Australian Category Theory Seminar

The **Australian Category Theory Seminar** (also know as **AusCat**), previously called the **Sydney Category Theory Seminar**, is a weekly meeting on ([[higher category theory|higher]]) [[category theory]] held by the Centre of Australian Category Theory that has been held since 1970. The first seminar was probably held on the 23rd September, 1970.

> During 1970 Bob Walters, who had been a student of Bernhard Neumann at Canberra, and who had developed an interest in category theory, took a Lectureship at Sydney University. I went off with my family to spend the academic year 70/71 at Chicago, and Brian Day too had a post at Chicago for that year. While I was away, Ross and Bob began weekly meetings, along with my research student Geoffrey Lewis, to discuss category theory. This was the beginning of the Australian Category Seminar, which has met ever since for a half day each week. Using Fulbright funds at first to support visitors, we learnt that the Australian Research Council was willing to receive proposals from mathematicians. So we prepared such a proposal, and were successful, and have ever since had such support.
-[Kelly, 2007](#Kelly)

## Links

- [Seminar webpage](http://web.science.mq.edu.au/groups/coact/seminar/)
- [Blog post](https://rfcwalters.blogspot.com/2010/08/category-seminar-in-sydney.html) by [[RFC Walters]] regarding the first seminar
- [Seminar abstracts 1986](https://web.archive.org/web/20170210231156/https://dl.dropboxusercontent.com/u/92056191/Archive/seminars/sydney-category-seminar-abstracts/sydney_category_seminar_1986.pdf)
- [Seminar abstracts 1987](https://web.archive.org/web/20170210231156/https://dl.dropboxusercontent.com/u/92056191/Archive/seminars/sydney-category-seminar-abstracts/sydney_category_seminar_1987.pdf)
- [Seminar abstracts 1988](https://web.archive.org/web/20170210231156/https://dl.dropboxusercontent.com/u/92056191/Archive/seminars/sydney-category-seminar-abstracts/sydney_category_seminar_1988.pdf)

[[!redirects Sydney Category Theory Seminar]]

## References

* {#Kelly} [[G. Maxwell Kelly]], _The beginnings of category theory in Australia_, in: *Categories in Algebra, Geometry and Mathematical Physics*, Contemporary Mathematics **431** (2007) 1-6 [[doi:10.1090/conm/431](https://doi.org/10.1090/conm/431)]

category: reference


autoequivalence type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

\tableofcontents

## Definition

In [[dependent type theory]], given a [[type]] $A$, the **autoequivalence type** of $A$ is the [[equivalence type]] between $A$ and $A$ itself, $\mathrm{Aut}(A) \coloneqq (A \simeq A)$. The elements ([[terms]]) of $\mathrm{Aut}(A)$ are called **autoequivalences** or **self-equivalences**.

## Properties
 {#Properties}

* $\mathrm{Aut}(A)$ is an [[infinity-group|$\infty$-group]].

* Given a [[univalent universe|univalent]] [[Russell universe]] $U$ and an element $A:U$, there is an equivalence between $\mathrm{Aut}(A)$ and the [[loop space type]] $\Omega(U, A)$, $\mathrm{ua}(A, A):\Omega(U, A) \simeq \mathrm{Aut}(A)$. $\Omega(U, A)$ is called the *[[automorphism infinity-group|automorphism $\infty$-group]]* in $U$ at $A$.

* The autoequivalence type on an [[n-truncation modality|$n$-truncated type]] $A$ is an [[n-group|$(n + 1)$-group]].

* If $A$ is a [[set]], then $\mathrm{Aut}(A)$ is the **[[symmetric group]]** on $A$ and is also written as $S_A$, $\Sigma_A$, or $\mathrm{Sym}(A)$, and the elements of $\mathrm{Aut}(A)$ are called **[[permutations]]**. Every autoequivalence type is a symmetric group in a [[dependent type theory]] with [[axiom K]] or [[uniqueness of identity proofs]].

* If $A$ is a [[mere proposition]], then $\mathrm{Aut}(A)$ is a [[contractible type]].

* Given types $A$ and $B$, there is a [[function]] $\mathrm{ae}_\mathrm{Aut}:(A \simeq B) \to (\mathrm{Aut}(A) \simeq \mathrm{Aut}(B))$.

### Deloopings

Given an [[infinity-group]] $A$, the [[delooping]] of $A$ is defined to be a [[higher inductive type]] $B(A)$ generated by equivalences $\delta_A:\mathrm{Aut}(B(A)) \simeq A$ and $\epsilon_A:B(\mathrm{Aut}(A)) \simeq A$, and identity
$$\kappa_A:\mathrm{ae}_\mathrm{Aut}(\epsilon_A) =_{\mathrm{Aut}(B(\mathrm{Aut}(A))) \simeq \mathrm{Aut}(A)} \delta_{\mathrm{Aut}(A)}$$

## See also

* [[equivalence type]]

* [[permutation]], [[symmetric group]]

* [[automorphism group]], [[automorphism infinity-group]]

* [[loop space type]] (for the equivalent for [[identity types]])

[[!redirects autoequivalence]]
[[!redirects autoequivalences]]

[[!redirects self-equivalence]]
[[!redirects self-equivalences]]

[[!redirects autoequivalence type]]
[[!redirects autoequivalence types]]

[[!redirects type of autoequivalences]]
[[!redirects types of autoequivalences]]


AUTOMATH

{ ----------------------------- pseudoterms ------------------------------1-- }

pterm : TYPE := PRIM

{ ----------------------------- judgements -------------------------------4-- }

* [A:pterm] type : TYPE := PRIM
* [A:pterm][B:pterm] eq_type : TYPE := PRIM
* [a:pterm][A:pterm] in : TYPE := PRIM
* [a:pterm][b:pterm][A:pterm] eq_in : TYPE := PRIM

{ ----------------------------- equality rules ---------------------------8-- }

* [A:pterm][a:pterm][b:pterm][c:pterm]
A * [B:pterm][C:pterm]

A * [_1:type(A)] refl_type : eq_type(A,A) := PRIM
B * [_1:eq_type(A,B)] sym_type : eq_type(B,A) := PRIM
C * [_1:eq_type(A,B)][_2:eq_type(B,C)] trans_type : eq_type(A,C) := PRIM
a * [_1:in(a,A)] refl : eq_in(a,a,A) := PRIM
b * [_1:eq_in(a,b,A)] sym : eq_in(b,a,A) := PRIM
c * [_1:eq_in(a,b,A)][_2:eq_in(b,c,A)] trans : eq_in(a,c,A) := PRIM

B * [a:pterm][_1:in(a,A)][_2:eq_type(A,B)] conv_in : in(a,B) := PRIM
a * [b:pterm][_1:eq_in(a,b,A)][_2:eq_type(A,B)]
  conv_eq_in : eq_in(a,b,B) := PRIM

{ ----------------------------- congruence rules -------------------------2-- }

* [A:pterm][a:pterm][b:pterm][C:[x,pterm]pterm]
  [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>C)][_3:eq_in(a,b,A)]
  cong_type : eq_type(C,C) := PRIM
C * [c:[x,pterm]pterm]
  [_1:type(A)][_2:[x,pterm][_,in(x,A)]in(<x>c,<x>C)][_3:eq_in(a,b,A)]
  cong : eq_in(c,c,C) := PRIM

{ ----------------------------- the empty type ---------------------------4-- }

* '0' : pterm := PRIM
* [a:pterm] R_0 : pterm := PRIM

* '0'_form : type('0') := PRIM
a * [C:[x,pterm]pterm]
  [_1:[x,pterm][_,in(x,'0')]type(<x>C)][_2:in(a,'0')]
  '0'_elim : in(R_0(a),C) := PRIM

{ ----------------------------- the unit type ----------------------------7-- }

* '1' : pterm := PRIM
* star : pterm := PRIM
* [c:pterm][a:pterm] R_1 : pterm := PRIM

* '1'_form : type('1') := PRIM
* '1'_intro : in(star,'1') := PRIM
a * [C:[x,pterm]pterm]
  [_1:[x,pterm][_,in(x,'1')]type(<x>C)][_2:in(a,'1')][_3:in(c,<star>C)]
  '1'_elim : in(R_1(c,a),C) := PRIM
c * [C:[x,pterm]pterm]
  [_1:[x,pterm][_,in(x,'1')]type(<x>C)][_2:in(c,<star>C)]
  '1'_eq : eq_in(R_1(c,star),c,<star>C) := PRIM

{ ----------------------------- the Booleans ----------------------------13-- }

* '2' : pterm := PRIM
* 1 : pterm := PRIM
* 2 : pterm := PRIM
* [c:pterm][d:pterm][a:pterm] R_2 : pterm := PRIM

* '2'_form : type('2') := PRIM
* '2'_intro_1 : in(1,'2') := PRIM
* '2'_intro_2 : in(2,'2') := PRIM
a * [C:[x,pterm]pterm]
  [_1:[x,pterm][_,in(x,'2')]type(<x>C)][_2:in(a,'2')]
  [_3:in(c,<1>C)][_4:in(d,<2>C)]
  '2'_elim : in(R_2(c,d,a),C) := PRIM
d * [C:[x,pterm]pterm]
  [_1:[x,pterm][_,in(x,'2')]type(<x>C)][_2:in(c,<1>C)][_3:in(d,<2>C)]
  '2'_eq_1 : eq_in(R_2(c,d,1),c,<1>C) := PRIM
  '2'_eq_2 : eq_in(R_2(c,d,2),d,<2>C) := PRIM

* [A:pterm][B:pterm][c:pterm][_1:type(A)][_2:type(B)][_3:in(c,'2')]
  '2'_elim_type : type(R_2(A,B,c)) := PRIM
B * [_1:type(A)][_2:type(B)]
  '2'_eq_type_1 : eq_type(R_2(A,B,1),A) := PRIM
  '2'_eq_type_2 : eq_type(R_2(A,B,2),B) := PRIM

{ ----------------------------- product types ----------------------------9-- }

* [A:pterm][B:[x,pterm]pterm] Pi : pterm := PRIM
A * [B':pterm] arrow : pterm := Pi(A,[x,pterm]B')
A * [b:[x,pterm]pterm] lambda : pterm := PRIM
* [f:pterm][a:pterm] app : pterm := PRIM

B * [C:[x,pterm]pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]eq_type(<x>B,<x>C)]
  Pi_cong : eq_type(Pi(A,B),Pi(A,C)) := PRIM
B * [b:[x,pterm]pterm][c:[x,pterm]pterm]
  [_1:type(A)][_2:[x,pterm][_,in(x,A)]eq_in(<x>b,<x>c,<x>B)]
  lambda_cong : eq_in(lambda(A,b),lambda(A,c),Pi(A,B)) := PRIM

B * [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
  Pi_form : type(Pi(A,B)) := PRIM
B * [b:[x,pterm]pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]in(<x>b,<x>B)]
  Pi_intro : in(lambda(A,b),Pi(A,B)) := PRIM
B * [a:pterm][f:pterm]
  [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)][_3:in(f,Pi(A,B))][_4:in(a,A)]
  Pi_elim : in(app(f,a),B) := PRIM
a * [b:[x,pterm]pterm]
  [_1:type(A)][_2:[x,pterm][_,in(x,A)]in(<x>b,<x>B)][_3:in(a,A)]
  Pi_eq {beta} : eq_in(app(lambda(A,b),a),b,B) := PRIM

{ ----------------------------- sum types -------------------------------11-- }

* [A:pterm][B:[x,pterm]pterm] Sigma : pterm := PRIM
* [a:pterm][b:pterm] pair : pterm := PRIM
a * pi_1 : pterm := PRIM
a * pi_2 : pterm := PRIM

B * [C:[x,pterm]pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]eq_type(<x>B,<x>C)]
  Sigma_cong : eq_type(Sigma(A,B),Sigma(A,C)) := PRIM

B * [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
  Sigma_form : type(Sigma(A,B)) := PRIM
B * [a:pterm][b:pterm]
  [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)][_3:in(a,A)][_4:in(b,B)]
  Sigma_intro : in(pair(a,b),Sigma(A,B)) := PRIM
B * [c:pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
  [_3:in(c,Sigma(A,B))]
  Sigma_elim_1 : in(pi_1(c),A) := PRIM
  Sigma_elim_2 : in(pi_2(c),<pi_1(c)>B) := PRIM
b * [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)][_3:in(a,A)][_4:in(b,B)]
  Sigma_eq_1 : eq_in(pi_1(pair(a,b)),a,A) := PRIM
  Sigma_eq_2 : eq_in(pi_2(pair(a,b)),b,B) := PRIM

{ ----------------------------- W types ----------------------------------8-- }

* [A:pterm][B:[x,pterm]pterm] W : pterm := PRIM
* [a:pterm][f:pterm] sup : pterm := PRIM
* [b:pterm][e:pterm] rec : pterm := PRIM

B * [C:[x,pterm]pterm][_1:type(A)][_2:[x,pterm][_,in(x,A)]eq_type(<x>B,<x>C)]
  W_cong : eq_type(W(A,B),W(A,C)) := PRIM

B * [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
  W_form : type(W(A,B)) := PRIM
B * [a,pterm][f:pterm]
  [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)][_3:in(a,A)]
  [_4:in(f,arrow(B,W(A,B)))]
  W_intro : in(sup(a,f),W(A,B)) := PRIM
B * [C:[z,pterm]pterm]
    [x:pterm][u:pterm]
    D : pterm := arrow(Pi(<x>B,[y,pterm]<app(u,y)>C),C)
C * [b:pterm][e:pterm]
  [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
  [_3:[z,pterm][_,in(z,W)]type(<z>C)]
  [_4:in(b,Pi(A,[x,pterm]Pi(arrow(<x>B,W),[u,pterm]D(x,u))))][_5:in(e,W)]
  W_elim : in(rec(b,e),<e>C) := PRIM
b * [a:pterm][f:pterm]
    g : pterm := lambda(B,[y,pterm]rec(b,app(f,y)))
  [_1:type(A)][_2:[x,pterm][_,in(x,A)]type(<x>B)]
  [_3:[z,pterm][_,in(z,W)]type(<z>C)]
  [_4:in(b,Pi(A,[x,pterm]Pi(arrow(<x>B,W),[u,pterm]D(x,u))))]
  [_5:in(a,A)][_6:in(f,arrow(B,W))]
  W_eq : eq_in(rec(b,sup(a,f)),app(app(app(b,a),f),g),C) := PRIM

{ ----------------------------- extensionality --------------------------10-- }

* [A:pterm][a:pterm][b:pterm] Eq : pterm := PRIM
b * [_1:type(A)][_2:in(a,A)][_3:in(b,A)] Eq_form : type(Eq(A,a,b)) := PRIM
b * [_1:eq_in(a,b,A)] Eq_intro : in(star,Eq(A,a,b)) := PRIM
b * [c:pterm][_1:in(c,Eq(A,a,b))] Eq_elim : eq_in(a,b,A) := PRIM
  Eq_eq : eq_in(c,star,Eq(A,a,b)) := PRIM

A * [B:pterm] EQ : pterm := PRIM
B * [_1:type(A)][_2:type(B)] EQ_form : type(EQ(A,B)) := PRIM
B * [_1:eq_type(A,B)] EQ_intro : in(star,EQ(A,B)) := PRIM
B * [c:pterm][_1:in(c,EQ(A,B))] EQ_elim : eq_type(A,B) := PRIM
  EQ_eq : eq_in(c,star,EQ(A,B)) := PRIM


Automath

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

\tableofcontents

## Idea

_Automath_ was historically the first [[logical framework]].
The goal of the Automath project, initiated by [[Nicolaas de Bruijn]], was to provide a tool for the
formalization of [[mathematics]] without [[foundations|foundational]] prejudice.

## Related concepts

* [[proof assistant]]

* [[dependent type theory]]

* [[Coq]], [[Agda]]

## References

* [The Automath Archive](http://www.win.tue.nl/automath/)

* [Automath Restaurant](https://cs.ru.nl/~freek/zfc-etc/) has examples of different foundations of mathematics encoded in Automath.

* [`aut`](http://www.cs.ru.nl/~freek/aut/) - An implementation in ANSI C.

On [[mathematical structures]] in Automath via [[type telescopes]]:

* {#Zucker1975} [[Jeffery Zucker]], *Formalization of Classical Mathematics in Automath*, Colloques Internationaux du Centre National de la Recherche Scientifique **249** (1975) 135-145 [[web](https://www.win.tue.nl/automath/archive/webversion/aut042/aut042.html), [pdf](https://www.win.tue.nl/automath/archive/pdf/aut042.pdf)]

  also in: Studies in Logic and the Foundations of Mathematics **133** (1994) 127-139 [doi:10.1016/S0049-237X(08)70202-7]

> [[Zucker (1975, §10.2)](#Zucker1975):] Now a general framework in which to view linear orders, or other algebraic  structures, has been proposed by de Bruijn. It uses the notion of “telescope”. [...] A telescope therefore functions like a “generalized $\sum$”.

* [[Nicolaas de Bruijn]], *Telescopic mappings in typed lambda calculus*, Information and Computation **91** 2 (1991) 189-204 [doi:10.1016/0890-5401(91)90066-B]

category: software


automatic differentiation

## Contents
* table of contents
{: toc}

## Idea
_Automatic differentiation_ (AD) is a technique for computing ([[transpose matrix|transposed]]) [[derivative|derivatives]] of functions implemented by computer programs, essentially by applying the chain-rule across program code.
It is typically the method of choice for computing derivatives in machine learning and scientific computing because of its efficiency and numerical stability.

## Forward and reverse automatic differentiation
AD works by calculating the (transposed) derivative of a composite program in terms of the (transposed) derivatives of the parts, by using the chain-rule. The distinction between derivatives and transposed derivatives leads to the main distinction in automatic differentiation modes:

* _forward mode AD_: this implements the [[tangent bundle]] functor $T$ and calculates derivatives by a forward pass to calculate function values (primals), followed by another forward pass to calculate derivative values (tangents); these two forward passes can be interleaved into a single forward pass for efficiency.

* _reverse mode AD_: this implements the [[cotangent bundle]] functor $T^*$ and calculates transposed derivatives by a forward pass to calculate function values (primals), followed by a _reverse_ pass (inverting the control flow of the original program) to calculate transposed derivative values (cotangents); these two passes cannot easily be interleaved due to their differing direction.

When calculating a derivative of a program that implements a function $f:R^n\to R^m$, reverse mode tends to be the more efficient algorithm if $n \gg m$ and forward mode tends to be more efficient if $n\ll m$.
Seeing that many tasks in machine learning and statistics require the calculation of derivatives (for use in gradient-based optimization or Monte-Carlo sampling) of functions $f:R^n\to R$ (e.g. probability density functions) for $n$ very large, reverse mode AD tends to be the most popular algorithm.

## Combinatory Homomorphic Automatic Differentiation (CHAD) - a categorical take on AD

Let us fix some class of categories with [[stuff]] $S$. We will call its members $S$-categories. For example, for our purposes, $S$ might include [[property|properties]] like

* closure under finite [[products]] (aka tuples, in programming-speak);
* [[cartesian closed category|Cartesian closure]] (aka higher-order functions);
* closure under finite [[coproducts]] (aka sum or variant types);

and [[structure]] like

* designated objects like $R$ and $Z$ (which we think of as real numbers and integers)
* designated morphisms like $sin:R\to R$, $cos: R\to R$, $(+): R\times R\to R$, $(*):R\times R\to R$ (which we think of as the corresponding mathematical operations).

The idea behind CHAD will be to view forward and reverse AD as the unique structure (stuff) preserving functor (homomorphism of $S$-categories) from the initial $S$-category $Syn$ to two suitably chosen $S$-categories $\Sigma_{CSyn}LSyn$ and $\Sigma_{CSyn}LSyn^{op}$.

### The source language
Consider the initial $S$-category $Syn$ (put differently, the S-properties category that is freely generated from $S$-structure).
We can think of this category as a programming language: its objects are types and its morphisms are programs modulo [[beta-equivalence]] and [[eta-equivalence]].
In fact, for a wide class of programming languages, we can find a suitable choice of stuff $S$, such that the programming language arises as the initial $S$-category.
We will refer to this category as the _source language_ of our AD transformation: its morphisms are the programs we want to differentiate.

For example, if we choose the property part of $S$ to consist of Cartesian closure and the structure part of $S$ consists of designated objects $R$ and $Z$ and morphisms $sin$, $cos$, $(+)$, and $(*)$, then $Syn$ is the simply typed [[lambda-calculus]] with base types $R$ and $Z$ and the primitive operations $sin$, $cos$, $(+)$, and $(*)$.

### The target language and deriving AD
Given a strictly [[indexed category]] $L:C^{op}\to Cat$, we can form its [[Grothendieck construction]] $\Sigma_C L$, which is a [[split fibration]] over $C$.
Similarly, we can take the [[opposite category]] $L^{op}:C^{op}\to Cat$ of $L$ and form $\Sigma_C L^{op}$ from that.

For most natural choices of stuff $S$, we can find elegant sufficient conditions on $C$ and $L$ that guarantee that $\Sigma_C L$ and $\Sigma_C L^{op}$ are both $S$-categories.
Let us call a strictly indexed category $L:C^{op}\to Cat$ satisfying these conditions a $S'$-category (where we think of $S'$ again as [[stuff]]).
We give the corresponding $S'$ for some examples for different choices of $S$:

* for the property of closure under finite products, we demand that $C$ has finite products and $L$ has finite ([[indexed]]) [[biproducts]] (or, equivalently, that $L$ has finite products and is enriched over [[commutative monoids]]); (this can be extended with conditions to guarantee Cartesian closure; see the CHAD paper below;)
* for the property of closure under finite coproducts, we demand that $C$ has finite coproducts and that $L$ is [[extensive]], in the sense that the canonical functors $L(0)\to 1$ and $L(A+B)\to L(A)\times L(B)$ are isomorphisms of categories;
* for structure like distinguished objects $R$, we need to have chosen distinguished objects $R'$ of $C$, $TR'$ (which we think of as the tangent bundle of $R'$)  of $L(R')$ and $T^* R'$ (the cotangent bundle of $R'$) of $L(R')$;
* for structure like distinguished morphisms $(*):R\times R\to R$, we need to have chosen distinguished morphisms $(*)'$ in $C(R'\times R', R')$ (the primal computation of $(*)$), $T(*)'$ in $L(R'\times R')(TR'\times TR', TR')$ (the derivative of $(*)$) and $T^*(*)'$ in $L(R'\times R')(TR', TR'\times TR')$ (the transposed derivative of $(*)$).

Let us write $LSyn:CSyn^{op}\to Cat$ for the initial $S'$-category.
We think of this category as the _target language_ of automatic differentiation, in the sense that the forward/reverse derivatives of programs (morphisms) in the source language $Syn$ with consist of an associated primal program that is a morphism in $CSyn$ and an associated tangent/cotangent program that is a morphism in $LSyn$/$LSyn^{op}$.
As a programming language, $LSyn$ is a [[linear dependent type theory]] over the Cartesian type theory $CSyn$.

Indeed, as for any $S'$-category $L:C^{op}\to Cat$, $\Sigma_C L$ and $\Sigma_C L^{op}$ are $S$-categories, it follows that $\Sigma_{CSyn} LSyn$ and $\Sigma_{CSyn} LSyn^{op}$ are, in particular, $S$-categories.
Seeing that $Syn$ is the initial $S$-category, we obtain unique morphisms of $S$-categories:

* forward AD: $Df:Syn\to \Sigma_{CSyn} LSyn$;
* reverse AD: $Dr:Syn\to \Sigma_{CSyn} LSyn^{op}$.

### Semantics of the source and target languages
The category $Set$ of [[sets]] and functions gives another example of an $S$-category, for a lot of choices of $S$ (if we choose the sets $[[R]]$ and functions $[[op]]$ that we would like to denote with the types $R$ and operations $op$ in the structure part of $S$).
Moreover, the strictly indexed category $Fam(CMon):Set^{op}\to Cat$ of [[families]] of [[commutative monoids]] tends to give an example of an $S'$-category.
By initiality of $Syn$ and $LSyn:CSyn^{op}\to Cat$, we obtain

* the semantics of the source language: the unique $S$-functor $[[-]]:Syn\to Set$;
* the semantics of the target language: the unique $S'$-functor $[[-]]:(CSyn,LSyn)\to (Set,Fam(CMon))$ (with a slight abuse of notation).

In particular, we see that source language programs $t\in Syn(A, B)$ get interpreted as a function $[[t]]\in Set([[A]], [[B]])$.
Similarly, programs  $t\in CSyn(A, B)$ in the target language with _Cartesian_ type get interpreted as a function $[[t]]\in Set([[A]], [[B]])$.
Finally, programs $t\in LSyn(A)(B,C)$ in the target language with _linear_ type get interpreted as a function $[[t]] \in Fam(CMon)([[A]])([[B]],[[C]])$:
families of monoid homomorphisms.

### Correctness of CHAD
We say that CHAD calculates the correct derivative (resp. transposed derivative) of a program $s$ if
the [[semantics]] $[[Df(s)]]$ of the program $Df(s)$ (resp. $Dr(s)$) equals the pair of the semantics $[[s]]$ of $s$ and the derivative $T[[s]]$ (resp. transposed derivative $T^*[[s]]$) of the semantics $[[s]]$ of $s$.
CHAD is correct in the sense that it calculates the correct (transposed) derivative of any composite (possibly higher-order) program between first-order types  (meaning: types built using only [[positive type]] formers), provided that it calculates the correct (transposed) derivatives of all primitive operations like $(*)$ that we used to generate the source language.
That is, CHAD is a valid way for compositionally calculating (transposed) derivatives of composite computer programs, as long as we correctly implement the derivatives for all primitive operations (basic mathematical functions like multiplication, addition, sine, cosine) in the language.

We can prove this by a standard [[logical relations]] argument, relating smooth curves to their primal and (co)tangent curves.
Viewed more abstractly, the proof follows automatically because the [[Artin gluing]] along a representable functor (like the hom out of the real numbers) of an $S$-category is itself again an $S$-category, for common most choices of $S$.

## Related concepts

* [[differential programming]]
* [[backpropagation]]

## References

Forward mode automatic differentiation was introduced by Robert Edwin Wengert in

* [[Robert Edwin Wengert]], A simple automatic derivative evaluation program. Communications of the ACM 7.8. 1964.

An early description of reverse mode automatic differentiation can be found in

* [[Bert Speelpenning]], Compiling fast partial derivatives of functions given by algorithms. Illinois Univ., Urbana (USA). Dept. of Computer Science. Technical report. 1980.

but it was already described earlier by others such as Seppo Linnainmaa:

* [[Seppo Linnainmaa]], The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors. Master's Thesis (in Finnish), Univ. Helsinki. 1970.

A categorical analysis of (non-interleaved) forward and reverse AD is given by

* [[Matthijs Vákár]], CHAD: Combinatory Homomorphic Automatic Differentiation. 2021. [arXiv preprint 2103.15776](https://arxiv.org/abs/2103.15776) [Haskell implementation](https://github.com/VMatthijs/CHAD; funded by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 895827)

* [[Fernando Lucatelli Nunes]], [[Matthijs Vákár]], CHAD for Expressive Total Languages. 2021. [arXiv preprint 2110.00446] (https://arxiv.org/abs/2110.00446; funded by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 895827)

A categorical analysis of (interleaved) forward mode AD for calculating higher order derivatives is given by

* [[Mathieu Huot]], [[Sam Staton]], [[Matthijs Vákár]], Higher Order AD of Higher Order Functions. 2021. [arXiv preprint 2101.06757](https://arxiv.org/abs/2101.06757; funded by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 895827)


automaton

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# Automata
* table of contents
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## Idea

An *automaton* is an abstract concept of machine, modelled as a collection of states and transitions between states, together with an assignment of some external behavior (typically input and/or output) to these transitions.  A quintessential example of an automaton is a vending machine, which can be in any number of different states (e.g., "READY", "INSERT 2 TOKENS", "OUT OF SERVICE", etc.), and will transition between these states in response to user input (e.g., from "READY" to "INSERT 2 TOKENS" after the user makes a selection, and from "INSERT 2 TOKENS" to "INSERT 1 TOKEN" after the user inserts a token).

Many different notions of automaton exist in the literature.  For now this article only considers one fairly basic notion taken from [[Joy of Cats]], although it is by no means the simplest nor the most general.

## Definition

A __deterministic, sequential, Moore automaton__ is formally definable (as in [[Joy of Cats]]) as a sextuple ($Q$, $\Sigma$, $Y$, $\delta$,  $q_{0}$, $y$), where $Q$ is the set of states,  $\Sigma$ and $Y$ are the sets of input symbols and output symbols, respectively,  $\delta$:  $\Sigma$ $\times$ $Q$ $\to$ $Q$ is the transition map, $q_{0}$ $\epsilon$ $Q$ is the initial state, and $y$: $Q$ $\to$ $Y$ is the output map.  [[morphism|Morphisms]] from an automaton ($Q$, $\Sigma$, $Y$, $\delta$,  $q_{0}$, $y$) to an automaton ($Q$′, $\Sigma$′, $Y$′, $\delta$′,  $q_{0}$′, $y$′) are triples ($f_{Q}$, $f_{\Sigma}$, $f_{Y}$) of [[function|functions]] $f_{Q}:  Q \to Q\prime$,  $f_{\Sigma}: \Sigma \to \Sigma\prime$,  and $f_{Y}:  Y \to Y\prime$ satisfying the following conditions:

(i) preservation of transition:
    $\delta\prime$($f_{\Sigma}$($\sigma$), $f_{Q}$($q$)) =      $f_{Q}$($\delta$($\sigma$, $q$)),

(ii) preservation of outputs: $f_{Y}$($y$($q$)) = $y$$\prime$($f_{Q}$($q$)),

(iii) preservation of initial state:
      $f_{Q}$($q_{0}$) = $q_{0}\prime$.

Note that in such an automaton, the outputs are determined by the current state alone (and do not depend directly on the input).

A morphism $f$ : ($Q$, $\delta$, $q_{0}$, $F$) $\to$ ($Q\prime$, $\delta\prime$, $q_{0}\prime$, $F\prime$) (called a __simulation__) is a function $f : Q \to Q\prime$ that preserves:

(i) the transitions, i.e., $\delta\prime$($\sigma$, $f$($q$)) = $f$($\delta$($\sigma$, $q$)),

(ii) the initial state, i.e., $f$($q_{0}$) = $q_{0}\prime$, and

(iii) the final states, i.e., $f[F] \subseteq F\prime$.

## The category of automata

There is a [[category]] $Aut$ whose [[object|objects]] are deterministic sequential Moore automata and whose [[morphism|morphisms]] are simulations.

## Examples

* [[Mealy machine]]

* [[Moore machine]]

* [[Conway's game of life]]

## Variants

There are several variant forms of automaton.  The above just gives a basic one. Others are treated in the entries:

* [[asynchronous automaton]];

* [[deterministic automaton]];

* [[nondeterministic automaton]].

There are tentative definitions of

*[[higher dimensional automaton]]

which take a more nPOV of automata theory.

## References

* [[Jiri Adamek]], [[Horst Herrlich]], [[George Strecker]], *[[Abstract and Concrete Categories]]*, John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories **17** (2006) 1-507 [[tac:tr17](http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html), [book webpage](http://katmat.math.uni-bremen.de/acc/)]

* [[Mark V. Lawson]], _[Finite automata](http://www.ma.hw.ac.uk/~markl/books.html)_, CRC Press, see also [here](http://www.ma.hw.ac.uk/~markl/teaching/AUTOMATA/kleene.pdf) for a shorter version in the form of Course Notes.

* [[Jiří Adámek]], *Free algebras and automata realizations in the language of categories*, Commentationes Mathematicae Universitatis Carolinae **15**.4 (1974) 589-602 [[eudml:16649](https://eudml.org/doc/16649)]

* [[Jiří Adámek]], Věra Trnková, *Automata and algebras in categories* **37** Springer (1990) [[ISBN:9780792300106](https://link.springer.com/book/9780792300106)]

* Liang-Ting Chen, Henning Urbat, _A fibrational approach to automata theory_, [arxiv/1504.02692](http://arxiv.org/abs/1504.02692)

An early discussion of automata via [[string diagrams]] in the [[Cartesian monoidal category]] of [[finite sets]]:

* {#Hotz65} [[Günter Hotz]], _Eine Algebraisierung des Syntheseproblems von Schaltkreisen_, EIK, Bd. 1, (185-205), Bd, 2, (209-231) 1965 ([part I](https://www.magentacloud.de/lnk/LiPMlYfh), [part II](https://www.magentacloud.de/lnk/YivslUWJ), [[HotzSchaltkreise.pdf:file]])

Discussion of [[non-deterministic automata]] as 1-dimensional [[defect QFT|defect]] [[TQFTs]]:

* Paul Gustafson, [[Mee Seong Im]], Remy Kaldawy, [[Mikhail Khovanov]], Zachary Lihn, *Automata and one-dimensional TQFTs with defects* [[arXiv:2301.00700](https://arxiv.org/abs/2301.00700)]

Review:

* [[Mee Seong Im]], *Correspondence between automata and one-dimensional Boolean topological theories and TQFTs*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#ImMar2023), video:[YT](https://www.youtube.com/watch?v=q5kynYxwTmk)]

category:computer science

[[!redirects automata]]
[[!redirects automatons]]

[[!redirects Aut]]

[[!redirects finite state automaton]]
[[!redirects finite state automatons]]
[[!redirects finite state automata]]

[[!redirects cellular automaton]]


automorphic form

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## Idea
 {#Idea}

Quite generally, _automorphic forms_ are suitably well-behaved [[functions]] on a [[quotient]] space $K\backslash X$ where $K$ is typically a [[discrete group]], hence suitable functions on $X$ which are [[invariant]] under the [[action]] of a discrete group. The precise definition has evolved a good bit through time.

[[Henri Poincaré]] considered [[analytic functions]] invariant under a discrete infinite group of [[fractional linear transformations]] and called them _[[Fuchsian functions]]_ (after his advisor [[Lazarus Fuchs]]).

More generally, automorphic forms in the modern sense are suitable functions on a [[coset space]] $K \backslash G$, hence functions on [[groups]] $G$ which are [[invariant]] with respect to the [[action]] of the [[subgroup]] $K \hookrightarrow G$. The archetypical example here are [[modular forms]] regarded as functions on $K\backslash PSL(2,\mathbb{R})$ where $K$ is a [[congruence subgroup]], and for some time the terms "modular form" and "automorphic form" were used essentially synonymously, see [below](#ModularForms). Based on the fact that a [[modular form]] is a [[section]] of some [[line bundle]] on the [[moduli stack of elliptic curves]], [[Pierre Deligne]] defined an automorphic form to be a section of a line bundle on a [[Shimura variety]].

By pullback of functions the linear space of such functions hence constitutes a [[representation]] of $G$ and such representations are then called _automorphic representations_ (e.g. [Martin 13, p. 9](#Martin13)) , specifically so if $G = GL_n(\mathbb{A}_K)$ is the [[general linear group]] with [[coefficients]] in a [[ring of adeles]] of some [[global field]] and $K = GL_n(K)$. This is the subject of the _[[Langlands program]]_. There one also considers [[unramified]] such representations, which are constituted by functions that in addition are invariant under the action of $GL_n$ with coefficients in the [[integral adeles]], see [below](#InNumberTheory).

### Modular forms as classical automorphic forms on $PSL(2,\mathbb{R})$
 {#ModularForms}

By a standard definition, a _[[modular form]]_ is a [[holomorphic function]] on the [[upper half plane]] $\mathfrak{H}$ satisfying a specified transformation property under the [[action]] of a given [[congruence subgroup]] $\Gamma$ of the [[modular group]] $G = PSL(2,\mathbb{Z})$ (e.g. [Martin 13, definition 1](#Martin13), [Litt, def. 1](#Litt)).

But the [[upper half plane]] is itself the [[coset]] of the [[projective linear group]] $G = PSL(\mathbb{R})$ by the subgroup $K = Stab_G(\{i\}) \simeq SO(2)/\{\pm I\}$

$$
  f\colon \mathfrak{H} \simeq PSL(2,\mathbb{R})/K
  \,.
$$

In view of this, one finds that every modular function $f \colon \mathfrak{H} \to \mathbb{C}$ lifts to a function

$$
  \tilde f \colon \Gamma\backslash PSL(2,\mathbb{R})
  \longrightarrow \mathbb{C}
  \,,
$$

hence to a function on $G$ which is actually _invariant_ with respect to the $\Gamma$-action ("automorphy"), but which instead now satisfies some transformation property with respect to the action of $K$, as well as some well-behavedness property

This $\tilde f$ is the incarnation as an _automorphic function_ of the modular function $f$ (e.g. [Martin 13, around def. 3](#Martin13), [Litt, section 2](#Litt)). For emphasis these automorphic forms on $PSL(2,\mathbb{R})$ equivalent to modular forms are called _classical modular forms_.

This is where the concept of automorphic forms originates (for more on the history see e.g. [this MO comment](http://mathoverflow.net/a/124785/381) for the history of terminology) [and this one](http://mathoverflow.net/a/21556/381).

### Modular forms as adelic automorphic forms on $GL(2,\mathbb{A})$
 {#ModularFormsAdAdelicAutomorphicForms}

Where by the [above](#ModularForms) an ordinary [[modular form]] is equivalently a suitably periodic function on $SL(2,\mathbb{R})$, one may observe that the [[real numbers]] $\mathbb{R}$ appearing as [[coefficients]] in the latter are but one of many [[p-adic number]] completions of the [[rational numbers]]. Hence it is natural to consider suitably periodic functions on $SL(2,\mathbb{Q}_p)$ of all these completions at once. This means to consider functions on $SL(2,\mathbb{A})$, for $\mathbb{A}$ the [[ring of adeles]]. These are the _adelic automorphic forms_. They may be thought of as subsuming ordinary modular forms for all [[level structures]].
(e.g. [Martin 13, p. 8](#Martin13), also [Goldfeld-Hundley 11, lemma 5.5.10](#GoldfeldHundley11), [Bump, section 3.6](#Bump), [Gelbhart 84, p. 22](#Gelbhart84)): we have

$$
  \Gamma \backslash PSL(2,\mathbb{R})
  \simeq
  Z(\mathbb{A}) GL_2(\mathbb{Q})\backslash GL_2(\mathbb{A})/ GL_2(\mathbb{A}_{\mathbb{Z}})
  \,,
$$

where $\mathbb{A}_{\mathbb{Z}}$ are the [[integral adeles]]. (The [[double coset]] on the right is analogous to that which appears in the [[Weil uniformization theorem]], see the discussion there and at _[[geometric Langlands correspondence]]_ for more on this analogy.)

This leads to the more general concept of _adelic automorphic forms_ [below](#InNumberTheory).

### General adelic automorphic forms
 {#InNumberTheory}

More generally, for the [[general linear group]] $G = GL_n(\mathbb{A}_F)$, for any $n$ and with [[coefficients]] in a [[ring of adeles]] $\mathbb{A}_F$ of some [[number field]] $F$, and for the subgroup $GL_n(F)$, then sufficiently well-behaved functions on $GL_n(F)\backslash GL_n(\mathbb{A}_F)$ form [[representations]] of $GL_n(\mathbb{A}_{F})$ which are called _[[automorphic representations]]_. Here "well-behaved" typically means

1. **finiteness** -- the functions [[invariant]] under the [[action]] of the [[maximal compact subgroup]] [[span]] a [[finite number|finite]] [[dimension|dimensional]] [[vector space]];

1. **central character** -- the action by the [[center of a group|center]] is is controled by (...something...);

1. **growth** -- the functions are [[bounded functions]];

1. **cuspidality** -- (...)

(e.g. [Frenkel 05, section 1.6](#Frenkel05), [Loeffler 11, page 4](#Loeffler11), [Martin 13, definition 4](#Martin13), [Litt, def.4](#Litt)).

(These conditions are not entirely set in stone, they are being varied according to application (see e.g. [this MO comment](http://mathoverflow.net/a/66598/381))).

In particular one considers subspaces of "[[unramified]]" such functions, namely those which are in addition trivial on the subgroup of $GL_n$ of the [[integral adeles]] $\mathcal{O}_F$ ([Goldfeld-Hundley 11, def. 2.1.12](#GoldfeldHundley11)). This means that that unramified automorphic representations are spaces of functions on a [[double

coset]] of the form

$$
  GL_n(F)\backslash GL_n(\mathbb{A}_F) / GL_n(\mathcal{O}_F)
  \,.
$$

See at _[[Langlands correspondence]]_ for more on this. Such double cosets are [[analogy|analogous]] to those appearing in the [[Weil uniformization theorem]] in [[complex analytic geometry]], an analogy which leads to the conjecture of the [[geometric Langlands correspondence]].

### Dirichlet characters

For the special case of $n = 1$ in the discussion of adelic automorphic forms [above](#InNumberTheory), the group

$$
  GL_1(\mathbb{A}_F) = (\mathbb{A}_F)^\times = \mathbb{I}_F
$$

is the [[group of ideles]] and the quotient

$$
  GL_1(F) \backslash GL_1(\mathbb{A}_F)
  =
  F^\times \backslash (\mathbb{A}_F)^\times
$$

is the [[idele class group]]. Automorphic forms in this case are effectively [[Dirichlet characters]] in disguise... ([Goldfeld-Hundley 11, theorem 2.1.9](#GoldfeldHundley11)).

## Properties

### Function field analogy

[[!include function field analogy -- table]]

### Application in string theory

In [[string theory]] [[partition functions]] tend to be automorphic forms for [[U-duality]] groups. See the [references below](#ReferencesInStringTheory)

## Related entries

* [[modular form]], [[topological modular form]], [[topological automorphic form]]

* [[automorphic L-function]]

* [[Langlands duality]]

## References

### General

Introductions and surveys include

* {#Deligne} [[Pierre Deligne]], _Fromed Modulaires et representations de $GL(2)$_ ([](http://publications.ias.edu/sites/default/files/Number21.pdf))

* {#Gelbhart84} [[Stephen Gelbart]], starting on p. 20 (196) of _An elementary introduction to the Langlands program_,  Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 ([web](http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/))

* Nolan Wallach, _Introductory lectures on automorphic forms_ ([pdf](http://math.ucsd.edu/~nwallach/luminy-port2.pdf))
[[!redirects automorphic forms]]

* E. Kowalski, section 3 of  _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))

* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

* {#Bump} Daniel Bump, _Automorphic forms and representations_

* {#Loeffler11} David Loeffler, _Computing with algebraic automorphic forms_, 2011 ([[LoefflerAutomorphic.pdf:file]])

* {#Martin13} [[Kimball Martin]], _A brief overview of modular and automorphic forms_,2013 [pdf](http://www2.math.ou.edu/~kmartin/papers/mfs.pdf)

* {#Litt} [[Daniel Litt]], _Automorphic forms notes, part I_ ([pdf](http://math.stanford.edu/~dlitt/Talks/automorphicformspt1.pdf))

* Denis Trotabas, Modular Forms and Automorphic Representations (2010) ([pdf](http://math.stanford.edu/~conrad/modseminar/pdf/L10.pdf))

* Werner Muller, Spectral theory of automorphic forms (2010) ([pdf](http://www.math.uni-bonn.de/people/mueller/skripte/specauto.pdf))

* {#Miyake76} Toshitsune Miyake's _Modular Forms_ 1976 (English version 1989) ([review pdf](projecteuclid.org/euclid.bams/1183556263))

Review in the context of the [[geometric Langlands correspondence]] is in

* {#Frenkel05} [[Edward Frenkel]], _Lectures on the Langlands Program and Conformal Field Theory_, in _Frontiers in number theory, physics, and geometry II_, Springer Berlin Heidelberg, 2007. 387-533. ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

The generalization of theta functions to [[automorphic forms]] is due to

* [[André Weil]], _Sur certaines groups d'operateur unitaires_, Acta. Math.  111 (1964), 143-211

see [Gelbhart 84, page 35 (211)](Langlands+program#Gelbhart84) for review.

Further developments here include

* {#Kudla77} [[Stephen Kudla]], _Relations between automorphic forms produced by theta-functions_, in _Modular Functions of One Variable VI_, Lecture Notes in Math. 627, Springer, 1977, 277–285.

* {#Kudla78} [[Stephen Kudla]], _Theta functions and Hilbert modular forms_,Nagoya Math. J. 69 (1978) 97-106

* {#Stopple95} [[Jeffrey Stopple]], _Theta and $L$-function splittings_, Acta Arithmetica LXXII.2 (1995) ([pdf](http://matwbn.icm.edu.pl/ksiazki/aa/aa72/aa7221.pdf))

### In string theory
  {#ReferencesInStringTheory}

The relation between [[string theory]] on [[Riemann surfaces]] and automorphic forms was first highlighted in

* {#Witten88} [[Edward Witten]], _Quantum field theory, Grassmannians, and algebraic curves_, Comm. Math. Phys. Volume 113, Number 4 (1988), 529-700 ([Euclid](http://projecteuclid.org/euclid.cmp/1104160350))

See also

* {#GRV10} [[Michael Green]], Jorge G. Russo, Pierre Vanhove, _Automorphic properties of low energy string amplitudes in various dimensions_ ([arXiv:1001.2535](http://arxiv.org/abs/1001.2535))

[[!redirects automorphic forms]]

[[!redirects automorphic representation]]
[[!redirects automorphic representations]]

[[!redirects Fuchsian function]]
[[!redirects Fuchsian functions]]


automorphic L-function

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## Idea
 {#Idea}

An _automorphic L-function_  $L_\pi$ is an [[L-function]] built from an [[automorphic representation]] $\pi$, generalizing

1. how the [[Mellin transform]] takes an [[automorphic form]] to a [[zeta function]];

1. (nonabelian generalization) how a [[Dirichlet L-function]] $L_\chi$ is associated to a [[Dirichlet character]] $\chi$ (which is an automorphic form on the (abelian) [[idele group]]).

In analogy to how [[Artin reciprocity]] implies that to every 1-dimensional [[Galois representation]] $\sigma$ there is a [[Dirichlet character]] $\chi$ such that the [[Artin L-function]] $L_\sigma$ equals the [[Dirichlet L-function]] $L_\chi$, so the [[conjecture|conjectured]] [[Langlands correspondence]] says that to every $n$-dimensional [[Galois representation]] $\sigma$ there is an [[automorphic representation]] $\pi$ such that the automorphic L-function $L_\pi$ equals the [[Artin L-function]] $L_\sigma$.

## Related concepts

* [[Rankin-Selberg method]]

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

* Wikipedia, _[Automorphic L-function](http://en.wikipedia.org/wiki/Automorphic_L-function)_

* {#Gelbhart84} [[Stephen Gelbart]], starting on p. 26 (202) of _An elementary introduction to the Langlands program_,  Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 ([web](http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/))

* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

* F. Shahidi, _Automorphic $L$-functions: A Survey_ in Laurent Clozel, [[James Milne]] (eds.) _Automorphic forms, Shimura varieties and $L$-functions_, volume I ([pdf](www.jmilne.org/math/articles/1990aT.pdf))

[[!redirects automorphic L-functions]]


automorphism

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## Definition

An __automorphism__ of an [[object]] $x$ in a [[category]] $C$ is an [[isomorphism]] $f : x \to x$.  In other words, an automorphism is an [[endomorphism]] that is an [[isomorphism]].

## Automorphism group

Given an object $x$, the automorphisms of $x$ form a [[group]] under [[composition]], the __automorphism group__ of $x$, which is a submonoid of the [[endomorphism monoid]] of $x$:
$$
  Aut_C(x) = End_C(x) \cap Iso(C) = Iso_C(x,x)
,$$
which may be written $Aut(x)$ if the category $C$ is understood.  Up to equivalence, every group is an automorphism group; see [[delooping]].

## Examples

(...)

* [[permutation|Permutations]] are automorphisms in [[FinSet]].

* [[automorphism of a vertex operator algebra]]

## Related concepts

* **automorphism group**

  * [[inner automorphism group]]

  * [[outer automorphism group]]

  * [[automorphism Lie group]]

* [[automorphism 2-group]]

* [[automorphism ∞-group]]

## References

Discussion of automorphism groups [[internalization|internal]] to [[sheaf toposes]] ("automorphism sheaves"):

* [[Simon Henry]], [MO:a/262687](https://mathoverflow.net/a/262687/381)

* Robert Friedman, John W. Morgan, §2.1 in: *Automorphism sheaves, spectral covers, and the Kostant and Steinberg sections* [[arXiv:math/0209053](https://arxiv.org/abs/math/0209053)]

[[!redirects automorphisms]]
[[!redirects automorphism group]]
[[!redirects automorphism groups]]

[[!redirects group of automorphisms]]
[[!redirects groups of automorphisms]]


automorphism 2-group

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## Automorphism 2-groups

For $C$ any [[2-category]] and $c \in C$ any object of it, the category $Aut_C(c) \subset Hom_C(c,c)$ of auto-equivalences of $c$ and invertible 2-morphisms between these is naturally a [[2-group]], whose group product comes from the horizontal composition in $C$.

If $C$ is a [[strict 2-category]] there is the notion of strict [[automorphism 2-group]]. See there for more details on that case.

For instance if $C = Grp_2 \subset Grpd$ is the 2-category of [[group]] obtained by regarding groups as one-object [[groupoid]]s, then for $H \in Grp$ a group, its automorphism 2-group obtained this way is the strict 2-group

$$
  AUT(H) := Aut_{Grp_2}(H)
$$

corresponding to the [[crossed module]] $(H \stackrel{Ad}{\to} Aut(H))$, where $Aut(H)$ is the ordinary [[automorphism group]] of $H$.

## Inner automorphism 2-groups

See [[inner automorphism 2-group]].

## Related concepts

* [[group]], [[∞-group]],

* [[automorphism group]], [[automorphism ∞-group]],

* [[center]], [[center of an ∞-group]]

* [[outer automorphism group]], [[outer automorphism ∞-group]]

[[!redirects automorphism 2-groups]]


automorphism infinity-group

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## Definition

### Externally

Let $\mathcal{C}$ be an [[(∞,1)-category]]. Let $X \in \mathcal{C}$ be an [[object]].

As a [[discrete ∞-groupoid|discrete]] [[∞-group]] the **automorphism $\infty$-group** of $X$ is the [[monomorphism in an (∞,1)-category|sub-]][[∞-groupoid]]

$$
  Aut(X) \hookrightarrow \mathcal{C}(X,X)
$$

of the [[derived hom space]] of [[morphisms]] in $\mathcal{C}$ from $X$ to itself, on those that are [[equivalence in an (∞,1)-category|equivalences]].

This is an [[∞-group]] in [[∞Grpd]],

$$
 Aut(X)\in Grp(\infty Grpd)
  \,.
$$

### Internally

Let $\mathcal{C}$ be a [[cartesian closed (∞,1)-category]] (for instance an [[(∞,1)-topos]]). Write

$$
  [-,-] : \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}
$$

for the [[internal hom]]. Then for $X \in \mathcal{C}$ an object, the _internal automorphism $\infty$-group_ is the [[subobject]]

$$
  \mathbf{Aut}(X) \hookrightarrow [X,X]
$$

of the [[internal hom]] on those morphism that are [[equivalence in an (infinity,1)-category|equivalences]].

In the special case that $\mathcal{C}$ is an [[∞-topos]], the [[delooping]] $\mathbf{B}\mathbf{Aut}(X)$ of the internal automorphism $\infty$-group is equivalently the [[∞-image]]

$$
  * \to \mathbf{B}\mathbf{Aut}(X) \hookrightarrow Obj
$$

of the morphism

$$
  * \stackrel{\vdash X}{\to} Obj
$$

to [[generalized the|the]] [[object classifier]], that modulates $X$ (the "name" of $X$).

### In homotopy type theory
 {#InHomotopyTypeTheory}

Let $\mathcal{C}$ be an [[(∞,1)-topos]]. Then its [[internal language]] is [[homotopy type theory]]. In terms of this the object $X \in \mathcal{C}$ is a [[type]] ([[homotopy type]]). In the type theory syntax the internal automorphism $\infty$-group $\mathbf{Aut}(X)$ then is (as a type, without yet the group structure)

$$
  \vdash (X \stackrel{\simeq}{\to} X) : Type
  \,,
$$

the [[subtype]] of the [[function type]] on the [[equivalence in homotopy type theory|equivalences]]. Its [[delooping]] $\mathbf{B}\mathbf{Aut}(X)$ is

$$
  \vdash \; \left(\sum_{Y : Type} [Y = X]\right) \colon Type
  \,,
$$

where on the right we have the [[dependent sum]] over one argument of the [[bracket type]]/[[inhabited type|(-1)-truncation]] $[X = Y] = isInhab(X = Y)$ of the [[identity type]] $(X = Y)$.

The equivalence of this definition to the previous one is essentially equivalent to the [[univalence axiom]].

## Examples

### In a 1-category

If $\mathcal{C}$ happens to be a [[1-category]] then the external automorphism $\infty$-group of an object is the ordinary [[automorphism group]] of that object.

If $\mathcal{C}$ happens to be a 1-[[topos]], then the internal automorphism $\infty$-group is the traditional automorphism [[group object]] in the topos. Etc.

### Of $\infty$-groups

For $G \in \infty Grp(\mathcal{X})$ an [[∞-group]] there is the direct automorphism $\infty$-group $Aut(G)$. But there is also the [[delooping]] $\mathbf{B}G \in \mathcal{X}$ and _its_ automorphism $\infty$-group.

Sometimes (for instance in the discussion of [[∞-gerbes]]) one considers

$$
  AUT(G) := Aut(\mathbf{B}G)
$$

and calls this the automorphism $\infty$-group of $G$.

For instance when $G$ is an ordinary [[group]], $AUT(G)$ is the [[2-group]] discussed at [[automorphism 2-group]].

## Related concepts

* There may be the [[stuff, structure, property|structure]] of an [[∞-Lie group]] on $Aut(F)$. The corresponding [[∞-Lie algebra]] is an [[automorphism ∞-Lie algebra]].

* [[group]], [[∞-group]],

* [[automorphism group]], [[automorphism 2-group]]

* [[autoequivalence type]]

* [[center]], [[center of an ∞-group]]

* [[outer automorphism group]], [[automorphism 2-group]], [[outer automorphism ∞-group]]

* [[rigidification of a stack]]

[[!redirects automorphism ∞-group]]
[[!redirects automorphism ∞-groups]]

[[!redirects autoequivalence ∞-group]]
[[!redirects autoequivalence ∞-groups]]

[[!redirects ∞-automorphism]]
[[!redirects ∞-automorphisms]]

[[!redirects automorphism infinity-groups]]

[[!redirects automorphism n-group]]
[[!redirects automorphism n-groups]]


automorphism infinity-Lie algebra

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#### $\infty$-Lie theory
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#### Rational homotopy theory
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[[!include differential graded objects - contents]]
=--
=--
=--

> ... under construction ...

#Contents#
* table of contents
{:toc}

## Idea

The _automorphism $\infty$-Lie algebra_ $aut(\mathfrak{g})$ of an [[∞-Lie algebra]] $\mathfrak{g}$ -- or dually $aut(CE(\mathfrak{g}))$ of the corresponding [[Chevalley-Eilenberg algebra]] -- has in degree $k$ the [[derivations]] on $CE(\mathfrak{g})$ of degree $-k$. The [[higher Lie algebra]] version of the [[automorphism Lie algebra]] of an ordinary [[Lie algebra]].

In terms of [[rational homotopy theory]] $aut(\mathfrak{g})$ is a model for the rationalization of the  group of [[automorphisms]]s of the [[rational space]] $\exp(\mathfrak{g})$ corresponding to $CE(\mathfrak{g})$ under the [[Sullivan construction]].

## Definition

Let $A := (\wedge^\bullet \mathfrak{a}^*, d_A)$ be a [[semifree dga|semifree]] [[dg-algebra]] of [[finite type]].

Notice that for $\phi : A \to A$ a [[derivation]] of degree $-k$ and $\lambda : A\to A$ another derivation of degree $-l$ the [[commutator]]

$$
  [\phi,\lambda] := \phi \circ \lambda - \lambda \circ \phi : A \to A
$$

is itself a derivation, of degree $-(k+l)$. In particular, since the [[differential]] $d_A : A \to A $ is itself a derivation of degree +1, we have that

$$
  d_A \phi := [d_A, \phi] : A \to A
$$

is a derivation of degree $-(k+1)$.

+-- {: .un_defn}
###### Definition
**(automorphism $\infty$-Lie algebra)**

The [[∞-Lie algebra]] $aut(A)$ is the [[dg-Lie algebra]] which

* in degree $-k$ for $k \gt 0$ has the derivations $\phi : A \to A$ of degree $-k$;

* in degree $0$ the derivations that commute with the differential $d_A$

* whose differential $\delta_{aut(A)} := [d_A,-]$ is given by the commutator with the differential of $A$;

* whose Lie bracket is the commutator $[\phi,\lambda] = \phi \circ \lambda - \lambda \circ \phi$.

=--

## Properties

### Automorphism group {#AutomorphismGroup}

For stating the fundamental theorem about $aut(\mathfrak{g})$ below we need some facts about the ordinary [[automorphism group]] of a dg-algebra $A$.

(...)

(See chapter 6 of [Sullivan](#Sullivan)).

### Classifying space for $Aut(X)$-principal bundles
 {#ClassifyingSpaces}

Let $X$ be a [[rational space]] whose [[Sullivan model]] is $\mathfrak{g}$, $X \simeq \exp(\mathfrak{g})$. Let $aut'(\mathfrak{g}) \subset aut(\mathfrak{g})$ be the sub dg-algebra of the automorphism $\infty$-Lie algebra on the maximal nilpotent ideal in degree 0. Let $G(X)$ be the maximal reductive group of genuine automorphisms of $CE(\mathfrak{g})$ (see [above](#AutomorphismGroup)).

Then the [[rational space]]

$$
  \exp(aut'(\mathfrak{g}))/G(X)
  \simeq
  B Aut (X)
$$

is the [[classifying space]] for $Aut(X)$-[[principal bundle]]s, i.e. for [[bundle]]s with typical fiber $X$.

## Examples

* The [[inner derivation Lie 2-algebra]] $inn(\mathfrak{g})$ is the full subalgebra of the automorphism $\infty$-Lie algebra on the _inner_ derivations of the [[Chevalley-Eilenberg algebra]] of a [[Lie algebra]] $\mathfrak{g}$.

## Related concepts

* [[automorphism Lie algebra]]

* [[deformation theory]]

* [[tangent complex]]

* [[cotangent complex]]

## References

The general definition of $aut(\mathfrak{g})$ is the topic of p. 313 (45 of 63) and following in

* [[Dennis Sullivan]], _Infinitesimal computations in topology_ Publications Mathématiques de l'IHÉS, 47 (1977) ([numdam](http://www.numdam.org/item?id=PMIHES_1977__47__269_0)) {#Sullivan}

The automorphism group $Aut(A)$ of a dg-algebra is discussed in paragraph 6 there. Few details on proofs are given there. Only recently in

* [[Andrey Lazarev]], [[Jonathan Block]], ...

a detailed proof is given.

Concrete computations of $aut(\mathfrak{g})$ for some classes of [[rational space]]s $X = \exp(\mathfrak{g})$ can be found for instance in

* Samual Bruce Smith, _The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces_ , Journal of Pure and Applied Algebra
Volume 160, Issues 2-3, 25 June 2001, Pages 333-343

[[!redirects automorphism ∞-Lie algebra]]
[[!redirects automorphism dg-Lie algebra]]

[[!redirects automorphism infinity-Lie algebras]]
[[!redirects automorphism ∞-Lie algebras]]
[[!redirects automorphism dg-Lie algebras]]

[[!redirects automorphism L-infinity algebra]]
[[!redirects automorphism L-∞ algebra]]
[[!redirects automorphism L-infinity algebras]]
[[!redirects automorphism L-∞ algebras]]


automorphism Lie group

The [[automorphism group]] $Aut(G)$ of a [[Lie group]] $G$ itself canonically inherits the structure of a Lie group. Its [[Lie algebra]] is the [[derivation Lie algebra]] of the Lie algebra $\mathfrak{g}$ of $G$: $Lie(Aut(G)) \simeq \mathfrak{der}(\mathfrak{g})$.

[[!redirects automorphism Lie groups]]


automorphism of a vertex operator algebra

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#### String theory
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[[!include string theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Given a [[vertex operator algebra]] $\mathcal{V}$ ([[VOA]]) or [[super vertex operator algebra]] ([[sVOA]]), or more generally a full [[2d CFT]] (of which the [[VOA]] is the local and chiral data) or [[2d SCFT]] (hence a "[[2-spectral triple]]") one may ask (as for any [[object]] in any [[category]]) for its [[automorphisms]], hence the [[homomorphisms]]

$$
  \mathcal{V}
   \overset{\simeq}{\longrightarrow}
  \mathcal{V}
$$

in the corresponding [[category]] of sVOA-s/[[2d SCFTs]], from $\mathcal{V}$ to itself, which are [[invertible morphism|invertible]] and hence constitute a _[[symmetry]]_ of $\mathcal{V}$.

## In perturbative string vacua and Connes-Lott models

If $\mathcal{V}$ is a [[direct sum|direct summand]] of a [[2d CFT]]/[[2-spectral triple]] encoding a [[perturbative string theory vacuum]], such as, typically, a [[rational 2d CFT]] such as a [[Gepner model]] encoding a "[[non-geometric string vacuum|non-geometric]]" [[KK-compactification]], then the automorphisms of $\mathcal{V}$ are the [[formal dual|formal duals]] to [[symmetries]] of that [[KK-compactification]]-[[fiber]] space (see the references [below](#AsSymmetriesOfStringCompactifications))

In the point-particle limit where the [[2d SCFT]]/[[2-spectral triple]]  becomes an ordinary [[spectral triple]] (see [there](2-spectral+triple#References)) this hence reduces to the automorphisms of internal algebras as discussed in [[Connes-Lott-Chamseddine-Barrett models]].

(...)

## Examples

### Moonshine

[[moonshine]]-examples:

* The [[Conway group]] $CO_{0}$ is the [[automorphism group|group of]] [[automorphisms of a super VOA]] of the unique chiral [[number of supersymmetries|N=1]] [[super vertex operator algebra]] of [[central charge]] $c = 12$ without fields of [[conformal weight]] $1/2$

  ([Duncan 05](#Duncan05), see also [Paquette-Persson-Volpato 17, p. 9](#PaquettePerssonVolpato17))

* similarly, there is a super VOA, the _[[Monster vertex operator algebra]]_, whose [[automorphism group|group of]] of [[automorphisms of a VOA]] is the [[monster group]]

  ([Frenkel-Lepowski-Meurman 89](#FrenkelLepowskiMeurman89), [Griess-Lam 11](#GriessLam11))

## References

### General

(...)

### As symmetries of non-geometric string compactifications
 {#AsSymmetriesOfStringCompactifications}

Automorphisms of [[vertex operator algebras]] regarded as [[symmetries]] of non-geometric [[perturbative string theory vacua]] (e.g. [[Gepner models]]):

* {#HullIsraelSarti17} [[Chris Hull]], Dan Israel, Alessandra Sarti, _Non-geometric Calabi-Yau Backgrounds and K3 automorphisms_, JHEP11(2017)084 ([arXiv:1710.00853](https://arxiv.org/abs/1710.00853))

### Moonshine automorphism groups

* {#FrenkelLepowskiMeurman89} [[Igor Frenkel]], [[James Lepowsky]], Arne Meurman, _Vertex operator algebras and the monster_, Pure and Applied Mathematics __134__, Academic Press, New York 1998. liv+508 pp. [MR0996026](http://www.ams.org/mathscinet-getitem?mr=996026)

* {#Duncan05} John F. Duncan, _Super-moonshine for Conway's largest sporadic group_ ([arXiv:math/0502267](https://arxiv.org/abs/math/0502267))

* {#GriessLam11} [[Robert Griess]] Jr., Ching Hung Lam, _A new existence proof of the Monster by VOA theory_ ([arXiv:1103.1414](https://arxiv.org/abs/1103.1414))

* {#KachruPaquetteVolpato16} [[Shamit Kachru]], [[Natalie Paquette]], [[Roberto Volpato]], _3D String Theory and Umbral Moonshine_ ([arXiv:1603.07330](http://arxiv.org/abs/1603.07330))

* {#PaquettePerssonVolpato16} [[Natalie Paquette]], Daniel Persson, [[Roberto Volpato]], _Monstrous BPS-Algebras and the Superstring Origin of Moonshine_ ([arXiv:1601.05412](http://arxiv.org/abs/1601.05412))

* [[Miranda Cheng]], Sarah M. Harrison, [[Roberto Volpato]], Max Zimet, _K3 String Theory, Lattices and Moonshine_ ([arXiv:1612.04404](https://arxiv.org/abs/1612.04404))

* {#PaquettePerssonVolpato17} [[Natalie Paquette]], Daniel Persson, Roberto Volpato, _BPS Algebras, Genus Zero, and the Heterotic Monster_ ([arXiv:1701.05169](https://arxiv.org/abs/1701.05169))

* [[Shamit Kachru]], Arnav Tripathy, _The hidden symmetry of the heterotic string_ ([arXiv:1702.02572](https://arxiv.org/abs/1702.02572))

[[!redirects automorphisms of a vertex operator algebra]]
[[!redirects automorphisms of vertex operator algebras]]

[[!redirects automorphism of a VOA]]
[[!redirects automorphisms of a VOA]]
[[!redirects automorphisms of VOAs]]

[[!redirects automorphism of a super vertex operator algebra]]
[[!redirects automorphisms of a super vertex operator algebra]]
[[!redirects automorphisms of super vertex operator algebras]]

[[!redirects automorphism of a super VOA]]
[[!redirects automorphisms of a super VOA]]
[[!redirects automorphisms of super VOAs]]

[[!redirects automorphism of an sVOA]]
[[!redirects automorphisms of an sVOA]]
[[!redirects automorphisms of sVOAs]]

[[!redirects automorphism of a 2d conformal field theory]]
[[!redirects automorphisms of a 2d conformal field theory]]
[[!redirects automorphisms of 2d conformal field theories]]

[[!redirects automorphism of a 2d CFT]]
[[!redirects automorphisms of a 2d CFT]]
[[!redirects automorphisms of 2d CFT]]

[[!redirects automorphism of a 2d super-conformal field theory]]
[[!redirects automorphisms of a 2d super-conformal field theory]]
[[!redirects automorphisms of 2d super-conformal field theories]]

[[!redirects automorphism of a 2d sCFT]]
[[!redirects automorphisms of a 2d sCFT]]
[[!redirects automorphisms of 2d sCFT]]

[[!redirects automorphism of a 2-spectral triple]]
[[!redirects automorphisms of a 2-spectral triple]]
[[!redirects automorphisms of 2-spectral triples]]


automorphism of free group

It appears that the group $Aut(F_n)$ of automorphisms of the free group $F_n$ on $n$ letters and especially the group $Out(F_n)$ of outer automorphisms of the free group on $n$ letters appear often at sometimes unexpected places in geometry, theory of operads, deformation theory, study of moduli spaces etc. Some of the occurences are expressed in terms of Kontsevich's [[graph homology]], ribbon diagrams and alike.

Some of the authors who wrote extensively on related issues include Karen Vogtmann ([homepage](http://www.math.cornell.edu/~vogtmann/papers/papers.html)). Cf. also articles by [[Leila Schneps]] etc.


automorphism of the complex numbers

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[automorphism]] of the [[field]] of [[complex numbers]].

Beyond the two evident ones, the [[identity]] and [[complex conjugation]], there is an infinitude of "wild" automorphisms, which, however, are not [[continuous functions|continuous]] and not [[constructive mathematics|constructive]] (their existence requires [[Zorn's lemma]]).

## References

* H. Kestelman, _Automorphisms of the Field of Complex Numbers_, Proceedings of the London Mathematical Society, Volume s2-53, Issue 1, 1951 ([doi:10.1112/plms/s2-53.1.1](https://doi.org/10.1112/plms/s2-53.1.1))

* Paul B. Yale, _Automorphisms of the Complex Numbers_, Mathematics Magazine
Mathematics Magazine Vol. 39, No. 3 (May, 1966), pp. 135-141 ([doi:10.2307/2689301](https://doi.org/10.2307/2689301))

* MathOverflow, _[Wild automorphisms of the complex numbers](https://math.stackexchange.com/q/412010/58526)_

[[!redirects automorphisms of the complex numbers]]


autopoiesis

## References

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

* [[Erich Jantsch]], _[[The Self-Organizing Universe]]_


auxiliary field

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#### Fields and quanta
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[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] an "auxiliary field" is a [[field (physics)|field]] [[type]] that is introduced on top of the [[field (physics)|fields]] in the [[field theory]] genuinely of interest, in order to bring that field theory of actual interest into another form -- usually while not actually changing it up to [[equivalence]] -- which lends itself better to certain purposes.

## Examples

For example under suitable conditions a field theory with [[constraints]] may be formulated equivalently as another field theory without explicit constraints, but with an auxiliary [[Lagrange multiplier]] field which does induce that constraint after all, but indirectly so via its [[equation of motion]].

In [[BV-BRST formalism]], a powerful generalization of this idea of [[Lagrange multiplier]] is the usage of auxiliary fields that allow to define "[[gauge fixing fermions]]" to implement [[gauge fixing]] via their [[equations of motion]].

For instance in the [[quantization of Yang-Mills theory]] the _[[Nakanishi-Lautrup field]]_ (and its "[[antighost field]]") are auxiliary fields that are introduced in order to indirectly induce [[gauge fixing]] to [[Lorentz gauge]]. The [[BV-BRST formalism]] also makes precise how exactly this introduction of auxiliary fields does not change the field theory, up to [[equivalence]]: the auxiliary fields change the [[BV-BRST complex]] (which effectivley defines the field theory) only up to a [[quasi-isomorphism]], i.e. the relevant notion of [[equivalence]] in this context.

In fact even before it comes to [[gauge fixing]], [[BV-BRST formalism]] introduces a variety of auxiliary fields: the [[ghosts]], the [[ghosts-of-ghosts]], etc., and the [[antifields]].

[[!redirects auxiliary fields]]

[[!redirects auxiliary field bundle]]
[[!redirects auxiliary field bundles]]


Avogadro constant

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#### Physics
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=--

In macroscopic [[thermodynamics|thermodynamic]] systems one deals with a large [[number]] of microscopic particles. The number of particles is, for 'human sized' systems (e.g. engines, rooms), roughly on the order of the _Avogadro constant_ (or 'Avogadro\'s number' in historical usage). This constant is used as a scaling factor between macroscopic and microscopic observations.

More precisely, **Avogadro's number** is defined as the number of [[atoms]] in $0.012$ [[kg]] of the [[carbon]] [[isotope]] $\mathrm{C}^12$. This is roughly the [[real number]]

$$
  6.02214179(30) \times 10^{23} \;\in\; \mathbb{R}
  \,.
$$

An amount of any chemical substance which contains precisely this number of [[molecule]]s is called a **mole** of substance. So in particular, by definition, a mole of $\mathrm{C}^{12}$ is precisely of weight $12 \,\mathrm{g}$.

Attempts to define the SI unit of mass, i.e. the kilogram, in terms of invariant and basic physical concepts generally involve trying to measure Avogadro's number at increasing levels of accuracy. As with the speed of light, once the measurement of this derived quantity (taking the kilogram as the mass of certain reference kilogram cylinders) reaches a certain threshold of accuracy, it will make sense to specify Avogadro's number exactly and then derive the units of mass.

Of course the choice of $12 \,\mathrm{g}$ of $\mathrm{C}^{12}$ is entirely arbitrary and just serves as some example of a typical laboratory-scale macroscopic quantity.

Accordingly, later it was found useful to have different such macroscopic reference quantities, for instance the _pound mole_ in addition to the (standard) gram mole. For exchanging between these it was found useful to introduce [[unit]]s and treat 'amount of substance' as a [[dimensional analysis|dimension]].

Then the **Avogadro constant** is taken to be

$$
  N_A = 6.02214179(30) \times 10^{23} \,mol^{-1}
  \,.
$$

(Strictly speaking this is $N_A = 1 \in \mathbb{R}$ in [[natural unit]]s.)

[[!redirects Avogadro constant]]
[[!redirects Avogadro's constant]]

[[!redirects Avogadro number]]
[[!redirects Avogadro's number]]

[[!redirects mole]]
[[!redirects moles]]


Awais Shaukat

## Selected writings

On the [[ordinary homology]] of [[configuration spaces of points]] in a [[surface]] as a [[group representation]] of the [[mapping class group]]:

* [[Christian Blanchet]], [[Martin Palmer]], [[Awais Shaukat]], *Heisenberg homology on surface configurations* [[arXiv:2109.00515](https://arxiv.org/abs/2109.00515)]

* [[Awais Shaukat]], [[Christian Blanchet]], *Weakly framed surface configurations, Heisenberg homology and Mapping Class Group action*, Archiv der Mathematik **120** (2023) 99–109 [[arXiv:2206.11475](https://arxiv.org/abs/2206.11475), [doi:10.1007/s00013-022-01793-3](https://doi.org/10.1007/s00013-022-01793-3)]

category: people


Awodey's proposal

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#### Type theory
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#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### Statement

In [Awodey 09](#Awodey09), [Awodey 10](#Awodey10) was first expressed
the idea that [[dependent type theory]] with intensional [[identity types]] ([[Martin-Löf dependent type theory]]), viewed as [[homotopy type theory]], is in similar relation to the concept of [[(∞,1)-toposes]] as [[extensional type theory]] is to the ordinary concept of [[toposes]] (as discussed at [[relation between type theory and category theory]]).

From [Awodey 09, p. 13](#Awodey09), [Awodey 10, p. 15](#Awodey10):

> The homotopy interpretation of Martin-Löf type theory into Quillen model categories, and the related results on type-theoretic constructions of higher groupoids, are analogous to basic results interpreting extensional type theory and higher-order logic in (1-)toposes, and clearly indicate that the logic of higher toposes, and therewith of higher homotopy theory, is a form of intensional type theory.

{#ConciseReStatement} A concise re-statement would be that:

1. the [[internal logic of an (∞,1)-topos|internal logic of (∞,1)-toposes]] is [[univalence axiom|univalent]] [[homotopy type theory]]

   (though there is fine print involved, e.g. the [[initiality conjecture]]);

1. there is a [[model of type theory in an (infinity,1)-topos|model of (univalent) homotopy type theory in any $(\infty,1)$-topos]]

   (this version has a proof, see [below](#Proof));

1. [[homotopy type theory]] is *[[synthetic homotopy theory]]*

   (this may be read as a suggestive colloquial version of the previous statement, remaining vague on whether [[univalence axiom|univalence]] is considered or not).

Following this suggestion, the weaker form of this idea, ignoring the [[univalence|univalent]] [[type universe]] and relating to the broader class of [[locally Cartesian closed (∞,1)-categories]], was stated more concretely as a conjecture in [Joyal 11](#Joyal11). For more precision see [Kapulkin-Lumsdaine 16, p. 9](#KapulkinLumsdaine16).

Roughly, this is about the following table of correspondences (for more see at *[[relation between type theory and category theory]]*):

| [[internal logic]]/[[type theory]] | [[higher category theory|higher]] [[category theory]] |
|-----|--------|
| [[type theory]] | [[locally Cartesian closed categories]] |
| [[homotopy type theory]] | [[locally Cartesian closed (∞,1)-categories]] |
| [[homotopy type theory]] with [[univalence|univalent]] [[type universes]] | [[elementary (∞,1)-toposes]] |

Fore more precision see [Kapulkin-Lumsdaine 16, p. 9](#KapulkinLumsdaine16).

### Proof
 {#Proof}

A [[proof]] of the weaker version of the conjecture, in form of the statement that every [[locally presentable (∞,1)-category|locally presentable]] [[locally Cartesian closed (∞,1)-category]] is presented by a suitable [[type theoretic model category]] which provides [[categorical semantics]] for [[homotopy type theory]], was proven in [Shulman 12, Example 2.16](#Shulman12), following [Cisinski 12](#Cisinski12).

Generalizing this to a proof of the full conjecture required finding "strict" models for the [object classifier](object+classifier#DetailsObjClassf) by strict [[type universes]]. A series of article ([Shulman 12](#Shulman12), [Shulman 13](#Shulman13)) showed that this is possible in an increasing class of special cases.

A proof of the general case was finally announced in [Shulman 19](#Shulman19).

For more see at _[[model of type theory in an (infinity,1)-topos]]_.

## References

### Statement

The idea is due to

* {#Awodey09} [[Steve Awodey]], _Homotopy and Type Theory_, grant proposal project description ([pdf](https://ncatlab.org/homotopytypetheory/files/proposal2009.pdf))

* {#Awodey10} [[Steve Awodey]], *Type theory and homotopy*, in: Dybjer P., Lindström S., Palmgren E., Sundholm G. (eds.),  *Epistemology versus Ontology*, Springer (2012) 183-201 $[$[arXiv:1010.1810](http://arxiv.org/abs/1010.1810), [doi:10.1007/978-94-007-4435-6_9](https://doi.org/10.1007/978-94-007-4435-6_9)$]$

A pronounced statement of the weaker version was highlighted in

* {#Joyal11} [[André Joyal]], _Remarks on homotopical logic_ Oberwolfach (2011) ([pdf](http://hottheory.files.wordpress.com/2011/06/report-11_2011.pdf#page=19))

and stated more precisely in

* {#KapulkinLumsdaine16} [[Chris Kapulkin]], [[Peter Lumsdaine]], *The homotopy theory of type theories*, Advances in Mathematics **337** (2018) 1-38 $[$[arXiv:1610.00037](https://arxiv.org/abs/1610.00037), [doi:10.1016/j.aim.2018.08.003](https://doi.org/10.1016/j.aim.2018.08.003)$]$

### Proof
 {#ReferencesProof}

The proof of the weaker version of Awodey's conjecture (that every [[locally Cartesian closed (∞,1)-category]] has a presentation by a suitable [[type-theoretic model category]] which provides [[categorical semantics]] for [[homotopy type theory]]) is due, independently, to

* {#Shulman12} [[Michael Shulman]], Example 2.16 of *Univalence for inverse diagrams and homotopy canonicity*, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( _From type theory and homotopy theory to Univalent Foundations of Mathematics_ ) June 2015 ([arXiv:1203.3253](https://arxiv.org/abs/1203.3253), [doi:/10.1017/S0960129514000565](https://doi.org/10.1017/S0960129514000565))

following

* {#Cisinski12} [[Denis-Charles Cisinski]], [blog comment](http://golem.ph.utexas.edu/category/2012/05/the_mysterious_nature_of_right.html#c041306) (2012)

The proof of the stronger version (including [[univalence|univalent]] [[type universes]] modelling [object classifier](object+classifier#DetailsObjClassf) of [[(∞,1)-toposes]]) was found for the special case of [[(∞,1)-presheaf (∞,1)-toposes]] over [[elegant Reedy categories]] in

* [Shulman 12](#Shulman12)

* {#Shulman13} [[Michael Shulman]], _The univalence axiom for elegant Reedy presheaves_, Homology, Homotopy and Applications Volume 17 (2015) Number 2 ([arXiv:1307.6248](https://arxiv.org/abs/1307.6248), [doi:10.4310/HHA.2015.v17.n2.a6](http://dx.doi.org/10.4310/HHA.2015.v17.n2.a6))

A general proof was announced in

* {#Shulman19} [[Michael Shulman]], slides 5 to 10 of: _Semantics of higher modalities_, talk at _[Geometry in Modal HoTT (2019)](https://felix-cherubini.de/modal-workshop.html)_ ([pdf slides](http://home.sandiego.edu/~shulman/papers/cmu2019b.pdf), [video recording](https://www.youtube.com/watch?v=Wcpi1vVMrCs))

and appeared in

* {#Shulman19} [[Michael Shulman]], _All $(\infty,1)$-toposes have strict univalent universes_ ([arXiv:1904.07004](https://arxiv.org/abs/1904.07004))

It is reviewed in:

* {#Riehl22} [[Emily Riehl]], *On the $\infty$-topos semantics of homotopy type theory*, lecture at *[Logic and higher structures](https://conferences.cirm-math.fr/2689.html)* CIRM (Feb. 2022) $[$[pdf](https://emilyriehl.github.io/files/semantics.pdf), [[Riehl-InfinityToposSemantics.pdf:file]]$]$

[[!redirects Awodey proposal]]

[[!redirects Awodey conjecture]]
[[!redirects Awodey's conjecture]]

[[!redirects Awodey's conjecture]]
[[!redirects Awodey's conjecture]]


Axel Kleinschmidt

* [webpage](http://www.aei.mpg.de/~axkl/)

## Selected writings

On [[U-duality]] and [[diffeomorphisms]] in [[exceptional generalized geometry]]:

* [[David Berman]], [[Martin Cederwall]], [[Axel Kleinschmidt]], [[Daniel C. Thompson]], _The gauge structure of generalised diffeomorphisms_, High Energ. Phys. 2013, 64 (2013) [[arXiv:1208.5884](http://arxiv.org/abs/1208.5884)]

On [[E9]] [[exceptional field theory]]:

* Guillaume Bossard, Franz Ciceri, Gianluca Inverso, [[Axel Kleinschmidt]], [[Henning Samtleben]], _$E_9$ exceptional field theory II. The complete dynamics_ ([arXiv:2103.12118](https://arxiv.org/abs/2103.12118))

On [[E11]]-[[exceptional field theory]]:

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _On supersymmetric E11 exceptional field theory_ ([arXiv:1907.02080](https://arxiv.org/abs/1907.02080))

* [[Guillaume Bossard]], [[Axel Kleinschmidt]], [[Ergin Sezgin]], _A master exceptional field theory_ ([arXiv:2103.13411](https://arxiv.org/abs/2103.13411))

On maximal [[gauged supergravity|gauged]] [[D=2 supergravity]] via [[KK-reduction]] of [[D=11 supergravity]]:

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Maximal $D=2$ supergravities from higher dimensions* [[arXiv:2309.07232](https://arxiv.org/abs/2309.07232)]

* [[Guillaume Bossard]], [[Franz Ciceri]], [[Gianluca Inverso]], [[Axel Kleinschmidt]], *Consistent truncation of eleven-dimensional supergravity on $S^8 \times S^1$* [[arXiv:2309.07233](https://arxiv.org/abs/2309.07233)]

## Related entries

* [[U-duality]], [[Kac-Moody group]]

category: people


Axel Ljungström

* [institute page](https://www.su.se/english/profiles/axlj4439-1.450268)

## Selected writings and talks

On [[ordinary cohomology in homotopy type theory]], specifically on [[integral cohomology]]:

* [[Guillaume Brunerie]], [[Axel Ljungström]], [[Anders Mörtberg]], *Synthetic Integral Cohomology in Cubical Agda*, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) **216** (2022) $[$[doi:10.4230/LIPIcs.CSL.2022.11](https://doi.org/10.4230/LIPIcs.CSL.2022.11)$]$

  > (in [[cubical type theory|cubical]] [[Agda]])

On [[homotopy groups of spheres in homotopy type theory]], specifically on the [[first stable homotopy group of spheres]]:

* {#Ljungstroem22} [[Axel Ljungström]], *The Brunerie Number Is -2* (June 2022) [[blog entry](https://homotopytypetheory.org/2022/06/09/the-brunerie-number-is-2/)]

* [[Axel Ljungström]], *Calculating a Brunerie Number*, [[Homotopy Type Theory Electronic Seminar Talks]], 20 October 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Ljungstrom-2022-10-20-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=MIMeQ88YMKI))

Implementation of [[ordinary cohomology|ordinary]]$\;$[[cohomology rings]] in [[cubical agda]]:

* [[Thomas Lamiaux]], [[Axel Ljungström]], [[Anders Mörtberg]], _Computing Cohomology Rings in Cubical Agda_, CPP 2023: Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs (2023) [[arxiv:2212.04182] (https://arxiv.org/abs/2212.04182), [doi:10.1145/3573105.3575677](https://doi.org/10.1145/3573105.3575677)]

category: people

[[!redirects Axel Ljungstrom]]
[[!redirects Axel Ljungstroem]]


Axel Osmond

* [institute page](https://www.irif.fr/users/osmond/index)

## Selected writings

On the notion of [[spectrum (geometry)|spectrum in geometry]] known as the [[Diers spectrum]]:

* [[Axel Osmond]], _On Diers theory of Spectrum I: Stable functors and right multi-adjoints_, ([arXiv:2012.00853](https://arxiv.org/abs/2012.00853))

  > (on [[multi-adjoints]])

* [[Axel Osmond]], _On Diers theory of Spectrum II: Geometries and dualities_, ([arXiv:2012.02167](https://arxiv.org/abs/2012.02167))

* [[Axel Osmond]], _The general construction of Spectra_, [arXiv:2102.01259](https://arxiv.org/abs/2102.01259).

category: people


axio-dilaton

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###Context###
#### String theory
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[[!include string theory - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[type IIB supergravity]] the [[dilaton]]/[[string coupling constant]] $g_s$ and the degree-1 [[RR-field]] $C_0$ of the [[D(-1)-brane]] combine into a [[complex number|complex]] function

$$
  \tau \coloneqq C_0 + i/ g_s
$$

which transforms under the [[modular group]] $SL(2,\mathbb{Z})$ [[S-duality]] $\subset$ [[U-duality]] as

$$
  \left(
  \array{
    a & b
    \\
    c & d
  }
  \right)
  :
  \tau
    \mapsto
  \frac{a \tau + b}{c \tau + d}
  \,.
$$

This combined field is also called the **axio-dilaton**.

## Properties

* In [[F-theory]] the variation of the axio-dilaton determines an [[elliptic fibration]] which encodes non-perturbative [[vacua]] of type IIB string theory.


axiom

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

# Contents
* table of contents
{: toc}

## Idea

An _axiom_ is a [[proposition]] in [[logic]] that a given [[theory]] requires to be [[true]]: every [[model]] of the theory is required to make the axiom hold true. The sense however is that an axiom is a _basic_ true proposition, used to prove other true propositions (the [[theorems]]) in the theory.

## Definition

Given a [[language]] $L$ (perhaps specified by a [[signature (in logic)|signature]]: a collection of [[types]], [[function symbols]] and [[relation symbols]]), a [[theory]] is the collection of assertions which are derivable (using the rules of [[deduction]] of the ambient [[logic]] or [[deductive system]]) from a given set of assertions, called **axioms** of the theory.  In other words, a theory is generated from a set of axioms, by starting with those axioms and applying [[rule of inference|rules of deduction]], much as terms in an algebraic system may be generated from a set of basic terms by applying operations. Axioms should therefore be considered as _presenting_ a theory; different axiom sets may well give the same theory.

In terms of a [[deductive system]], axioms can be regarded as "rules with zero hypotheses".  The form of such axioms depends on the details of the deductive system used: it could be [[natural deduction]], [[sequent calculus]], a [[Hilbert system]], etc.  If we take sequent calculus, for instance, then any collection of [[sequents]] written in the given language $L$

$$
  \vec{\phi} \vdash_{\vec x} \vec{\psi}
$$

(asserting that "If every [[proposition]] $\phi_i$ is [[true]] in [[context]] $\vec{x}$ then also some $\psi_i$ is/has to be true") can be taken as a collection of axioms for some theory. [[model|Models]] of the theory will then be those [[structure in model theory|structures]] of the language in which the axioms are interpreted as true statements. For example, a model of [[group theory]] is a structure in the language of groups for which the group theory axioms hold, which is (of course) a [[group]].

Assuming the deductive system is [[soundness theorem|sound]], every sequent which is the conclusion of a valid sequent deduction, starting from the axioms, will also be true in every model. And if the deductive system is also [[completeness theorem|complete]], then every sequent of the language which is true in every model will in fact be provable from the axioms.

## Examples
 {#Examples}

[[!include foundational axiom - contents]]

(...)

* [[pushout-product axiom]]

(...)

* [[Hilbert's sixth problem]] asks for an axiomatization of [[physics]].

## Related concepts

* [[axiom schema]]

[[!include mathematical statements --- contents]]

## References

For instance def. D1.1.6 in

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects axiom]]
[[!redirects axioms]]

[[!redirects axioms]]

[[!redirects axiomatization]]
[[!redirects axiomatizations]]


axiom K

Axiom K may refer to

* [[axiom K (type theory)]]
* [[axiom K (modal logic)]]

[[!redirects K axiom]]
[[!redirects Axiom K]]


axiom K (modal logic)

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### Context
#### Modalities, Closure and Reflection
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[[!include modalities - contents]]

=--
=--
=--

*Disambiguation: For axiom K as a principle of [[type theory]], see [[axiom K (type theory)]]*

# Contents
* table of contents
{: toc}

## Idea

In [[modal logic]], axiom K, named after [[Saul Kripke]], is a basic principle which almost all versions of propositional modal logic satisfy

$$K \colon \Box(p \to q) \to \Box p \to \Box q.$$

## Developments

It is possible to consider a variant in [[dependent type theory]] ([Spitters](#Spitters))

$$
Dependent\; K \colon \Box \Pi_{y:A} B(y) \to \Pi_{x: \Box A} \Box B [open\; x/y].
$$

## Related concepts

* [[K modal logic]]

## References

* {#Spitters} [[Bas Spitters]] et al., _Modal Dependent Type Theory and Dependent Right Adjoints_, ([slides](http://www.cs.au.dk/~spitters/floc.pdf))

Axiom K is discussed in Example 6.1.7 of:

* [[Daniel Gratzer]], *Syntax and semantics of modal type theory* ([pdf](https://jozefg.github.io/papers/phd-thesis.pdf))


axiom K (type theory)

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

> This entry is about the axiom K in [[type theory]]. For axiom K of [[modal logic]], see *[[axiom K (modal logic)]].

# Contents
* table of contents
{: toc}

## Idea

In [[type theory]], the _axiom K_ is an [[axiom]] that when added to [[intensional type theory]] turns it into [[extensional type theory]] --- or more precisely, what is called [[extensional type theory|here]] "propositionally extensional type theory". In the language of [[homotopy type theory]], this means that all types are [[h-sets]], accordingly axiom K is incompatible with the [[univalence axiom]].

Heuristically, the axiom asserts that each [[term]] of each [[identity type]] $Id_A(x,x)$ (of [[equivalences]] of a [[term]] $x \colon A$) is [[propositional equality|propositionally equal]] to the canonical [[reflexive relation|reflexivity]] equality proof $refl_x \colon Id_A(x,x)$.

See also at _[[set-level type theory]]_.

Axiom K can also be called **loop induction** or **self-identification induction** in parallel to [[path induction]].

## Statement

$$
  K
  \colon
  \underset{A \colon Type}{\prod}
  \underset{x \colon A}{\prod}
  \underset{P \colon Id_A(x,x) \to Type}{\prod}
  \left(
     P(refl_A x)
     \to
    \underset{h \colon Id_A(x,x)}{\prod} P(h)
  \right)
$$

If one doesn't have [[type universes]] in the type theory, then axiom K has to be expressed as an [[inference rule]], and thus is called the **K-rule**:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, p:\mathrm{Id}_A(x,x) \vdash P(x, p) \; \mathrm{type}}{\Gamma \vdash K_A:\prod_{x:A} P(x, \mathrm{refl}_A(x)) \to \prod_{h:\mathrm{Id}_A(x,x)} P(x, h)}$$

### Using the circle type

One can, instead of using elements $x:A$ and self identifications $p:\mathrm{Id}_A(x,x)$, use a function $p:S^1 \to A$ from the [[circle type]] $S^1$ to express axiom K:

$$
  K^{\prime}
  \colon
  \underset{A \colon Type}{\prod}
  \underset{x \colon A}{\prod}
  \underset{P \colon (S^1 \to A) \to Type}{\prod}
  \left(
     P(\lambda i:S^1.x)
     \to
    \underset{p:S^1 \to A}{\prod} P(p)
  \right)
$$

This states that the function type $S^1 \to A$ is a [[positive copy]] of $A$, and is equivalent to the other formulation of axiom K through the recursion principle of the [[circle type]].

If one doesn't have [[type universes]] in the type theory, then axiom K has to be expressed as an [[inference rule]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, p:S^1 \to A \vdash P(p) \; \mathrm{type}}{\Gamma \vdash K_A^{\prime}:\prod_{x:A} P(\lambda i:S^1.x) \to \prod_{p:S^1 \to A} P(p)}$$

## Properties

### Relation to $S^1$-localization

The negative analogue of axiom K is the [[axiom of S1-localization|axiom of $S^1$-localization]], which states that
$$\mathrm{const}_{A, S^1} \equiv \lambda x:A.\lambda i:S^1.x:A \to (S^1 \to A)$$
is an [[equivalence of types]] or a definitional [[isomorphism]].

\begin{theorem}
Suppose that every type $A$ is definitionally $S^1$-local.

Then $S^1 \to A$ is a [[positive copy]] of $A$: given any type $A$, and any type family $C(p)$ indexed by loops $p:S^1 \to A$ in $A$, and given any dependent function $t:\prod_{x:A} C(\lambda i:S^1.x)$ which says that for all elements $x:A$, there is an element of the type defined by substituting the constant loop of $x:A$ into $C$, $C(\lambda i:S^1.x)$, one can construct a dependent function $K_A(t):\prod_{z:S^1 \to A} C(z)$ such that for all $x:A$, $K_A(t, \lambda i:S^1.x) \equiv t(x):C(\lambda i:S^1.x)$.
\end{theorem}

\begin{proof}
$K_A(t)$ is defined to be

$$K_A(t) \equiv \lambda p:S^1 \to A.t(\mathrm{const}_{A, S^1}^{-1}(p))$$

and by the computation rules of loop types as negative copies, one has that for all $x:A$,

$$\mathrm{const}_{A, S^1}^{-1}(\lambda i:\mathbb{I}.x) \equiv x$$

and so by definition of $\mathrm{ind}_{S^1 \to A}(t)$ and the judgmental congruence rules for substitution, one has

$$K_A(t, \lambda i:S^1.x) \equiv t(\mathrm{const}_{A, S^1}^{-1}(\lambda i:S^1.x)) \equiv t(x)$$
\end{proof}

\begin{theorem}
Suppose that for all types $A$, $S^1 \to A$ is a [[positive copy]] of $A$ through the function
$$\mathrm{const}_{A, S^1} \equiv \lambda x:A.\lambda i:S^1.x:A \to (S^1 \to A)$$

Then Streicher's axiom K is true: given a type $A$ and given a type family $C(x, P)$ indexed by $x:A$ and $p:x =_A X$, and a dependent function $t:\prod_{x:A} C(x, \mathrm{refl}_A(x))$, one can construct a dependent function

$$K_A(t):\prod_{x:A} \prod_{p:x =_A X} C(x, p)$$

such that for all $x:A$,

$$K_A(t, x, \mathrm{refl}_A(x)) \equiv t(x)$$
\end{theorem}

\begin{proof}
By the induction principle of [[positive copies]] on the type family $C(f(\mathrm{base}), \mathrm{ap}_f(\mathrm{loop}))$ indexed by $f:S^1 \to A$, we can construct a dependent function

$$K_A^{\prime}(t):\prod_{f:\mathbb{I} \to A} C(f(\mathrm{base}), \mathrm{ap}_f(\mathrm{loop}))$$

such that for all $x:A$,

$$K_A^{\prime}(t, \lambda i:S^1.x) \equiv t(x):C(x, \mathrm{refl}_A(x))$$

since by definition of constant function and reflexivity, one has

$$(\lambda i:\mathbb{I}.x)(\mathrm{base}) \equiv x \quad \mathrm{ap}_{\lambda i:\mathbb{I}.x}(\mathrm{loop}) \equiv \mathrm{refl}_A(x)$$

We define

$$K_A(t, x, p) \equiv K_A^{\prime}(t, \mathrm{rec}_{S^1}^A(x, p))$$

since by circle recursion one has a path $\mathrm{rec}_{S^1}^A(x, p):S^1 \to A$ such that

$$\mathrm{rec}_{S^1}^{A}(x, p)(\mathrm{base}) \equiv x \quad \mathrm{ap}_{\mathrm{rec}_{S^1}^{A}(x, p)}(\mathrm{base}, \mathrm{base}, \mathrm{loop}) \equiv p$$
\end{proof}

One has the following analogies between localization at a specific type and the type theoretic letter rule that it proves:

| localization rule | type theoretic letter rule |
|-------------------|----------------------------|
| [[I-localization|$\mathbb{I}$-localization]] | [[J-rule]] |
| [[S1-localization|$S^1$-localization]] | [[K-rule]] |

### Computational behavior

Unlike its logical equivalent [[axiom UIP]], axiom K can be endowed with computational behavior: $K(A,x,P,d,refl_A x)$ computes to $d$.  This gives a way to specify a computational propositionally extensional type theory.

This sort of computational axiom K can also be implemented with, and is sufficient to imply, a general scheme of function definition by pattern-matching.  This is implemented in the proof assistant [[Agda]].  (The flag `--without-K` alters Agda's pattern-matching scheme to a weaker version appropriate for [[intensional type theory]], including [[homotopy type theory]].)

## Related concepts

* [[axiom UIP]]

* [[axiom of S1-localization|axiom of $S^1$-localization]]

* [[J-rule]], [[interval type localization]]

## References

The axiom K was introduced in

* [[Thomas Streicher]], _Investigations into intensional type theory_ Habilitation thesis (1993) ([pdf](http://www.mathematik.tu-darmstadt.de/~streicher/HabilStreicher.pdf))

For a review and discussion of the implementation in [[Coq]], see

* Pierre Corbineau, _The K axiom  in Coq (almost) for free_ ([pdf](http://coq.inria.fr/files/adt-2fev10-corbineau.pdf))

Discussion in the context of [[homotopy type theory]] is in

* [[Univalent Foundations Project]], _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

around theorem 7.2.1

[[!redirects Streicher axiom K]]
[[!redirects Streicher's axiom K]]
[[!redirects Streicher\'s axiom K]]
[[!redirects Axiom K (type theory)]]

[[!redirects K rule]]
[[!redirects Streicher's K rule]]
[[!redirects Streicher\'s K rule]]
[[!redirects K rule (type theory)]]

[[!redirects K-rule]]
[[!redirects Streicher's K-rule]]
[[!redirects Streicher\'s K-rule]]
[[!redirects K-rule (type theory)]]

[[!redirects self-identity induction]]
[[!redirects loop induction]]
[[!redirects self-identification induction]]
[[!redirects self-equality induction]]

[[!redirects self-identity elimination]]
[[!redirects loop elimination]]
[[!redirects self-identification elimination]]
[[!redirects self-equality elimination]]

[[!redirects self-identity computation]]
[[!redirects loop computation]]
[[!redirects self-identification computation]]
[[!redirects self-equality computation]]