched category theory]] the explicit indexing over the set of morphisms has to be replaced by something that makes sense in an [[enriched category]].

To that end, observe that we have a canonical isomorphism (of sets, still)

$$
  \prod_{{(c_1 \stackrel{f}{\to} c_2)} \in Mor(C)} F(c_1)
  \simeq
  \prod_{c_1,c_2 \in Obj(C)} F(c_1)^{C(c_1,c_2)}
  \,.
$$

If we write for the [[hom-set]] instead

$$
 [C(c_1,c_2), F(c_1)] := F(c_1)^{C(c_1,c_2)}
$$

with $[-,-]$ the [[internal hom]] in [[Set]], then the expression starts to make sense in any $V$-[[enriched category]].

Still equivalently but suggestively rewriting the above, we now obtain the limit over $F$ as the [[equalizer]] of

$$
  \prod_{c \in Obj(C)}
  F(c)
  \stackrel{\stackrel{\rho}{\to}}{\stackrel{\lambda}{\to}}
  \prod_{c_1,c_2 \in Obj(C)}
  [C(c_1,c_2),F(c_1)]
  \,,
$$

where in components

$$
  \rho_{c_1, c_2} : F(c_1) \to [C(c_1,c_2), F(c_1)]
$$

is the [[adjunct]] of

$$
  C(c_1, c_2) \to * \to [F(c_1), F(c_1)]
$$

(with the last map the [[adjunct]] of $Id_{F(c_1)}$) and where

$$
  \lambda_{c_1, c_2} : F(c_2) \to [C(c_1,c_2), F(c_1)]
$$

is the [[adjunct]] of

$$
  F_{c_1, c_2} : C(c_1, c_2) \to [F(c_2), F(c_1)]
  \,.
$$

So for definiteness, the equalizer we are looking at is that of

$$
  \rho := \prod_{c_1, c_2 \in C} \rho_{c_1,c_2}\circ pr_{F(c_1)}
$$

and

$$
  \lambda := \prod_{c_1, c_2 \in C} \lambda_{c_1,c_2}\circ pr_{F(c_2)}
$$

This way of writing the [[limit]] clearly suggests that it is more natural to have $\lambda$ and $\rho$ on equal footing. That leads to the following definition.

##### Enriched ends over $V$-valued functors as equalizers

For $V$ a [[symmetric monoidal category]], $C$ a $V$-[[enriched category]] and
$F \colon C^{op} \times C \to V$ a  $V$-[[enriched functor]],
the **end** of $F$ is the [[equalizer]]

$$
  \int_{c \in C}
  F(c,c)
    \longrightarrow
  \prod_{c \in Obj(C)}
  F(c,c)
   \underoverset
    {\underset{\lambda}{\longrightarrow}}
    {\overset{\rho}{\longrightarrow}}
    {}
  \prod_{c_1,c_2 \in Obj(C)}
    [C(c_1,c_2),F(c_1,c_2)]
  \label{endeq}
$$

with $\rho$ in components given by

$$
  \rho_{c_1, c_2} \colon F(c_1,c_1) \longrightarrow [C(c_1,c_2), F(c_1,c_2)]
$$

being the [[adjunct]] of

$$
  F(c_1,-) \colon  C(c_1, c_2) \longrightarrow [F(c_1,c_1), F(c_1,c_2)]
$$

and

$$
  \lambda_{c_1, c_2} \colon F(c_2,c_2) \longrightarrow [C(c_1,c_2), F(c_1,c_2)]
$$

being the [[adjunct]] of

$$
  F(-,c_2) \colon C(c_1, c_2) \longrightarrow [F(c_2,c_2), F(c_1,c_2)]
  \,.
$$

This definition manifestly exhibits the **end as the equalizer of the left and right action** encoded by the [[distributor]] $F$.

Dually, the coend of $F$ is the [[coequalizer]]

$$
\coprod_{c_1,c_2} C(c_2,c_1) \otimes F(c_1,c_2)\, \rightrightarrows\,
\coprod_c F(c,c)\,\to\,
\int^c F(c,c)
\label{coendcoeq}
$$

with the parallel morphisms again induced by the two actions of $F$.

#### End as a weighted limit

The end for $V$-functors with values in $V$ serves, among other things, to define [[weighted limits]], and weighted limits in turn define ends of bifunctors with values in more general $V$-categories.

For $C$ and $D$ both $V$-categories and $F : C^\op \times C \to D$ an $V$-[[enriched functor]], the **end** of $F$ is the [[weighted limit]] of $F$

$$
  \int_{c \in C} F(c,c)
  \coloneqq
  \{Hom_C, F\} =
  lim^{Hom_C} F
  \,,
$$

with weight $Hom_C : C^{op} \times C \to V$.  The **coend** of $F$ is the colimit

$$
\int^{c \in C} F(c,c) \coloneqq Hom_{C^{op}} \ast F =  \colim^{Hom_{C^{op}}} F
$$
of $F$ weighted by the hom functor of $C^{op}$.

#### Connecting the two definitions

If $C$ is a $V$-category, then the hom functor $C(-,-) \colon C^{op} \times C \to V$ is the [[coequalizer]] in

$$
\coprod_{c,c'} C(-,c) \times C(c,c') \times C(c',-)
\,\rightrightarrows\,
\coprod_c C(-,c) \times C(c,-) \,\to\, C(-,-)
$$

It is also a general fact (see e.g. [Kelly, ch. 3](#Kelly)) that weighted (co)limits are cocontinuous in their weight: that is,
$$ \{W \ast V, F\} \cong \{W, \{V-, F\}\}$$
and
$$ (W \ast V) \ast G \cong W \ast (V \ast G)$$
This implies that $\{-,F\}$ takes the coequalizer above to an equalizer, which, after some fiddling with the [[Yoneda lemma]], turns out to be isomorphic to (eq:endeq).
Similarly, $(- \ast F)$ takes the analogous coequalizer presentation of $C^{op}(-,-)$ to (eq:coendcoeq).

## Properties {#Properties}

### $Set$-enriched coends as ordinary colimits {#SetCoendsAsColimits}

Let the enriching category be $\mathcal{V} = $ [[Set]]. We describe a special way in this case to express ends/coends that give [[weighted limits]]/colimits in terms of ordinary (co)limits over categories of elements.

Consider

* $C$ a $Set$-enriched category/[[locally small category]]
  [[tensoring|tensored]] over [[Set]];

* $D$ be a [[small category]];

* $F : D \to C$ a functor;

* $W : D^{op} \to Set$ another functor;

* $el W$ the [[category of elements]] of $W$.

\begin{proposition}\label{CoendAsColimitOverCategoryOfElements}
**(coend as [[colimit]] over [[category of elements]])** \linebreak
There is a [[natural isomorphism]] in $C$

$$
  \int^{d \in D} W(d) \cdot F(d)
  \simeq
  \lim_{\to}(
    (el W)^{op} \to D \stackrel{F}{\to} C
  )
$$

between the coend, as indicated, and the [[colimit]] over the [[opposite category|opposite]] of the [[category of elements]] of $W$.

\end{proposition}

This is equation (3.34) in ([Kelly](#Kelly)) in view of (3.70).

+-- {: .num_cor #ConPres}
###### Corollary
Any [[continuous functor]] preserves ends, and any cocontinuous functor preserves coends. In particular, for functors $F: D^{op} \times D \to C$ and $c \in C$, we have the isomorphisms
$$
\begin{aligned}
  C(\int^x F(x, x), c) &\cong \int_x C(F(x, x), c)\\
  C(c, \int_x F(x, x)) &\cong \int_x C(c, F(x, x))
\end{aligned}
$$
=--

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

If $W = D(-,e)$ is a [[representable functor]], then

$$
  (el W)^{op} = D/e
$$

is the [[over category]] over the representing object $e$. This has a [[terminal object]], namely $(e \stackrel{Id}{\to} e$). Therefore

$$
  \lim_\to( D/e \to D \stackrel{F}{\to} C) \simeq F(e)
  \,.
$$

Since this is natural in $e$, the above proposition asserts a [[natural isomorphism]]

$$
  F(-) \simeq \int^{k \in D} D(k,-) \cdot F(k)
  \,.
$$

This statement is sometimes called the [[co-Yoneda lemma]].
=--

### Commutativity of ends and coends

Ordinary [[limit]]s commute with each other, if both limits exist separately. The analogous statement does hold for ends and coends. Since there it looks like the commutativity of two integrals, it is called the _Fubini theorem_ for ends (for instance [Kelly, p. 29](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf#page=29)).

\begin{proposition}\label{Fubini}
**(Fubini theorem for ends)**
\linebreak
Let $\mathcal{V}$ be a [[symmetric monoidal category]]. Let $\mathcal{A}$ and $\mathcal{B}$ be small
$\mathcal{V}$-[[enriched categories]].

Let

$$
  T \colon (\mathcal{A} \otimes \mathcal{B})^{op} \otimes
      (\mathcal{A} \otimes \mathcal{B})
      \to
      \mathcal{V}
$$

be a $\mathcal{V}$-[[enriched functor]]. Then:

If for all object $B,B' \in \mathcal{B}$ the end $\int_{A \in \mathcal{A}} T(A,B,A,B')$ exists, then

$$
  \int_{(A,B) \in \mathcal{A} \otimes \mathcal{B}}
   T(A,B,A,B)
  \simeq
  \int_{A \in \mathcal{A}}
  \int_{B \in \mathcal{B}}
  T(A,B,A,B)
$$

if either side exists. In particular, since $\mathcal{A} \otimes \mathcal{B} \simeq \mathcal{B} \otimes \mathcal{A}$ this implies that

$$
  \int_{B \in \mathcal{B}}
  \int_{A \in \mathcal{A}}
  T(A,B,A,B)
  \simeq
  \int_{A \in \mathcal{A}}
  \int_{B \in \mathcal{B}}
  T(A,B,A,B)
$$

if either side exists.
\end{proposition}

## Examples

### Natural transformations

+-- {: .num_prop #NatTrans}
###### Proposition
Let $F, G: C \to D$ be [[functors]] between two categories, and let $[C, D] (F, G)$ be the set of [[natural transformations]] between them. Then we have
$$
  [C, D] (F, G) = \int_{c \in C} D(F(c), G(c)).
$$
=--

+-- {: .proof}
###### Proof
An element of $\int_{c \in C} D(F(c), G(c))$ is by definition a collection $\tau_c: F(c) \to G(c)$ of morphisms in $D$ such that for any morphism $f: c \to d$ in $C$, the following square commutes:
$$
\array{
F(c) & \overset{F(f)}{\to} & F(d)\\
^\mathllap{\tau_c}\downarrow & & \downarrow^\mathrlap{\tau_d}\\
G(c) & \underset{G(f)}{\to} & G(d)
}
$$
which is by definition a natural transformation $F \to G$.
=--

### Enriched functor categories
In light of Proposition \ref{NatTrans}, we can define the natural transformations object for [[enriched functors]] as an end:

For $C$ and $D$ both $V$-[[enriched category|enriched categories]], the $V$-[[enriched functor category]] $[C,D]$ is the $V$-[[enriched category]] whose

* objects are $V$-[[enriched functors]] $F : C \to D$;

* [[hom-objects]] in $V$ are given by the end-formula $[C,D](F,G) := \int_{c \in C} D(F(c), G(c))$.

+-- {: .num_example}
###### Example
For $V = Set$ this reproduces of course the ordinary [[functor category]].
=--

+-- {: .num_example}
###### Example
For $V = \mathbb{R}_{\geq 0}\cup \{\infty\}$ with the monoidal product given by addition, a $V$-enriched category $X$ is a [[metric space]], with the distance between points $x, y \in X$ given by $X(x, y)$. Given two metric spaces $X, Y$ and maps $f, g: X \to Y$, the distance between the maps is
\[
  [X, Y](f, g) = \int_{x \in X} Y(f(x), g(x)) = \sup_{x \in X} Y(f(x), g(x)).
\]
=--

### Kan extension

If the $V$-[[enriched category]] $D$ is [[copower|tensor]]ed over $V$, then the (left) [[Kan extension]] of a [[functor]] $F : C \to D$ along a functor $p : C \to B$ is given by the coend

$$
  Lan F : b \mapsto  \int^{c \in C} hom(p(c),b) \cdot F(c)
  \,.
$$

See [[Kan extension]] for more details.

### Geometric realization

A special case of the example of Kan extension is that of [[geometric realization]].

Very generally, geometric realization is the left Kan extension of a functor $F : C \to D$ along the [[Yoneda embedding]] $Y : C \to [C^{op},V]$.

The "geometric realization" of an object $X \in [C^{op},V]$ with respect to $F$ is then the coend

$$
  |X| := \int^{c \in C} F(c) \cdot hom(Y(c),X)
  \simeq \int^{c \in C} F(c) \cdot X(c)
  \,,
$$

where the last step on the right uses the [[Yoneda lemma]].

More specifically, traditionally this is thought of as applying to the case where $C = \Delta$ is the [[simplex category]] and where $F : \Delta \to Top$ regards the abstract $n$-[[simplex]] as the standard simplex as a [[topological space]].

### Tensor product of functors

If $S : C^\op \to D$ and $T : C \to D$ are functors, their [[tensor product of functors|tensor product]] is the coend
$$ S \otimes_C T = \int^c S(c) \otimes T(c), $$
where the tensor product on the right hand side refers to some [[monoidal structure]] on $D$.

## (Co)end calculus
The formal properties of (co)ends in Propositions \ref{ConPres}, \ref{Fubini} and \ref{NatTrans} allow us to prove certain results by [[abstract nonsense]].

+-- {: .num_example}
###### Example
Let $F: C^op \to Set$ be a functor. We prove the [[co-Yoneda lemma]], that
$$
  F(c) \simeq \int^{c' \in C} C(c,c')\times F(c')
$$
We perform the following manipulations, where each isomorphism is natural:
$$
\begin{aligned}
  Set (\int^{c' \in C} C(c,c')\times F(c'), y) &\simeq \int_{c' \in C} Set (C(c,c')\times F(c'), y)\\
  &\simeq \int_{c' \in C} Set (C(c, c'), Set(F(c'), y))\\
  &\simeq [C, Set] (C(c, -), Set(F(-), y))\\
  &\simeq Set(F(c), y).
\end{aligned}
$$
So by the [[Yoneda lemma]], we have
$$
  F(c) \simeq \int^{c' \in C} C(c,c')\times F(c').
$$
=--

For more examples see e.g. [Loregian (2021)](#Loregian21).

## Related concepts

* [[lax biend]]

* [[(∞,1)-end]]

[[!include homotopy-homology-cohomology]]

## References

The notion of (co)ends as introduced in

* [[Nobuo Yoneda]], §4 of: *On ext and exact sequences* (PhD thesis), Journal of the Faculty of Science, Section I. **8** University of Tokyo (1960) 507–576 [[pdf](http://alpha.math.uga.edu/~lorenz/YonedaExtExactSequences.pdf), [CiNii:naid/500000325773](https://ci.nii.ac.jp/naid/500000325773)]

An early account with an eye towards application in [[geometric realization of simplicial topological spaces]]:

* {#MacLane70} [[Saunders MacLane]], Section 2 of: _The Milgram bar construction as a tensor product of functors_,  In: F.P. Peterson  (eds.) *The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod's Sixtieth Birthday*, Lecture Notes in Mathematics *168*,  Springer 1970 ([doi:10.1007/BFb0058523](https://doi.org/10.1007/BFb0058523), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0058523.pdf))

Textbook accounts:

* {#Kelly82} [[Max Kelly]], _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

  * ends of $V$-valued bifunctors are discussed in section 2.1

  * the enriched functor category that they give rise to is discussed in section 2.2;

  * enriched [[weighted limits]] in terms of enriched functor categories are in section 3.1

  * the end of general $V$-enriched functors in terms of weighted limits is in section 3.10 .

* [[Francis Borceux]], Def. 6.6.8 in: *[[Handbook of Categorical Algebra]]*, Vol. 2:  *Categories and Structures*, Encyclopedia of Mathematics and its Applications **50** Cambridge University Press (1994) ([doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865))

* [[Emily Riehl]], §4.1 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

* {#Loregian21} [[Fosco Loregian]], *Coend calculus*, Cambridge University Press (2021) [[arXiv:1501.02503](http://arxiv.org/abs/1501.02503), [doi:10.1017/9781108778657](https://doi.org/10.1017/9781108778657), ISBN:9781108778657)]

See also:

* [Ends](http://golem.ph.utexas.edu/category/2014/01/ends.html), $n$-Category Café discussion.

Application in [[conformal field theory]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], _Coends in conformal field theory_ ([arXiv:1604.01670](https://arxiv.org/abs/1604.01670))

[[!redirects end]]
[[!redirects ends]]
[[!redirects coend]]
[[!redirects coends]]

[[!redirects Fubini theorem for ends]]
[[!redirects Fubini theorem for coends]]

[[!redirects enriched end]]
[[!redirects enriched ends]]


end compactification

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### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# End compactification
* table of contents
{: toc}

## Idea

Whereas the [[one-point compactification]] of a (sufficiently [[nice topological space|nice]]) [[topological space]] adjoins only a single [[point at infinity]], the _end compactification_ [[Freudenthal 1931](#Freudenthal31)] adjoins one point for each [[connected component]] of infinity.

## Definition

The definition was originally given only for sufficiently [[nice topological space|nice]] topological spaces (the [[hemicompact space|hemicompact]] ones).  The general definition is a bit more complicated.  We will give three versions.

### For hemicompact spaces only

Let $X$ be a [[topological space]], and suppose that $X$ is [[hemicompact space|hemicompact]]; this means that there exists an [[infinite sequence]] $n \mapsto K_n$ of [[compact subspaces]] of $X$ with $K_n \subseteq K_{n+1}$ such that every compact subspace of $X$ is contained in at least one (hence in almost all) of the $K_i$.

Consider the [[connected components]] of the [[complements]] $X \setminus K_i$.  An __end__ of $X$ is an infinite sequence that chooses one such connected component for each $i$.  Remarkably, the set of ends is independent of the sequence $K$ chosen (up to [[natural bijection]]).

The __end compactification__ of $X$ has, as its [[underlying set]], the [[disjoint union]] of the underlying set of $X$ and the set of ends.  Its [[topological structure|topology]] is generated (from a [[topological base|base]]) by the topology of $X$ and, for each end $e = (U_1,U_2,\ldots)$, the [[open sets]] $V \cup \{e\}$ whenever $V$ is open in $X$ and $U_i \subseteq V$ for some (hence almost every) $i$.

### Abstract

Let $X$ be a [[topological space]], and consider the [[poset]] $Comp(X)$ of [[compact subspaces]] of $X$, ordered by [[inclusion]].  For each compact subspace $K$, consider its [[complement]] $X \setminus K$, and consider the [[set]] $\Pi_0(X \setminus K)$ of its [[connected components]].  For each inclusion $K \hookrightarrow K'$, we have a [[function]] $\Pi_0(X \setminus K') \to \Pi_0(X \setminus K)$.  This defines a [[contravariant functor]] from $Comp(X)$ to [[Set]]; its [[limit]] is the __set of ends__ of $X$.

For the [[topological structure|topology]], each compact subspace $K$ defines a topological space $K \uplus \Pi_0(X \setminus K)$; here, the points of $\Pi_0(K \setminus K)$ are all [[isolated point|isolated]].  For each inclusion $K \hookrightarrow K'$, we have a [[continuous map]] $K' \uplus \Pi_0(X \setminus K') \to K \uplus \Pi_0(X \setminus K)$; it sends $x$ to itself if $x \in K$, and $x$ to the connected component $[x] \in \Pi_0(X \setminus K)$ if $x \in K' \setminus K$.  This defines a [[contravariant functor]] from $Comp(X)$ to [[Top]]; its [[limit]] is the __end compactification__ of $X$.

### Concrete

Let $X$ be a [[topological space]].  An __end__ of $X$ assigns, to each [[compact subspace]] $K$ of $X$, a [[connected component]] $e_K$ of its [[complement]] $X \setminus K$, in such a way that $e_{K'} \subseteq e_K$ whenever $K \subseteq K'$.  The __end compactification__ of $X$ has, as its [[underlying set]], the [[disjoint union]] of the underlying set of $X$ and the set of ends.  Its [[topological structure|topology]] is generated (from a [[topological base|base]]) by the topology of $X$ and, for each end $e\colon K \mapsto e_K$, the [[open sets]] $V \cup \{e\}$ whenever $V$ is open in $X$ and $e_K \subseteq V$ for some compact subspace $K$.

## Examples

A [[compact space]] has no ends, hence is its own end compactification.  The converse (that a space with no ends must be compact) seems to require the [[axiom of choice]] (although [[excluded middle]] and [[dependent choice]] suffice for hemicompact spaces).

The end compactification of the [[real line]] is the [[extended real number]] line segment; the ends are $\infty$ and $-\infty$.  But the [[complex plane]] has only one end; its end compactification is the [[Riemann sphere]] (the same as its [[one-point compactification]]).

## Applications

Ends are important in [[proper homotopy theory]].

According to [Peschke 1990](#Peschke90), [Freudenthal 1931](#Freudenthal31) was led to his theory of ends by the following observation. For a space $X$, consider a path-connected family $F \subseteq Homeo(X)$ containing the identity $1_X$. Let $K \subseteq X$ be compact, and let $U$ be a connected component of $X \setminus K$. Then for all $f \in F$, it may be shown $f(U) \setminus U$ is contained in a compact subset of $X$. The upshot is that $f$ extends to a homeomorphism on the end compactification that is *pointwise fixed* on the ends. If in addition for each pair $(x, y) \in X^2$ there is $f \in F$ with $f(x) = y$, then there is a severe constraint on the ends; in particular Freudenthal showed the following.

+-- {: .num_theorem}
###### Theorem
A path-connected topological group has at most two ends.
=--

For example, it follows that the space obtained by removing two points from $\mathbb{R}^3$ cannot be given a topological group structure.

## Related concepts

* [[compactification]]

  * [[One-point compactification]]

  * [[Stone–Cech compactification]]

  * **End compactification

  * [[Bohr compactification]]

## References

The original text:

* {#Freudenthal31} [[Hans Freudenthal]], _Über die Enden topologischer Räume und Gruppen_, Mathematische Zeitschrift **33** (1931) 692-713 [[doi:10.1007/BF01174375](https://doi.org/10.1007/BF01174375), [pdf](http://dspace.library.uu.nl/bitstream/handle/1874/7437/1930-freudenthal-dissertatie.pdf?sequence=1)]

Further discussion:

* [[Hans Freudenthal]], *Neuaufbau der Endentheorie*, Annals of Mathematics Second Series **43** 2 (1942) 261-279 [[doi:10.2307/1968869](https://doi.org/10.2307/1968869), [jstor:1968869](https://www.jstor.org/stable/1968869)]

* [[Hans Freudenthal]], *Über die Enden diskreter Räume und Gruppen*, Commentarii Mathematici Helvetici **17** (1944) 1–38 [[doi:10.1007/BF02566233](https://doi.org/10.1007/BF02566233)]

* {#Peschke90}  [[Georg Peschke]], _The Theory of Ends_, Nieuw Archief voor Wiskunde, 8 (1990), 1-12 [[pdf](https://www.ualberta.ca/~gepe/pdf/Peschke_TheoryOfEnds.pdf), [[Peschke-Ends.pdf:file]]]

See also:

* Wikipedia, *End (topology)*

[[!redirects end compactification]]
[[!redirects end compactifications]]

[[!redirects topological end]]
[[!redirects topological ends]]

[[!redirects end of a topological space]]
[[!redirects ends of a topological space]]
[[!redirects ends of topological spaces]]


endofunction

#Contents#
* table of contents
{:toc}

## Idea

An __endofunction__ (or __self-mapping__ or other variation) is an [[endomorphism]] in [[the category of sets]], that is a [[function]] from a [[set]] $X$ to $X$ itself.  This is the typical meaning of the word 'function' in some contexts, such as high-school algebra (where $X$ is the set of [[real numbers]], although there we actually use [[partial functions]]) or when distinguishing from the [[higher-order logic|higher-order]] concepts of [[functional]] and [[operator]].

## Related concepts

* [[fixed point]]

* [[endomorphism]]

* [[endofunctor]]

[[!redirects endofunction]]
[[!redirects endofunctions]]
[[!redirects endomap]]
[[!redirects endomaps]]
[[!redirects endomapping]]
[[!redirects endomappings]]
[[!redirects self-function]]
[[!redirects self-functions]]
[[!redirects self-map]]
[[!redirects self-maps]]
[[!redirects self-mapping]]
[[!redirects self-mappings]]


endofunctor

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

## Definition

A [[functor]] from a [[category]] to itself is called an **endofunctor**.

Given any category $C$, the [[functor category]]

$$End(C) = C^C $$

is called the **endofunctor category** of $C$.  The objects of $End(C)$ are endofunctors $F: C \to C$, and the morphisms are [[natural transformations | natural transformation]] between such endofunctors.

## Properties

### Monoidal structure

The endofunctor category is a [[strict monoidal category|strict]] [[monoidal category]], thanks to our ability to compose endofunctors:

$$\circ : End(C) \times End(C) \to End(C)$$

The unit object of this monoidal category is the identity functor from $C$ to itself:

$$1_C \in End(C) $$

### Monoids

A [[monoid]] in this endofunctor category is called a [[monad]] on $C$.

## Related concepts

* [[algebra over an endofunctor]]

* [[coalgebra over an endofunctor]]

* [[pointed endofunctor]]

[[!redirects endofunctors]]
[[!redirects endofunctor category]]
[[!redirects endofunctor categories]]


endomorphism

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#### Monoid theory
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=--
=--

# Contents
* table of contents
{: toc}

## Definition

An __endomorphism__ of an [[object]] $x$ in a [[category]] $C$ is a [[morphism]] $f : x \to x$.

An endomorphism that is also an [[isomorphism]] is called an [[automorphism]].

## Properties

Given an object $x$, the endomorphisms of $x$ form a [[monoid]] under [[composition]], the __endomorphism monoid__ of $x$:
$$
  End_C(x) = Hom_C(x,x)
,$$
which may be written $End(x)$ if the category $C$ is understood.  Up to equivalence, every monoid is an endomorphism monoid; see [[delooping]].

An endomorphism monoid is a special case of a monoid structure on an [[end]] construction. Let $d:D\to C$ be a diagram in $C$, where $C$ is a [[monoidal category]] (in the case above the monoidal structure is the [[cartesian product]] and $d$ is a constant diagram from the initial category). One defines $End(d)$ as an object in $C$, equipped with a natural transformation $a: End(d) \otimes d \to d$ which is universal in the sense that for all objects $Z \in C$, and any natural transformation $f: Z \otimes d \to d$ there exists a unique morphism $g: Z \to End(d)$ such $a \circ (g \otimes d) = f: Z \otimes d \to d$.

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

If the universal object $(End(d),a)$ exists then there is a unique structure of an [[internalization|internal]] monoid $\mu: End(d) \otimes End(d) \to End(d)$, such that the map $a: End(d) \otimes d \to d$ is an [[action]].
=--

+-- {: .num_prop}
###### Proposition
In a [[cartesian monoidal category]] $C$, if an endomorphism monoid $End(c)$ for an object $c: 1 \to C$ exists and is [[commutative monoid|commutative]], then $c$ is a [[subterminal object]].
=--

+-- {: .proof}
###### Proof
Let $k: c \to End(c)$ correspond to first projection $\pi_1: c \times c \to c$. Then the composition

$$c \times c \stackrel{k \times k}{\to} End(c) \times End(c) \stackrel{comp}{\to} End(c)$$

(where $comp$ denotes internal composition) may be computed to be $k \pi_1$, corresponding to first projection $\pi_1: c \times c \times c \to c$. Thus, assuming commutativity of $End(c)$ and letting $\sigma$ generally denote a symmetry map, consideration of the diagram

$$\array{
c \times c & \stackrel{k \times k}{\to} & End(c) \times End(c) & \stackrel{comp}{\to} & End(c) \\
\mathllap{\sigma} \downarrow & & \mathllap{\sigma} \downarrow & & \downarrow \mathrlap{id} \\
c \times c & \stackrel{k \times k}{\to} & End(c) \times End(c) & \stackrel{comp}{\to} & End(c)
}$$

leads to the conclusion that $k \pi_1 = k \pi_1 \sigma$, or $\pi_1 = \pi_2: c \times c \times c \to c$. We easily conclude $\pi_1 = \pi_2: c \times c \to c$, which forces equality $f = g$ for any two maps $f, g: d \to c$, so that the unique map $!: c \to 1$ is a [[monomorphism]].
=--

## Related entries

* **endomorphism**, [[automorphism]]

* [[endomorphism monoid]], [[endomorphism ring]], [[endomorphism monoid object]]

* [[center of a category]], [[center of an additive category]]

* [[endomorphism operad]]

* [[endofunction]]

* [[endofunctor]]

* [[EI-category]]

[[!redirects endomorphism]]
[[!redirects endomorphisms]]

[[!redirects endomorphism monoid]]
[[!redirects endomorphism monoids]]


endomorphism dg-Lie algebra

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## Definition

Let $Ch(\mathcal{A})$ be a [[category of chain complexes]] in a [[category]] $\mathcal{A}$ that is a [[closed monoidal category]]. For instance the category of chain complexes in $\mathcal{A} = $ [[Vect]]${}_k$.

For $V \in Ch$ any [[object]] -- any [[chain complex]] -- write

$$
  end(V) := [V,V] \in Ch
$$

for the [[internal hom]] object of morphisms from $V$ to $V$. This is the chain complex which in degree $n$ is

$$
  end(V)_n = \bigoplus_{i \in \mathbb{Z}} Hom_{\mathcal{A}}(V_i, V_{i+n})
$$

and whose [[differential]]

$$
  \delta_{end(V)} : end(V)_n \to end(V)_{n-1}
$$

is given by

$$
  \delta_{end(V)} (V_i \stackrel{f}{\to} V_{i+1})
  =
  (V_i \stackrel{f}{\to} V_{i+n} \stackrel{\delta_V}{\to} V_{i+n-1})
  - (-1)^i
  (V_{i+1} \stackrel{\delta_V}{\to} V_i \stackrel{f}{\to} V_{i+n})
  \,.
$$

This complex naturally carries the structure of an [[internalization|internal]] [[Lie algebra]], hence of a [[dg-Lie algebra]] with Lie bracket given by the graded [[commutator]] for $f \in end(V)_k$ and $g \in end(V)_l$

$$
  [f,g] = f \circ g - (-1)^{k \cdot l} g \circ f
  \,.
$$

This

$$
  \mathfrak{gl}(V) := (end(V), [-,-])
$$

is the **endomorphism dg-Lie algebra** of $V$. Or, since dg-Lie algebras are special cases of [[L-∞ algebra]]s: the **endomorphism $L_\infty$-algebra** of $V$.

## Representation

A [[representation]] of an [[L-∞ algebra]] $\mathfrak{g}$ on a chain complex $V$ is a morphism of $L_\infty$-algebras

$$
  \rho : \mathfrak{g} \to \mathfrak{gl}(V)
  \,.
$$

Sometimes this is called a _representation up to homotopy_ .

## Examples

* For $V$ a [[vector space]] regarded as a [[chain complex]] concentrated in degree-0, $\mathfrak{gl}(V)$ is the ordinary [[general linear Lie algebra]] of $V$.

## References

See for instance the paragraph above theorem 5.4 in

* [[Tom Lada]], [[Martin Markl]], _Strongly homotopy Lie algebras_ ([arXiv:9406095](http://arxiv.org/abs/hep-th/9406095))

[[!redirects endomorphism L-∞ algebra]]
[[!redirects endomorphism L-∞ algebras]]

[[!redirects endomorphism L-infinity algebra]]
[[!redirects endomorphism L-infinity algebras]]

[[!redirects endomorphism infinity-Lie algebra]]


endomorphism Lie algebra

#Contents#
* table of contents
{:toc}

## Definition

Given a [[vector space]] $V$, then its _endomorphism Lie algebra_ $\mathfrak{end}(V)$ is the [[Lie algebra]] whose elements are the [[linear maps]] $\phi \colon V \to V$ and whose Lie bracket is their [[commutator]]

$$
  [\phi_1, \phi_2] = \phi_1 \circ \phi_2 - \phi_2 \circ \phi_1
  \,.
$$

A Lie algebra homomorphisms $\mathfrak{g} \to \mathfrak{end}(V)$ is a Lie action or _[[Lie algebra representation]]_ of $\mathfrak{g}$ on $V$.

## Related concepts

* [[general linear Lie algebra]]

[[!redirects endomorphism Lie algebras]]


endomorphism monoid object

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#### Enriched category theory
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[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $V$ be a [[monoidal category]]. In a $V$-[[enriched category]], the [[hom-object]] of [[endomorphisms]] on an [[object]] is a [[monoid object]] in $V$, the **endomorphism monoid object**.

Important examples include

* $V$ is [[Set]], where the endomorphism monoid objects are [[endomorphism monoids]],
* $V$ is [[CMon]], where the endomorphism monoid objects are endomorphism [[rigs]],
* $V$ is [[Ab]], where the endomorphism monoid objects are [[endomorphism rings]],
* $V$ is $R$-[[Mod]] and $R$ a [[commutative ring]], where the endomorphism monoid objects are [[endomorphism algebra|endomorphism $R$-algebras]]. For $M$ a [[free module|free $R$-module]] with [[finite set|finite]] [[rank]], the endomorphism $R$-algebra on $M$ is isomorphic to a [[matrix algebra|matrix $R$-algebra]].
* $V$ is [[Top]], where the endomorphism monoid objects are endomorphism [[topological monoids]],

## See also

* [[endomorphism]]

* [[endomorphism monoid]], [[endomorphism ring]], **endomorphism monoid object**

* [[endomorphism operad]]

[[!redirects endomorphism monoid object]]
[[!redirects endomorphism monoid objects]]

[[!redirects endomorphism rig]]
[[!redirects endomorphism rigs]]


endomorphism operad

# Endomorphism operad

* table of contents
{: toc}

## Idea

Just as every [[object]] in a [[category]] has a [[monoid]] of [[endomorphisms]], every object of a [[monoidal category]], or more generally a [[multicategory]], has an [[operad]] of endomorphisms.  This includes not just ordinary endomorphisms $X\to X$, but also higher-ary "endomorphisms" like $X\otimes X\otimes X\to X$.

## Definition

The **endomorphism operad** of an object $X$ in a monoidal category $C$ is the full sub-[[multicategory]] of the [[representable multicategory]] $Rep(C)$ associated to $C$ on the single object $X$.  (Note that an non-symmetric [[operad]] is, essentially by definition, a one-object multicategory.)  More generally, we can consider any one-object full subcategory of a multicategory to be an "endomorphism operad".  This can also be generalized to [[symmetric monoidal categories]] and [[symmetric operads]], and also to other kinds of [[generalized multicategories]].

## Properties

### Algebras

The structure of an [[algebra over an operad]] on an object $A \in C$ over an operad $P$ is equivalently a [[morphism]] of operads

$$
 \rho : P \to End(A)
$$

## Related concepts

* [[endomorphism]]

* [[endomorphism monoid]], [[endomorphism ring]], [[endomorphism monoid object]]

* [[representable multicategory]]

* [[operad of set operations]]

[[!redirects endomorphism operads]]


endomorphism ring

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=--

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* table of contents
{:toc}

## Definition

In an [[Ab]]-[[enriched category]] an [[endomorphism monoid]] is a [[monoid]] in [[Ab]] and hence a [[ring]]: the **endomorphism ring**.

In a [[linear category]] this is an [[associative algebra]] over the ground ring, the _endomorphism algebra_.

## Related concepts

* [[endomorphism]]

* [[endomorphism monoid]], **endomorphism ring**, [[endomorphism monoid object]]

* [[endomorphism operad]]

## References

On endomorphism algebras in ([[dagger compact closed category|dagger]]-)[[compact closed categories]] (as [[Frobenius algebras]]):

* {#Vicary11} [[Jamie Vicary]], Lemma 3.17 in: *Categorical Formulation of Finite-Dimensional Quantum Algebras*, Communications in Mathematical Physics **304** (2011) 765–796 [[doi:10.1007/s00220-010-1138-0](https://doi.org/10.1007/s00220-010-1138-0)]

* Peter Hines, Thm. 50 in: *On strict extensional reflexivity in compact closed categories* [[arXiv:2202.08130](https://arxiv.org/abs/2202.08130)]

[[!redirects endomorphism rings]]

[[!redirects endomorphism algebra]]
[[!redirects endomorphism algebras]]


energy

> this entry needs attention

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# Energy
* table of contents
{: toc}

## Idea

In [[physics]], the __energy__ of a [[physical system]] is one of its most basic [[observable]] properties, an [[extensive quantity]] roughly measuring how much stuff is in the system.

In [[experiment]] it is often measured in [[units]] of [[electronvolt]].

The term 'energy' is also used in [[Riemannian geometry]] and [[statistics]] in vague analogy with the concept from physics.

## Conservation

As a system [[time evolution|evolves]] without interacting with its environment, its energy remains constant.  If different systems [[interaction|interact]], then they may exchange energy, but the total energy remains constant; equivalently, we may combine the interacting systems into a supersystem whose energy is the sum of the energies of its parts, and this supersystem's energy is conserved.

Through [[Noether's theorem]], conservation of energy corresponds to [[time]] invariance.  If a system\'s equations of motion are *not* invariant with time, then the system will not exhibit energy conservation, and physically we conclude that the system must be interacting with something else.

In [[continuum physics]], conservation of energy may be treated locally: the change in energy in any region over a period of time must equal the total flux of energy through the region\'s boundary over that time.  In [[general relativity]], this makes sense (in general) only as a [[differential equation]] involving both the [[stress-energy tensor]] and the [[Levi-Civita connection]].

## Relationship to mass

Energy is distinguished from [[mass]] in various ways:

*  __Relativistic mass__ is the same as energy, although conventionally measured in different [[unit of measurement|units]].  We have $E = m c^2$ in conventional units, or simply $E = m$ in units where the [[speed of light]] is $c = 1$.

*  __[[rest mass|Rest mass]]__ (the usual meaning of 'mass' in modern physics) is the minimum energy inherent in a system; whereas energy is relative, rest mass is absolute.  Here we have
   \[ E = m c^2 \sqrt{1 - \frac {v^2} {c^2}} \label{emcg} \]
   in conventional units, or
   \[ E = m \sqrt{1 - v^2} \label{emg} \]
   if $c = 1$.

*  In the non-relativistic limit, energy splits into two independently conserved quantities: mass (relativistic or rest, it makes no difference) and energy proper.  That is, expanding (eq:emcg) as a [[power series]] in $v$, we have
   $$ E = m c^2 + \frac 1 2 m v^2 + \frac 3 8 m \frac {v^4} {c^2} + \cdots ;$$
as $c \to \infty$, this splits into the quantities $m$, $(1/2) m v^2$, $(3/8) m v^4$, etc.  The first of these is the non-relativistic mass, the second is the [[kinetic energy]] $K$, the next is simply $3 K^2 / 2 m$, and all subsequent terms are also determined by $m$ and $K$.

   However, the kinetic energy $K$ is not the *total* energy in non-relativistic physics.  Any entities with a rest mass of $0$ (or approaching zero in the limit) will still have energy and so must be treated separately from kinetic energy (which is proportional to mass).  Non-relativistically, these may be viewed as [[field theory|fields]] with energy of their own, or (if the field strengths are derived entirely from interacting massive bodies) this energy may be attributed to the massive bodies in the system as [[potential energy]].  Thus the total energy of the system is $E = K + U$, where $K$ is total kinetic energy and $U$ is total potential or field energy.

## In special relativity

In relativistic physics, energy is relative, the timelike part of a [[4-vector]] quantity whose spacelike parts comprise [[linear momentum]].  If we also take into account the distribution of energy and momentum across space, this give a symmetric second-rank [[tensor]], the [[stress-energy tensor]].

Even in non-relativistic physics, there is an analogy between energy and linear momentum.  In [[Hamiltonian mechanics]], both energy and linear momentum are examples of [[generalized momentum|generalised momenta]]; energy corresponds to [[time]] while linear momentum corresponds to position in [[physical space|space]].  However, the special treatment accorded to time and energy in the usual formulation of Hamiltonian mechanics obscures this; we also have (oddly) that the precise generalised momentum corresponding to the time variable $t$ is not the total energy $H$ but $-H$ instead.

This analogy from [[classical mechanics]] also applies to [[quantum mechanics]] and is particularly evident in Schrödinger's original formulation of his quantum theory.

[[!include plane waves -- table]]

## In general relativity

In [[general relativity]], the concept of total energy of a system does not make sense, except in certain special cases.  General relativity is a [[field theory]] in which energy appears only as one of $10$ (or $({n \atop 2})$ in $n$ [[dimensions]] of [[spacetime]]) independent components of the [[stress-energy tensor]].  Mathematically, it makes no sense to integrate just this component over a region of space to determine a total energy in that region.  This makes it difficult to interpret general relativity in frameworks such as [[Hamiltonian mechanics]] and [[thermodynamics]] where energy is a primary concept.  This is called the [[problem of time]] (since energy and [[time]] are conjugate observables and the problem occurs with both).

## Energy scales

[[energy]] $\,$ [[scales]]:

\begin{center}
<img src="https://ncatlab.org/nlab/files/EnergyScales.jpg" width="600">
\end{center}

> graphics grabbed from [here](http://sten.astronomycafe.net/the-particle-desert/)

## Related concepts

* [[rest energy]]

* [[binding energy]]

* [[Dirichlet energy]], [[harmonic map]]

## References

Serious references discussing energy in general relativity include

* [[Edward Witten]], _A new proof of the positive energy theorem_, Comm. Math. Phys. __80__ (1981), no. 3, 381–402, [MR83e:83035](http://www.ams.org/mathscinet-getitem?mr=626707), [euclid](http://projecteuclid.org/getRecord?id=euclid.cmp/1103919981)
* A. Trautman, _Conservation laws in general relativity. In: Gravitation: an introduction to current research_, Witten, L. (ed.). New York: Wiley 1962;
* Misner, C., Thome, K.S., Wheeler, J.A.: Gravitation. San Francisco: W. H. Freeman 1973;
* S. Weinberg, _Gravitation and cosmology_, New York: Wiley 1972
* R. Schoen, S. T. Yau, Comm. Math. Phys. 65 (1979), no. 1, 45--76; MR80j:83024

[[!redirects energy]]
[[!redirects energies]]

[[!redirects relativistic mass]]
[[!redirects relativistic masses]]


energy condition

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* table of contents
{:toc}

## Idea

The theory of [[classical field theory|classical]] [[gravity]] ([[general relativity]]) a priori and straightforwardly admits the coupling to gravity of ([[matter]]-)[[field (physics)|fields]] with any covariant [[local Lagrangian]]. The resulting [[Einstein equations]] will always exist and will equate the [[Einstein tensor]] with the given [[stress-energy tensor]] of the matter fields.

While that hence gives consistent theories in themselves -- a set of [[differential equations]] --, not all choices of matter Lagrangians are physically reasonable. One consistency condition for physical reasonable choices is for instance that from far away any non-vanishing matter distribution should behave as a single ball of positive mass. An _energy condition_ is an extra condition imposed on a [[stress-energy tensor]] that is meant to reflect physical plausibility conditions of this kind.

There are various conditions of various strengths considered in the literature. One is called the _dominant energy condition_ and that indeed implies the above asymptotic  positivity of mass distributions, this is the content of the _[[positive energy theorem]]_.

## Related concepts

* [[phantom dark energy]]

## References

See also:

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

Discussion of the [[averaged null energy condition]]  for [[conformal field theory]] on [[de Sitter spacetime]] and [[anti de Sitter spacetime]]:

* Felipe Rosso, _Global aspects of conformal symmetry and the ANEC in dS and AdS_ ([arXiv:1912.08897](https://arxiv.org/abs/1912.08897))

See also:

* Taha A Malik, Rafael Lopez-Mobilia, _Proof of the quantum null energy condition for free fermionic field theories_ ([arXiv:1910.07594](https://arxiv.org/abs/1910.07594))

* Philipp Stanzer, _Numerical Relativity, Holography and the Quantum Null Energy Condition_ ([arXiv:2009.07040](https://arxiv.org/abs/2009.07040))

[[!redirects energy conditions]]

[[!redirects null energy condition]]
[[!redirects null energy conditions]]

[[!redirects dominant energy condition]]


Engeler's lemma

# Contents #
* table of contents
{:toc}

## Idea

Engeler's lemma is one of a number of set-theoretic propositions connected with [[axiom of choice|choice principles]] and [[prime ideal theorem|prime ideal]]/[[maximal ideal theorem|maximal ideal]] theorems. A key phrase for such set-theoretic propositions is "systems of finite character"; related results include the [[Teichmüller-Tukey lemma]].

## Statement and proof

+-- {: .num_theorem}
###### Theorem
**(Engeler's lemma)**
Let $X$ be a [[set]], and let $F$ be a collection of [[partial functions]] $f: X \rightharpoonup 2 = \{0, 1\}$:

$$X \hookleftarrow A \stackrel{f}{\to} 2$$

where $A = \dom(f)$ is the [[domain]] of $f$. Suppose the following 3 conditions are satisfied:

1. For every $f: \dom(f) \to 2$ in $F$ and finite subset $S \subseteq \dom(f)$, the restriction $Res_S(f): S \to 2$ belongs to $F$;

2. If $f: X \rightharpoonup 2$ is any partial function such that $Res_S(f)$ belongs to $F$ for every [[finite set|finite]] subset $S \subseteq \dom(f)$, then $f$ belongs to $F$;

3. Every finite subset $S \subseteq X$ is the domain of some $f \in F$.

Then, assuming the [[ultrafilter theorem|ultrafilter principle]] (UP), there exists a [[total function]] $f \in F$.
=--

+-- {: .proof}
###### Proof
We abuse notation by identifying partial functions $f: X \rightharpoonup \{0, 1\}$ with total functions $X \to 3 = \{0, 1, 2\}$, defining $\dom(f)$ for $f: X \to 3$ in terms of the corresponding partial function: $\dom(f) = \{x \in X: f(x) \neq 2\}$. For $f: X \to 3$, the notation $Res_S(f)$ means we regard $f$ as a partial function, obtain another partial function by restriction, and consider this a function $X \to 3$.

Thus $F$ defines a subspace $F \subseteq 3^X$. Under [[Tychonoff theorem|UP]], the [[product]] space $3^X$ is [[compact Hausdorff space|compact Hausdorff]]. We claim $F$ is [[closed subset|closed]]. For suppose $f: X \to 3$ is in the closure of $F$. This means that for any finite $A \subseteq X$ there exists $g \in F$ with $f|_A = g|_A$. In particular, for any finite $B \subseteq \dom(f)$ there exists $g \in F$ such that $f|_B = g|_B$. In particular $g(x) \neq 2$ for all $x \in B$, so $B \subseteq \dom(g)$, and so $Res_B(f) = Res_B(g)$ belongs to $F$ by condition 1. Since $Res_B(f)$ belongs to $F$ for all finite $B \subseteq \dom(f)$, we have that $f \in F$ by condition 2. Thus $F$ is closed, and therefore compact.

Now we claim that the set $T = \{f \in F: \forall_{x \in X} f(x) \neq 2\}$ is [[inhabited set|inhabited]], which in the partial function interpretation means $F$ contains a total function. For, the family of closed sets $C_x = F \cap \{f \in 3^X: f(x) \neq 2\}$ satisfies the finite intersection property, by condition 3. Thus $T = \bigcap_{x \in X} C_x$ is inhabited by compactness of $F$.
=--

## References

* Marcel Erné, _Prime Ideal Theorems and systems of finite character_, Commentationes Mathematicae Universitatis Carolinae, Vol. 38, No. 3 (1997), 513-536. ([web](http://dml.cz/dmlcz/118949))


enhanced factorisation system

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#### Factorization systems
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#### Category theory
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#Contents#
* table of contents
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## Definition

An **enhanced factorisation system** on a [[2-category]] is an orthogonal [[factorisation system on a 2-category]] such that, for any invertible 2-cell $\alpha : t e \Rightarrow m s$

\begin{tikzcd}
	A & B \\
	C & D
	\arrow["e", from=1-1, to=1-2]
	\arrow["t", from=1-2, to=2-2]
	\arrow["s"', from=1-1, to=2-1]
	\arrow["m"', from=2-1, to=2-2]
	\arrow["\alpha"{description}, draw=none, from=1-2, to=2-1]
\end{tikzcd}

there is a unique pair of a 1-cell $r : B \to C$ and invertible 2-cell $\beta : t \Rightarrow m r$ such that $r e = s$ and $\beta e = \alpha$.

## References

Enhanced factorisation systems were defined in

* [[Stephen Lack]]. _Codescent objects and coherence_. Journal of Pure and Applied Algebra **175** 1-3 (2002) 223-241 [doi:10.1016/S0022-4049(02)00136-6]

attributed to [[Max Kelly]].

[[!redirects enhanced factorisation systems]]
[[!redirects enhanced factorization system]]
[[!redirects enhanced factorization systems]]


enhanced gauge symmetry

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#Contents#
* table of contents
{:toc}

## Idea

Generally, an _enhanced gauge symmetry_ refers to the appearance of [[gauge theory]] with a larger [[gauge group]] than may be superficially apparent.

### On coincident D-branes

A famous case is the enhanced gauge symmetry in the [[Chan-Paton gauge fields]] of coincident [[D-branes]]. A priori each D-brane carries a [[complex line bundle]]/[[circle principal bundle]], but as $n$ of them coincide the [[gauge group]] is expected to enhance from $(U(1))^n$ to the [[unitary group]] $U(n)$ (or the [[special unitary group]] $SU(n)$).

Mathematically this gauge enhancement on D-branes is modeled by the [[Baum-Douglas geometric cycle]] model for [[K-homology]] ([Reis-Szabo 05, def. 1.5 and p. 16](#ReisSzabo05), [Szabo 08, p. 4](#Szabo08)): One such cycle is given by a submanifold $W \overset{\phi}{\hookrightarrow} X$ of [[spacetime]] $X$ equipped with [[spin^c structure]] and with a [[complex vector bundle]] $E \to X$, and given two such cycles $(W,\phi,E_1)$ and $(W,\phi,E_2)$ with the same underlying manifold ("coincident D-brane [[worldvolume]]") then their formal [[direct sum]] is identified with the single cycle carrying the [[direct sum of vector bundles]]

$$
  (W,\phi, E_1) + (W, \phi, E_2)
  \;\sim\;
  (W, \phi, E_1 \oplus E_2)
  \,.
$$

If here $E_i$ has [[rank]] $n_i$ and hence [[structure group]] $U(n_i)$, then the [[direct sum of vector bundles]] $E_1 \oplus E_2$ has [[structure group]] the [[product group]] $U(n_1) \times U(n_2)$.

In particular the sum of $n$ coincident cycles each carrying a line bundle yields a rank-$n$ vector bundle with structure group $(U(1))^n$. In this sense then a cycle carrying a general rank $n$-vector bundle, hence with structure group $U(n)$, may be thought of as a "gauge enhancement" of making $n$ D-branes coincide.

### From massless BPS states

In the context of the realization of [[super Yang-Mills theories]] as [[effective field theories]] of [[superstring theory]], [[BPS branes]] in the string background becomes BPS states of the SYM theory. These become massless as the [[cycles]] on which the BPS charge is supported shrink away (at boundary points of the given [[moduli space]]). In this case these additional massless states may appear as additional gauge bosons in the compactified gauge theory which thereby develops a larger gauge group. Typically a previously abelian gauge group becomes non-abelian this way.

Examples include [[KK-compactification]] of [[M-theory]] on [[K3]] fibers and on [[G2-manifolds]] with [[ADE singularities]], corresponding to [[F-theory]] on [[elliptic fibrations]] making [[K3]]-fibrations with singular fibers, corresponding to [[heterotic string theory]] on singular elliptic fibrations. See at

* [[M-theory lift of gauge enhancement on D6-branes]]

* _[[M-theory on G2-manifolds]]_ the section _[Nonabelian gauge groups and chiral fermions at orbifold singularities](M-theory+on+G2-manifolds#EnhancedGaugeGroups)_;

* _[[F-theory]]_ the section _[F-brane scan](F-theory#Fbranescan)_

* _[[duality between F-theory and heterotic string theory]]_.

### For MK6 branes at ADE-singularities

The [[blow-up]] of an [[ADE-singularity]] is given by a [[union]] of [[Riemann spheres]] that touch each other such as to form the shape of the [[Dynkin diagram]] whose A-D-E label corresponds to that of the given [[finite subgroup of SU(2)]].

This statement is originally due to ([duVal 1934 I, p. 1-3 (453-455)](#duVal1934I)). A description in terms of [[hyper-Kähler geometry]] is due to [Kronheimer 89a](#Kronheimer89a).

Quick survey of this fact is in [Reid 87](#Reid87), a textbook account is [Slodowy 80](#Slodowy80).

{#InStringTheory} In [[string theory]] this fact is interpreted in terms of [[gauge enhancement]] of the [[M-theory]]-lift of coincident [[black brane|black]] [[D6-branes]] to an  [[MK6]] at an ADE-singularity ([Sen 97](enhanced+gauge+symmetry#Sen97)):

> graphics grabbed from [HSS18](http://ncatlab.org/schreiber/show/Equivariant+homotopy+and+super+M-branes)

See at _[[M-theory lift of gauge enhancement on D6-branes]]_ for more.

### On M9-Branes

In [[Horava-Witten theory]] there is supposed to be gauge enhancement over the [[M9-branes]] such as to identify their [[worldvolume]] theory with [[E8]]-[[heterotic string theory]].

Under the [[duality in string theory|duality]] between [[M-theory]] and [[type IIA string theory]] the M9-brane may be identified with the [[O8-plane]]:

> from [GKSTY 02, section 3](#GKSTY02)

This may be used to understand the [[gauge enhancement]] to [[E8]]-[[gauge groups]] at the [[heterotic string theory|heterotic]] boundary of [[Horava-Witten theory]]:

> from [GKSTY 02, section 3](#GKSTY02)

## Related concepts

* [[spontaneously broken symmetry]]

## References

### On coincident D-branes
  {#ReferencesOnCoincidentDBranes}

The idea that on $N$ coincident D-branes there is gauge enhancement to $U(N)$-gauge field theory is due to

* {#Witten95} [[Edward Witten]], section 3 of _Bound States Of Strings And $p$-Branes_, Nucl.Phys.B460:335-350, 1996 ([arXiv:hep-th/9510135](https://arxiv.org/abs/hep-th/9510135))

There, this is called an "obvious guess" (first line on p. 8). Subsequently,  most authors cite this obvious guess as a fact; for instance the review

* {#Myers03} [[Robert Myers]], section 3 of _Nonabelian Phenomena on D-branes_, Class.Quant.Grav. 20 (2003) ([arXiv:hep-th/0303072](https://arxiv.org/abs/hep-th/0303072))

Discussion in terms of [[boundary conformal field theory]] is in

* {#FuchsKasteLercheLutkenSchweigert00} [[Jürgen Fuchs]], P. Kaste, [[Wolfgang Lerche]], C. Lutken, [[Christoph Schweigert]], J. Walcher, _Boundary Fixed Points, Enhanced Gauge Symmetry and Singular Bundles on K3_, Nucl.Phys.B598:57-72, 2001 ([arXiv:hep-th/0007145](https://arxiv.org/abs/hep-th/0007145))

By actual computation in [[open string field theory]] "convincing evidence" for the nonabelian [[Yang-Mills theory]] on coincident D-branes was found, numerically, in (see bottom of p. 22)

* {#ColettiSigalovTaylor03} Erasmo Coletti, Ilya Sigalov, [[Washington Taylor]], _Abelian and nonabelian vector field effective actions from string field theory_, JHEP 0309 (2003) 050 ([arXiv:hep-th/0306041](https://arxiv.org/abs/hep-th/0306041))

Similar numerical derivation, as well as exact derivation at zero momentum, is in

* [[Nathan Berkovits]], [[Martin Schnabl]], _Yang-Mills Action from Open Superstring Field Theory_, JHEP 0309 (2003) 022 ([arXiv:hep-th/0307019](https://arxiv.org/abs/hep-th/0307019))

The first full derivation seems to be due to

* {#Lee16} [[Taejin Lee]], _Covariant open bosonic string field theory on multiple D-branes in the proper-time gauge_, Journal of the Korean Physical Society December 2017, Volume 71, Issue 12, pp 886–903 ([arXiv:1609.01473](https://arxiv.org/abs/1609.01473))

which is surveyed in

* {#Lee17} [[Taejin Lee]], _Deformation of the Cubic Open String Field Theory_, Phys. Lett. B 768 (2017) 248 ([arXiv:1701.06154](https://arxiv.org/abs/1701.06154))

That on [[D0-branes]] this reproduces the [[BFSS matrix model]] and on [[D(-1)-branes]] the [[IKKT matrix model]] is shown in

* [[Taejin Lee]], _Covariant Open String Field Theory on Multiple D$p$-Branes_ ([arXiv:1703.06402](https://arxiv.org/abs/1703.06402))

Discussion of gauge enhancement on coincident D-branes in terms of [[Baum-Douglas geometric cycles]] for [[K-homology]] is in

* {#ReisSzabo05} Rui Reis, [[Richard Szabo]], _Geometric K-Homology of Flat D-Branes_ ,Commun.Math.Phys. 266 (2006) 71-122, Journal of the Korean Physical Society December 2017, Volume 71, Issue 12, pp 886–903 ([arXiv:hep-th/0507043](https://arxiv.org/abs/hep-th/0507043))

reviewed in

* {#Szabo08} [[Richard Szabo]], _D-branes and bivariant K-theory_, Noncommutative Geometry and Physics 3 1 (2013): 131. ([arXiv:0809.3029](http://arxiv.org/abs/0809.3029))

Derivation via [[rational parameterized stable homotopy theory]] applied to [[schreiber:The brane bouquet]] is in

* [[Vincent Braunack-Mayer]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Gauge enhancement of Super M-Branes]]_, Comm. Math. Phys. **371** (2019) 197-265 [[arXiv:1806.01115](https://arxiv.org/abs/1806.01115), [doi:10.1007/s00220-019-03441-4](https://doi.org/10.1007/s00220-019-03441-4)]

### On M-branes at ADE-singularities

The picture of [[M-theory lift of gauge enhancement on D6-branes]] is due to

* {#Sen97} [[Ashoke Sen]], Section 2, _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))

Review of the standard lore of gauge enhancement in M-theory includes

* {#Barrett06} Adam B. Barrett, section 2.3 (following [Acharya-Gukov 04](#AcharyaGukov04)) of _M-Theory on Manifolds with $G_2$ Holonomy_, 2006 ([arXiv:hep-th/0612096](http://arxiv.org/abs/hep-th/0612096))

(...)

### General

Original articles include

* [[Edward Witten]], section 4.6 of _[[String Theory Dynamics In Various Dimensions]]_, Nucl.Phys.B443:85-126,1995 ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

* {#HullTownsend95} [[Chris Hull]], [[Paul Townsend]], _Enhanced Gauge Symmetries in Superstring Theories_, Nucl.Phys. B451 (1995) 525-546 ([arXiv:hep-th/9505073](http://arxiv.org/abs/hep-th/9505073))

* [[Paul Aspinwall]], _Enhanced Gauge Symmetries and K3 Surfaces_, Phys.Lett. B357 (1995) 329-334 ([arXiv:hep-th/9507012](http://arxiv.org/abs/hep-th/9507012))

* [[Chris Hull]], _Duality, Enhanced Symmetry and Massless Black Holes_, in Proceedings of Strings'95: Future Perspectives in String Theory (World Scientific, 1996, I. Bars et al. eds.), p. 230 ([pdf](http://physics.usc.edu/Strings95/Proceedings/pdf/hull.pdf))

* M. Bershadsky, [[Ken Intriligator]], [[Shamit Kachru]], [[David Morrison]], V. Sadov, [[Cumrun Vafa]], _Geometric Singularities and Enhanced Gauge Symmetries_, Nucl.Phys.B481:215-252,1996 ([arXiv:hep-th/9605200](http://arxiv.org/abs/hep-th/9605200))

* [[Philip Candelas]], Eugene Perevalov, Govindan Rajesh, _Toric Geometry and Enhanced Gauge Symmetry of F-Theory/Heterotic Vacua_, Nucl.Phys. B507 (1997) 445-474 ([arXiv:hep-th/9704097](http://arxiv.org/abs/hep-th/9704097))

* [[Mirjam Cvetic]], [[Chris Hull]], _Wrapped Branes and Supersymmetry_, Nucl.Phys.B519:141-158,1998 ([arXiv:hep-th/9709033](http://arxiv.org/abs/hep-th/9709033))

* {#AcharyaGukov04} [[Bobby Acharya]], [[Sergei Gukov]], _M theory and Singularities of Exceptional Holonomy Manifolds_, Phys.Rept.392:121-189,2004 ([arXiv:hep-th/0409191](http://arxiv.org/abs/hep-th/0409191))

* {#HalvorsonMorrison15} [[James Halverson]], [[David Morrison]], _On Gauge Enhancement and Singular Limits in $G_2$ Compactifications of M-theory_ ([arXiv:1507.05965](http://arxiv.org/abs/1507.05965))

* [[Neil Lambert]], Miles Owen, _Charged Chiral Fermions from M5-Branes_ ([arXiv:1802.07766](https://arxiv.org/abs/1802.07766))

Discussion via duality of [[M9-branes]] to [[O-planes]] is in

* {#GKSTY02} E. Gorbatov, V.S. Kaplunovsky, J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, section 3 of _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))

[[!redirects enhanced gauge symmetries]]

[[!redirects gauge enhancement]]
[[!redirects gauge enhancements]]


enhanced triangulated category

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#Contents#
* table of contents
{:toc}

## Idea

Any [[stable (∞,1)-category]] $C$ gives rise to a canonical [[triangulated category]] $ho(C)$, its [[homotopy category of an (∞,1)-category|homotopy category]].  Much of the information contained in $C$ is still present in $ho(C)$, so that in many situations it can be enough to work at the level of [[triangulated categories]].  However, in general various problems with [[triangulated categories]] require one to work with some sort of higher structure lying above $ho(C)$.  Such a structure is often called an _enhancement_ of the [[triangulated category]] $ho(C)$.

{#ExamplesOfNotions} Examples of enhanced triangulated categories are, in order of increasing information:

* [[stable derivators]],
* [[pretriangulated dg-categories]] or [[pretriangulated A-∞-categories]],
* and [[stable (∞, 1)-categories]] themselves.

## History

After [[Jean-Louis Verdier]] started the development of the theory of triangulated categories in his 1963 thesis, it was quickly recognized that some sort of enhancements were necessary to get the full picture.  For this purpose, [[Alexandre Grothendieck]] suggested the notion of [[derivator]] in the early 1980's, while [[Alexei Bondal]] and [[Mikhail Kapranov]] developed the notion of [[pretriangulated dg-category]] in 1990.  In 2006 [[Jacob Lurie]] developed the notion of [[stable (∞,1)-category]].

## Problems with triangulated categories

* non-functoriality of the [[mapping cone]],
* more generally, non-existence of [[homotopy colimits]] and [[homotopy limits]],
* ...

(to be expanded on)

## Comparison results

In [[pretriangulated dg-category#Cohn13|(Cohn 13)]] it is shown that the [[(∞,1)-category]] of  [[idempotent complete (infinity,1)-category|idempotent complete]] [[pretriangulated dg-categories]] over a [[commutative ring]] $k$ is equivalent to the [[(∞,1)-category]] of [[stable (∞,1)-category|stable]] [[k-linear (∞,1)-categories]].  See [[pretriangulated dg-category]] for details.

In [[derivator#Renaudin|(Renaudin 2006)]] it is shown that the [[2-category]] of locally presentable [[derivators]] is equivalent to the [[localization]] of the [[2-category]] of [[combinatorial model categories]] at the [[Quillen equivalences]].  [[combinatorial model categories|Combinatorial model categories]] are equivalent to [[locally presentable (∞,1)-categories]] in a sense made precise there.  However, a comparison between the stability conditions in the derivator and (∞,1)-category settings seems still to be open.

## Related concepts

* [[triangulated category]]

* [[stable derivator]]

* [[pretriangulated dg-category]], [[pretriangulated A-infinity category]]

* [[stable (∞,1)-category]]

## References

* {#BondalKapranov91} [[Alexei Bondal]], [[Mikhail Kapranov]], _Enhanced triangulated categories_, Mat. Sb. (Russian) 181:5 (1990), 669–683 [link](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=1196); Engl. transl. in Sbornik Math. 70:1 (1991)  93–107 [doi:10.1070/SM1991v070n01ABEH001253](https://doi.org/10.1070/SM1991v070n01ABEH001253) [MR1055981](https://www.ams.org/mathscinet-getitem?mr=1055981)

* [[Valery Lunts|Valery A. Lunts]], [[Dmitri Orlov|Dmitri O. Orlov]], _Uniqueness of enhancement for triangulated categories_, J. Amer. Math. Soc. 23 (2010), no. 3, 853–908

* [[Benjamin Antieau]], _On the uniqueness of infinity-categorical enhancements of triangulated categories_, [arxiv/1812.01526](https://arxiv.org/abs/1812.01526)

Brief overview and list of further references:

* {#SchwedeHausmann19} [[Stefan Schwede]], [[Markus Hausmann]], _Enhancements of triangulated categories_, Graduate Seminar Topology, Bonn 2019-2020 ([[Schwede_EnhancedSeminar.pdf:file]])

[[!redirects enhanced triangulated categories]]

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[[!redirects DG enhancements]]

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enriched (infinity,1)-category

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#### Enriched category theory
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#Contents#
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## Idea

The notion of _enriched $(\infty,1)$-category_ is the generalization of the notion of _[[enriched category]]_ from [[category theory]] to [[(∞,1)-category theory]]. For $\mathcal{V}$ a [[monoidal (∞,1)-category]], a _$\mathcal{V}$-enriched $(\infty,1)$-category_ $C$ is

* a [[set]]/[[class]] of [[objects]] $Obj(C)$;

* for every tuple $(a \in C,b \in C)$ an object $C(a,b) \in \mathcal{V}$ "of [[morphisms]]" from $a$ to $b$ in $C$;

* for each sequence $(a_i \in C)_{i = 0}^n$ of objects in $C$ an object $C(a_0, \cdots, a_n) \in \mathcal{V}$ "of sequences of composable morphisms _and_ their composites";

* such that these composites exist essentially uniquely and satisfy [[associativity]] in a [[coherence|coherent]] fashion.

One way to make this precise in a general abstract way should be to define $C$ to be a [[∞-algebra over an (∞,1)-operad]] in $\mathcal{V}^{\otimes}$ over [[Assoc]]${}_{Obj(C)}$, the $Obj(C)$-colored version of the [[associative operad]];

$$
  C \in Alg_{Assoc_{Obj(C)}}(\mathcal{V}^\otimes)
  \,.
$$

For $\mathcal{V} \in$ [[∞Grpd]], this should be [[equivalence of (∞,1)-categories|equivalent]] to ordinary [[(∞,1)-categories]]. This is for instance in ([Lurie, def. 4.2.1.12](#Lurie)).

A construction that should be a model for this notion in terms of a [[model category]] presentation for $\mathcal{V}$ is discussed in ([Simpson](#Simpson)). For the case that $\mathcal{V} = $ [[∞Grpd]] [[presentable (∞,1)-category|presented]] by the standard [[model structure on simplicial sets]] this reproduces the notion of _[[Segal categories]]_. (See there for further details and references.) The iteration of this construction yields [[Segal n-categories]], a model for [[(∞,n)-categories]].

Once a model category $V$ for $\mathcal{V}$ has been chosen, one can consider [[semi-strict infinity-category|semi-strict]] $\infty$-enrichments given by ordinary $V$-[[enriched categories]] equipped with a notion of weak equivalence that remembers that these are presentations for enriched $(\infty,1)$-categories. See also _[[enriched homotopical category]]_.

## Examples
 {#Examples}

A [[stable (∞,1)-category]] is naturally enriched in the [[(∞,1)-category of spectra]].

More generally, for $R$ an [[E-∞ ring]] then an $R$-[[linear (∞,1)-category]] is naturally enriched in $R$-[[∞-modules]]. (This includes the previous case for $R$ the [[sphere spectrum]].)

A [[closed monoidal (∞,1)-category]] is naturally enriched over itself.

([Gepner-Haugseng 13](#GepnerHaugseng13))

## Related concepts

* [[enriched (∞,1)-functor]]

* [[Segal category]], [[Segal n-category]]

* [[enriched homotopical category]], [[weak enrichment]]

* [[internal category in an (∞,1)-category]]

* [[table - models for (∞,1)-operads]]

* [[enriched type]]

## References

* {#Simpson} [[Carlos Simpson]], _[[Homotopy Theory of Higher Categories]]_ ([arXiv:1001.4071](http://arxiv.org/abs/1001.4071))

* {#Lurie} [[Jacob Lurie]], _[[Higher Algebra]]_

* {#GepnerHaugseng13} [[David Gepner]], [[Rune Haugseng]], _Enriched ∞-categories via non-symmetric ∞-operads_ ([arXiv:1312.3178](http://arxiv.org/abs/1312.3178))

* Hadrian Heine, _An equivalence between enriched ∞-categories and ∞-categories with weak action_, [arxiv](https://arxiv.org/abs/2009.02428), 2020

* [[John D. Berman]], *Enriched infinity categories I: enriched presheaves* ([arXiv:2008.11323](https://arxiv.org/abs/2008.11323))

Further discussion of [[(infinity,n)-categories]] as $\infty$-categories enriched in $(\infty,n-1)$-categories is (via [[Theta-spaces]]) in

* {#BergnerRezk} [[Julie Bergner]], [[Charles Rezk]], _Comparison of models for $(\infty,n)$-categories_, I ([arXiv:1204.2013](http://arxiv.org/abs/1204.2013))

* {#BergnerRezk14} [[Julie Bergner]], [[Charles Rezk]], _Comparison of models for $(\infty,n)$-categories_, II ([arXiv:1406.4182](http://arxiv.org/abs/1406.4182))

See also

* {#Haugseng15} [[Rune Haugseng]], _Bimodules and natural transformations for enriched ∞-categories_ ([arXiv:1506.07341](http://arxiv.org/abs/1506.07341))

On [[(infinity,1)-colimits|$\infty$-colimits]] and [[Day convolution]] in the context of [[enriched (infinity,1)-categories|enriched $\infty$-categories]]:

* [[Vladimir Hinich]], *Colimits in enriched ∞-categories and Day convolution*, Theory and Applications of Categories **39** 12 (2023) 365-422 [[tac:39-12](http://www.tac.mta.ca/tac/volumes/39/12/39-12abs.html), [arXiv:2101.09538](https://arxiv.org/abs/2101.09538)]

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

[[!redirects enriched (infinity,1)-categories]]


enriched (∞,1)-functor

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#### Enriched category theory
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#Contents#
* table of contents
{:toc}

## Idea

The canonical [[homomorphisms]] between [[enriched (∞,1)-categories]] are called _enriched (∞,1)-functors_. This is the generlization to [[(∞,1)-category theory]] of the concept of [[enriched functors]] in [[category theory]].

## Examples

* A _[[spectral Mackey functor]]_ is an enriched $(\infty,1)$-functor, out of a [[Burnside category]], enriched over the [[(∞,1)-category of spectra]].

## References

See the references at _[[enriched (∞,1)-category]]_.

[[!redirects enriched (∞,1)-functors]]

[[!redirects enriched (infinity,1)-functor]]
[[!redirects enriched (infinity,1)-functors]]


enriched adjoint functor

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#Contents#
* table of contents
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## Idea

The concept of _enriched adjoint functors_ is the generalization of that of [[adjoint functors]] ([[adjunctions]] in [[Cat]]) from [[category theory]] to [[enriched category theory]] ([[adjunctions]] in [[VCat]]).

## Definition

\begin{definition}\label{EnrichedAdjunction}
**(enriched adjoint functors)**
\linebreak
For $\mathcal{V}$ a [[closed monoidal category|closed]] [[symmetric monoidal category]] with all [[limits]] and [[colimits]] (a [[cosmos for enrichment]]), let $\mathcal{C}$, $\mathcal{D}$ be a [[pair]] of $\mathcal{V}$-[[enriched categories]].

Then an _enriched [[adjoint pair]] of $\mathcal{V}$-[[enriched functors]]_ or _$\mathcal{V}$-enriched adjunction_ between them

$$
  \mathbf{C}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\; \bot \;\;}
  \mathbf{D}
$$

is a [[pair]] of $\mathcal{V}$-[[enriched functors]] as shown, such that the following equivalent conditions hold ([Kelly, §1.11](#Kelly))

* there exists an [[adjunction]] between them when regarded as [[1-morphisms]] in the [[2-category]] [[VCat|$\mathcal{V}Cat$]], namely $V$-[[enriched natural transformations]]

  $\eta \,\colon\, Id_{\mathbf{D}} \longrightarrow R \circ L$ (the [[adjunction unit]])
  and
  $\epsilon \,\colon\, R \circ L \longrightarrow id_{\mathbf{C}}$ (the [[adjunction counit]])

  such that the following *[[zig-zag law]]* holds in [[VCat|$\mathcal{V}Cat$]]:

\begin{tikzcd}[row sep=10pt]
  \mathbf{D}
  \ar[rr, bend left=40, "{\mathrm{id}}",
      "{\ }"{name=s1, swap}
  ]
  \ar[r, "L"{description}, "{\ }"{name=t1, pos=.9}]
  \ar[
    from=s1,
    to=t1,
    Rightarrow,
    "{ \eta }"{swap}
  ]
  &
  \mathbf{C}
  \ar[rr, bend right=40, "{\mathrm{id}}"{swap},
    "{\ }"{name=t2}
  ]
  \ar[r, "R"{description}]
  &
  \mathbf{D}
  \ar[r, "L"{description}, "{\ }"{swap, name=s2, pos=.1}]
  \ar[
    from=s2,
    to=t2,
    Rightarrow,
    "{ \epsilon }"
  ]
  &
  \mathbf{C}
  &=&
  \mathbf{D}
  \ar[r, bend left=40, "{L}", "{\ }"{name=s3, swap}]
  \ar[r, bend right=40, "{\mathrm{id}}"{swap}, "{\ }"{name=t3}]
  \ar[from=s3, to=t3, Rightarrow, "{\mathrm{id}}"]
  &
 \mathbf{C}
\end{tikzcd}

\begin{tikzcd}[row sep=10pt]
  \mathbf{C}
  \ar[rr, bend right=40, "{\mathrm{id}}"{swap},
      "{\ }"{name=s1}
  ]
  \ar[r, "R"{description}, "{\ }"{name=t1, pos=.9, swap}]
  \ar[
    from=t1,
    to=s1,
    Rightarrow,
    "{ \epsilon }"{swap, pos=.3}
  ]
  &
  \mathbf{D}
  \ar[rr, bend left=40, "{\mathrm{id}}",
    "{\ }"{name=t2, swap}
  ]
  \ar[r, "L"{description}]
  &
  \mathbf{C}
  \ar[r, "R"{description}, "{\ }"{name=s2, pos=.1}]
  \ar[
    from=t2,
    to=s2,
    Rightarrow,
    "{ \eta }"{pos=.8}
  ]
  &
  \mathbf{D}
  &=&
  \mathbf{C}
  \ar[r, bend left=40, "{R}", "{\ }"{name=s3, swap}]
  \ar[r, bend right=40, "{R}"{swap}, "{\ }"{name=t3}]
  \ar[from=s3, to=t3, Rightarrow, "{\mathrm{id}}"]
  &
 \mathbf{D}
\end{tikzcd}

* {#ViaHomIsomorphism} there is a $\mathcal{V}$-[[enriched natural isomorphism|enriched natural]] [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) between [[enriched hom-functors]], of the form

$$
  \mathbf{C}\big(L(-),-\big)
  \;\simeq\;
  \mathbf{D}\big(-,R(-)\big)
  \,.
$$
\end{definition}
If the [[adjunction unit]] and [[adjunction counit]] are (enriched natural) *[[isomorphisms]]* then this is called an *enriched adjoint equivalence*.

(e.g. [Kelly, §1.11](#Kelly), [Borceux 94, Def. 6.7.1](#Borceux94))

The [[2-functor]] $\mathcal{V}$-$Cat \rightarrow {Cat}$ that sends an enriched category to its underlying ordinary category (i.e. the [[change of enrichment]] to the [[terminal category]] along $\mathcal{V} \to \ast$) sends a pair of enriched adjoint functor to an ordinary pair of [[adjoint functors]] and sends an enriched adjoint equivalence to an [[adjoint equivalence]].

## Examples

\begin{example}
  A pair of [[Set]]-enriched adjoint functors is an ordinary pair of [[adjoint functors]].
\end{example}

\begin{example}
 A pair of [[truth values]]-enriched adjoint functors is equivalently known as a *[[Galois connection]]*.
\end{example}

\begin{example}
\label{StrictAdjointTwoFunctors}
  With *[[Cat]]* denoting the [[1-category]] of [[small category|small]] [[strict categories]] equipped with its [[cartesian monoidal category|cartesian monoidal]] [[structure]] (via forming [[product categories]]), a pair of [[Cat]]-enriched adjoint functors is also known as a pair *[[strict adjoint 2-functors]]*.
\end{example}

\begin{example}
  Assuming the [[axiom of choice]] in the underlying [[set theory]], every Dwyer-Kan [[simplicial groupoid]] (i.e. [[sSet]]-[[enriched groupoid]]) is [[sSet]]-enriched equivalent to a [[disjoint union]] of simplicial [[delooping groupoids]] of [[simplicial groups]] --- see the discussion [there](simplicial+groupoid#RelationToSimplicialGroups)
\end{example}

## Related concepts

* [[enriched Quillen adjunction]]

  * [[simplicial Quillen adjunction]]

* [[monoidal adjunction]]

## References

The notion is due to

* [[Max Kelly]], §3 in: *Adjunction for enriched categories*, in: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106**, Springer (1969) [[doi:10.1007/BFb0059145](https://doi.org/10.1007/BFb0059145)]

Review:

* {#Kelly} [[Max Kelly]], section 1.11 of: _Basic Concepts of Enriched Category Theory_, Cambridge University Press, Lecture Notes in Mathematics **64** (1982),  Republished in: Reprints in Theory and Applications of Categories, **10** (2005) 1-136 [[tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

* {#Borceux94} [[Francis Borceux]], Def. 6.7.1 of: _[[Handbook of Categorical Algebra]] Vol 2_, Cambridge University Press (1994)

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enriched bicategory

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#Contents#
* table of contents
{:toc}

## Idea

One notion often called an __enriched bicategory__ is the same as that of an [[enriched category]], but where the enriching context $\mathcal{V}$ is allowed to be generalized from a [[monoidal category]] to a [[monoidal bicategory]], while suitably weakening the [[associativity]] and [[unitality]] conditions on the enrichment.  Thus, it has a collection of objects with [[hom-objects]] $C(x,y)\in\mathcal{V}$.  It may also naturally be called a **pseudo enriched category**.

A different notion that is also sometimes called an **enriched bicategory** is that of a [[bicategory]] enriched over a [[monoidal category|monoidal 1-category]] $V$ (which must be at least [[braided monoidal category|braided]]) at the level of 2-cells only.  Thus it has a collection of objects, with 1-morphisms between the objects, and for any parallel 1-morphisms $f,g\colon x\to y$, a hom-object $C(x,y)(f,g) \in V$.  This can be identified with a $(V Cat)$-enriched bicategory in the previous sense, so on this page we focus on the former, more general, definition.

## Definition

For $\mathcal{V}$ a [[monoidal bicategory]], a  $\mathcal{V}$-enriched (bi)category $C$ consists of

* a collection of [[object]]s;

* for every ordered pair $(x,y)$ of objects a [[hom-object]] $C(x,y) \in \mathcal{V}$

* for every ordered triple $(x,y,z)$ a [[composition]] morphism of the form

  $$
    comp_{x,y,z} : C(x,y)\otimes C(y,z) \to C(x,z)
  $$

  in $\mathcal{V}$

* for every object $x$ an [[identity]] morphism

  $$
    i_x : I \to C(x,x)
  $$

  from the tensor unit of $\mathcal{V}$;

* a [[natural isomorphism|natural 2-ismorphism]]

  $$
    \alpha : comp(Id \otimes comp) \Rightarrow comp(comp \otimes Id)
  $$

  called the [[associator]]

* similarly left and right [[unitor]]s

* such that some more or less evident [[coherence]] conditions hold (see the references).

## Examples

* When $\mathcal{V} = Cat$, a $\mathcal{V}$-enriched bicategory is just a plain [[bicategory]].

* When $\mathcal{V}$ is an ordinary [[monoidal category]], a $\mathcal{V}$-enriched bicategory is just an ordinary [[enriched category]].

* When $\mathcal{V}$ is the cartesian monoidal 2-category of bicategories, pseudo [[2-functors]], and [[icons]], then a $\mathcal{V}$-enriched bicategory is an [[iconic tricategory]].

* When $\mathcal{V}$ is the cartesian monoidal 2-category of [[fully faithful functors]], then a $\mathcal{V}$-enriched bicategory is a weak [[F-category]].

## References

Bicategories enriched in monoidal 2-categories first appeared in:

* Syméon Bozapalides, _Théorie formelle des bicatégories_, PhD thesis, 1976.

* Syméon Bozapalides, _Bicatégories relatives à une 2-catégorie multiplicative_, [[CRAS]], 1976.

The definition appeared independently in:

* S. M. Carmody, _Cobordism Categories_, PhD thesis, University of Cambridge, 1995.

A definition also appeared in

* [[Steve Lack]], _The algebra of distributive and extensive categories_ , PhD thesis, University of Cambridge, 1995.

Forcey has studied the combinatorics of [[polytope]]s associated to enrichment and [[higher category theory|higher categories]] in detail. See for example

* S. Forcey, _Quotients of the Multiplihedron as Categoried Associahedra_ ,  ([arXiv:0803.2694](http://arxiv.org/abs/0803.2694)).

The definition is reviewed in

* [[Alex Hoffnung]], _Notes on enrichment_, ([[Hoffnung_enrichment.pdf:file]]),

and in Chapter 7 of

* [[Alex Hoffnung]], _Foundations of Categorified Representation Theory_ ([pdf](https://math.ucr.edu/home/baez/thesis_hoffnung.pdf))

and studied further in

* [[Richard Garner]] and [[Mike Shulman]], *Enriched categories as a free cocompletion*, [arXiv](http://arxiv.org/abs/1301.3191)

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