expectation value

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[probability theory]] the _expectation value_ of a _[[random variable]]_ or _[[observable]]_ is to be thought of as the _[[mean]]_ value of that variable/observable under the given [[probabilities]].

Taking the concept of expectation value as the primary concept ([Whittle 92](#Whittle92)) leads to _[[quantum probability theory]]_.

## Definition

For $(X, \mu)$ a [[measure space]] of finite total measure $\int_X \mu$ and for $f$ an [[measurable function]] on $X$, a _[[random variable]]_, then its **expectation value** is

$$
  \langle f\rangle
  \coloneqq
  \frac{\int_X f \cdot \mu}{\int_X \mu}
 \,.
$$

In terms of the [[probability measure]] $\mu_P \coloneqq \frac{1}{\int_X \mu} \mu$ this is simply the [[integral]]

$$
  \langle f\rangle
  =
 \int_X f \cdot \mu_P
  \,.
$$

## In terms of probability monads

For classical probability (not quantum), spaces equipped with a notion of expectation value can be modeled as [[algebra over a monad|algebras]] over a [[probability monad]].
See [[probability monad#algebras_expectation_values|probability monad - algebras]] for more.

## Related concepts

* [[random variable]]

* [[correlation function]]

* [[moment]]

* [[vacuum expectation value]]

* [[Feynman diagram]]

* [[probability monad]]

## References

* {#Whittle92} [[Peter Whittle]], _Probability via expectation_, Springer 1992

[[!redirects expectation values]]


experimental alternative definition of adjunction

This page is a result of the following question originally asked at [[adjunction]]:

+-- {: .query}
[[Eric]]: Is it true that given categories $C$ and $D$ that functors $F:C\to D$ and $G:D\to C$ form an adjunction if the following diagram commutes for all morphisms $f:x\to y$ in $C$?

$$
 \array{
  x &\stackrel{f}{\to}& y
  \\
  \uparrow && \uparrow
  \\
  G\circ F(y)
  &\stackrel{G\circ F(f)}{\leftarrow}&
  G\circ F(x)
 }
$$

If so, is that an "if and only if" so that you can define adjunctions this way? If so, this would, in my opinion, be the most digestible way to define adjunctions for "scientists and engineers".
=--

#Contents#

+-- {: .standout}
This page is an informal/speculative discussion of an alternative (yet hopefully equivalent) definition of [[adjunction]].
=--

* tic
{: toc}

##Attempt #2

### Definition (Experimental)

##Attempt #1

### Definition (Experimental)

+-- {: .standout}
>Note the **Crash!** below. This definition is not correct. I think something like this should work, but I need to think about it.
=--

Given [[categories]] $C$ and $D$, a pair of [[functors]] $F:C\to D$ and $G:D\to C$ form an **adjunction** if for every morphism $f:x\to y$ in $C$, there are component morphisms
$$\eta_{y x}: G\circ F(x)\to y$$
and
$$\eta_{x y}: G\circ F(y)\to x$$
such that the following diagram commutes
$$
 \array{
  x &\stackrel{f}{\to}& y
  \\
  \uparrow && \uparrow
  \\
  G\circ F(y)
  &\stackrel{G\circ F(f)}{\leftarrow}&
  G\circ F(x)
 }.
$$

Note that since $G\circ F:C\to C$ is an [[endofunctor]], there are component morphisms
$$\alpha_x: x\to G\circ F(x)$$
and
$$\alpha_y: y\to G\circ F(y)$$
such that the following diagram also commutes:
$$
 \array{
  x &\stackrel{f}{\to}& y
  \\
  \downarrow && \downarrow
  \\
  G\circ F(x)
  &\stackrel{G\circ F(f)}{\rightarrow}&
  G\circ F(y)
 }.
$$

Putting the two commuting squares together, we have a commuting square with commuting (upward) diagonals giving rise to additional commuting triangles.

These commuting diagrams give rise to the following relations:

* $\eta_{y x}\circ \alpha_x = f$

* $\alpha_y\circ\eta_{y x} = G\circ F(f)$

* $[G\circ F(f)]\circ\alpha_x = \alpha_y\circ f$

* $\eta_{x y}\circ [G\circ F(f)]\circ\alpha_x = 1_x$

* $f\circ\eta_{x y}\circ\alpha_y = 1_y.$

Note: I'm not sure if these are independent or not (need to count how many independent equations we get!), but it is instructive to note:

$$\eta_{x y}\circ \alpha_y\circ\eta_{y x}\circ\alpha_x = 1_x$$

and

$$\eta_{y x}\circ\alpha_x\circ\eta_{x y}\circ \alpha_y = 1_y$$

which means

$$\eta_{y x}\circ\alpha_x = \left(\eta_{x y}\circ \alpha_y\right)^{-1}.$$

**CRASH!**

I'll need to check, but unless I made some algebra mistake, this last statement might kill the whole idea. Since
$$f = \eta_{y x}\circ\alpha_x,$$
then the above statement says $f$ has an inverse.

### Response from Toby

You need to say more than just that *there are* morphisms that make these diagrams commute.  Whether $F \vdash G$ is an adjunction is not actually a property but a structure; we need to say *how* they are an adjunction.  That is, you have to pick the family of morphism $G(F(x)) \to y$ in advance.

But then you can make it work, I believe ... although I\'m pretty sure that you need more commutative diagrams than the one that you\'ve drawn.  That is, an __adjunction__ $\eta$ from $F$ to $G$ consists of a family of maps $\eta_f\colon G(F(x)) \to y$, one for each morphism $f\colon x \to y$, such that

*  a bunch of diagrams commute.

I have to think about whether this works and what diagrams you need.

[[Eric]]: Thanks Toby. I would be surprised if it were this simple, but I drew some pictures based on the diagram in Goldblatt, and I think this works.

[[Eric]]: Note, I had some typos in the original version of the diagram, but it should be correct now. Or it least it now matches the diagram on my paper :)

>I\'m pretty sure that you need more commutative diagrams than the one that you\'ve drawn.

We get another commuting square from the fact that $G\circ F$ is an endofunctor. I've now drawn that above. When you put the two together, it gives you a bunch of additional commuting triangles, but everything follows from the one first simple diagram.

[[Eric]]: Hi Toby (and anyone else who may stop by). Note the **Crash!** above. The definition cannot be correct as stated. I probably won't give up though. I think instead of saying the diagram commutes, I need to say something about "commuting up to a 2-morphism" or "there is a 2-morphism such that" blah blah.


experimental alternative definition of functor

+-- {: .standout}
Good news! The twists and turns on this page have culminated in [Functors and generalized elements](http://ncatlab.org/nlab/show/functor#OnGeneralizedElements). From there, I drew the diagram

Staring at this diagram resulted in the following:

***

##Definition

Given categories $C$ and $D$, a **functor** $F:C\to D$ is a map that sends each object $x$ in $C$ to an object $F(x)$ in $D$ and each morphism $f:x\to y$ in $C$ to a morphism $F(f):F(x)\to F(y)$ in $D$ such that each commuting diagram in $C$ maps to a commuting diagram in $D$.

***

This has now been included in the definition of [[functor]].

Note this is still essentially the same thing as conveyed in my earlier picture

requiring the square commutes, but is done in a way that avoids introducing the non-standard morphism-like components $\alpha_x:x\to F(x)$.

In other words, if we stopped worrying about what $\alpha_x:x\to F(x)$ really was and simply accepted that the indicated square commutes, it implies automatically that any commuting diagram in $A$ maps to a commuting diagram in $B$.

=--

#Contents#

+-- {: .standout}
This page is an informal/speculative discussion of an alternative (yet hopefully equivalent) definition of [[functor]]. It first appeared as a discussion at [[functor]] itself, but was subsequently moved here.
=--

* tic
{: toc}

##Discussion 3.0##

I think I may have found a way to express this idea without resorting to non-standard morphism-like maps $x\to F(x)$ or cographs.

+-- {: .standout}
##Definition
Given categories $A$ and $B$, a map $F:A\to B$ is a functor if for every morphism $f:x\to y$ in $A$
$$F(f) F(x) = F(f(x)).$$
=--
Let's check...

$$\array{
F(g\circ f) F(x)
&= F(g\circ f(x)) \\
&= F(g)\circ F(f(x)) \\
&= F(g)\circ F(f) F(x).}$$

Since this is true for all morphisms $f:x\to y$ and $g:y\to z$, we have

$$F(g)\circ F(f) = F(g\circ f)$$

as required.

Next,

$$F(1_A) F(x) = F(1_A(x)) = F(x)$$

as required.

Therefore, it seems to work.

Note this is still essentially the same thing as conveyed in my earlier picture

requiring the square commutes, but is done in a way that avoids introducing the non-standard morphism-like components.

##Discussion 2.0##

After Discussion 1.0 below, it seems that the original intention of this page was to define a [[functor]] in terms of a [[cograph of a functor|cograph]]. Given a functor, it is straightforward to construct its cograph. However, it may be possible to do the reverse, i.e. start with a cograph and infer the functor that gives rise to that cograph.

Is it possible to define a functor in terms of cograph? If so, how?

##Discussion 1.0##

###Definition###
Given small categories $A$ and $B$ define their **disjoint union** $A\sqcup B$ to be the category with
$$Obj(A\sqcup B) = Obj(A)\sqcup Obj(B)$$
and
$$Hom_{A\sqcup B}(x,y) =
  \left\{
    \begin{aligned}
      Hom_A(x,y) & if x,y \in A \\
      Hom_B(x,y) & if x,y \in B \\
      \emptyset & otherwise
    \end{aligned}
  \right.
$$
There are two **inclusion maps**
$$i_A:A\to A\sqcup B\quad\text{and}\quad i_B:B\to A\sqcup B$$
defined for any morphism $f:a\to b$ in $A$ and morphism $g:c\to d$ in $B$ by
$$i_A(a) = a\sqcup\emptyset,\quad i_A(b) = b\sqcup\emptyset,\quad\text{and}\quad i_A(f) = f\sqcup\emptyset$$
and
$$i_B(c) = \emptyset\sqcup\c,\quad i_B(d) = \emptyset\sqcup d,\quad\text{and}\quad i_B(g) = \emptyset\sqcup g.$$

###Definition###
Given categories $A$, $B$ and inclusion maps $i_A:A\to A\sqcup B$, $i_B:B\to A\sqcup B$, a **functor** is a map $F:A\to B$ together with component morphisms $\alpha_x:i_A(x)\to i_B\circ F(x)$ and $\alpha_y:i_A(y)\to i_B\circ F(y)$ for any morphism $f:x\to y$ in $A$ such that the following diagram commutes:
$$
  \array{
    i_A(x)
    &
    \stackrel{i_A(f)}{\to}
    &
    i_A(y)
    \\
    \mathllap{\scriptsize{\alpha_x}}\downarrow
    &&
    \downarrow\mathrlap{\scriptsize{\alpha_y}}
    \\ i_B\circ F(x)
    &
    \stackrel{i_B\circ F(f)}{\to} & i_B\circ F(y)
  }
$$

***

[[nlab:Todd Trimble|Todd]]: If $A \sqcup B$ means the disjoint union of $A$ and $B$, then there are no morphisms of the form $i_A(x) \to i_B(F(x))$.

[[Eric]]: Hi Todd. I saw your comment after I added the figure above. I am probably not expressing myself clearly. I want to get $Obj(A)$ and $Obj(B)$ into the same category while preserving all morphisms and then ADD components $\alpha_x$ and $\alpha_y$. Is that kind of thing allowed?

I think the picture expresses what I'm trying to do, but maybe I'm converting that into the wrong formulas.

[[Eric]]: The idea is that by requiring every such square to commute, you automatically get $F(g\circ f) = F(g)\circ F(f)$ and $F(Id_x) = Id_{F(x)}$. Or so I think...

[[Eric]]: Not to mention, this _looks_ more like a natural transformation, so a pattern is more apparent.

[[Todd Trimble|Todd]]: What I was saying is that wherever these $\alpha_x$ are supposed to live, it's can't be in $A \sqcup B$, because in $A \sqcup B$, there simply are no morphisms of that form.

Your picture suggests that you have in mind some category $C$ into which $A$ and $B$ embed; it can't be $A \sqcup B$ since your $\alpha_x$ simply don't exist there. But it's some $C$ in which such morphisms $\alpha_x$ exist, let's say. Then it looks like your notion of functor involves the following things:

* Embeddings (full embeddings?) $i_A: A \to C$, $i_B: B \to C$ into some category $C$.

* A rule which assigns to each object $x$ of $A$ an object $F(x)$ of $B$, and to each morphism $f: x \to y$ of $A$ a morphism $F(f): F(x) \to F(y)$ of $B$. In other words, a map $F: A \to B$ between the underlying directed graphs of $A$ and $B$.

* A rule which assigns to each object $x$ of $A$ a morphism $\alpha_x: i_A(x) \to i_B(F(x)$, making that square commute.

Okay, so far these conditions don't ensure functoriality: that $F(g \circ f) = F(g) \circ F(f)$ and $F(1_x) = 1_{F(x)}$. You're going to have to add more conditions to make that inference. For example, suppose $C$ has just one morphism from each object $i_A(a)$ to each object of $i_B(b)$. Then commutativity of those squares is automatic, for any directed graph morphism $F: A \to B$. Insofar as not all graph morphisms are functorial, you can't infer functoriality.

You could ask that all the $\alpha_x$ be isomorphisms. Then commutativity plus that condition would give functoriality, and your concept amounts to that of ordinary functor $F$ together with an ordinary natural isomorphism $i_A \overset{\sim}{\to} i_B \circ F$.

But the trouble with all this is that I have no idea what $C$ is supposed to be! It can't be $A \sqcup B$. Would it be something constructed in terms of $A$ and $B$ (and if so, what)? If it's just _any_ old category into which $A$ and $B$ embed and for which such isos $\alpha_x$ exist (and compatible with the $F(f)$ via commutative diagrams), then I have problems with that too.

[[Eric]]; Hi Todd. I think I'm confused because with the standard definition of functor $F:A\to B$ no one ever complains about writing $F(x)\in B$, so in a way, isn't $F$ an _arrow_ from $x\in A$ to $F(x)\in B$? We also write $F(f)\in B$, which makes me think $F$ is a bunch of 1-arrows and 2-arrows. I mean, doesn't the picture make sense? Why is it so hard to convert the picture to a mathematical statement?

: _Toby_:  Eric, here you seem to be conflating these two arrows: $\to$ and $\mapsto$.  At the level of functions between sets, if $A$ and $B$ are sets and $f$ is a function from $A$ to $B$, then I write $f\colon A \to B$ or $A \overset{f}\to B$.  Then if $x$ is an element of $A$ and $y$ is an (the) element of $B$ such that $y = f(x)$, then I write $f\colon x \mapsto y$ or $x \overset{f}\mapsto y$.  Or without bothering with $y$, I write $f\colon x \mapsto f(x)$ or $x \overset{f}\mapsto f(x)$.  (Actually, this is best when '$x$' is a variable and '$f(x)$' is a formula in terms of that variable; for example, if $f$ is the squaring function, then I write $f\colon x \mapsto x^2$ or $x \overset{f}\mapsto x^2$, and this serves to *define* $f$.)

: Actually, you can find a lot of literature where the same symbol '$\rightarrow$' is used for both $\to$ and $\mapsto$, but they are still different concepts.

: Of course, what the [[cograph]] gets you is that it turns $\mapsto$ into a special case of $\to$.  But they are still different; $f\colon A \to B$ is an arrow in the category of sets or (in the categorified case) in the $2$-category of categories, while $x \overset{f}\mapsto y$ is (only in the categorified case) an arrow in the cograph of $f$.

>>[[Eric]]: Hmm.. but we ARE talking about cographs :) At least I was accidentally :)

[[Urs Schreiber]]: concerning the nature of $C$: as I suggested before, it does make sense to take this to be the [[cograph of a functor]]. And indeed, in some situations it is useful to _define_ the notion of functor in terms of the notion of cograph. In [[Higher Topos Theory]] the notion of [[adjoint (infinity,1)-functor]] is defined entirely in terms of cographs of functors.

[[Eric]]: Thanks Urs. Now I just need to parse what you say into something I can understand, but it is encouraging. In fact, quoting myself from earlier in the discussion:

>It seems like maybe I am trying to work backwards, i.e. define a functor via a cograph.

When I tried to grok "cograph" on my personal web [[ericforgy:Natural Transformation]], I stripped out the line including the functor and with Toby's help, realized what I (think I) was talking about was the disjoint union of categories.

I guess, my plan of attack now is to see how we can define functor via cograph. For some reason, that makes sense to me.

[[Todd Trimble|Todd]]: Yes, that's true, the cograph is fine. I thought that Eric had wanted a $C$ which would uniformly work, independently of which functor $F$ was being considered. Instead, we construct the cograph $C$ in terms of $F$, and everything is for the best in this best of all possible worlds. And we get the picture Eric was aiming for.

[[Eric]]: Neat. Now, if we (I'll try, but I'm not confident and wouldn't mind if someone else did it!) can write down a succinct definition of functor in terms of cographs with a picture like the one above (if not that one), we could add it to [[functor]].

[[Eric]]: By the way, here is something I wrote on my personal web [[ericforgy:Natural Transformation]]:

>[[Urs Schreiber]]: at [[cograph]] it is indicated how that concept encodes the notion of [[profunctor]] which is a bit more general than that of [[functor]].

>[[Eric]]: Oh! That is pretty! Thanks. I like this diagram
$$
  \array{
    R &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    X \times Y &\stackrel{\chi_R}{\to}& \{0,1\}
  }
$$
>I need to think about it, but a vale of fog has been lifted. More fog remains, but thanks!

***

###Definition###
Given small categories $A$ and $B$ define their **disjoint union** $A\sqcup B$ to be the category with
$$Obj(A\sqcup B) = Obj(A)\sqcup Obj(B)$$
and
$$Hom_{A\sqcup B}(x,y) =
  \left\{
    \begin{aligned}
      Hom_A(x,y) & if x,y \in A \\
      Hom_B(x,y) & if x,y \in B \\
      \emptyset & otherwise
    \end{aligned}
  \right.
$$
There are two **inclusion maps**
$$i_A:A\to A\sqcup B\quad\text{and}\quad i_B:B\to A\sqcup B$$
defined for any morphism $f:a\to b$ in $A$ and morphism $g:c\to d$ in $B$ by
$$i_A(a) = a\sqcup\emptyset,\quad i_A(b) = b\sqcup\emptyset,\quad\text{and}\quad i_A(f) = f\sqcup\emptyset$$
and
$$i_B(c) = \emptyset\sqcup\c,\quad i_B(d) = \emptyset\sqcup d,\quad\text{and}\quad i_B(g) = \emptyset\sqcup g.$$

***

I intentionally did not use the word "functor" above even though that is really the [[coproduct]] in [[Cat]]. The reason is that I want to use this disjoint union to _define_ a functor. The following is a possible starting point:

###Definition###
Given categories $A$, $B$ and inclusion maps $i_A:A\to A\sqcup B$, $i_B:B\to A\sqcup B$, a **functor** is a map $F:A\to B$ that assigns morphisms $\alpha_x:i_A(x)\to i_B\circ F(x)$ and $\alpha_y:i_A(y)\to i_B\circ F(y)$ for any morphism $f:x\to y$ in $A$ such that the following diagram commutes:
$$
  \array{
    i_A(x)
    &
    \stackrel{i_A(f)}{\to}
    &
    i_A(y)
    \\
    \mathllap{\scriptsize{\alpha_x}}\downarrow
    &&
    \downarrow\mathrlap{\scriptsize{\alpha_y}}
    \\ i_B\circ F(x)
    &
    \stackrel{i_B\circ F(f)}{\to} & i_B\circ F(y)
  }
$$

***

###Discussion###

[[Eric]]: Motivated by some discussion over at [[natural transformation]], I was wondering if the following alternative definition of functor holds water:

>###Definition##
>Given categories $A$, $B$, a **functor** is a map $F:A\to B$ that assigns maps $\alpha_x:x\to F(x)$ and $\alpha_y:y\to F(y)$ for any morphism $f:x\to y$ in $A$ such that the following diagram commutes:
$$
  \array{
    x
    &
    \stackrel{f}{\to}
    &
    y
    \\
    \mathllap{\scriptsize{\alpha_x}}\downarrow
    &&
    \downarrow\mathrlap{\scriptsize{\alpha_y}}
    \\ F(x)
    &
    \stackrel{F(f)}{\to} & F(y)
  }
$$

If so, it has a certain beauty that appeals to me. Thanks!

Urs: that diagram seems to be meaningful only if $A = B$, right? In that case, this is a functor $F : A \to A$ together with a natural transformation from the identity functor on $A$ into $F$.

[[Eric]]: Yeah. It needs some work. Of course I don't want $A=B$, but I would like to be able to introduce morphisms from $A$ to $B$. Could I form some kind of category union or something? The picture is kind of neat. Each diagram in $A$ "hovers" above a diagram in $B$. If it can be ironed out, it might be an efficient definition.

Urs: now you are looking for the notion [[cograph of a functor]] (you should make us write out more details at that entry!)

[[Eric]]: **Gulp!** Ok. But is there a term for the (small) category $C = A\cup B$ where $C_0 = A_0\cup B_0$ and $C_1 = A_1\cup B_1$?

_Toby_:  You could just call that the _union_ of the two categories.  But it\'s not really a meaningful concept unless you already have $A$ and $B$ as [[subcategory|subcategories]] of some ambient category $D$, or something like that.  To ask whether an element of $A_0$ matches an element an element of $B_0$ (and similarly for $A_1$ and $B_1$) is [[evil]].

Alternatively, you could take it as understood that the sets $A_0$ and $B_0$ are disjoint, (and similarly for $A_1$ and $B_1$).  Then you get the _disjiont union_ of the two categories, and that is an important concept; it is the [[coproduct]] in [[Cat]].  And that is one of the ingredients of the [[cograph of a functor]], so probably it\'s what you want.

[[Eric]]: It seems like maybe I am trying to work backwards, i.e. define a functor via a cograph.

[[Eric]]: Ok! I think I got it. Does this hold water?

***

###Definition###
Given categories $A$, $B$ and inclusion maps $i_A:A\to A\sqcup B$, $i_B:B\to A\sqcup B$, a **functor** is a map $F:A\to B$ that assigns morphisms $\alpha_x:i_A(x)\to i_B\circ F(x)$ and $\alpha_y:i_A(y)\to i_B\circ F(y)$ for any morphism $f:x\to y$ in $A$ such that the following diagram commutes:
$$
  \array{
    i_A(x)
    &
    \stackrel{i_A(f)}{\to}
    &
    i_A(y)
    \\
    \mathllap{\scriptsize{\alpha_x}}\downarrow
    &&
    \downarrow\mathrlap{\scriptsize{\alpha_y}}
    \\ i_B\circ F(x)
    &
    \stackrel{i_B\circ F(f)}{\to} & i_B\circ F(y)
  }
$$
It follows that $i_A$ and $i_B$ are functors.

Note that a map $F:A\to B$ is a functor if the map $\alpha:i_A\Rightarrow i_B\circ F$ is a natural transformation.

***

Urs: to be frank, I don't get it! Also: what is it you are really after? The concept of a functor is entirely non-mysterious, it seems. What are you dissatisfied with, with the standard definition?

[[Eric]]: I enjoyed fiddling with [[natural transformations]] recently. I hadn't thought about them much. The introduction of "components" was kind of interesting. I also liked that "sheet transformation", which I now think of as a some kind of 3-morphism. So I was wondering if we could introduce "components" to define functors (which I now think of as 2-morphisms). Then I wondered if we could continue a pattern downward all the way to (-2)-categories.

The _pattern_ natural transformation $\to$ functor $\to$ morphism is not 100% clear to me. Each individually "makes sense", but I'd like to see a clearer pattern how they're related.

Basically, I'm trying to understand

* (-2)-morphisms
* (-1)-morphisms
* 0-morphisms
* 1-morphisms (morphisms)
* 2-morphisms (functors?)
* 3-morphisms (natural transformations? sheet transformations?)
* Etc.

in some consistent way.

For example, with categories $A$ and $B$ we have a [[functor category]] $[A,B]$ which is related to (if not equal to) a 2-category. How about sets $A$ and $B$ thought of as 0-categories? We could form a "morphism category" $[A,B]$ that is related (if not equal to) a 1-category. Then I can imagine a "truth category" $[True,True]$ that is related (if not equal) to a 0-category. Etc etc. This is all pretty elementary stuff I'm sure.

In a nutshell, I'm not saying there is anything wrong the definitions there, but I'm trying to understand them in a different way if possible.

##Reference

* See all [[experimental alternative definition of adjunction]].

[[!redirects functor (discussion)]]


experimental mathematics

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

#Contents#
* table of contents
{:toc}

## Idea

What is called _experimental mathematics_ is the exploration of [[mathematics]] and the possibe [[truth]] of [[theorems]] not by general [[proof]], but by checking special cases using [[computer programs]].

## Examples

### Riemann hypothesis

A general [[proof]] of [[Riemann hypothesis]] remains famously open, but computer experiment so far confirms it, see [there](Riemann+hypothesis#ReferencesComputerChecks).

### Volume conjecture

A general [[proof]] of the [[volume conjectures]] remains open (though various special cases do have proof), but computer experiment (such as using [[SnapPy]]) so far confirms (and motivates) these conjectures.

### Image of beta

The [[image]] of the canonical [[homomorphism]] $\beta$ from the [[Burnside ring]] of a [[finite group]] to the [[representation ring]] over some [[field]] is partly known only by computer experiment, see [there](permutation+representation#WhenAllVirtualLinearRepsAreVirtualPermutationReps).

## Related concepts

* [[experimental observation]]

## References

* David H. Bailey, Jonathan M. Borwein, _Experimental Mathematics: Examples, Methods and Implications_, Notices of the AMS, Volume 52, Number 5, 502-514. ([pdf](https://www.ams.org/notices/200505/fea-borwein.pdf))

* David H. Bailey, Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke, Victor H. Moll, _Experimental Mathematics in Action_. ([pdf](https://math.dartmouth.edu/archive/m56s13/public_html/BaileyBorweinetal2006book_Experimental_Mathematics_in_Action.pdf))

* Herbert S. Wilf, _Mathematics: An Experimental Science_. ([pdf](https://www.math.upenn.edu/~wilf/website/Mathematics_AnExperimentalScience.pdf))

See also

* Wikipedia, _[Experimental mathematics](https://en.wikipedia.org/wiki/Experimental_mathematics)_

[[!redirects computer experiment]]


experimental observation

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

#Contents#
* table of contents
{:toc}

## Examples

[[!include experiments -- contents]]

## Related entries

* [[measurement]], [[quantum measurement]]

* [[theory (physics)]], [[coordination]]

* [[observable universe]]

* [[standard model of particle physics]], [[standard model of cosmology]]

* [[empiricism]]

## Relation to mathematical proof

In ([Jaffe-Quinn 93, p. 2](JaffeQuinn93)) it was claimed that

> the role of rigorous [[proof]] in mathematics is functionally [[analogy|analogous]] to the role of [[experiment]] in the natural sciences

Responses to that claim may be found [here](http://arxiv.org/abs/math/9404229).

## Related concepts

* [[precision experiment]]

* [[local observables]], [[spacetime support]]

* [[computer experiment]]

## References

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

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

Discussion relating experiment to mathematical [[proof]] includes

* {#JaffeQuinn93} [[Arthur Jaffe]], [[Frank Quinn]], _"Theoretical Mathematics": Towards a cultural synthesis of mathematics and theoretical physics_, Bulletin of the AMS, Volume 29,Number 1, July 1993 ([arXiv:math/9307227](http://arxiv.org/abs/math/9307227))

[[!redirects experimental observations]]

[[!redirects experiment]]
[[!redirects experiments]]
[[!redirects observation]]
[[!redirects observations]]


experiments -- contents

**[[experiments]]**

* [[Michelson-Morley experiment]]

  ([[speed of light]])

* [[LHC]]

  ([[particle physics]], [[Higgs field]])

* [[LHCb]], [[Belle experiment]], [[BaBar experiment]]

  ([[B-mesons]], [[flavour anomalies]], [[V_cb puzzle]], [[pentaquark]])

* [[RHIC]]

  ([[quark-gluon plasma]])

* [[Super-Kamiokande]], [[ANITA experiment]]

  ([[neutrinos]], [[proton decay]])

* [[LIGO]]

  ([[gravitational waves]])

* [[EDGES]]

  ([[hydrogen line]] in [[cosmic microwave background]])

* [[Chandra Observatory]]

* [[Event Horizon Telescope]]

  ([[black holes]])

* [[James Webb Space Telescope]]

  ([[galaxies]], [[cosmology]])


explicit conversion

\tableofcontents

## Idea

In [[type theory]], there are many ways to represent the notion of conversion of [[terms]] from one to another in the syntax. Usually, this is represented by untyped or typed [[judgmental equality]] as a [[judgment]] $a \equiv b$ or $a \equiv b:A$ respectively. **Explicit conversion** is the representation of conversion as [[terms]] of a [[type]] $p:a \equiv b$.

In [[objective type theory]] and other [[weak type theories]] which don't have a judgmental equality as a judgment, explicit conversion is used for [[definitions]] and for [[beta-conversion]] and [[eta-conversion]], and is represented by [[identifications]] $p:a =_A b$ for [[elements]] and by [[equivalence of types|equivalences]] $e:A \simeq B$ for [[types]].

## Related concepts

* [[judgmental equality]]

## References

* [[Floris van Doorn]], [[Herman Geuvers]], and [[Freek Wiedijk]], *Explicit convertibility proofs in pure type systems*. Proceedings of the Eighth ACM SIGPLAN international workshop on Logical frameworks & meta-languages: theory & practice, ACM. 2013, pp. 25–36 ([pdf](https://www.cs.ru.nl/~herman/PUBS/ExplicitPTS.pdf))

* [[Theo Winterhalter]], *Formalisation and Meta-Theory of Type Theory* ([github](https://github.com/TheoWinterhalter/phd-thesis/releases/tag/v1.2.1))


explicit formulae (L-function)

## References

* Wikipedia, _[Explicit formulae (L-function)](http://en.wikipedia.org/wiki/Explicit_formulae_%28L-function%29)_

[[!redirects explicit formula (L-function)]]

[[!redirects explicit formulae for L-functions]]


explicit mathematics

# Explicit mathematics

* table of contents
{: toc}

## Idea

**Explicit mathematics** is a programme initiated by [[Solomon Feferman]] which is related to [[constructive mathematics]].

It is also a range of foundational logical systems based on the [[logic of partial terms]]. Starting from a first-order logic of partial terms axiomatizing [[combinatory algebra]], a second sort of _classes_ is added. Sometimes, the alternative names _classifications_ or _types_ are used.

The classes of explicit mathematics are not like the classes of [[set theory]], but also not like types of [[type theory]]. They are like the former in that they represent certain collections of the underlying first-order model, but they are like the latter in that they have _names_ that can be operated on to construct new ones.

The systems come in flavors with either intuitionistic or [[classical logic]]. On benefit is that they allow for smooth axiomatization of strong notions of [[universes]], see below.

The primary systems are called $T_0$ (the classical variant) and $T_0^i$ (the intuitionistic variant).

In the 1980s there was a proof assistant and program extraction tool called PX based on a version of $T_0$ relative to the Lisp programming language.

## The basic theory of operations and numbers

The first-order part of the systems treat an applicative universe in the sense of combinatory algebra. It includes constants $\mathrm{k}$ and $\mathrm{s}$ (combinators), $\mathrm{p}$, $\mathrm{p}_0$, and $\mathrm{p}_1$ (pairing and projection), $0$, $\mathrm{s}_N$, and $\mathrm{p}_N$ (zero, successor, and predecessor), and $\mathrm{d}_N$ (definition by cases on the natural numbers), and possibly further applicative constants.

There is one binary function symbol ${\cdot}$ for application (usual just indicated by juxtaposition), one unary relation symbol $\downarrow$ for being defined, one unary relation symbol $\mathrm{N}$ for natural numbers, as well the binary equality relation.

In the logic of partial terms, we use the usual abbreviation $s \simeq t \leftrightarrow ({s\downarrow} \wedge {t\downarrow} \to s=t)$. The basic axioms for the operations and natural numbers are:

1. $\mathrm{k} a b = a$,
1. $\mathrm{s} a b \downarrow \wedge \mathrm{s} a b c \simeq a c(b c)$,
1. $\mathrm{p}_0(a,b)=a \wedge \mathrm{p}_1(a,b) = b$,
1. $0\in N \wedge (\forall x\in N)(\mathrm{s}_N x \in N)$,
1. $(\forall x\in N)(\mathrm{s}_N x \ne 0 \wedge \mathrm{p}_N(\mathrm{s}_N x)=x)$,
1. $(\forall x\in N)(x\ne0 \to \mathrm{s}_N(\mathrm{p}_N x) = x)$,
1. $a\in N \wedge b\in N \wedge a=b \to \mathrm{d}_N u v a b = u$,
1. $a\in N \wedge b\in N \wedge a\ne b \to \mathrm{d}_N u v a b = v$.

From the first two axioms we get lambda-definability and a fixed-point combinator.

To these axioms we add the scheme of induction over the natural numbers to obtain the theory $BON + (\mathrm{L}{-}Ind_N)$. This is proof-theoretically equivalent to first-order Peano arithmetic, $PA$, via interpretations both ways. We can interpret $BON + (\mathrm{L}{-}Ind_N)$ in $PA$ by coding operations as indices for recursive functions using Kleene's $T$ and $U$ predicates.

## Elementary comprehension

When we go from the purely applicative theory to the theories of classes and names, we add a second sort (for classes) as well as several new constants, a binary relation symbol $\in$ (for membership) and a binary relation symbol for names $\Re$, where $\Re(s, U)$ means that $s$ is a name for the class $U$. We use the abbreviation $\Re(s) :\leftrightarrow \exists U \Re(s,U)$, indicating that $s$ is a name.

The basic axioms regarding classes and names state that every class has a name, that there are no homonyms, and that $\Re$ respects extensional equality.

The names of classes corresponding to elementary comprehension are generated from $nat$ (natural numbers) and $id$ (the equality relation) using $co$ (complements), $int$ (intersections), $dom$ (domains), and $inv$ (inverse images). [These are for the classical systems. The intuitionistic systems are slightly different. Todo: look this up]

Using the abbreviation $s\dot\in t :\leftrightarrow \exists U(\Re(t,U) \wedge s\in U)$, the axioms are:

1. $\Re(nat) \wedge \forall x(x \dot\in nat \leftrightarrow N(x))$,
1. $\Re(id) \wedge \forall x(x \dot\in id \leftrightarrow \exists y(x = (y,y)))$,
1. $\Re(a) \to \Re(co(a)) \wedge \forall x(x \dot\in co(a) \leftrightarrow \neg x\dot\in a)$,
1. $\Re(a) \wedge \Re(b) \to \Re(int(a,b)) \wedge \forall x(x \dot\in int(a,b) \leftrightarrow x \dot\in a \wedge x \dot\in b)$,
1. $\Re(a) \to \Re(dom(a)) \wedge \forall x(x \dot\in dom(a) \leftrightarrow \exists y((x,y) \dot\in a))$,
1. $\Re(a) \to \Re(inv(a,f)) \wedge \forall x(x \dot\in inv(a,f) \leftrightarrow f x \dot\in a)$.

Again we add some induction over the natural numbers to obtain theories $EC + (\mathrm{C}{-}Ind_N)$ (induction axiom for classes) and $EC + (\mathrm{L}{-}Ind_N)$ (induction schema for all formulas). These correspond proof-theoretically to the subsystems of second-order arithmetic $ACA_0$ and $ACA$, respectively.

## Join and inductive generation

The _join_ of classes gives disjoint unions of classes, and thus corresponds to the $\Sigma$-types of type theory. The inductive generation operation provides accesible parts of classes coding binary relations.

[todo: the axioms for join and inductive generation.]

The theory $T_0$ has elementary comprehension, join, inductive generation, and full induction over the natural numbers. It has the same proof-theoretic strength as [[CZF]] plus $REA$, the [[regular extension axiom]]. Further equivalences are listed at [[ordinal analysis]].

## Universes in explicit mathematics

[todo]

## References

* [Introduction (draft)](https://math.stanford.edu/~feferman/papers/DraftIntroFEM.pdf)

* Solomon Feferman, [How a Little Bit goes a Long Way: Predicative Foundations of Analysis](http://home.inf.unibe.ch/~ltg/em_bibliography/feferman13.pdf), notes prepared 1977-1981 with a new introduction from 2013.

* [Foundations of explicit mathematics](https://web.archive.org/web/20190108170710/http://www.ltg.unibe.ch/research/Foundations%20of%20Explicit%20Mathematics)

* [Bibliography of explicit mathematics](http://home.inf.unibe.ch/~ltg/em_bibliography/)

* Reinhard Kahle and Anton Setzer, [An extended predicative definition of the Mahlo universe](http://www.cs.swan.ac.uk/~csetzer/articles/kahleSetzerExtendedPredicativeMahloPohlersFestschrift.pdf), In: Ways of Proof Theory, pp. 315–340. Ontos Mathematical Logic, vol. 2, 2010.

* Susumu Hayashi and Hiroshi Nakano, [PX: a computational logic](https://dl.acm.org/citation.cfm?id=62010), MIT Press, 1988.


explicit substitution

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Explicit substitutions
* table of contents
{: toc}

## Idea

Ordinary [[substitution]] is a meta-level notion, i.e. an operation *on* [[syntax]].  By contrast, *explicit substitutions* incorporate a notion of substitution as *part* of a syntax. The original calculus of explicit substitutions was introduced by Abadi, Cardelli, Curien and Lévy.

## Definition/Example

A paradigmatic (and in some sense universal) example is given by adding explicit substitutions to [[simply typed lambda calculus]].  For this we add a new term constructor

$$M\langle x\coloneqq N\rangle$$

denoting internally "the substitution of $N$ for $x$ in $M$".  That this is a term constructor means, for instance, that $x\langle x\coloneqq y\rangle$ is a term distinct from $y$; whereas for the ordinary notion of substitution as an *operation*, $x[x\coloneqq y]$ would be a notation for the result of the operation of substituting the term $y$ for the variable $x$ in the term $x$, which is precisely the term $y$.

The relation between $x\langle x\coloneqq y\rangle$ and $y$ is recovered by modifying the [[reduction]] relation.  We modify the usual [[beta-reduction]] rule $(\lambda x.M)(N) \to M[x\coloneqq N]$ to use an explict substitution instead:

$$(\lambda x.M)(N) \to M\langle x\coloneqq N\rangle$$

and then give reduction rules that mimic the *inductive definition* of the usual *operation* of substitution on terms:

$$\begin{aligned}
x\langle x\coloneqq N\rangle &\to N\\
y \langle x\coloneqq N\rangle &\to y \quad (x\neq y)\\
(M_1 M_2)\langle x\coloneqq N\rangle &\to (M_1 \langle x\coloneqq N\rangle)(M_2 \langle x\coloneqq N\rangle) \\
(\lambda y.M)\langle x\coloneqq N\rangle &\to \lambda y. (M\langle x\coloneqq N\rangle) \quad (x\neq y \;\text{and}\; y \;\text{not free in}\; N)
\end{aligned}$$

(We are being somewhat cavalier about [[variables]]; one can also define and implement explicit substitutions using [[de Bruijn indices]].)

## Combining substitutions and strong normalization

Note that there is no rule for reducing a term of the form $(M \langle x\coloneqq N\rangle) \langle y\coloneqq P\rangle$.  Instead we have to first reduce $M \langle x\coloneqq N\rangle$ until its "head" is no longer an explicit substitution, and then we can apply reduction rules for the "outer" substitution $\langle y\coloneqq P\rangle$.  We might be tempted to assert a rule such as

$$ (M \langle x\coloneqq N\rangle) \langle y\coloneqq P\rangle \to (M\langle y\coloneqq P\rangle) \langle x\coloneqq N\langle y\coloneqq P\rangle \rangle$$

but such a rule is undesirable because it destroys the [[strong normalization]] property of the reduction system: although every term still *can* be reduced to a [[normal form]], there are also infinite reduction sequences that simply interchange pairs of substitutions forever.

One might hope that this could be avoided by introducing "explicit simultaneous substitutions" with a reduction rule

$$ (M \langle x\coloneqq N\rangle) \langle y\coloneqq P\rangle \to M \langle x\coloneqq N, y\coloneqq P\rangle. $$

Apparently for a long time it was hoped that this would work, but [Mellies](#Mellies) showed that it doesn't: explicit simultaneous substitutions with a rule of this sort also yield infinite reduction sequences.  Note that Mellies worked with a de Bruijn version, so his rules look somewhat different; a version of the counterexample using named variables can be found in [Bloo-Rose](#BlooRose), along with a proof that without any way to "combine substitutions" strong normalization is preserved.

## Categorical Models of Explicit Substitutions

The original calculus of explicit substitutions of Abadi et al, considered as a type theory, can be given sound and complete categorical models, in terms of indexed categories.

This explicit substitutions construction is parametric in the kind of logic that the terms stand for. The Calculus of Constructions ([Ritter] (#Ritter)), Intuitionistic Propositional Logic and Intuitionistic Linear Logic ([Ghani2000] (#Ghani2000)) and propositional Constructive Necessity ([Ghani1998] (#Ghani1998)) can be given formalizations and implementations in terms of explicit substitutions calculi. These constructions help to show the correctness of the implementation of the associated type theories.

## The linear substitution calculus

In 2010 a new version of a *linear substitution calculus* was introduced by Delia Kesner and Beniamino Accattoli, which according to Damiano Mazza is based on two fundamental ideas:

> *Explicit substitutions in the $\lambda$-calculus come from [[linear logic]]*: if you write down a term calculus for the $\{\multimap,!\}$ fragment of ILL, however you do it, you're going to see let binders pop up, and these behave very much like explicit substitutions, in the sense that normalization needs commuting conversions. Now, if you take Girard's (call-by-name) embedding of the $\lambda$-calculus in ILL, you see that $\beta$-reduction decomposes exactly along the lines of what an ES calculus would do, with the explicit substitutions corresponding to the let binders for exponentials. While I think that this had been noticed immediately after the introduction of linear logic, Delia and Beniamino took it a step further and designed an ES calculus (the structural $\lambda$-calculus, a precursor to the LSC) by mimicking the behavior of ILL terms which are in the image of Girard's embedding. The upshot is that, since Girard's embedding preserves SN, so does the structural $\lambda$-calculus: linear logic is the "right" way of implementing explicit substitutions.

> *Explicit substitutions should behave like boxes in proof nets*: the let binders mentioned above correspond to what are known as "exponential boxes" in proof nets. Cut-elimination in proof nets has the amazing property (from the rewriting viewpoint) that it does not need commuting conversions. In terms of explicit substitutions, this means that ESs should not need to "move" through a term in order to be applied, one should be able to apply them just where they are created. Rewriting should happen "remotely" or, as they say, "at a distance". Beniamino and Delia noticed that there's a consistent way of generalizing redexes in a term calculus with ESs (no need for proof nets!) so that they are transparent to the presence of ESs. Combine this with the above idea (that ESs should behave like in linear logic, i.e., be "linear", substituting one occurrence at a time) and you obtain the LSC.

The LSC, just as the previous versions of explicit substitutions calculi, has found applications in

> **abstract machines**: Beniamino Accattoli, Pablo Barenbaum, and Damiano Mazza showed how the LSC may be related in a very sharp way to a host of abstract machines, for any reduction strategy you can think of (call by name, call by value, call by need). Each strategy in the LSC "distills" an abstract machine, in the sense that the rewriting rules of the LSC are in bijection with the actual computational transitions of the machines, while the "administrative" transitions (needed by the machine to navigate through the term in search for the next redex) sort of "evaporate" and become structural equivalences in the LSC (they relate syntactically different terms which are behaviorally equal). In particular, the LSC gives a slick formulation of (weak) call-by-need.

In fact the formalization of abstract machines is given as the original motivation of Abadi et al.

The LSC also has applications in:

> **complexity**: for decades, people had been wondering whether $\beta$-reduction is a reasonable cost model for [[computational complexity]].  Most were convinced that the answer was "no", because $\beta$-reduction may make an arbitrary number of duplications in just one step.  Suprisingly, Beniamino Accattoli and Ugo Dal Lago showed that the answer is in fact "yes", and their proof uses the LSC in a crucial way, in particular its capability of succinctly expressing computation thanks to sharing.

## References
* Martín Abadi, Luca Cardelli, Pierre-Louis Curien, and Jean-Jacques Lévy. *Explicit Substitutions*. J. Funct. Program., 1(4):375–416, 1991. doi:10.1017/S0956796800000186.

* [[Paul-André Melliès]], *Typed $\lambda$-calculi with explicit substitution may not terminate*.  In M. Dezani (ed.), Int. Conf. on Typed Lambda Calculus and Applications, Lecture Notes in Computer Science, 1995.

* Roel Bloo and Kristoffer H. Rose, *Preservation of Strong Normalisation in Named Lambda Calculi with Explicit Substitution and Garbage Collection*.  In CSN-95: Computer Science in the Netherlands 1995, [citeseer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.5026)
 {#BlooRose}

* Eike Ritter, *Categorical abstract machines for higher-order typed lambda calculi*, Theoretical Computer Science, vol 136, p.125–162, 1994.

* Neil Ghani, Valeria de Paiva and Eike Ritter, *Linear Explicit Substitutions*, Journal of the IGPL, Vol, 8(1), pp.7-31, 2000.

* Neil Ghani, Valeria de Paiva and Eike Ritter, *Explicit substitutions for constructive necessity*, In Larsen, Kim G. and Skyum, Sven and Winskel, Glynn (eds.), Automata, Languages and Programming, 1998.

The linear substitution calculus is discussed in:

* Beniamino Accattoli, _An abstract factorization theorem for explicit substitutions_, [link](https://hal.archives-ouvertes.fr/hal-00780344)

* Beniamino Accattoli, Eduardo Bonelli, Delia Kesner, Carlos Lombardi, _A nonstandard standardization theorem_, [link](https://dl.acm.org/doi/10.1145/2535838.2535886)

* Beniamino Accattoli, Pablo Barenbaum, Damiano Mazza, _Distilling Abstract Machines_, [arxiv](https://arxiv.org/abs/1406.2370)

* Beniamino Accattoli and Ugo Dal Lago, _(Leftmost-outermost) beta reduction is invariant, indeed_, [lmcs](https://lmcs.episciences.org/1627/pdf)

[[!redirects explicit substitutions]]
[[!redirects linear substitution calculus]]
[[!redirects LSC]]


Exploding a Category > history

< [[Exploding a Category]]
[[!redirects Exploding a Category/history]]
[[!redirects Exploding a Category -- history]]


exponent of a group

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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
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=--
=--

# Contents #
* table of contents
{:toc}

## Definition

The *exponent* of a [[group]] $G$, denoted $\exp(G)$, is equivalently

* the least [[natural number]] $n$ such that $g^n = 1$, the [[identity element]], for all $g \in G$,

* hence the [[least common multiple]] of the [[order of a group element|orders]] of the group elements.

Here "least" should, by all rights, refer to the *divisibility order* on $\mathbb{N}$, where $m \leq n$ means $m$ divides $n$. Notice that with this poset structure, $\mathbb{N}$ is a [[complete lattice]] with bottom element $1$ and top element $0$; under this convention, the exponent always exists as the least common multiple of the orders of elements, which is their join ([[supremum]]) in the lattice, and is $0$ if (for example) $G$ is a [[torsion|torsion-free group]].

## Properties

If $G$ is a [[finite group]] of [[order of a group|order]] $n$, then $\exp(G)$ divides $n$ since $g^n = 1$ for all $g \in G$.

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

If $G$ is a [[finite group|finite]] [[abelian group]] and $\exp(G) = ord(G)$, then $G$ is [[cyclic group|cyclic]].

=--

+-- {: .proof}
###### Proof
Of course $ord(G)$ can't be $0$, so $e = \exp(G)$ here has a prime factorization
$e = p_1^{r_1} p_2^{r_2} \ldots p_k^{r_k}$. Since $e$ is the least common multiple of the orders of elements, the exponent $r_i$ is the maximum multiplicity of $p_i$ occurring in orders of elements; any element realizing this maximum will have order divisible by $p_i^{r_i}$, and some power $y_i$ of that element will have order exactly $p_i^{r_i}$. Then $y = \prod_i y_i$ will have order $e = ord(G)$ by the following lemma and induction, so that powers of $y$ exhaust all $ord(G)$ elements of $G$, i.e., $y$ generates $G$ as desired.
=--

+-- {: .num_lemma}
###### Lemma
If $m, n$ are relatively prime and $x$ has order $m$ and $y$ has order $n$ in an abelian group, then $x y$ has order $m n$.
=--

+-- {: .proof}
###### Proof
Suppose $(x y)^k = x^k y^k = 1$. For some $a, b$ we have $a m - b n = 1$, and so $1 = x^{k a m} y^{k a m} = y^{k a m} = y^k y^{k b n} = y^k$. It follows that $n$ divides $k$. Similarly $m$ divides $k$, so $m n = lcm(m, n)$ divides $k$, as desired.
=--

+-- {: .num_prop}
###### Proposition
If $G$ is a finite group, then the prime factors of $\exp(G)$ coincide with the prime factors of $ord(G)$.
=--

+-- {: .proof}
###### Proof
Since $\exp(G)$ divides $ord(G)$, every prime factor of $\exp(G)$ is a factor of $ord(G)$. If $p$ is a prime factor of $ord(G)$, then by the [[Cauchy group theorem]], $G$ has an element of order $p$, and thus $p$ divides $\exp(G)$.
=--

## Examples

* If $\exp(G) = 2$, then $G$ is abelian and is a [[vector space]] over the field $\mathbb{F}_2$.

* Similarly, if $G$ is abelian and its exponent is a prime $p$, then $G$ is a vector space over $\mathbb{F}_p$.

* A finitely generated group of exponent $3$, $4$, or $6$ must be finite. On the other hand, it is not known whether a group of exponent $5$ generated by $2$ elements must be finite.

These last facts are part of the lore of the celebrated Burnside problem. The *free Burnside group* $B(m, n)$ of exponent $n$ with $m$ generators is presented by $\langle x_1, \ldots, x_m\; |\; w^n = 1 \; for\; all\; words\; w\; in \; x_1, \ldots, x_m \rangle$, and one formulation of the original Burnside problem was whether $B(m, n)$ is finite. It is now known that the answer is negative for all $m \gt 1$ and odd $n \gt 665$, and there is a similar result for certain even $n$ as well.

* For a fixed prime $p$, a *Tarski monster* for $p$ is a finitely generated infinite group where every proper nontrivial subgroup is cyclic of order $p$. Such monsters exist for sufficiently large $p$, forming a class of dramatic counterexamples to the Burnside conjecture; they are of course of exponent $p$.

* It was shown by Efim Zelmanov that there are only finitely many finite groups with $m$ generators and exponent $n$ (restricted Burnside problem). It was largely on the strength of this work that he was awarded the Fields Medal (1994).

## References

See also

* [[Groupprops]], _[Exponent of a group](https://groupprops.subwiki.org/wiki/Exponent_of_a_group)_

* Wikipedia, _Burnside problem_, [web](https://en.wikipedia.org/wiki/Burnside%27s_problem)


exponentiable topos

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#### Topos Theory
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[[!include topos theory - contents]]
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#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

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

## Idea

An **exponentiable topos** is a generalization of the notion of an exponentiable [[locale]] and can be viewed as a (topological) "space" $X$ that behaves well with respect to the construction of mapping spaces $Y^X$.

## Definition

+-- {: .num_defn}
###### Definition
A [[Grothendieck topos]] $\mathcal{E}$ is called _exponentiable_ (in the [[2-category]] of Grothendieck toposes $GrTop$) if the 2-functor ${}_{-}\times\mathcal{E}$ has a right 2-adjoint $(_{-})^\mathcal{E}$.
=--

## Remarks

* More concretely, $\mathcal{E}$ is exponentiable if there exists a functor $(_{-})^\mathcal{E}$ such that for all toposes $\mathcal{F},\mathcal{G}$ the category $Hom(\mathcal{F}\times\mathcal{E},\mathcal{G})$ is (naturally) equivalent as a category to $Hom(\mathcal{F},\mathcal{G}^\mathcal{E})$.

* By setting $\mathcal{F}=\mathcal{S}$ , the base topos, one sees from $Hom(\mathcal{S},\mathcal{G}^\mathcal{E})=Hom(\mathcal{S}\times\mathcal{E},\mathcal{G})=Hom(\mathcal{E},\mathcal{G})$ that $\mathcal{G}^\mathcal{E}$ is indeed a "mapping space" whose [[point of a topos|points]] are the [[geometric morphism|geometric morphisms]] $\mathcal{E}\to\mathcal{G}$.

* The concept generalizes to [[higher topos theory]] (cf. [Anel 2015](#A15), [Anel-Lejay 2018](#AJ18), [Lurie 2018](#Lurie18)).

## Properties

Interestingly, in the category of locales exponentiability of a locale $X$ hinges on the existence of the single exponential $S^X$ where $S$ is the [[Sierpinski space]]: $Y^X$ exists for all $Y$ iff $S^X$ exists.

In $GrTop$ the [[classifying topos for the theory of objects|object classifier]] $\mathcal{S}[\mathbb{O}]$ takes over the role of the Sierpinski space and we have the following

+-- {: .num_prop}
###### Proposition
A [[Grothendieck topos]] $\mathcal{E}$ is exponentiable iff the [[exponential object|exponential]] $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ exists.

=--

This result is due to Johnstone-Joyal ([1982](#JJ82), p.282) and occurs as theorem 4.3.1 of Johnstone ([2002, vol.1 p.433](#J02)).

The following theorem pursues this analogy and generalizes a result of [[Martin Hyland]] on locales ([1981](#Hyland81)).

+-- {: .num_prop}
###### Theorem
A [[Grothendieck topos]] $\mathcal{E}$ is an exponentiable object in the 2-category of Grothendieck toposes and [[geometric morphisms]] iff $\mathcal{E}$ is a [[continuous category]].

=--

This result is due to Johnstone-Joyal ([1982](#JJ82)) and occurs as theorem 4.4.5 of Johnstone ([2002, p.748](#J02)).

Exponentiability is a local property:
+-- {: .num_prop}
###### Proposition
1. If a [[Grothendieck topos]] $\mathcal{E}$ is exponentiable so is $\mathcal{E}/X$ for any object $X\in\mathcal{E}$.
2. If $\mathcal{E}/X$ is exponentiable and  $X\to 1$ is an epimorphism, then $\mathcal{E}$ is exponentiable as well.

=--

This occurs as lemma 4.2 in Johnstone-Joyal ([1982](#JJ82), p.281).

Continuity also leaves a lattice-theoretic trace:

+-- {: .num_prop}
###### Proposition
If a [[Grothendieck topos]] $\mathcal{E}$ is exponentiable then the lattice of [[subobject|subobjects]] of any object $X\in\mathcal{E}$ is [[continuous lattice|continuous]].

=--

This occurs as lemma 5.1 in Johnstone-Joyal ([1982](#JJ82), p.287).

## Examples

* Since [[locally finitely presentable categories]] are [[continuous category|continuous]] and [[coherent topos|coherent toposes]] are locally finitely presentable (cf. Johnstone ([2002, p.915](#J02))) it follows that _coherent toposes are exponentiable_. This can be viewed as an avatar of the fact that (locally) compact topological spaces behave well with respect to mapping spaces.

* By the same reasoning all functor categories $Set^{\mathcal{C}}$ for  $\mathcal{C}$ a [[small category]] are exponentiable since they are [[locally finitely presentable category|locally finitely presentable]]. This includes in particular all [[presheaf toposes]] on small categories.

* An example of the latter is the [[Sierpinski topos]] $\mathcal{S}^2$. Here the exponential $\mathcal{F}^{\mathcal{S}^2}$ classifies the [[theory of model homomorphisms|theory of $\mathbb{T}$-model homorphisms]] (with $\mathbb{T}$ the [[geometric theory]] classified by $\mathcal{F}$) i.e. the theory $\mathbb{T}^2$ such that $Mod_{\mathbb{T}^2}(\mathcal{E})=Mod_\mathbb{T}(\mathcal{E})^2$.

##Remarks on duality

It is worthwhile to muse a bit about $\mathcal{S}[\mathbb{O}]^\mathcal{E}$:

First of at all, it is [[injective topos|injective]] since $\mathcal{S}[\mathbb{O}]$ is injective and $(_-)^\mathcal{E}$ preserves inclusions.

This and the various universal properties of the toposes involved imply that an exponentiable topos
$$\mathcal{E}\cong Hom(\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{S}\times\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{S},\mathcal{S}[\mathbb{O}]^\mathcal{E})$$
is (up to equivalence) the _category of points of an injective topos_.

Now consider an arbitrary topos $\mathcal{E}$ classifying a geometric theory $\mathbb{T}$.

Then from the [[geometric theory#FunctorialDefinition|functorial perspective on logic]] the 2-functor $\mathbb{T}^*:GrTop^{op}\to CAT$ assigning to $\mathcal{F}$ the category of models $\mathbb{T}^*(\mathcal{F}):=\mathcal{F}\times\mathcal{E}=\mathcal{F}\times\mathcal{S}[\mathbb{T}]$ is called the **dual theory** of $\mathbb{T}$.

It need not be geometric i.e. have a [[classifying topos]] but when it is, $\mathbb{T}$ being called _dualizable_ in that case, the following

$$\mathbb{T}^*(\mathcal{F})=\mathcal{F}\times\mathcal{E}\cong Hom(\mathcal{F}\times\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{F},\mathcal{S}[\mathbb{O}]^\mathcal{E})$$

shows that $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ has precisely the properties required for $\mathcal{S}[\mathbb{T}^*]$.

In other words, $\mathbb{T}^*$ is geometric (and classified by $\mathcal{S}[\mathbb{O}]^\mathcal{E}$) precisely iff $\mathcal{E}=\mathcal{S}[\mathbb{T}]$ is exponentiable!

Compare also the remarks of Anel ([2015](#A15)) on the $\infty$-case.

### Examples

* Let $0$ be the [[initial topos]] that classifies the _inconsistent theory_ $\mathbb{T}^\emptyset_1$ over the empty signature. Then from $\mathcal{S}[\mathbb{O}]^0=\mathcal{S}$ follows that the _dual_ of the inconsistent theory is $\mathbb{T}_1^{\emptyset\ast}=\mathbb{T}_{\emptyset}^{\emptyset}$ with $\mathbb{T}_{\emptyset}^{\emptyset}$ the [[empty theory]] (over the empty signature).

* From $\mathcal{S}[\mathbb{O}]^\mathcal{S}=\mathcal{S}[\mathbb{O}]$ follows in turn that the _double dual_ of the inconsistent theory, or, in other words, the dual of the empty theory, is $\mathbb{T}_1^{\emptyset\ast\ast}=\mathbb{T}^{\emptyset\ast}_\emptyset=\mathbb{O}$ with $\mathbb{O}$ the [[theory of objects]]. (More generally, the dual theory of any dualizable theory is itself dualizable.)

* Let $\mathcal{S}^2$ be the [[Sierpinski topos]] that classifies [[subterminal object|subterminal objects]] or the [[localic topos|theory of completely prime filters]] of the frame of opens of the [[Sierpinski space]]. Since $\mathcal{E}\times \mathcal{S}^2\cong \mathcal{E}^2$ it follows that
  $$\mathcal{E}^2=Hom(\mathcal{E}^2,\mathcal{S}[\mathbb{O}])=Hom(\mathcal{E}\times\mathcal{S}^2,\mathcal{S}[\mathbb{O}])= Hom(\mathcal{E},\mathcal{S}[\mathbb{O}]^{\mathcal{S}^2})=Hom(\mathcal{E},\mathcal{S}[\mathbb{O}^2])\quad ,$$
  i.e. the dual theory is the [[theory of model homomorphisms|theory of morphisms]] $\mathbb{O}^2$.

### The theory of sheaves on a coherent site

Recall, that given a small site $(\mathcal{C},J)$ there is a [[geometric theory]] $\mathbb{F}_J$ called the [[theory of flat functors|theory of J-continuous flat functors]] that (albeit uneconomically) axiomaticizes the theory classified by $Sh(\mathcal{C},J)$ using the objects and morphisms in $\mathcal{C}$ for its signature. When $Sh(\mathcal{C},J)$ is exponentiable this theory is dualizable, and, in the case the site $(\mathcal{C},J_c)$ is [[coherent site|coherent]] i. e. $\mathcal{C}$ has finite limits and $J_c$ is generated by finite covering families, the following theory $\mathbb{S}$ called **the theory of sheaves** on $(\mathcal{C},J_{c})$ axiomatizes the dual theory $\mathbb{F}_{J_{c}}^\ast$ (in an equally uneconomic way):

The signature of $\mathbb{S}$ has a type symbol $A'$ for every object $A\in\mathcal{C}$ and a function symbol $f':B'\to A'$ for every morphism $f:B\to A$ in $\mathcal{C}^{op}$. The axioms are given by the following schemata

* $\top\vdash_x i'(x)=x$ for all identity morphisms $i$.

* $\top\vdash_x f'(x)=h'(g'(x))$ for all composable triples $f=g\circ h$ in $\mathcal{C}$.

* $\bigwedge_{i=1}^{n} (\alpha_i'(x)=a_i'(x'))\vdash_{x,x'} x=x'$ for each generating finite $J_{c}$-covering family $(Y_i\overset{\alpha_i}{\to} X|i=1,\dots, n)$.

* $\bigwedge_{i,j}(\beta_{ij}'(y_i)=\gamma_{ij}'(y_j))\vdash_{y_i,y_j}\exists x\big ( \bigwedge_{i=1}^{n}\alpha_i'(x)=y_i\big )$ for each generating finite $J_{c}$-covering family $(Y_i\overset{\alpha_i}{\to} X|i=1,\dots, n)$ and pullback

  $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
  \array{
    Z_{ij} &\overset{\beta_{ij}}{\longrightarrow}& Y_i
    \\
    {}^{\mathllap{\gamma_{ij}}}
    \downarrow
    & &
    \downarrow^{\mathrlap{\alpha_i}}
    \\
    Y_j
      &\underset{\alpha_j}{\longrightarrow}&
    X
  }
$ $\qquad \qquad.$

The models of $\mathbb{S}$ in a Grothendieck topos $\mathcal{F}$ are "sheaves of $\mathcal{F}$-objects" on $(\mathcal{C},J_{c})$ i.e. functors $\mathcal{C}^{op}\to\mathcal{F}$ satisfying the diagrammatic form of the sheaf axioms but

$$Mod_\mathbb{S}(\mathcal{F})=\mathcal{F}\times Sh(\mathcal{C},J_{c})=Hom(\mathcal{F}\times Sh(\mathcal{C},J_{c}), \mathcal{S}[\mathbb{O}])=Hom(\mathcal{F},\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})})$$

i.e. $\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})}$ is the classifying topos for $\mathbb{S}$. (For the details cf. Johnstone [1977](#J77), p.248f).

In particular, since $\mathbb{S}$ is a [[coherent logic|coherent theory]] this implies that $\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})}$ is also a [[coherent topos]]. From

$$Hom(\mathcal{S},\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})})=Mod_\mathbb{S}(\mathcal{S})=\mathcal{S}\times {Sh(\mathcal{C},J_{c})}=Sh(\mathcal{C},J_c)$$

it follows then that every coherent topos $Sh(\mathcal{C},J_c)$ is the _category of points_ of a coherent topos.

## Ramifications

The notion of a [[tiny object]] in a cartesian closed category suggests the following

+-- {: .num_defn}
###### Definition
An exponentiable Grothendieck topos $\mathcal{E}$ is called _tiny_ (or _infinitesimal_) if the 2-functor $(_{-})^\mathcal{E}$ has a right 2-adjoint $(_{-})_\mathcal{E}$.
=--

## Related entries

* [[exponential object]]

* [[continuous category]]

* [[reflexive object]]

* [[injective topos]]

* [[ind-object]]

* [[classifying topos for the theory of objects]]

* [[cartesian closed 2-category]]

* [[convenient category of spaces]]

* [[Verdier duality]]

## References

* [[Mathieu Anel]], _Toposes are commutative rings_ , talk at _[Topos à l'IHES](https://indico.math.cnrs.fr/event/747/)_,  IHES Paris, Nov. 25-27, 2015 ([video recording](https://www.youtube.com/watch?v=wG5MZqj_JK8), at min 52:47 ff.).

* [[Mathieu Anel]], Damien Lejay, _Exponentiable Higher Toposes_ , arXiv:1802.10425 (2018). ([abstract](https://arxiv.org/abs/1802.10425))

* {#Blass84}[[Andreas Blass]], _The interaction of category theory and set theory_ , Cont. Math. **30** (1984) pp.5-29. ([draft](http://www.math.lsa.umich.edu/~ablass/interact.pdf))

* {#Hyland81}[[Martin Hyland]], _Function spaces in the category of locales_ , Springer LNM **871** (1981) pp.264-281.

* {#J77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Also available as Dover reprint Mineola 2014; pp.209-210, 248-249)

* {#J02}[[Peter Johnstone]], _Sketches of an Elephant vols. 1&2_ , CUP 2002. (sections B4.3 pp.432-438, C4.4 pp.745-754)

* {#JJ82}[[Peter Johnstone]], [[André Joyal]], _Continuous categories and exponentiable toposes_ ,  JPAA **25** (1982) pp.255-296.

* {#Lurie18}[[Jacob Lurie]], _Spectral Algebraic Geometry_ , ms. Harvard University 2018. (section 21.1.6)

* [[Susan Niefield]], _Exponentiable Morphisms: posets, spaces, locales and Grothendieck toposes_ , TAC **8** (2001) pp.16-32. ([abstract](http://www.tac.mta.ca/tac/volumes/8/n2/8-02abs.html))

* [[Myles Tierney]], _Forcing Topologies and Classifying Toposes_ , pp.211-219 in Heller, Tierney (eds.), _Algebra, Topology and Category Theory_  , Academic Press New York 1976. (p.211, 218)

[[!redirects Exponentiable topos]]
[[!redirects exponential topos]]


exponential exact sequence

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## Idea

In the context of [[complex analytic geometry]], the term "exponential exact sequence" typically referes to the [[short exact sequence]]

$$
  0 \to ker(\exp) \longrightarrow \mathbb{G}_{a} \stackrel{\exp(\tfrac{i}{\hbar}(-))}{\longrightarrow} \mathbb{G}_m \to 0
$$

given by the [[exponential map]] $\exp(\tfrac{i}{\hbar}(-))$ from the [[additive group]] to the [[multiplicative group]]. Here $\hbar$ is any element of $\mathbb{R}^\times$ ("[[Planck's constant]]") but is traditionally set either to $1$ or to $1/2 \pi$.

Hence more explicitly over the [[complex numbers]] this is

$$
  0 \to \hbar2\pi\mathbb{Z} \longrightarrow \mathbb{C} \stackrel{\exp(\tfrac{i}{\hbar}(-))}{\longrightarrow} \mathbb{C}^\times \to 0
  \,,
$$

where $\mathbb{C}$ denotes the complex numbers as the additive [[abelian group]] and where $\mathbb{C}^\times = \mathbb{C} - \{0\}$ is the [[group of units]] of the [[ring]] structure of the complex numbers.

Often this is considered and displayed [[slice topos|relative]] to a [[complex analytic space]] $X$, where in terms of the [[structure sheaf]] $\mathcal{O}_{X}$ it reads

$$
  0 \to Lconst(\mathbb{Z}) \longrightarrow \mathcal{O}_X \stackrel{\exp(\tfrac{i}{\hbar}(-))}{\longrightarrow} \mathcal{O}_X^\times \to 0
  \,.
$$

In this form the sequence is then also called the _exponential sheaf sequence_.

## Properties

### Long sequence in cohomology
 {#LongSequenceInCohomology}

The [[connecting homomorphisms]] of the [[long exact sequence in cohomology]] induces by the exponential exact sequence

$$
  H^n(-,\mathbb{G}_m) \longrightarrow H^{n+1}(-,\mathbb{Z})
$$

encode the canonical [[characteristic classes]] of [[line n-bundles]].

* For $n=1$ this is the [first Chern class in complex analytic geometry](first+Chern+class#InComplexAnalyticGeometry) defined on the [[Picard group]] of [[holomorphic line bundles]];

* for $n = 2$ this is the complex analytic-analog of the [[Dixmier-Douady class]] of [[holomorphic 2-line bundles]], defined on the [[bigger Brauer group]];

and so on.

### In logarithmic geometry
 {#InLogarithmicGeometry}

In [[algebraic geometry]] there is no exponential sequence, the closest analogs being the [[Kummer sequence]] and the [[Artin-Schreier sequence]]. But in [[logarithmic geometry]] there is again a kind of exponential sequence (e.g. [Ogus 01, chapter IV, remark 1.1.7](#Ogus01), [Brylinski 94, page 15](#Brylinski94)). Compare also the sequences in [Kato-Nakayama 99, section 1.4](#KatoNakayama99).

## Related concepts

* [[Kummer exact sequence]]

* [[Artin-Schreier exact sequence]]

* [[Kummer-Artin-Schreier-Witt exact sequence]]

* [[Bockstein homomorphism]]

## References

* Wikipedia, _[Exponential sheaf sequence](http://en.wikipedia.org/wiki/Exponential_sheaf_sequence)_

Discussion in [[real analytic geometry]]:

* {#Huisman02} [[Johannes Huisman]], _The exponential sequence in real algebraic geometry and Harnack's Inequality for proper reduced real schemes_, Communications in Algebra, Volume 30, Issue 10, 2002 ([pdf](http://pageperso.univ-brest.fr/~huisman/rech/publications/exphi.pdf))

Discussion in [[logarithmic geometry]]

* {#Brylinski94} [[Jean-Luc Brylinski]], _Holomorphic gerbes and the Beilinson regulator_, Astérisque 226 (1994): 145-174 ([[Brylinski94.pdf:file]])

* {#Ogus01} [[Arthur Ogus]], _Lectures on logarithmic algebraic geometry_, TeXed notes, 2001, [pdf](http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf)

* {#KatoNakayama99} [[Kazuya Kato]], Chikara Nakayama, _Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $\mathbb{C}$_ Kodai Math. J.
Volume 22, Number 2 (1999), 161-186.
([ProjectEuclid](http://projecteuclid.org/euclid.kmj/1138044041))

[[!redirects exponential exact sequences]]

[[!redirects exponential sequence]]
[[!redirects exponential sequences]]

[[!redirects exponential short exact sequence]]
[[!redirects exponential short exact sequences]]


exponential ideal

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## Definition

A class $I$ of [[object]]s in a [[cartesian closed category]] $C$ is called an **exponential ideal** if whenever $Y\in I$ and $X\in C$, the [[exponential object]] $Y^X$ is in $I$.

## Properties

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

If $I \hookrightarrow C$ is a [[reflective subcategory]], then it is an exponential ideal if and only if its [[reflector]] $C\to I$ preserves [[finite products]].

=--

This appears for instance as ([Johnstone, A4.3.1](#Johnstone)); it can also been seen as a consequence of [[Day's reflection theorem]]. See also at _[[reflective subuniverse]]_.  Note that in this case $I$ is itself a [[cartesian closed category]], since being a reflective subcategory it is also closed under finite products.

## References

The relation of exponential ideals to [[reflective subcategories]] is discussed in section A4.3.1 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
  {#Johnstone}

[[!redirects exponential ideals]]


exponential law for parameterized topological spaces -- references

### Exponential law for parameterized topological spaces
 {#ParameterizedTopology}

On [[exponential law for spaces|exponential objects]] ([[internal homs]]) in [[slice categories]] of ([[compactly generated topological spaces|compactly generated]]) [[topological spaces]] -- see at *[[parameterized homotopy theory]]*):

* {#Booth70} [[Peter I. Booth]], *The Exponential Law of Maps I*, Proceedings of the London Mathematical Society **s3-20** 1 (1970) 179-192 $[$[doi:10.1112/plms/s3-20.1.179](https://doi.org/10.1112/plms/s3-20.1.179)$]$

* {#Booth71} [[Peter I. Booth]], *The exponential law of maps. II*, Mathematische Zeitschrift **121** (1971) 311–319 $[$[doi:10.1007/BF01109977](https://doi.org/10.1007/BF01109977)$]$

* {#BoothBrown78a} [[Peter I. Booth]], [[Ronnie Brown]], *Spaces of partial maps, fibred mapping spaces and the compact-open topology*, General Topology and its Applications **8**  2 (1978) 181-195 $[$doi:10.1016/0016-660X(78)90049-1$]$

* {#BoothBrown78b} [[Peter I. Booth]], [[Ronnie Brown]], *On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps*, General Topology and its Applications **8** 2 (1978) 165-179 $[$doi:10.1016/0016-660X(78)90048-X$]$

* {#Lewis85} [[L. Gaunce Lewis, Jr.]],  §1 of:  *Open maps, colimits, and a convenient category of fibre spaces*, Topology and its Applications **19** 1 (1985) 75-89 $[$doi.org/10.1016/0166-8641(85)90087-2$]$

And with an eye towards [[parameterized homotopy theory]]:

* [[Ioan Mackenzie James]]: §II.9 in: *Fibrewise topology*, Cambridge Tracts in Mathematics, Cambridge University Press (1989) $[$ISBN:9780521360906$]$

* {#MaySigurdsson06} [[Peter May]], [[Johann Sigurdsson]], §1.3.7-§1.3.9 in: _[[Parametrized Homotopy Theory]]_,  Mathematical Surveys and Monographs, vol. 132, AMS 2006   ([ISBN:978-0-8218-3922-5](https://bookstore.ams.org/surv-132), [arXiv:math/0411656](https://arxiv.org/abs/math/0411656), [pdf](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf))


exponential law for spaces

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## Idea

When in a [[convenient category of topological spaces]], e.g. [[compactly generated space]]s, the category is [[cartesian closed category|cartesian closed]], so that there is an [[adjunction]] between the mapping space and the cartesian product in that category.  For general [[topological space]]s there is no globally defined adjunction, but we can instead characterize exactly which spaces are exponentiable.

## Exponentiable spaces

For $C$ a [[finitely complete category|category with finite products]], recall that an object $c$ is **[[exponentiable object|exponentiable]]** if the functor $c \times -: C \to C$ has a [[right adjoint]], usually denoted $(-)^c: C \to C$.

+-- {: .un_theorem}
###### Theorem (Exponentiability, I)
Let [[Top]] be the category of all topological spaces. An object $X$ of $Top$ is exponentiable if and only if $X \times -: Top \to Top$ preserves [[coequalizer]]s, or equivalently [[quotient space]]s.
=--

This functor always preserves [[coproducts]], so this condition is equivalent to saying that $X \times -$ preserves all small [[colimits]].  This is then equivalent to exponentiability by the [[adjoint functor theorem]].

This condition, however, is not really any more explicit.  More interesting is to characterize the exponentiable spaces in terms of a point-set-topological condition.

### Core-compactness

For open subsets $U$ and $V$ of a topological space $X$, we write $V\ll U$ to mean that any open cover of $U$ admits a finite subcover of $V$; this is read as **$V$ is relatively compact under $U$** or **$V$ is way below $U$**.  We say that $X$ is **core-compact** if for every open neighborhood $U$ of a point $x$, there exists an open neighborhood $V$ of $x$ with $V\ll U$.  In other words, $X$ is core-compact iff for all open subsets $V$, we have $V = \bigcup \{ U | U\ll V \}$.  This says essentially the same thing as saying that the open-set lattice of $X$ is a [[continuous lattice]], which yields the corresponding definition for [[locales]].

+-- {: .un_theorem}
###### Theorem (Exponentiability, II)
An object $X$ of $Top$ is exponentiable if and only if it is core-compact.
=--

If $X$ is [[Hausdorff space|Hausdorff]], then core-compactness is equivalent to [[locally compact space|local compactness]]; thus in particular all locally compact Hausdorff spaces are exponentiable.  In fact, local compactness implies exponentiability even without the Hausdorff condition, if [[locally compact space|local compactness]] is defined appropriately (for every point the compact neighborhoods form a neighborhood basis). For this reason, that core-compactness is also called **quasi local compactness**.

When $X$ is core-compact, we can explicitly describe the exponential topology on $Y^X$ (whose points are continuous maps $f: X \to Y$).  It is generated by [[topological basis|subbasis]] elements $O_{U,V}$, for $U$ an open subset of $X$ and $V$ an open subset of $Y$, where a continuous map $f\colon X \to Y$ belongs to $O_{U,V}$ iff $U\ll f^{-1}(V)$:

$$ O_{U, V} = \{f \in Y^X: U \ll f^{-1}(V)\} .$$

If $X$ and $Y$ are Hausdorff, then this topology on $Y^X$ coincides with the [[compact-open topology]].

### In terms of convergence {#ViaConvergence}

Exponentiable (i.e. core-compact) spaces can also be characterized in terms of [[ultrafilter]] convergence.  Recall that a topological space can equivalently be defined as a [[relational beta-module|lax algebra for the ultrafilter monad]] $U$ on the [[(1,2)-category]] [[Rel]] of sets and relations.  In other words, it consists of a set $X$ and a relation $R\colon U X \to X$ called "convergence", such that $id_X \subseteq R \circ \eta$ and $R\circ U R \subseteq R\circ \mu$, where $\eta$ and $\mu$ are the unit and multiplication of the ultrafilter monad, regarded as relations.  In the paper

* [[Claudio Pisani]], _Convergence in exponentiable spaces_, [TAC](http://www.tac.mta.ca/tac/volumes/1999/n6/5-06abs.html)

it is shown that a space is exponentiable (i.e. core-compact) if and only if we have equality in the multiplication law $R\circ U R = R\circ \mu$.

Some intuition for this characterization can be obtained as follows.  Consider the standard non-locally-compact space, the rationals $\mathbb{Q}$ as a subspace of the reals $\mathbb{R}$.  Suppose that $x$ is a rational number and that $y_n$ is a sequence of irrationals converging to $x$.  Then for each $n$ we can find a sequence $z^n_m$ of rationals which converges to $y_n$; hence the $z^n_m$ form a "sequence of sequences" which "globally converges" to $x$ in $\mathbb{Q}$, i.e. which are related to $x$ by the composite relation $R\circ \mu$, but for which does not converge elementwise to an intermediate sequence which in turn converges to $x$, i.e. it is not related to $x$ by the relation $R \circ U R$.  It turns out that when generalized to ultrafilter convergence, this sort of behavior exactly characterizes what it means to fail to be (quasi) locally compact.

## General exponential laws

If $X$ is exponentiable, then the exponential law gives us an isomorphism of *sets* $Map(Y,B^X) \cong Map(X\times Y,B)$ for any other spaces $B$ and $Y$.  If $Y$ is also exponentiable, then the [[Yoneda lemma]] yields from this a homeomorphism $B^{X\times Y} \cong (B^X)^Y$.  However, we can also say some things in general without all spaces involved being exponentiable.

We now agree to denote by $Map(X,Y)=X^Y$ the space of continuous maps $X\to Y$ in the [[compact-open topology]].

+-- {: .un_theorem}
###### Theorem (Exponential law)
Let
$X,Y,B$ be [[topological spaces]]. For any $f\in B^{X\times Y}$,
the formula
$$
[(\theta f)(y)](x) = f(x,y)
$$
defines a [[continuous map]] $\theta f:Y\to B^X$
which we call the map adjoint to $f$, or the [[adjunct]] of $f$.

The adjunction map
$$
\theta : Map(X\times Y,B)\to Map(Y,B^X), \,\,\,\,\,\,\,\theta:f\mapsto \theta f
$$
is a [[one-to-one function]], and if $X$ is [[locally compact space|locally compact and Hausdorff]] then $\theta$ is a [[bijection]].
Independently from that assumption on $X$, if $Y$ is
[[Hausdorff space|Hausdorff]], then $\theta$ is continuous in the [[compact-open topology]]
$$
\theta : B^{X\times Y}\to (B^X)^Y.
$$

If both assumptions (on $X$ and $Y$) are satisfied, then
$\theta$ is not only a continuous bijection, but also open, hence a [[homeomorphism]].
=--

## Based exponential laws

There is also a version for based (= [[pointed space|pointed]]) topological spaces. The [[cartesian product]] then needs to be replace by the [[smash product]] of the based spaces. Regarding that the maps preserve the base point, the adjunction map $\theta$ induces the adjunction map

 $$\theta_*:Map_*(X\wedge Y,B)\to Map_*(Y,Map_*(X, B))$$

where the mapping space $Map_*$ for based spaces is the subspace of the usual mapping space, in the compact-open topology, which consists of the mappings preserving the base point.

It appears that $\theta_*$ is again one-to-one and continuous, and it is bijective if $X$ is locally compact Hausdorff. If $X, Y$ are compact Hausdorff then $\theta_*$ is a homeomorphism.

## Related concepts

* [[exponential object]], [[mapping space]]

* [[compactly generated topological space]]

## References

### General
 {#GeneralReferences}

* [[C. R. F. Maunder]], §6.2 in: *Algebraic Topology*, Cambridge University Press, Cambridge (1970, 1980) $[$[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/maunder.pdf)$]$

* [[Peter Johnstone]], §7.4 of: _[[Stone Spaces]]_, Cambridge Studies in Advanced Mathematics **3**, Cambridge University Press (1982, 1986) $[$[ISBN:9780521337793](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/stone-spaces?format=PB&isbn=9780521337793)$]$

* Eva Lowen-Colebunders and Günther Richter, _An elementary approach to exponential spaces_, Applied Categorical Structures **9** (2001) 303–310 $[$[doi:10.1023/A:1011268007097](https://doi.org/10.1023/A:1011268007097), [MR](http://www.ams.org/mathscinet-getitem?mr=1836256)$]$

* [[Martín Escardó]], [[Reinhold Heckmann]], *Topologies of spaces of continuous functions* (2001) $[$[pdf](http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf), [[EscardoHeckmann-TopologiesOnFunctionSpaces.pdf:file]]$]$

* [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], §1.3 of: *Algebraic topology from a homotopical viewpoint*, Springer Science & Business Media, 2008 $[$[doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586), [toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf)$]$

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exponential map

> For the concept in [[category theory]] see at _[[exponential object]]_.

***

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#### Analysis
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#### Algebra
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# Exponential maps
* table of contents
{: toc}

## Idea

The exponential [[function]] of classical [[analysis]] given by the [[series]],
\[ \label{series} \exp x \coloneqq \sum_{i = 0}^{\infty} \frac{x^i}{i!} ,\]
is the solution of the [[differential equation]]
$$ f' = f $$
with initial value $f(0) = 1$.

It is also the limit of the following [[sequence]] of [[polynomial functions]]

$$f_n(x) \coloneqq \left(1 + \frac{x}{n}\right)^n$$

$$\exp x \coloneqq \lim_{n \to \infty} f_n(x)$$

This classical function is defined on the [[real line]] (or the [[complex plane]]).  To generalise it to other [[manifolds]], we need two things:

*  by its nature, the argument of the function should be a [[tangent vector]]; so in the classical function $\mathbb{R} \to \mathbb{R}$, the [[source]] $\mathbb{R}$ is really the [[tangent space]] to the [[target]] $\mathbb{R}$ at the point $\exp 0 = 1$.
*  We need a [[covariant derivative]] to tell us what $f'$ means.

So in the end we have, for any [[point]] $p$ on a [[differentiable manifold]] $M$ with an [[affine connection]] $\Del$, a map $\exp_p\colon T_p M \to M$, which is defined at least on a [[neighbourhood]] of $0$ in the [[tangent space]] $T_p M$.

Note that $p$ here comes from the initial value $\exp_p 0 = p$; we usually take $p = 1$ when we work in a [[Lie group]], but otherwise we are really generalising the classical exponential function $x \mapsto p \exp x$; every solution to $f' = f$ takes this form.

Classically, there are some other functions called 'exponential'; given any nonzero real (or complex) number $b$, the map $x \mapsto b^x$ (or even $x \mapsto p\, b^x$) is also an exponential map.  Using the [[natural logarithm]], we can define $b^x$ in terms of the natural exponential map $\exp$:
$$ b^x \coloneqq \exp (x \ln b) .$$
So while $b$ is traditionally called the 'base', it is really the number $\ln b$ that matters, or even better the operation of multiplication by $\ln b$.  This operation is an [[endomorphism]] of the real line (or complex plane), and every such endomorphism takes this form for some nonzero $b$ (and some branch of the natural logarithm, in the complex case).  So we see that this generalised exponential map is simply the [[composite]] of the natural exponential map after a linear endomorphism.

## Definition

See also at _[[flow of a vector field]]_.

### With respect to an affine connection

Let $M$ be a [[differentiable manifold]], let $\Del$ be an [[affine connection]] on $M$, and let $p$ be a [[point]] in $M$.  Then by the general theory of [[differential equations]], there is a unique [[maximal partial function|maximally]] defined [[partial function]] $\exp_p$ from the [[tangent space]] $T_p M$ to $M$ such that:

*  $\Del \exp_p = \exp_p$ and
*  $\exp_p(0) = 1$.

This function is the __natural exponential map on $M$ at $p$ relative to $\Del$__.  We have $\exp_p\colon U \to M$, where $U$ is some [[neighbourhood]] of $0$ in $T_p M$.  If $M$ is [[complete manifold|complete]] (relative to $\Del$), then $U$ will be all of $T_p M$.

Let $M$ be a [[Riemannian manifold]] (or a [[pseudo-Riemannian manifold]]) and let $p$ be a point in $M$.  Then $M$ may be equipped with the [[Levi-Civita connection]] $\Del_{lc}$, so we define the __natural Riemannian exponential map on $M$ at $p$__ to be the natural exponential map on $M$ at $p$ relative to $\Del_{lc}$.

Given any [[endomorphism]] $\phi\colon T_p M \to T_p M$, we can also consider the __exponential map on $M$ at $p$ relative to $\Del$ with logarithmic base $\phi$__, which is simply $x \mapsto \exp_p \phi(x)$.  We say 'logarithmic base' since a classical exponential function with base $b$ corresponds to an exponential function whose logarithmic base is multiplication by $\ln b$.

### Via geodesics

Recall that a [[geodesic]] is a [[curve]] on a manifold whose [[velocity]] is constant (as measured along that curve relative to a given affine connection).  Working naïvely, we may write
$$ \gamma' = v ,$$
pretend that this is a differential equation for a function $\gamma\colon \mathbb{R} \to \mathbb{R}$, and take the solution
$$ \gamma(t) = p \exp t x ,$$
where $p$ is given by the initial value $\gamma(0) = p$.  We recognise this as being, morally, $\exp_p t x$.  This suggests (although we need more work for a proof) the following result:

Let $M$ be a [[differentiable manifold]], let $\Del$ be an [[affine connection]] on $M$, and let $p$ be a [[point]] in $M$.  Given a [[tangent vector]] $x$ at $p$, there is a unique [[maximal geodesic]] $\gamma$ on $M$ tangent to $x$ at $p$.  If $\gamma(1)$ is defined (which it will be whenever $M$ is [[complete manifold|complete]] and may be in any case), we have $exp_p x = \gamma(1)$.  In any case, we have $\exp_p (t x) = \gamma(t)$ for sufficiently small $t$.

## In real algebras

Let $\mathbb{R}$ be any [[sequentially Cauchy complete]] [[Archimedean field|Archimedean]] [[ordered field|ordered]] [[field]], and let $F:\mathbb{R}\mathrm{Alg} \to \mathbb{R}\mathrm{Vect}$ be the [[forgetful functor]] which takes an $\mathbb{R}$-algebra $A$ to its underlying $\mathbb{R}$-[[vector space]] $F(A)$. Given any $\mathbb{R}$-algebra $A$ whose underlying $\mathbb{R}$-[[vector space]] $F(A)$ is finite-dimensional, each element of $A$ could be expressed as a [[linear combination]], a finite sum of basis vectors. Then one could define the exponential function $\exp:A \to A$ as either

$$\exp(x) \coloneqq \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n$$

or

$$\exp(x) \coloneqq \lim_{n \to \infty} \sum_{i = 0}^{n} \frac{x^i}{i!}$$

since according to the [[algebraic limit theorem]], limits distribute over finite sums.

In particular, this is how exponential functions are defined in [[Clifford algebras]] and [[matrix algebras]]. However, the exponential functions in non-commutative algebras are not [[abelian group homomorphisms]], because multiplication is not commutative while addition is commutative.

### In the dual numbers

The [[dual number]] real algebra $\mathbb{D} \coloneqq \mathbb{R}[\epsilon]/\epsilon^2$ has a notion of exponential function, which is the solution to the functional equation $\exp(x + \epsilon) = \exp(x) (1 + \epsilon)$ with $\exp(0) = 1$.

## In arbitrary Archimedean ordered fields

In general, Archimedean ordered fields which are not [[sequentially Cauchy complete]] do not have an exponential function. Nevertheless, the exponential map is still guaranteed to be a [[partial function]], because every Archimedean ordered field is a [[Hausdorff space]] and thus a [[sequentially Hausdorff space]]. Thus, an axiom could be added to an Archimedean ordered field $F$ to ensure that the exponential partial function is actually a total function:

**Axiom of exponential function:** _For all elements $x \in F$, there exists a unique element $\exp(x) \in F$ such that for all positive elements $\epsilon \in F_+$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \in \mathbb{N}$, if $n \geq N$, then $-\epsilon \lt \left(1 + \frac{x}{n}\right)^{n} - \exp(x) \lt \epsilon$ (or equivalently, $-\epsilon \lt \left(\sum_{i = 0}^{n} \frac{x^i}{i!}\right) - \exp(x) \lt \epsilon$)._

There is another axiom which uses the fact that derivatives of functions are well defined in the [[ordered local ring|ordered local]] [[local Artin algebra|Artin $F$-algebra]] $F[\epsilon]/\epsilon^2$ by the equation $f(x + \epsilon) = f(x) + f'(x) \epsilon$:

**Axiom of exponential function:** _Let $F[\epsilon]/\epsilon^2$ be the ordered local Artin $F$-algebra, with non-zero non-positive non-negative [[nilpotent element]] $\epsilon \in F[\epsilon]/\epsilon^2$ where $\epsilon^2 = 0$ and canonical $F$-algebra homomorphism $h:F \to F[x]/x^2$. There exists a unique function $\exp:F \to F$ and a function $\exp':F[\epsilon]/\epsilon^2 \to F[\epsilon]/\epsilon^2$ such that for every element $x \in F$, $h(\exp(x)) = \exp'(h(x))$, $\exp'(x + \epsilon) = \exp'(x) + \exp'(x) \epsilon$, and $\exp'(0) = 1$._

## In constructive mathematics

In classical mathematics, one could prove that the [[modulated Cantor real numbers]] $\mathbb{R}_C$ are [[sequentially Cauchy complete]] and equivalent to the [[HoTT book real numbers]] $\mathbb{R}_H$. However, in constructive mathematics, the above cannot be proven; while the HoTT book real numbers $\mathbb{R}_H$ are still sequentially Cauchy complete, the modulated Cantor real numbers $\mathbb{R}_C$ in general cannot be proven to be sequentially Cauchy complete. In particular, this means that the sequences

$$\left(1 + \frac{x}{n}\right)^{n} \quad \mathrm{or} \quad \sum_{i = 0}^{n} \frac{x^i}{i!}$$

do not have a limit for all modulated Cantor real numbers $x \in \mathbb{R}_C$. However, the sequences, by definition of $\mathbb{R}_C$, do have a limit for all [[rational numbers]] $x \in \mathbb{Q}$; this means that one could restrict the [[domain]] of the exponential function to the rational numbers $\exp:\mathbb{Q} \to \mathbb{R}_C$, and define it in the usual manner:

* For all rational numbers $x \in \mathbb{Q}$, there exists a unique modulated Cantor real number $\exp(x) \in \mathbb{R}_C$ such that for all positive rational numbers $\epsilon \in \mathbb{Q}_+$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \in \mathbb{N}$, if $n \geq N$, then $-\epsilon \lt \left(1 + \frac{x}{n}\right)^{n} - \exp(x) \lt \epsilon$ (or equivalently, $-\epsilon \lt \left(\sum_{i = 0}^{n} \frac{x^i}{i!}\right) - \exp(x) \lt \epsilon$).

## In Lie groups {#exp_of_Lie_groups}

Note: this section is under repair.

The classical exponential function $\exp \colon \mathbb{R} \to \mathbb{R}^*$ or $\exp \colon \mathbb{C} \to \mathbb{C}^*$ satisfies the fundamental property:

+-- {: .num_prop}
###### Proposition
The function $\exp \colon \mathbb{C} \to \mathbb{C}^*$ is a homomorphism taking addition to multiplication:
$$\exp(x + y) = \exp(x) \cdot \exp(y)$$
=--

+-- {: .proof}
###### Proof
A number of proofs may be given. One rests on the combinatorial [[binomial theorem|binomial identity]]

$$(x + y)^n = \sum_{j + k = n} \frac{n!}{j! k!} x^j y^k$$

(which crucially depends on the fact that multiplication is [[commutative monoid|commutative]]), whereupon

$$\array{
\sum_{n \geq 0} \frac{(x+y)^n}{n!} & = & \sum_{n \geq 0} \sum_{j + k = n} \frac1{j!} \frac1{k!} x^j y^k \\
 & = & (\sum_{j \geq 0} \frac{x^j}{j!}) \cdot (\sum_{k \geq 0} \frac{y^k}{k!}) \\
 & = & \exp(x) \cdot \exp(y)
}$$

An alternative proof begins with the observation that $f = \exp$ is the solution to the system $f' = f$, $f(0) = 1$. For each $y$, the function $g_1 \colon x \mapsto f(x) f(y)$ is a solution to the system $g' = g$, $g(0) = f(y)$, as is the function $g_2 \colon x \mapsto f(x + y)$. Then by uniqueness of solutions to ordinary differential equations (over connected domains; see, e.g., [here](http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem)), $g_1 = g_2$, i.e., $f(x + y) = f(x)f(y)$ for all $x, y$.[^fine]
=--

[^fine]: A previous edit offered even more detail: "An alternative proof begins with the premise that each solution of the ordinary differential equation $g' = 0$ is locally constant. Suppose $c$ is a complex number. As $\exp' = \exp$, we find that $(\exp(x) \exp(c - x))' = \exp(x) \exp(c - x) + \exp(x) (-\exp(c-x)) = 0$. Hence, by the premise and the connectedness of the domain of $\exp$ (either ${\mathbb{R}}$ or ${\mathbb{C}}$), we obtain $\exp(x)\exp(c - x) = \exp(0)\exp(c)$. The initial condition $\exp(0) = 1$ then yields $\exp(x)\exp(c - x) = \exp(c)$. The result follows by setting $c = x + y$."

Let $M$ be [[Lie group]] and let $\mathfrak{g}$ be its [[Lie algebra]] $T_1 M$, the tangent space to the [[identity element]] $1$.  Then $M$ may be equipped with the canonical left-invariant connection $\Del_l$ or the canonical right-invariant connection $\Del_r$.  It turns out that the natural Riemannian exponential maps on $M$ at $1$ relative to $\Del_l$ and $\Del_r$ are the same; we define this to be the __natural Lie exponential map on $M$ at the identity__, denoted simply $\exp$.  Several nice properties follow:

*  $\exp$ is defined on all of $\mathfrak{g}$.
*  $\exp \colon \mathfrak{g} \to G$ is a [[smooth map]].
*  If $\rho \colon \mathbb{R} \to \mathfrak{g}$ is a smooth homomorphism from the additive group $\mathbb{R}$ (i.e., if $\rho$ is an $\mathbb{R}$-linear map, uniquely determined by specifying $X = \rho(1)$), then $\exp \circ \rho_X \colon \mathbb{R} \to G$ is a smooth homomorphism.
*  For $X, Y \in \mathfrak{g}$, if $[X, Y] = 0$, then the restriction of $\exp \colon \mathfrak{g} \to G$ to the subspace spanned by $X$ and $Y$ is a smooth homomorphism to $G$. In particular, $\exp \colon \mathfrak{g} \to G$ is a homomorphism if $\mathfrak{g}$ is abelian (e.g., if $G$ is a commutative Lie group).
*  $\exp$ is [[surjection|surjective]] (a [[regular epimorphism]]) if $G$ is [[connected space|connected]] and [[compact space|compact]] (and also in some other situations, such as the classical cases where $G$ is $]0,\infty[$ or $\mathbb{C} \setminus \{0\}$). See [this post](https://terrytao.wordpress.com/2011/06/25/two-small-facts-about-lie-groups/) by Terence Tao, Proposition 1; see also the first comment which indicates an alternative proof based on the fact that [[maximal tori]] in $G$ are all conjugate to one another. Note also that the exponential map might not be surjective if the compactness assumption is dropped, as in the case of $G = SL_2(\mathbb{R})$ or $SL_2(\mathbb{C})$, both of which are connected; see [here](http://math.stackexchange.com/questions/643216/non-surjectivity-of-the-exponential-map-to-sl2-mathbbc) for instance.
*  If $G$ is [[compact space|compact]], then it may be equipped with a [[Riemannian metric]] that is both left and right invariant (see Tao's post linked in the previous remark); then the Lie exponential map is the same as the Riemannian exponential map at $1$.
*  If $G$ is a [[matrix Lie group]], then $\exp$ is given by the classical series formula (eq:series).

(to be expanded on)

## Logarithms

A __[[logarithm]]__ is a [[local section]] of an exponential map.

## Related concepts

* [[plane wave]]

* [[flow of a vector field]]

* [[Euler number]], [[e]]

* [[logarithm]]

* [[Euler's formula]]

* [[exponential exact sequence]]

* [[Hamiltonian flow]]

* [[exponential modality]]

* [[exponential ring]]

* [[one-parameter semigroup]]

## References

Discussion in [[constructive analysis]] of the exponential function on [[real numbers]]:

* {#Bishop} [[Errett Bishop]], §7 in: *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

* [[Errett Bishop]], [[Douglas Bridges]] §7 in: *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

See also:

* Wikipedia, [exponential map (Lie theory)](https://en.wikipedia.org/wiki/Exponential_map_%28Lie_theory%29), [derivative of the exponential map](https://en.wikipedia.org/wiki/Derivative_of_the_exponential_map), [exponential map (Riemannian geometry)](https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry))

* Springer [[eom]]: [exponential mapping](https://www.encyclopediaofmath.org/index.php/Exponential_mapping)

An extensive treatment for the general exponential map for an affine connection, for exponential map for Riemannian manifolds and the one for Lie groups is

* Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces

Specifically for [[Lie groups]], a different detailed treatment of the exponential map is in

* [[M M Postnikov]], _Lie groups and Lie algebras_, Geometry V

Some nice historical notes are in

* Wilfried Schmid, _Poincare and Lie groups_, Bull. Amer. Math. Soc. 6:2, 1982 [pdf](http://www.ams.org/journals/bull/1982-06-02/S0273-0979-1982-14972-2/S0273-0979-1982-14972-2.pdf)

Discussion in [[point-free topology]]:

* Ming Ng, [[Steve Vickers]], *Point-free Construction of Real Exponentiation*, Logical Methods in Computer Science **18** 3 (2022) [[doi:10.46298/lmcs-18(3:15)2022](https://doi.org/10.46298/lmcs-18%283%3A15%292022), [arXiv:2104.00162](https://arxiv.org/abs/2104.00162)]

* [[Steve Vickers]], *The Fundamental Theorem of Calculus point-free, with applications to exponentials and logarithms* [[arXiv:2312.05228](https://arxiv.org/abs/2312.05228)]

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