affine connection

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# Contents
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## Definition

An _affine connection_ $\nabla$ on a [[smooth manifold]] $M$ is a [[connection on a bundle|connection]] on the [[frame bundle]] $F M$ of $M$, i.e., the [[principal bundle]] of [[frame of a vector space|frames]] in the [[tangent bundle]] $T M$.

The components of the local [[Lie-algebra valued 1-form]] of an affine connection are called [[Christoffel symbols]].

## Related concepts

* [[connection on a bundle]]

  * [[parallel transport]], [[holonomy]]

* [[principal connection]]

  * [[affine connection]], [[Levi-Civita connection]], [[Cartan connection]], [[symplectic connection]]

* [[connection on a 2-bundle]]

* [[connection on an infinity-bundle]]

  * [[higher parallel transport]]

## References

A coordinate-free treatment first appeared in

* [[Harley Flanders]], _Development of an extended exterior differential calculus_.  Transactions of the American Mathematical Society 75:2 (1953), 311–311.  [doi](https://doi.org/10.1090/s0002-9947-1953-0057005-8).

* wikipedia [affine connection](http://en.wikipedia.org/wiki/Affine_connection)

* eom [affine connection](http://eom.springer.de/a/a010950.htm)

* А.П. Норден, _Пространства аффинной связности_, 1976, [djvu](http://gen.lib.rus.ec/get?nametype=orig&md5=05EC4513F05BA7E7F05195880858AA29)

[[!redirects affine connection]]
[[!redirects affine connections]]


affine Grassmannian

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## Idea

According to [wikipedia](https://en.wikipedia.org/wiki/Affine_Grassmannian),

> the affine Grassmannian of an algebraic group $G$ over a field $k$ is an [[ind-scheme]] which can be thought of as a flag variety for the loop group $G(k((t)))$

The affine Grassmannian is ind-representable.
Affine Grassmannian of $SL_n$ admits embedding into [[Sato Grassmanian]].

## Definitions

### The $\GL_{n}$ case

\begin{definition}(Definition 1.1.1 of [#Zhu2016](#Zhu2016))
Let $k$ be a field and let $R$ be a $k$-algebra. An _$R$-family of lattices in $k((t))^{n}$_ is a finitely generated projective submodule $\Lambda$ of $R((t))^{n}$ such that $\Lambda \otimes_{R[[t]]}R((t))=R((t))^{n}$.
\end{definition}

\begin{definition}(Definition 1.1.2 of [#Zhu2016](#Zhu2016))
The _affine Grassmannian $\Gr_{\GL_{n}}$ for $\GL_{n}$_ is the presheaf that assigns to every $k$-algebra $R$ the set of $R$-families of lattices in $k((t))^{n}$.
\end{definition}

### The general case

\begin{definition}(Section 1.2 of [#Zhu2016](#Zhu2016))
Let $k$ be a field and let $G$ be an affine $k$-group. If $R$ is a $k$-algebra, let $D_{R}=\Spec k[[t]]\widehat{\times}\Spec R$ and let $D^{*}_{R}=\Spec k((t))\widehat{\times} \Spec R$. Let $\mathcal{E}^{0}$ be the trivial $G$-torsor over $D_{R}$.

The _affine Grassmannian $\Gr_{G}$ of $G$_ is the presheaf that assigns to every $k$-algebra $R$ the set of pairs $(\mathcal{E},\beta)$, where $\mathcal{E}$ is a $G$-torsor on $D_{R}$ and $\beta:\mathcal{E}\vert_{D^{*}_{R}}\xrightarrow{\simeq}\mathcal{E}^{0}\vert_{D^{*}_{R}}$ is a trivialization.
\end{definition}

## Related

* [[geometric Langlands correspondence]], [[Ran space]]
* [[G-torsor]]
* [[Grassmannian]], [[flag variety]]

## References
* Evgeny Feigin, Michael Finkelberg, Markus Reineke, _Degenerate affine Grassmannians and loop quivers_, [http://arxiv.org/abs/1410.0777](http://arxiv.org/abs/1410.0777)

* {#Zhu2016} Xinwen Zhu, _An introduction to affine Grassmannians and the geometric Satake  equivalence_, [arXiv:1603.05593](http://arxiv.org/abs/1603.05593v2).

* [[Bhargav Bhatt]], [[Peter Scholze]], _Projectivity of the Witt vector affine Grassmannian_, Invent. math. 209, 329-423 (2017) [doi](https://doi.org/10.1007/s00222-016-0710-4) [arXiv:1507.06490](https://arxiv.org/abs/1507.06490)

* Alexander Schmitt, _Affine flag manifolds and principal bundles_, Trends in Mathematics, Springer 2010

category: algebra, algebraic geometry


affine group

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## Definition

Given an [[affine space]] underlying a [[vector space]] $V$, then its _affine group_ is its group of [[automorphisms]], i.e. the [[semidirect product group]]

$$
  Aff(V) = V \rtimes GL(V)
$$

of the [[translation group]] of $V$ and the [[general linear group]] of $V$.

## Related concepts

* [[affine symplectic group]]

* [[Galilei group]]

* [[Poincare group]]

[[!redirects affine groups]]

[[!redirects affine transformation]]
[[!redirects affine transformations]]


affine group scheme

> Distinguish from the notion of a group of affine transformations (of an affine space) or the (general) affine group.

## Definition

An __affine group scheme__ is a [[group object]] (hence a [[group scheme]]) [[internalization|internal]] to the [[category]] of [[affine schemes]].

## Related notions

If $k$ is a [[commutative ring]], the category of affine group $k$-schemes is [[opposite category|opposite]] to the category of commutative [[Hopf algebra|Hopf $k$-algebras]].

By an __affine algebraic group__, one additionally assumes that the underlying scheme is over an [[algebraic variety]] (in particular reducible) and over a field. The narrower notion then agrees with the notion of a [[linear algebraic group]].

[Waterhouse](#Waterhouse)12.1 The Lie algebra of the affine group $k$-scheme $X$ is the Lie algebra of left-invariant $k$-linear derivations $T:\mathcal{O}(X)\to\mathcal{O}(X)$ of the Hopf algebra of regular functions of $X$, which are also morphisms of right comodules: $\Delta\circ T = (id\otimes T)\circ\Delta:\mathcal{O}(X)\to\mathcal{O}(X)\otimes\mathcal{O}(X)$
(one says also that $T$ is left-invariant).

This Lie algebra is also isomorphic to the Lie algebra of $k$-linear $k$-valued derivations of $\mathcal{O}(X)$.

## Examples

* The *[[additive group]] scheme* (viewed as a group valued functor on the opposite of the category of affine schemes) assigns to a commutative ring its additive group.

## Literature

Comprehensive textbooks include

* [[James Milne|James S. Milne]], _Basic theory of affine group schemes_, [pdf](https://www.jmilne.org/math/CourseNotes/AGS.pdf)

* {#Waterhouse} William C. Waterhouse, _Introduction to affine group schemes_, Graduate texts in mathematics 66, 1979

category: algebra, algebraic geometry

[[!redirects affine group schemes]]

[[!redirects affine algebraic group]]
[[!redirects affine algebraic groups]]


affine Lie algebra

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## Idea

__Affine Lie algebras__ (sometimes: _[[current algebras]]_) are the most important class of [[Kac-Moody Lie algebras]]. They should be viewed as tangent Lie algebras to the [[loop groups]], with a correction term which is sometimes related to quantization/quantum anomaly.

These affine Lie algebras appear in [[quantum field theory]] as the [[current algebras]] in the [[WZW model]] as well as in its "chiral halfs", as such for instance in the [[heterotic string]] [[2d CFT]].

## Related concepts

* [[Kac-Weyl character]]

* [[quantum affine algebra]]

* [[Borcherds algebra]]

* [[double affine Hecke algebra]]

* [[su(2)-anyon]]

## References

Textbook account

* [[Victor G. Kac]], *Infinite Dimensional Lie Algebras*,  Progress in Mathematics **44** Springer 1983 [[doi:10.1007/978-1-4757-1382-4](https://link.springer.com/book/10.1007/978-1-4757-1382-4)], Cambridge University Press (1990) [[doi:10.1017/CBO9780511626234](https://doi.org/10.1017/CBO9780511626234)]

Lecture notes:

* [[Minoru Wakimoto ]], *Lectures on Infinite-Dimensional Lie Algebra*, World Scientific (2001) [[doi:10.1142/4269](https://doi.org/10.1142/4269)]

* David Hernandez, _An introduction to affine Kac-Moody algebras_ (2006) ([pdf](http://www.ctqm.au.dk/events/2006/October/Week42/Masterclassnotes.pdf))

* {#Gordon08} [[Iain Gordon]], _Infinite-dimensional Lie algebras_ (2008/9) ([pdf](http://www.maths.ed.ac.uk/~igordon/LA1.pdf)

* {#Wassermann11} [[Antony Wassermann]], _Kac-Moody and Virasoro algebras_, course notes (2011) ([pdf](https://www.dpmms.cam.ac.uk/~ajw/course11.pdf))

The standard textbook on [[loop groups]] is

* A. Pressley, [[Graeme Segal]], _Loop groups_, Oxford Univ. Press 1988

The relation to [[quantum physics|quantum]] [[physics]] ([[WZW model]]) is highlighted in:

* S. Kass, R. V. Moody, J. Patera, _Affine Lie Algebras, Weight Multiplicities, and Branching Rules _

* Louise Dolan, _The Beacon of Kac-Moody symmetry for physics_, Notices of the AMS 1995 ([pdf](http://www.ams.org/notices/199512/dolan.pdf))

and specifically a review in the context of the [[Witten genus]] is in

* [[Kefeng Liu]], section 2.2 of _On modular invariance and rigidity theorems_, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 ([EUCLID](http://projecteuclid.org/euclid.jdg/1214456221), [pdf](http://www.math.ucla.edu/~liu/Research/loja.pdf))

The famous quote by Kac is from:

* [[Victor Kac]], _The idea of locality_ ([q-alg/9709008](http://arxiv.org/abs/q-alg/9709008))

> It is a well kept secret that the theory of Kac-Moody algebras has been a disaster. True, it is a generalization of a very important object, the simple finite-dimensional Lie algebras, but a generalization too straightforward to expect anything interesting from it. True, it is remarkable how far one can go with all these ei's, fi's and hi's. Practically all, even most difficult results of finite-dimensional theory, such as the theory of characters, Schubert calculus and cohomology theory, have been extended to the general set-up of Kac-Moody algebras. But the answer to the most important question is missing: what are these algebras good for? Even the most sophisticated results, like the connections to the theory of quivers, seem to be just scratching the surface.

> However, there are two notable exceptions. The best known one is, of course, the theory of affine Kac-Moody algebras. This part of the Kac-Moody theory has deeply penetrated many branches of mathematics and physics. The most important single reason for this success is undoubtedly the isomorphism of affine algebras and central extensions of loop algebras, often called current algebras. The second notable exception is provided by [[Borcherds' algebras]] which are roughly speaking the spaces of physical states of certain chiral algebras.

Relation to [[modular forms]]:

* [[Victor G. Kac]], [[Dale H. Peterson]], *Affine Lie algebras and Hecke modular forms*,  Bull. Amer. Math. Soc. (N.S.) **3** 3 (1980) 1057-1061 [[bams:1183547694](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-3/issue-3/Affine-Lie-algebras-and-Hecke-modular-forms/bams/1183547694.full)]

* [[Victor G. Kac]], [[Dale H. Peterson]], *Infinite-dimensional Lie algebras, theta functions and modular forms*, Advances in Mathematics **53** 2 (1984) 125-264 [doi:10.1016/0001-8708(84)90032-X]

review:

* [[Ian G. MacDonald]], *Affine Lie algebras and modular forms*, Séminaire Bourbaki: vol. 1980/81, exposés 561-578, Séminaire Bourbaki, no. 23 (1981), Exposé no. 577 [[numdam:SB_1980-1981__23__258_0](http://www.numdam.org/book-part/SB_1980-1981__23__258_0/)]

* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular and conformal invariance constraints in representation theory of affine algebras*, Advances in Mathematics **70** 2 (1988) 156-236 [doi:10.1016/0001-8708(88)90055-2, [spire:275458](https://inspirehep.net/literature/275458)]

On non-integrable but "admissible" [[irreps]] of [[affine Lie algebras]]:

* [[Victor G. Kac]], [[Minoru Wakimoto]], *Modular invariant representations of infinite-dimensional Lie algebras and superalgebras*,  PNAS **85** 14 (1988)  4956-4960 [[doi:10.1073/pnas.85.14.4956](https://doi.org/10.1073/pnas.85.14.4956)]

* [[Victor G. Kac]], [[Minoru Wakimoto]], *Classification of modular invariant representations of affine algebras*, p. 138-177 in V. G. Kač (ed.): *Infinite dimensional Lie algebras and groups*, Advanced series in Mathematical physics **7**, World Scientific (1989) [[pdf](https://math.mit.edu/~kac/not-easily-available/admissible.pdf),  [cds:268092](https://cds.cern.ch/record/268092)]

For the special case $\mathfrak{g} = $ [[sl(2)|$\mathfrak{sl}(2, \mathbb{C})$]] the formula for the "admissible" weights is made explicit in

* [[Boris Feigin]], [[Fyodor Malikov]], *Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level*, p. 357-405 in: Loday, Stasheff, Voronov (eds.) *Operads: Proceedings of Renaissance Conferences*,  Contemporary Mathematics **202** , AMS (1997) [[arXiv:q-alg/9511011](https://arxiv.org/abs/q-alg/9511011), [ams:conm-202](https://bookstore.ams.org/conm-202)]

[[!redirects affine Lie algebras]]

[[!redirects affine Kac-Moody Lie algebra]]
[[!redirects affine Kac-Moody Lie algebras]]


affine line

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## Idea

For every [[Lawvere theory]] $T$ containing the theory of [[abelian group]]s [[Isbell duality|Isbell dual]] [[sheaf topos]] over formal duals of $T$-algebras contains a canonical [[line object]] $\mathbb{A}^1$.

For $T$ the theory of commutative [[ring]]s this is called the _affine line_ .

## Definition

### Affine line

Let $k$ be a [[ring]], and $T$ the [[Lawvere theory]] of [[associative algebra]]s over $k$, such that the category of [[algebras over a Lawvere theory]] $T Alg = Alg_k$ is the [[category]] of $k$-algebras.

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

The canonical $T$-[[line object]] is the **affine line**

$$
  \mathbb{A}_k := Spec(F_T(*))  = Spec (k[t])
  \,.
$$

=--

Here the [[free functor|free]] $T$-algebra on a single generator $F_T(*)$ is the [[polynomial algebra]] $k[t] \in Alg_k$ on a single generator $* = t$ and $Spec k[t]$ may be regarded as the corresponding object in the [[opposite category]] $Aff_k := Alg_k^{op}$ of [[affine scheme]]s over $Spec  k$.

### Multiplicative group
 {#MultiplicativeGroup}

The [[multiplicative group]] object in $Ring^{op}$ corresponding to the affine line -- usually just called the **multiplicative group** -- is the [[group scheme]] denoted $\mathbb{G}_m$

* whose underlying affine scheme is

  $$
    (\mathbb{A}^1 - \{0\}) := Spec \left(k[t,t^{-1}]\right)
    \,,
  $$

  where $k[t,t^{-1}]$ is the [[localization]] of the ring $k[t]$ at the element $t = (t-0)$.

* whose multiplication operation

  $$
    \cdot \mathbb{G}_m \times \mathbb{G}_m \to \mathbb{G}_m
  $$

  is the morphism in $Ring^{op}$ corresponding to the morphism in [[Ring]]

  $$
    k[t_1,t_1^{-1}] \otimes_k k[t_2, t_2^{-1}] \leftarrow k[t,t^{-1}]
  $$

  given by $t \mapsto t_1 \cdot t_2$;

* whose unit map $Spec k \to Spec k[t,t^{-1}]$ is given by

  $$
    t \mapsto 1
  $$

* and whose inversion map $Spec k[t,t^{-1}] \to Spec[t,t^{-1}]$ is given by

  $$
    t \mapsto t^{-1}
    \,.
  $$

Therefore for $R$ any [[ring]] a morphism

$$
  Spec R \longrightarrow \mathbb{G}_m
$$

is equivalently a ring homomorphism

$$
  R \leftarrow k[t,t^{-1}]
$$

which is equivalently a choice of multiplicatively invertible element in $R$. Therefore

$$
  Hom(Spec R , \mathbb{G}_m) \simeq R^\times = GL_1(R)
$$

is the [[group of units]] of $R$.

### Additive group
 {#AdditiveGroup}

The [[additive group]] in $Ring^{op}$ corresponding to the affine line -- usually just called the **additive group** -- is the [[group scheme]] denoted $\mathbb{G}_a$

* whose underlying object is $\mathbb{A}^1$ itself;

* whose addition operation $\mathbb{G}_a \times \mathbb{G}_a \to \mathbb{G}_a$ is dually the ring homomorphism

  $$
    k[t_1] \otimes_k k[t_2] \leftarrow k[t]
  $$

  given by

  $$
    t \mapsto t_1 + t_2
    \,;
  $$

* whose unit map is given by

  $$
    t \mapsto 0
    \,;
  $$

* whose inversion map is given by

  $$
    t \mapsto -t
    \,.
  $$

### Group of roots of unity

The group of $n$th [[roots of unity]] is

$$
  \mu_n = Spec(k[t](t^n -1))
  \,.
$$

This sits inside the [[multiplicative group]] via the [[Kummer sequence]]

$$
  \mu_n \longrightarrow \mathbb{G}_m \stackrel{(-)^n}{\longrightarrow}\mathbb{G}_m
 \,.
$$

## Properties
 {#Properties}

### Grading
 {#Grading}

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

Let $R$ be a commutative $k$-algebra. There is a [[natural isomorphism]] between

* $\mathbb{Z}$-[[graded algebra|gradings]] on $R$;

* $\mathbb{G}_m$-[[action]]s on $Spec R$.

=--

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

For the first direction, let $R$ be a $\mathbb{Z}$-[[graded algebra|graded commutative algebra]]. Then $X = Spec R$ comes with a $\mathbb{G}$-action given as follows: the action morphism

$$
 \rho :  X \times \mathbb{G}_m \to X
$$

is dually the ring homomorphism

$$
  R \otimes_k \mathbb{Z}[t,t^{-1}] \leftarrow R
$$

defined on homogeneous elements $r$ of degree $n$ by

$$
  r \mapsto r \cdot t^n
  \,.
$$

The action property

$$
  \array{
     X \times \mathbb{G}_m \times \mathbb{G}_m &\stackrel{Id \times \cdot}{\to}& X \times \mathbb{G}
     \\
     {}^{\mathllap{\rho} \times Id}\downarrow && \downarrow^{\mathrlap{\rho}}
     \\
     X \times \mathbb{G}_m &\stackrel{\rho}{\to}& X
  }
$$

is equivalently the equation

$$
  r (t_1)^n \cdot (t_2)^n = r (t_1 \cdot t_2)^n
$$

for all $n \in \mathbb{Z}$. Similarly the [[unitality]] of the action is the equation

$$
  (1)^n = 1
  \,.
$$

Conversely, given an action of $\mathbb{G}_m$ on $Spec R$ we have some morphism

$$
  R[t,t^{-1}] \leftarrow R
$$

that sends

$$
  r \mapsto \sum_{n \in \mathbb{Z}} r_n t^n
  \,.
$$

By the action property we have that

$$
  \sum_n r_n (t_1 t_2)^n = \sum_{n,k} (r_n)_k t_1^n t_2^k
  \,.
$$

Hence

$$
  (r_n)_k =
  \left\{
    \array{
       r_n & if \; n = k
       \\
       0 & otherwise
    }
  \right.
$$

and so the morphism gives a decomposition of $R$ into pieces labeled by $\mathbb{Z}$.

One sees that these two constructions are [[inverse]] to each other.

=--

### Étale homotopy type

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

For $k$ a [[field]] of [[characteristic]] 0, then the affine line $\mathbb{A}^1_k$ has a [[contractible homotopy type|contractible]] [[étale homotopy type]] . This is no longer the case in [[positive number|positive]] [[characteristic]].

=--

([HSS 13, section 1](#HSS13))

### Internal formulation
 {#InternalFormulation}

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

Let $X$ be a [[scheme]] and $Sh(Sch/X)$ the [[Zariski site|big Zariski topos]] associated to $X$. Denote by $\mathbb{A}^1$ (the [[affine line]]) the [[ring object]] $T \mapsto \Gamma(T,\mathcal{O}_T)$, i.e. the functor represented by the $X$-scheme $\mathbb{A}^1_X \coloneqq X \times Spec(\mathbb{Z}[t])$. Then:

* $\mathbb{A}^1$ is [[internal logic|internally]] a [[local ring]].

* $\mathbb{A}^1$ is internally a [[field]] in the sense that any nonzero element is invertible.

* Internally, any [[function]] $f : \mathbb{A}^1 \to \mathbb{A}^1$ is a [[polynomial]] function, i.e. of the form $f(x) = \sum_i a_i x^i$ for some coefficients $a_i : \mathbb{A}^1$. More precisely,
$$ Sh(Sch/X) \models \forall f : [\mathbb{A}^1,\mathbb{A}^1]. \bigvee_{n \in \mathbb{N}} \exists a_0,\ldots,a_n : \mathbb{A}^1. \forall x : \mathbb{A}^1. f(x) = \sum_i a_i x^i. $$
Furthermore, these coefficients are uniquely determined.

=--

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

Since the internal logic is local, we can assume that $X = Spec(R)$ is affine. The interpretations of the asserted statements using the [[Zariski site#KripkeJoyal|Kripke?Joyal semantics]] are:

* Let $S$ be an $R$-algebra and $f, g \in S$ be elements such that $f + g = 1$. Then there exists a partition $1 = \sum_i s_i \in S$ such that in the localized rings $S[s_i^{-1}]$, $f$ or $g$ is invertible.

* Let $S$ be an $R$-algebra and $f \in S$ an element. Assume that any $S$-algebra $T$ in which $f$ is zero is trivial (fulfills $1 = 0 \in T$). Then $f$ is invertible in $S$.

* Let $S$ be an $R$-algebra and $f \in [\mathbb{A}^1,\mathbb{A}^1](S) = S[T]$ be an element. Then there exists a partition $1 = \sum_i s_i \in S$ such that in the localized rings $S[s_i^{-1}]$, $f$ is a polynomial with coefficients in $S[s_i^{-1}]$.

For the first statement, simply choose $s_1 \coloneqq f$, $s_2 \coloneqq g$.

For the second statement, consider the $S$-algebra $T \coloneqq S/(f)$.

The third statement is immediate, localization is not even necessary.

=--

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

Since the big Zariski topos is [[cocomplete category|cocomplete]] (being a [[Grothendieck topos]]), one can also get rid of the external [[disjunction]] and refer to the object $\mathbb{A}^1[X]$ of internal polynomials: The canonical ring homomorphism $\mathbb{A}^1[X] \to [\mathbb{A}^1,\mathbb{A}^1]$ (given by evaluation) is an [[isomorphism]].

=--

See also at _[[synthetic differential geometry applied to algebraic geometry]]_.

## Examples

### Projective space

The [[diagonal]] [[action]] of the multiplicative group on the [[product]] $\mathbb{A}^n := \prod_{i = 1 \cdots n} \mathbb{A}^1$ for $n \in \mathbb{N}$

$$
  \mathbb{A}^n \times \mathbb{G}_m \to \mathbb{A}^n
$$

is dually the morphism

$$
  k[t, t_1, \cdots, t_n] \leftarrow k[t_1, \cdots, t_n]
$$

given by

$$
  t_i \mapsto t \cdot t_i
  \,.
$$

This makes $k[t,\{t_i\}]$ the free [[graded algebra]] over $k$ on $n$ generators $t_i$ in degree 1. This is $\mathbb{N} \subset \mathbb{Z}$-graded. What is genuinely $\mathbb{Z}$-graded is

$$
  \mathcal{O} (\mathbb{A}^n - \{0\}) \simeq k[t_1, t_1^{-1}, \cdots, t_n, t_n^{-1}]
  \,.
$$

The quotient by the multiplicative group action

$$
  \mathbb{A} P^n_k := (\mathbb{A}^{n+1} - \{0\})/\mathbb{G}_m
$$

is the [[projective space]] over $k$ of [[dimension]] $n$.

### $\mathbb{A}^1$-homotopy theory

In [[A1-homotopy theory|A^1 homotopy theory]] one considers the [[reflective sub-(∞,1)-category|reflective]] [[localization of an (∞,1)-category|localization]]

$$
  Sh_\infty(C)_{\mathbb{A}^1}
  \stackrel{\leftarrow}{\hookrightarrow}
  Sh_\infty(C)
$$

of the [[(∞,1)-topos]] of [[(∞,1)-sheaves]] over a [[site]] $C$ such as the [[Nisnevich site]], at the morphisms of the form

$$
  p_1 : X \times \mathbb{A}^1 \to X
$$

that contract away cartesian factors of the affine line.

## Related concepts

* [[analytic affine line]]

* [[spectral affine line]]

* [[Tate sphere]]

## References

Discussion of [[étale homotopy type]] is in

* {#HSS13} Armin Holschbach, Johannes Schmidt, Jakob Stix, _Étale contractible varieties in positive characteristic_ ([arXiv:1310.2784](http://arxiv.org/abs/1310.2784))

[[!redirects affine lines]]


affine localization

A **[[localization functor]]** $Q^* : A\to B$ is **affine** if it has a right adjoint $Q_*$ (which is automatically fully faithful, i.e. the localized category if reflective), and which itself is having its own right adjoint $Q^!$. Therefore $Q^*\dashv Q_* \dashv Q^!$.

For example, every left [[Ore localization]] of [[ring]]s $i: R\to S^{-1}R$, induces a [[flat functor|flat]] affine localization functor $M\mapsto S^{-1}M = S^{-1}R\otimes_R M$.

A functor is an affine localization if it is an inverse image part of an  [[affine morphism]] with *full* direct image functor. Here affine morphism means an adjoint triple like above with *conservative* direct image functor; the conservativeness of an exact additive functor among abelian categories is equivalent to its faithfulness. While the notion of affine localization is fixed, the notion of an affine morphism given here has variants depending on the categorical context.

[[!redirects affine localization functor]]
[[!redirects affine localizations]]


affine logic

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

#Contents#
* table of contents
{:toc}

## Idea

In [[formal logic]], *affine logic* refers to [[substructural logics]] which omit the [[contraction rule]] but retain the [[weakening rule]]. Conversely, affine logic is [[linear logic]] adjoined with the [[weakening rule]].

In terms of [[categorical semantics]], affine logic is modeled by ([[symmetric monoidal category|symmetric]]) [[monoidal categories]] whose [[tensor unit]] $I$ is [[terminal object|terminal]], also known as *[[semicartesian monoidal categories]]*, an example of which is the category of [[affine spaces]], whence the name.

(In contrast, the category of [[linear spaces]], ie. [[vector spaces]], serves as [[categorical semantics]] for the multiplicative fragment of [[linear logic]] [eg. [Murfet 2014](linear+logic#Murfet14)], where also the [[weakening rule]] is dropped.)

## Categorical semantics

One might imagine that a more general notion of [[categorical semantics]] would be given by monoidal categories equipped with a [[natural transformation|natural]] (in $A$) family $A\to I$ of [[morphisms]] implementing [[weakening rule|weakening]] for each [[object]]. However, such an interpretation is in general badly behaved, unless one additionally requires these [[natural transformations]] to be *[[monoidal transformation|monoidal]]*, but it can be shown that this additional requirement already forces the [[tensor unit]] to be [[terminal object|terminal]] (specifically, this follows from the component at $I$ being the identity).

An example of the badly behaved case -- where the transformation is not monoidal, and the tensorial unit is not terminal -- is given by the category [[Rel]] of [[relations]], with cartesian product as tensor product (i.e., with $Rel$ as [[cartesian bicategory]]). Here a natural family of relations $A\to I$ is given by picking empty relations everywhere.  In the corresponding interpretation of affine logic, any weakening yields an empty relation, which contradicts intuitive principles like for example that "adding a dummy variable to a proof and then substituting a closed term" should not change the semantics.

## Examples

\begin{example}
  The substructural part of many forms of [[bunched logic]] are affine instead of [[linear type theory|linear]], sometimes inadvertently, ultimately due to former technical problems with formulating [[dependent linear types]] (see review in [Riley 2022 §1.7.2](dependent+linear+type+theory#Riley22Thesis)).
\end{example}

## Related concepts

* Affine **BCK** [[combinatory logic]]

## References

* [[Vaughan Pratt]],  *Re: Affine*, mailing list discussion  (1997) [[web](https://www.seas.upenn.edu/~sweirich/types/archive/1997-98/msg00135.html)]

* [[Alexei P. Kopylov]], *Decidability of Linear Affine Logic*, Information and Computation **164** 1 (2001) 173-198 [[doi:10.1006/inco.1999.2834](https://doi.org/10.1006/inco.1999.2834)]

* Gianluigi Bellin , *Two Paradigms of Logical Computation in Affine Logic?*,  in: *Logic for Concurrency and Synchronisation*, Trends in Logic **15** (2003) [[doi:10.1007/0-306-48088-3_3](https://doi.org/10.1007/0-306-48088-3_3)]

* {#Shulman22} [[Michael Shulman]], *Affine logic for constructive mathematics*, Bulletin of Symbolic Logic **28** 3 (2022) 327-386 [[arXiv:1805.07518](https://arxiv.org/abs/1805.07518), [doi:10.1017/bsl.2022.28](https://doi.org/10.1017/bsl.2022.28)]

See also:

* Wikipedia, *[Affine logic](https://en.wikipedia.org/wiki/Affine_logic)*

[[!redirects affine logics]]


affine modality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a suitable [[line object]] $\mathbb{A}^1$ in a suitable ambient [[(∞,1)-topos]], then there exists the [[cohomology localization]] at [[morphisms]] that induces [[equivalences]] in [[cohomology]] with [[coefficients]] in $\mathbb{A}^1$.

In this case the [[right adjoint]] to the reflector typically has the interpretation of producing [[spaces]] which are "affine" in that they are entirely characterized by their [[function algebras on ∞-stacks|function ∞-algebra]] with [[coefficients]] in $\mathbb{A}^1$.

Therefore in this case the [[localization]] [[modality]] deserves to be called the _affine modality_.

## Examples

Examples for this in [[higher geometry|higher]] [[algebraic geometry]] and [[synthetic differential geometry]] are discussed at _[[function algebras on ∞-stacks]]_ in the section _[Localization of the (∞,1)-topos at R-cohomology](function+algebras+on+infinity-stacks#Intrinsic)_.

## Related concepts

* [[n-truncation modality]], [[double negation modality]]

* [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]

* [[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]

[[!redirects affine modalities]]


affine monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context

#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Recall that for a [[monoidal monad]] (or equivalently [[commutative monad]]) on a [[monoidal category]], both the [[Kleisli category]] and the [[Eilenberg-Moore category]] inherit a [[monoidal category|monoidal structure]].

An *affine* monoidal monad, on a [[cartesian monoidal category]], preserves the monoidal unit (the [[terminal object]]), and so it makes the [[Kleisli category|Kleisli]] and [[Eilenberg-Moore category|Eilenberg-Moore categories]] [[semicartesian monoidal category|semicartesian]].

## Definition

Let $(C,\times,1)$ be a [[cartesian monoidal category]].
A [[commutative monad]] $(T,\mu,\eta)$ with [[monoidal monad|monoidal structure map]] $\nabla$ is called **affine** if any of the following equivalent conditions hold:

1. The unit map $\eta_1:1\to T1$ is an [[isomorphism]];
2. The following diagram commutes for all objects $A$ and $B$,
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
TA \times TB \ar{dr}[swap]{\mathrm{id}} \ar{r}{\nabla} & T(A\times B) \ar{d}{(T\pi_1,T\pi_2)} \\
& TA\times TB
\end{tikzcd}
where $\pi_1:A\times B\to A$ and $\pi_2:A\times B\to B$ are the [[product|product projection maps]].

(Note: This should be generalizable to monads on [[cartesian multicategories]].)

## Strongly and weakly affine monads

A strong monad on a cartesian monoidal category is called **strongly affine** ([Jacobs'16](#strongly_affine)) if and only if for all objects $A$ and $B$, the following diagram is a [[pullback]],
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
A\times TB \ar{r}{\sigma} \ar{d}{\pi_1} & T(A\times B) \ar{d}{T\pi_1} \\
A \ar{r}{\eta} & TA
\end{tikzcd}
where $\sigma$ denotes the [[strong monad|strength]] of the monad, and $\pi_1$ the product projection.

A commutative monad on a cartesian monoidal category is called **weakly affine** ([FGPT'23](#weaklymarkov)) if and only if any of the following equivalent conditions hold:

1. The [[internal monoid]] $T1$ (with its canonical monoid structure) is a [[group]];
2. For all objects $A$, $B$ and $C$, the [[monoidal functor|associativity diagram]] is a [[pullback]].
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large%
]
TA\times TB\times TC \ar{r}{\mathrm{id}\times\nabla} \ar{d}{\nabla\times\mathrm{id}} & TA\times T(B\times C) \ar{d}{\nabla} \\
T(A\times B)\times TC \ar{r}{\nabla} & T(A\times B\times C)
\end{tikzcd}

Every (commutative) strongly affine monad is affine, and every affine monad is weakly affine.

## Properties

* The [[Eilenberg-Moore category]] of an affine commutative monad is a [[semicartesian monoidal category]]. In particular, it is a model of [[affine logic]].

* The [[Kleisli category]] of an affine commutative monad is a [[Markov category]] ([Fritz'20](#fritzmarkov)). This makes affine commutative monads suitable as [[probability monads]], modelling [[joint and marginal distributions]].

Similarly:

* The [[Kleisli category]] of a strongly affine commutative monad is a [[Markov category#positivity_and_causality|positive Markov category]] ([FGGPS'23](#dilations)).

* The [[Kleisli category]] of a weakly affine commutative monad is a *weakly Markov category*, see [FGPT'23](#weaklymarkov).

## See also

* [[monoidal monad]], [[strong monad]], [[commutative monad]]
* [[Markov category]]
* [[joint and marginal distributions]]

## References

* [[Anders Kock]], _Bilinearity and cartesian closed monads_, Mathematica Scandinavica 29(2), 1971.

* [[Bart Jacobs]], _Semantics of weakening and contraction_, Annals of Pure and Applied Logic 69(1), 1994. ([full text](https://www.sciencedirect.com/science/article/pii/0168007294900205))

* {#strongly_affine} [[Bart Jacobs]], _Affine Monads and Side-Effect-Freeness_, Proceedings of CMCS, 2016. ([pdf](http://www.cs.ru.nl/B.Jacobs/PAPERS/side-effects.pdf))

* {#bimonoidal_monads} [[Tobias Fritz]], [[Paolo Perrone]], _Bimonoidal Structure of Probability Monads_, Proceedings of MFPS, 2018, ([arXiv:1804.03527](https://arxiv.org/abs/1804.03527))

* {#cd_categories} Kenta Cho, [[Bart Jacobs]], _Disintegration and Bayesian Inversion via String Diagrams_, Mathematical Structures of Computer Science 29, 2019. ([arXiv:1709.00322](https://arxiv.org/abs/1709.00322))

* {#fritzmarkov} [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics_, Advances of Mathematics 370, 2020. ([arXiv:1908.07021](http://arxiv.org/abs/1908.07021))

* {#dilations} [[Tobias Fritz]], Tomáš Gonda, Nicholas Gauguin Houghton-Larsen, [[Paolo Perrone]] and Dario Stein, _Dilations and information flow axioms in categorical probability_. Mathematical Structures in Computer Science, 2023. ([arXiv:2211.02507](https://arxiv.org/abs/2211.02507)).

* {#weaklymarkov} [[Tobias Fritz]], [[Fabio Gadducci]], [[Paolo Perrone]] and Davide Trotta, _Weakly Markov categories and weakly affine monads_, Proceedings of CALCO, LIPIcs 10, 2023. ([arXiv](https://arxiv.org/abs/2303.14049))

[[!redirects affine monads]]
[[!redirects strongly affine monad]]
[[!redirects strongly affine monads]]
[[!redirects weakly affine monad]]
[[!redirects weakly affine monads]]


affine morphism

# Affine morphisms
* table of contents
{: toc}

## Idea and definition

An affine morphism of [[schemes]] is a relative version of an [[affine scheme]]: given a scheme $X$, the canonical morphism $X \to Spec \mathbb{Z}$ is affine iff $X$ is an affine scheme. By the basics of [[spectrum|spectra]], every morphism of affine schemes $Spec S \to Spec R$ corresponds to a morphism $f^\circ\colon R \to S$ of [[rings]]. The affine morphisms of general schemes are defined as the ones which are locally of that form:

* a morphism $f\colon X\to Y$ of (general) schemes is **affine** if there is a [[cover]] of $Y$ (as a [[ringed space]]) by affines $U_\alpha$ such that $f^{-1} U_\alpha$ is an affine subscheme of $X$.

A seemingly stronger, but in fact equivalent, characterization follows: $f\colon X\to Y$ is affine iff for every affine $U \subset Y$, the [[inverse image]] $f^{-1}(U)$ is affine.

## Relative spectra and affine schemes

[[Grothendieck]] constructed a spectrum of a (commutative unital) [[commutative algebra|algebra]] in the category of quasicoherent $\mathcal{O}X$-modules. The result is a scheme over $X$; [[relative point of view|relative]] schemes of that form are called __relative affine schemes__.

## Functorial point of view

Now notice that a map of (associative) rings, possibly noncommutative (and possibly nonunital), induces an [[adjoint triple]] of functors $f^*\dashv f_*\dashv f^!$ among the categories of (say left) modules where $f^*$ is the extension of scalars, $f_*$ the restriction of scalars and $f^!\colon M \mapsto Hom_R(S,M)$ where the latter is an $R$-module via $(r x) (s) = x (s r)$. In particular, $f_*$ is exact.

In fact, if $f\colon X\to Y$ is a [[quasicompact morphism]] of schemes and $X$ is [[separated scheme|separated]], then $f$ is affine iff it is cohomologically affine, that is, the direct image $f_*$ is exact ([[Serre's criterion of affineness]], cf. EGA II 5.2.2, EGA IV 1.7.17).

An [[affine localization]] is a localization functor among categories of quasicoherent $\mathcal{O}$-modules which is also the inverse image functor of an affine morphism; or an abstraction of this situation.

See also [[monad in algebraic geometry]].

## Extensions

One can extend the notion of an affine morphism to [[algebraic spaces]], the noncommutative schemes of Rosenberg, Durov's generalized schemes, algebraic stacks and so on. The affinity is a local property so for algebraic stacks and the like one looks at the pullback to affine charts and checks if the resulting morphism is affine; for Durov's and Rosenberg's schemes one is basically generalizing the functorial criterium by definition. (more on this later)

## Literature

Some of the material is extracted from MathOverflow <http://mathoverflow.net/questions/15291/affine-morphisms-in-different-settings-coincide/58486>.

* R. Hartshorne, _Algebraic geometry_, exercise II.5.17

* [[A. L. Rosenberg]], _Noncommutative schemes_, Compositio Math. __112__ (1998) 93--125, [MR99h:14002](http://www.ams.org/mathscinet-getitem?mr=1622759), [doi](http://dx.doi.org/10.1023/A:1000479824211)

category: algebraic geometry, noncommutative geometry
[[!redirects affine morphism]]
[[!redirects affine morphisms]]
[[!redirects affine morphism of schemes]]
[[!redirects affine morphisms of schemes]]


affine scheme

[[!redirects affine schemes]]

#Contents#
* table of contents
{:toc}

## Definition

### General

An __affine scheme__ is a [[scheme]] that as a [[sheaf]] on the [[opposite category]] [[CRing]]${}^{op}$ of commutative [[ring]]s (or equivalently as a sheaf on the subcategory of finitely presented rings) is [[representable functor|representable]]. In a [[ringed space]] picture an affine scheme is a [[locally ringed space]] which is isomorphic to the [[prime spectrum]] of a commutative ring. Affine schemes form a [[full subcategory]] $Aff\hookrightarrow Scheme$ of the category of schemes.

The correspondence $Y\mapsto Spec(\Gamma_Y \mathcal{O}_Y)$ extends to a [[functor]] $Scheme\to Aff$. The __fundamental theorem on morphisms of schemes__ (see [below](#IsbellDuality)) says that there is a bijection

$$
  CRing(R, \Gamma_Y\mathcal{O}_Y) \cong Scheme(Y, Spec R).
$$

In other words, for fixed $Y$, and for varying $R$ there is a restricted functor

$$ Scheme(-,Y)|_{Aff^{op}} = h_Y|_{Aff^{op}} = h_Y|_{CRing} : CRing\to Set,$$

and the functor $Y\mapsto h_Y|_{CRing}$ from schemes to presheaves on $Aff$ is [[fully faithful functor|fully faithful]]. Thus the general schemes if defined as ringed spaces, indeed form a full subcategory of the category of presheaves on $Aff$. See at _[[functorial geometry]]_.

There is an analogue of this theorem for relative [[noncommutative scheme]]s in the sense of Rosenberg.

+-- {: .num_remark}
###### Remark
There is no similar equation the other way round, that is "$Ring(\Gamma_Y\mathcal{O}_Y, R) \cong Scheme(Spec R, Y)$". As a mnemonic, note that with ordinary Galois connections between power sets, one is always [[Galois connection#properties|homming into (not out of)]] the functorial construction. More geometrically, consider the example $Y = \mathbb{P}^n$ and $R = \mathbb{Z}$. Then the left hand side consists of all the $\mathbb{Z}$-valued points of $\mathbb{P}^n$ (of which there are many). On the other hand, the right hand side only contains the unique ring homomorphism $\mathbb{Z} \to \mathbb{Z}$, since $\mathcal{O}_{\mathbb{P}^n}(\mathbb{P}^n) \cong \mathbb{Z}$.

=--

### Relative affine schemes

A __relative affine scheme__ over a scheme $Y$ is a [[relative scheme]]  $f:X\to Y$ isomorphic to the spectrum of a (commutative unital) algebra $A$ in the category of quasicoherent $\mathcal{O}_Y$-modules; such a "relative" spectrum has been introduced by Grothendieck. It is characterized by the property that for every open $V\subset Y$ the inverse image $f^{-1}V\subset X$ is an open affine subscheme of $X$ isomorphic to $Spec(A(V))$ and such open affines glue in such a way that $f^{-1}V\hookrightarrow f^{-1}W$ corresponds to the restriction morphism $A(W)\to A(V)$ of algebras.

Relative affine scheme is a concrete way to represent an [[affine morphism]] of schemes.

## Properties

### Isbell duality
  {#IsbellDuality}

+-- {: .num_prop #AffineSchemesFullSubcategoryOfOppositeOfRings}
###### Proposition
**([[affine schemes]] form [[full subcategory]] of [[opposite category|opposite]] of [[rings]])**

The [[functor]]

$$
  \mathcal{O}
  \;\colon\;
  Schemes_{Aff}
    \longrightarrow
  Ring^{op}
$$

from affine schemes to their global [[rings of functions]] is a [[fully faithful functor]].

=--

(e.g. [Hartschorne 77, chapter II, prop. 2.3](#Hartschorne77))

+-- {: .num_remark}
###### Remark
**([[Isbell duality]] between [[geometry]] and [[algebra]])**

Prop. \ref{AffineSchemesFullSubcategoryOfOppositeOfRings} is the analog in [[algebraic geometry]] of similar statements of [[Isbell duality]] between [[geometry]] and [[algebra]], such as [[Gelfand duality]] or [[Milnor's exercise]].

=--

[[!include Isbell duality - table]]

### Affine Serre's theorem

 [[affine Serre's theorem|Affine Serre's theorem]]

Given a commutative unital ring $R$ there is an equivalence of categories
${}_R Mod\to Qcoh(Spec R)$ between the category of $R$-modules and the category of quasicoherent sheaves of $\mathcal{O}_{Spec R}$-modules given on objects by $M\mapsto \tilde{M}$ where $\tilde{M}$ is the unique sheaf such that the restriction on the principal Zariski open subsets is given by the localization $\tilde{M}(D_f) = R[f^{-1}]\otimes_R M$ where  $D_f$ is the principal Zariski open set underlying $Spec R[f^{-1}]\subset Spec R$, and the restrictions are given by the canonical maps among the localizations. The action of $\mathcal{O}_{Spec R}$ is defined using a similar description of $\mathcal{O}_{Spec R} = \tilde{R}$. Its right adjoint (quasi)inverse functor is given by the global sections functor $\mathcal{F}\mapsto\mathcal{F}(Spec R)$.

## Related concepts

* [[spectral topological space]]

## References

* {#Hartschorne66} Robin Hartshorne, _Algebraic geometry_, Springer 1977

* Demazure, Gabriel, _Algebraic groups_

For [[affine schemes]] in [[cubical type theory]], see:

* [[Anders Mörtberg]], [[Max Zeuner]], *A Univalent Formalization of Affine Schemes* ([arXiv:2212.02902](https://arxiv.org/abs/2212.02902))

* [[Max Zeuner]], *A univalent formalization of affine schemes*, 20 October 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Zeuner-2022-10-20-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=nLP7GjL1Buc))

category: algebraic geometry

[[!redirects Zariski duality]]


affine space

# Contents
* table of contents
{: toc}

## Idea

An __affine space__ or __affine linear space__ is a [[vector space]] that has forgotten its origin.  An __affine linear map__ (a morphism of affine spaces) is a linear map (a morphism of vector spaces) that need not preserve the origin.

Note that the 'linear functions' of elementary algebra ---the total functions whose graphs are lines--- are in fact (precisely) *affine* $\mathbb{R}$-linear maps from $\mathbb{R}$ to itself.  (Similarly, the 'linear relations' ---the relations whose graphs are lines--- are precisely the *[[projective space|projective]]* $\mathbb{R}$-linear maps.)

Alternatively, in [[algebraic geometry]], the terminology "$n$-dimensional affine space" $\mathbb{A}^n k$ (affine line, affine plane, etc.) over a [[field]] $k$ refers to, depending on context, the set $k^n$, or the set of maximal ideals of the [[polynomial algebra]] $k[x_1, \ldots, x_n]$ -- these definitions coinciding if $k$ is an [[algebraically closed field]] -- and typically considered as equipped with relevant extra structure such as a [[Zariski topology]] or, going even further, the [[locally ringed space]] structure adhering to the [[affine variety]] or [[affine scheme]] corresponding to the polynomial algebra $k[x_1, \ldots, x_n]$. Whatever the precise sense chosen, the idea is that an affine space $\mathbb{A}^n k$ is a setting in which the study of loci of polynomial equations, i.e. definable sets in the theory of [[commutative algebras]] over $k$, is carried out.

Most of this article concerns affine spaces in the sense of vector spaces that have forgotten their origins or identities; the algebraic geometry sense is very briefly touched upon in the section *Affine spaces as model spaces*.

## Definitions

### Summary

The definition of affine space can be made precise in various (equivalent) ways.  We give a name to some of the definitions for later reference.

*  An affine space is simply a vector space, but with different morphisms; an affine linear map is a function that is the difference between a linear map and a [[constant function]].

*  An affine space is a [[set]] equipped with an equivalence class of vector space structures, where two vector space structures are considered equivalent if the [[identity function]] is affine linear as a map from one structure to the other; whether a map between affine spaces is affine linear is independent of the representative vector space structures.

*  An affine space is a set $A$ together with a vector space $V$ and an [[action]] of (the additive group or _[[translation group]]_ of) $V$ on $A$ that makes $A$ into a $V$-[[torsor]] (over the point); an affine linear map is a $V$-equivariant map. For this point of view, see also [[zoranskoda:affine space]].

*  An affine space is a [[heap]] whose [[automorphism group]] is equipped with structure making it the additive group of a vector space; an affine linear map is a heap morphism.

*  An affine space is an [[inhabited set]] $A$ together with a vector space $V$ and a function $\Lambda\colon A \times A \to V$ (thought of $\Lambda(x,y) \coloneqq x - y$) that satisfies some equations; an affine linear map $A \to A'$ is a function equipped with a linear map $V \to V'$ relative to which it preserves subtraction (the "vector-valued difference" definition).

*  An affine space over the [[ground field]] $k$ is an [[inhabited set]] $A$ together with functions $\mu\colon A \times A \times A \to A$ (thought of as $\mu(x,y,z) \coloneqq x - y + z$) and $\Lambda_*\colon k \times A \times A \to A$ (thought of as $\Lambda_r(x,y) \coloneqq x - r x + r y$) that satisfy some equations; an affine linear map is a function that preserves these operations (the "two ternary operations" definition).

*  An affine space over $k$ is an inhabited set $A$ together with a function $\mu_*\colon k\times A\times A\times A\to A$ (thought of as $\mu_r(x,y,z) \coloneqq r x - r y + z$) that satisfies some equations; an affine linear map is a function that preserves this operation (the "one quaternary operation") definition.

*  Assuming that $2$ is invertible in the field $k$ (i.e. the [[characteristic]] of $k$ is not $2$), an affine space over $k$ is an inhabited set $A$ together with a function $\Lambda_*\colon k \times A \times A \to A$ that satisfies some equations; an affine linear map is a function that preserves this operation (the "one ternary operation" definition).

*  An affine space over the field $k$ is an inhabited set $A$ together with, for every [[natural number]] $n \geq 0$ and every $(n+1)$-tuple $(r_0,\dots,r_n)$ of elements of $k$ such that $r_0 + \dots + r_n = 1$, a function $\gamma_{r_0,\ldots,r_n}\colon A^{n+1}\to A$ (thought of as $\gamma_{r_0,\ldots,r_n}(x_0,\ldots,x_n) \coloneqq r_0 x_0 + \cdots + r_n x_n$), satisfying some equations; an affine linear map is a function that preserves these operations (the "[[unbiased]]" definition).

* An affine space over the field $k$ is a vector space $A'$ together with a surjective linear map $\pi:A'\to k$ (the "slice of $Vect$" definition).  The affine space itself (the set being regarded as equipped with affine-space structure) is the fiber $\pi^{-1}(1)$.

+-- {: .query}
[[Mike Shulman]]: I think there should also be a definition of the form "an affine space is a [[projective space]]" with a distinguished line called "infinity", which should also be equivalent to a "synthetic" description involving points and lines and incidence axioms.  This definition would not fix the field $k$ at the outset, but rather recover it synthetically using cross-ratios.  Accordingly, it ought to define an equivalent groupoid to the groupoid of pairs $(k,A)$ where $k$ is a field and $A$ is an affine space over $A$.  I don't know how one could recover the non-invertible affine transformations from it directly.
=--

+-- {: .query}
There should be another characterisation, which I don\'t quite see how to phrase, at least when $k = \mathbb{R}$, which is that an affine space is a manifold (perhaps Riemannian) that is sufficiently flat and unbounded in some sense.  ---Toby

[[Mike Shulman]]: It'd have to be at least Riemannian, otherwise you don't have enough structure.  I don't suppose it's enough to say that a (finitely generated) affine space is a Riemannian manifold isometric to some $\mathbb{R}^n$?

_Toby_:  I intended 'that is [...] in some sense' to include the possibility of structure that should be preserved by the morphisms.  Note that a Riemannian manifold is *too much* structure, although it allows a definition like the first one above.  (A Riemannian manifold isometric to some $\mathbb{R}^n$ is precisely a [[Euclidean space]].)  But really, I\'m hoping for some phrasing such that ‹isomorphic to some $\mathbb{R}^n$› actually becomes a (not too obvious) *theorem*.  I\'ll keep thinking about it.

[[Mike Shulman]]: Shouldn't a Riemannian manifold isometric to some $\mathbb{R}^n$ be a "Euclidean affine space" (a torsor over a Euclidean space)?  Seems that a Euclidean space would be a Riemannian manifold _equipped with_ an isometry to some $\mathbb{R}^n$.  It does seem like there should be a natural way to say this, but I don't know what it is.

_Toby_:  I guess that this depends on what you think 'Euclidean space' means; I\'ve known people to define it to *be* $\mathbb{R}^n$, but that seems quite ahistorical to me; I like that Urs calls such a thing [[Cartesian space]] instead.  Euclid did not have coordinates; he did not even have an origin, so a Euclidean space should be a heap rather than a group.  For my comment above, I would define a [[Euclidean space]] to be an affine inner product space; FWIW [Wikipedia agrees](https://secure.wikimedia.org/wikipedia/en/affine_space).  (However, Wikipedia doesn\'t go as far as I do when I claim that the inner product should be valued in an $\mathbb{R}$-line rather than in $\mathbb{R}$ itself; then again, I ignored that subtlety myself in my previous comment.)
=--

Clearly every vector space has an underlying affine space (and every linear map is affine linear), giving a [[forgetful functor]] $U:Vect \to Aff$.  Conversely, any affine space gives rise to a canonical vector space, sometimes called its space of _displacements_.  This is obvious from the definitions that involve a vector space as part of the structure, but a vector space can also be reconstructed from the other definitions as well, analogously to how a group can be reconstructed from a [[heap]].  This gives a functor $D:Aff\to Vect$ in the other direction.  One can verify that $D(U(V))\cong V$ and $U(D(A))\cong A$; the first isomorphism is [[natural isomorphism|natural]], but the second is not (otherwise $Vect$ and $Aff$ would be [[equivalent categories]], which they are not).

The category of affine spaces is almost a [[variety of algebras]], as can be seen from the last few definitions, except for the requirement that an affine space be inhabited.  To rectify this, sometimes one allows the [[empty set]] to be an affine space, although it does not have any particular vector space of displacements.  (See [heap#empty](heap#empty) for discussion.)

Note that there are a few different ways to think about the operations involved in the final three definitions (those not explicitly involving a vector space).  The operation $\mu\colon x,y,z \mapsto x - y + z$ is the same as the [[Mal'cev operation]] (i.e. [[heap]] structure) of the [[translation group|additive group]] of a vector space.  It can be viewed as the point completing a parallelogram with given vertices $x,y,z$, or equivalently as the result of adding $x$ and $z$, relative to a choice of $y$ as the origin.  The operation $\Lambda_*\colon r,x,y \mapsto x - r x + r y$ can be viewed as either a weighted average of $x$ and $y$ (i.e. as $(1-r)x + r y$) or as the result of multiplying the "displacement vector" $y-x$ by $r$, relative to the origin $x$ (i.e. as $x + r(y-x)$).

### Details and comparisons

The first few definitions, which explicitly involve a vector space, make no especial use of the fact that the vector space is a vector space rather than merely an abelian [[group]].  Thus, they are valid (and equivalent) in the more general context of [[torsors]] and [[heaps]].  They are also mostly complete as stated, except for the final one.

#### Vector-valued differences

In this definition, an affine space over a vector space $V$ is a set $A$ together with a "subtraction" function $\Lambda\colon A\times A\to V$, written $\Lambda\colon x,y \mapsto x-y$, such that:

* $\Lambda(x,x) = 0$, or $x-x = 0$, for any $x$ in $A$.
* $\Lambda(x,y) + \Lambda(y,z) = \Lambda(x,z)$, or $(x-y) + (y-z) = (x-z)$, for any $x,y,z$ in $A$.
* For any $x$ in $A$ and $v$ in $V$ there exists a unique $y$ in $A$ such that $\Lambda(y,x) = v$, or $y - x = v$.

If $y - x = v$, then we write $y = x + v$, which we can regard as an operation on $x$ and $v$ by the third axiom.  Hence we have $(x + v) - x = v$ and (by uniqueness) $x + (y - x) = y$, and also $x + 0 = x$ and $(x + v) + w = x + (v + w)$ by the first two axioms.  Thus, these axioms suffice to make $A$ into a [[torsor]] over the additive group of $V$ with the action $+$, which is one of the previous definitions given.

Note again that this would makes sense if $V$ is any group, not just the additive group of a vector space.

#### Two ternary operations

This definition is an affine version of the usual definition of a vector space in terms of addition and scalar multiplication.  However, in each case the affine operation needs to take an extra parameter.  In reading the following axioms it helps to think of $\mu(x,y,z)$ as "the sum of $x$ and $z$ relative to the basepoint $y$" and likewise $\Lambda_r(x,y)$ as "the product $r\cdot y$ relative to the basepoint $x$".

* $\mu(x,x,y) = y$ (identity for addition)
* $\mu(x,y,\mu(z,w,v)) = \mu(\mu(x,y,z),w,v)$ (associativity of addition)
* $\mu(x,y,z) = \mu(z,y,x)$ (commutativity of addition)
* $\Lambda_{r s}(x,y) = \Lambda_r(x, \Lambda_s(x,y))$ (associativity of scalar multiplication)
* $\Lambda_1(x,y) = y$ (identity for scalar multiplication)
* $\Lambda_0(x,y) = x$ (left zero for scalar multiplication)
* $\Lambda_{r+s}(x,y) = \mu(\Lambda_r(x,y), x, \Lambda_s(x,y))$ (left distributivity of scalar multiplication)
* $\Lambda_r(w, \mu(x,y,z)) = \mu(\Lambda_r(w,x), \Lambda_r(w,y), \Lambda_r(w,z))$ (right distributivity of scalar multiplication)

+-- {: .query}

_Joost_: Could it be that there is an axiom missing here ? One can go from Vector spaces to the 2 ternary operations definition and back, but I can't see that by starting with the two ternary operations definition, going to vectorspaces and back, you get the same $\Lambda$. I guess you need an extra axiom as $\mu(x,y,\Lambda_r(y,z))=\Lambda_r(x,\mu(x,y,z))$.

_Toby_:  Conceptually, there is something missing, which I\'ve inserted as the left zero property of scalar multiplication.  (The right zero property $\Lambda_r(x,x) = x$ follows from this using associativity of scalar multiplication with $s = 0$, same as with vector spaces.)  But I have to leave, and I haven\'t yet derived your axiom, even with this aid.

=--

#### One quaternary operation

This definition is an affine version of the less standard definition of a vector space in terms of a single operation $r,x,y\mapsto r\cdot x + y$.  Here an affine space over $k$ is a set $A$ together with a single operation $\mu\colon k\times A\times A\times A\to A$, written as $(r,x,y,z)\mapsto \mu_r(x,y,z)$ and thought of as the sum "$r\cdot x + z$ relative to the basepoint $y$," such that:

* $\mu_1(x,y,y) = x$
* $\mu_r(x,x,y) = y$
* $\mu_1(x,y,z) = \mu_1(z,y,x)$
* $\mu_r(\mu_s(x,y,z),w,v) = \mu_{r s}(x,y,\mu_r(z,w,v))$
* $\mu_{r+s}(x,y,z) = \mu_r(x,y,\mu_s(x,y,z))$

#### One ternary operation

In the affine case (in contrast to the vector space case), it turns out that if $2$ is invertible the "addition" $(x,y,z)\mapsto x-y+z$ can be recovered from the "scalar multiplication" $(r,x,y)\mapsto r x + (1-r)y$ by $\mu(x,y,z) = \Lambda_2(y,\Lambda_{1/2}(x,z))$.  Thus, in this case we can define an affine space over $k$ to be a set $A$ together with a single operation $\Lambda\colon k\times A\times A\to A$ such that the axioms for the two-ternary-operations definition are satisfied with this definition of $\mu$.

However, we can also simplify the requisite axioms in this presentation.  The following axioms are easier to state if we write $\Lambda_r(x,y)$ as $(1-r) x + r y$, or equivalently as $r x + s y$, where we require $r+s=1$ for the expression to be defined.

* Idempotence: $r x + s x = x$ whenever $r+s=1$.
* Commutativity: $r x + s y = s y + r x$ whenever $r+s=1$.
* Associativity: whenever $r+s+t=1$, whichever of the following three expressions are defined are equal:
  $$
  \array{
  r x + (1-r)\left(\frac{s}{1-r} y + \frac{t}{1-r} z\right)\\
  (1-s)\left(\frac{r}{1-s} x + \frac{t}{1-s} z\right) + s y\\
  (1-t)\left(\frac{r}{1-t} x + \frac{s}{1-t} y\right) + t z
  }
  $$
  The first is defined whenever $r\neq 1$, the second whenever $s\neq 1$, and the third whenever $t\neq 1$.  Since $k$ has characteristic $\neq 2$, we cannot have $r=s=t=1$ and $r+s+t=1$ at the same time, so at least one of these expressions is always defined.  We write $r x + s y + t z$ for the common value of whichever of them are defined.
* Cancellation: for any $r\in k$ and $x,y\in A$, we have $x + r y - r y = x$.

#### Unbiased definition

Let $Th_{vect}$ denote the [[Lawvere theory]] of $k$-vector spaces.  For any $n$, its $n$-ary operations are $n$-tuples $(r_1,\dots,r_n)\in k^n$ representing the [[linear combination]] operation $(x_1,\dots,x_n)\mapsto r_1 x_1 +\dots+ r_n x_n$.  Composition of operations is by substitution in the obvious way, and the identity operation is $(1)$.  A model of this theory is simply a vector space.  With this 'unbiased' definition, a vector space comes equipped with, for every integer $n\ge 0$ and $n$-tuple $(r_1,\dots,r_n)$ of elements of $k$, a function $V^n\to V$ (thought of as $(v_1,\dots,v_n)\mapsto r_1 v_1+\dots r_n v_n$), satisfying some axioms.

Let $Th_{aff}$ denote the subtheory of $Th_{vect}$ containing only those operations $(r_1,\dots,r_n)$ such that $r_1+\dots+r_n=1$; an **affine space** is a nonempty model of $Th_{aff}$.  (We have to observe that these are closed under the theory operations and thus define a subtheory.  Note that this excludes all zero-ary operations, so an affine space has no distinguished constants, and it also excludes all nonidentity unary operations.)  The basic operations $r_0x_0+\dots+r_n x_n$, when $r_0+\dots+r_n=1$, are called __affine (linear) combinations__ of elements of $A$.

The axioms for the unbiased definition are most straightforward to see by writing out the operations of $Th_{aff}$.  In particular, this includes "substitution" axioms of the form
$$
r_0(s_{00} x_{00} + \dots + s_{0m_0} x_{0m_0}) + \dots + r_n (s_{n0} x_{n0} + \dots + s_{n m_n} x_{n m_n}) = r_0 s_{00} x_{00} + \dots + r_n s_{n m_n} x_{n m_n}.
$$
However, it also includes "permutation" axioms of the form
$$
r_0 x_0 + \dots + r_n x_n = r_{\sigma 0} x_{\sigma 0} + \dots + r_{\sigma n} x_{\sigma n}
$$
and also "duplication" and "omission" axioms.  This Lawvere theory can be defined concisely as follows.  The Lawvere theory of vector spaces is the opposite of the category of finite-dimensional vector spaces; its operations are all [[linear combinations]].  The Lawvere theory for affine spaces is the sub-theory of this consisting of only the affine combinations.  (The Lawvere theory of vector spaces also has other interesting sub-theories, such as that consisting of _[[convex combinations]]_ whose algebras are abstract [[convex space]]s in one sense of the term.)  Note that the empty set is a model (algebra) of this Lawvere theory; an affine space is an *inhabited* model.

Given the unbiased definition in terms of a Lawvere theory, the previous three "biased" vector-space-free definitions can then be recovered by finding particular generating operations for the theory.  In particular, this Lawvere theory is generated by $2$-ary operations if $char(k)\neq 2$, and by $3$-ary ones if $char(k)=2$.  To wit, suppose given $(r_0,\dots,r_n)\in k^{n+1}$ with $n\ge 3$ such that $r_0+\dots+r_n=1$.  Suppose for the moment that the $r_i$ are not all $1$, and WLOG suppose that $r_0\neq 1$.  (Note that here we use the invariance under permutations.)  Then we have
$$
r_0 x_0 + \dots + r_n x_n = r_0 x_0 + (1-r_0)\left(\frac{1}{1-r_0} r_1 x_1 + \dots + \frac{1}{1-r_0}r_n x_n\right)
$$
so we have expressed the given $(n+1)$-ary operation in terms of a $2$-ary one and an $n$-ary one.  By induction, in this way we can express any $(n+1)$-ary operation in terms of $2$-ary ones (note that there is only one $1$-ary operation, namely the identity, and no $0$-ary ones) --- as long as we never hit a tuple where every $r_i=1$.  But since we always have the requirement $r_0+\dots+r_n=1$, this badness can only happen if the characteristic of $k$ is $n$.  Moreover, we still have
$$
x_0 + \dots + x_n = x_0 - x_1 + (2x_1 + x_2 + \dots + x_n)
$$
so we can still write this $(n+1)$-ary operation in terms of a $3$-ary one and an $n$-ary one.  So only if $n+1=3$ (i.e. $n=char(k)=2$) are we prevented from getting down to $2$-ary operations only, and in this case we can still get down to $3$-ary ones.  Finally, we observe that any $3$-ary operation can be written in terms of $2$-ary ones and the particular $3$-ary operation $x_0 - x_1 + x_2$:
$$
r_0 x_0 + r_1 x_1 + r_2 x_2 = \big(r_0 x_0 + (1-r_0) x_2\big) - x_2 + \big(r_1 x_1 + (1-r_1) x_2\big).
$$

#### Slice of $Vect$

Given an affine space $A$ (with any other definition), the corresponding $\pi:A'\to k$ is constructed as follows.  Let $A' = 1 \sqcup A$, where $\sqcup$ is the coproduct in affine spaces (akin to a simplicial [[join]]), $1$ is the [[terminal object|terminal]] affine space, and $\pi$ is the composite of $1 \sqcup !: 1 \sqcup A \to 1 \sqcup 1$ with a natural identification $\mu: 1 \sqcup 1 \cong k$.  Both $1 \sqcup !$ and $\mu$ which are morphisms of $Aff$ may be regarded as morphisms of $1 \downarrow Aff \simeq Vect$ (pointed affine spaces are vector spaces) if we let the first inclusion $i_0: 1 \to 1 \sqcup 1$ be the pointing of $1 \sqcup 1$ and $0: 1 \to k$ the pointing of $k$ and define $\mu$ by $\mu \circ i_0 = 0$, $\mu \circ i_1 = 1$ (the element $1 \in k$). (So $\mu$ is like two ends of a meter stick used to set up coordinates on the line $k$.)

Conversely, given $\pi:A'\to k$, the fiber $\pi^{-1}(1)$ naturally acquires a "vector-valued difference" affine space structure by simple subtraction in the vector space $A'$, where the vector space of displacements is $V = \pi^{-1}(0)$.

Note that this definition embeds the category $Aff$ of (inhabited) affine spaces fully-faithfully in the [[slice category]] $Vect/k$.  The objects of $Vect/k$ not in $Aff$ are those of the form $0:V\to k$, which form a category equivalent to $Vect$ itself.  Moreover, there are no morphisms from objects of $Aff$ to objects not in $Aff$; while by the above construction, a morphism from $0:V\to k$ to an affine space $\pi:A'\to k$ is just a map from $V$ to the vector space of displacements of $A$.  Hence, $Vect/k$ is equivalent to the (dual) [[cograph of a functor|cograph]] of $D:Aff\to Vect$.

### Closed monoidal structure {#ClosedMonoidal}

If we allow affine spaces to be empty, then they are the models of an [[algebraic theory]] $Th_{Aff}$.  Moreover, like $Th_{Vect}$, the theory $Th_{Aff}$ is a [[commutative theory]]. It follows that if $A, B$ are affine spaces, then the set $\hom(A, B)$ is closed under all affine space operations pointwise defined on the set of all functions from $A$ to $B$. This gives $Aff$ a [[closed category]] structure; on general grounds, it is in fact a [[symmetric monoidal closed category]]. The unit of this structure is the terminal or one-pointed affine space $1$, via the natural isomorphism $\hom(1, B) \cong B$. Thus $Aff$ is a closed [[semicartesian monoidal category]].

Analogous to the case of $Vect$, every affine space is a coproduct of copies of the monoidal unit: an affine space $A$ of dimension $n$ admits an affine basis, which amounts to an isomorphism $1 \sqcup 1 \sqcup \ldots \sqcup 1 \cong A$, represented by $n+1$ [*sic*] points of $A$. Such basis representations allow one to coordinatize spaces of maps $\hom(A, B) \cong B^{n+1}$, with dimension $(n+1)\dim(B)$. If one uses the first of the affine basis elements to give a pointing of the affine space (equivalent to a vector space structure), then the remaining affine basis elements provide a vector space basis, and in those coordinates every element $f \in \hom(A, B)$ may be written in matrix-vector form $f(x) = M x + b$, where again the space of such $(M, b)$ has dimension $m n + m$ if $m$ is the dimension of $B$. (There are also more 'unbiased' coordinate descriptions, not biased in favor of the first basis element playing the role of the origin.)

Similarly, we can coordinatize affine tensor products $A \otimes B$: the tensor distributes over coproducts (as it does in any symmetric monoidal closed category) and so

$$(\bigsqcup_{n+1} 1) \otimes (\bigsqcup_{m+1} 1) \cong \bigsqcup_{(m+1)(n+1)} 1 \otimes 1 \cong \bigsqcup_{(m+1)(n+1)} 1$$

with dimension $m n + m + n$. In other words, if $a_1, \ldots, a_{n+1}$ is an affine basis of $n$-dimensional $A$ and $b_1, \ldots, b_{m+1}$ a basis of $m$-dimensional $B$, then $A \otimes B$ has an affine basis consisting of the $m n + m + n + 1$ many elements $a_i \otimes b_j$.

The embedding $Aff \to Vect/k$ described in the previous subsection, sending $A$ to $1 \sqcup !: 1 \sqcup A \to 1 \sqcup 1$, is a [[strong monoidal functor]] (preserves the tensor product up to coherent isomorphism) if $Vect/k$ is endowed with the obvious tensor product acquired from $Vect$. Note that $Vect/k$ is the coreflection of $Vect$ from monoidal categories to [[semicartesian monoidal categories]]; the embedding $Aff \to Vect/k$ was in fact discovered by one of us in conjunction with this fact, and is the same as the functor induced by universality from the strong monoidal functor $Aff \to Vect$ given by $A \mapsto 1 \sqcup A$.

## Further Remarks

Every finitely-generated affine space is isomorphic to the $n$-fold [[direct sum]] $k^n$, where $k$ is the [[base field]] and $n$ is a [[natural number]] (possibly $0$).  In [[algebraic geometry]], an $n$-dimensional affine space is often denoted $\mathbb{A}^n$ and identified with $k^n$.  If one accepts the empty set as an affine space, then this is considered to have dimension $-1$ by convention (so $k^{-1} = \empty$).

The notion of affine space may be generalised to __affine module__ by replacing the vector space above by a [[module]] and the base field $k$ by a [[commutative ring]].  Then an affine module over the ring $\mathbb{Z}$ of [[integer]]s is precisely a commutative [[heap]], just like a module over $\mathbb{Z}$ is an [[abelian group]].  Note that the definition involving only one "scalar multiplication" operation works if and only if $2$ is invertible in $k$; it\'s not enough that $2 \ne 0$ in $k$.

+-- {: .query}
[[Mike Shulman]]: I haven't thought much about affine modules, but it seems likely to me that the "biased" module-free definitions won't be right any more, since the Lawvere theory needn't be generated by 2-ary or 3-ary operations (as far as I can see).  More explicitly, I don't immediately see how to write an operation like
$$(x_0,x_1,x_2,x_3) \mapsto 4x_0 - 6x_1 - 2x_2 + 5x_3$$
in terms of $A^3\to A$ and $\mathbb{Z}\times A^2\to A$, but it seems to me that this operation should still exist in an affine $\mathbb{Z}$-module.

[[Mike Shulman]]: Well that's rubbish isn't it.  The operation $A^3\to A$ is enough to give you a heap, hence an additive group, and then $\mathbb{Z}\times A^2\to A$ gives you the scalar multiplication.  And so
$$4x_0 - 6x_1 - 2x_2 + 5x_3 = \big((4x_0 - 3y) - y + (-6x_1+7y)\big) - y + \big((-2x_2+3y) - y + (5x_3 - 4y)\big)$$
for any $y$ at all.

_Toby_:  Right.  But I find an affine module of a *rig* to be a trickier concept.

[[Mike Shulman]]: Quite so.  Perhaps first one should look for a version of a [[heap]] corresponding to a [[monoid]]?

_Toby_:  Yes, that would be an affine $\mathbb{N}$-module.
=--

## Affine spaces as model spaces

Affine spaces typically serve as local models for more general kinds of spaces.

For instance a [[manifold]] is a [[topological space]] that is locally isomorphic to an affine space over the [[real numbers]].

Similarly, in [[algebraic geometry]] a [[scheme]] is locally isomorphic to an [[affine scheme]].

Therefore there are attempts to axiomatize properties of categories of affine spaces for the purpose of using these as model spaces for more complicated geometries. One such axiomatization is the notion of [[geometry (for structured (∞,1)-toposes)]]. and in particular that of [pregeometry](/nlab/show/geometry+(for+structured+(infinity%2C1%29-toposes%29#Pregeometry).

## Related concepts

* the [[automorphism group]] of an affine space is an _[[affine group]]_

* [[Cartesian space]]

* [[polydisk]]

* [[projective space]], [[conical space]]

* [[relative affine n-space]]

## References

Textbook accounts:

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §8 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

See also:

* [[Aurelio Carboni]], _Categories of Affine Spaces_ , JPAA **61** (1989) pp.243-250.

* [[Aurelio Carboni]], [[George Janelidze]], _Modularity and Descent_ , JPAA **99** (1995) pp.255-265.

* Michel Thiébaud, _Modular Categories_ , pp.386-400 in _Proc. Como conference - Category Theory_ , LNM **1488** Springer Heidelberg 1991.

[[!redirects affine space]]
[[!redirects affine spaces]]
[[!redirects affine vector space]]
[[!redirects affine vector spaces]]
[[!redirects affine linear space]]
[[!redirects affine linear spaces]]
[[!redirects affine-linear space]]
[[!redirects affine-linear spaces]]

[[!redirects affine function]]
[[!redirects affine functions]]
[[!redirects affine linear function]]
[[!redirects affine linear functions]]
[[!redirects affine-linear function]]
[[!redirects affine-linear functions]]
[[!redirects affine map]]
[[!redirects affine maps]]
[[!redirects affine linear map]]
[[!redirects affine linear maps]]
[[!redirects affine-linear map]]
[[!redirects affine-linear maps]]

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affine symplectic group

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

#Contents#
* table of contents
{:toc}

## Definition

Given a [[symplectic vector space]] $(V,\omega)$, then its _affine symplectic group_ $ASp(V,\omega)$ (or _inhomogeneous sympelctic group_ $ISp(V,\omega)$) is equivalently

* the [[intersection]] $ASp(V,\omega) = Aff(V)\times_{Diff(V)} Sympl(X,\omega)$ of the [[affine group]] of the [[affine space]] $V$ and the [[symplectomorphism group]] of the [[symplectic manifold]] $(X,\omega)$, i.e.the  [[group]] of all those [[affine transformations]] which preserve the [[symplectic form]] $\omega$;

* the [[semidirect product]] $ASp(V,\omega) = V \rtimes Sp(V,\omega)$ of the [[symplectic group]] acting on $V$ regarded as the [[translation group]] over itself.

The further restriction to [[linear functions]] gives the [[symplectic group]] proper.

## Properties

### Extensions

There is a [[circle group|circle]] [[group extension]] $ESp(V,\omega)$ of the affine symplectic group -- the _[[extended affine symplectic group]]_ -- given by restricting the [[quantomorphism group]] of $(V,\omega)$ to affine transformations. The further restriction of that to elements coming from [[translation group|translations]] is the [[Heisenberg group]]  $Heis(V,\omega)$.

$$
  \array{
     Heis(V,\omega)
     &\hookrightarrow&
     ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega)
     \\
     \downarrow && \downarrow && \downarrow
     \\
     V
     &\hookrightarrow&
     ASp(V,\omega)
     &\hookrightarrow&
     HamSympl(V,\omega)
  }
$$

## Related concepts

* [[Poincare group]]

## References

Review includes

* {#Low12} Stephen G. Low, section 1 of _Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group_, J. Math. Phys. 55, 022105 (2014) ([arXiv:1207.6787](http://arxiv.org/abs/1207.6787))

[[!redirects affine symplectic groups]]


affine variety

#Contents#
* table of contents
{:toc}

## Idea

__Affine__ $k$-variety is a locus of zeros of a set of polynomials in the affine $n$-dimensional space $\mathbf{A}^n_k$. Usually $k$ is taken to be a field.

## Definition

Given a field $k$, an __affine $k$-variety__ is a [[maximal spectrum]] (= set of [[maximal ideals]]) of a finitely generated [[noetherian ring|noetherian]] (commutative unital) $k$-[[commutative algebra]] without [[nilpotent element|nilpotents]], equipped with the [[Zariski topology]]; the algebra can be recovered as the coordinate ring of the variety; this correspondence is an equivalence of categories, if the morphisms are properly defined.

  Affine varieties can be embedded as closed subvarieties into an [[affine space]] (in the sense of algebraic geometry). As topological spaces affine varieties are [[noetherian space|noetherian]].

## Properties

### Cohomology
 {#Cohomology}

For $X$ an affine variety then its [[abelian sheaf cohomology]]
with [[coefficients]] in the [[structure sheaf]] satisfies

$$
  H^{\bullet \geq 1}(X,\mathcal{O}_X) = 0
  \,.
$$

The converse requires in addition some finiteness condition. ([Ballico 08](#Ballico08)).

## Related concepts

* [[affine scheme]]

* [[spectrum of a commutative ring]]

* [[Stein space]], [[Cartesian space]]

## References

* {#Ballico08} E. Ballico, _A characterization of affine varieties_ 2008

[[!redirects affine varieties]]


affinoid

#Contents#
* table of contents
{:toc}

## Idea

In [[analytic geometry]], an _affinoid_ is a sub-[[space]] of a unit [[polydisc]], formally dual to an [[affinoid algebra]] (see there for more). These are the basic spaces out of which [[analytic spaces]] are built by gluing.

## References

* Wikipedia, _[Rigid analytic space](http://en.wikipedia.org/wiki/Rigid_analytic_space)_

* Johannes Nicaise, Chenyang Xu, Tony Yue Yu, _The non-archimedean SYZ fibration_, [arxiv/1802.00287](https://arxiv.org/abs/1802.00287)

> We construct non-archimedean SYZ fibrations for maximally degenerate Calabi-Yau varieties, and we show that they are affinoid torus fibrations away from a codimension two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt-models along one-dimensional strata.

[[!redirects affinoids]]


affinoid algebra

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#### Analytic geometry
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _affinoid algebra_ is a local model for [[analytic spaces]] in [[analytic geometry]] ([[rigid analytic geometry]]).

## Definition

Let $K$ be a [[complete normed field|complete]] [[non-archimedean valued field]].

As a [[ring]], a __standard affinoid algebra__ (or __Tate algebra__) $T_{n,K}$ is the subring of the ring of [[formal power series]] in $K[ [x_1, \ldots, x_n] ]$ consisting of all strictly [[convergence|converging]] series $ c= \sum_I c_I x^I$, that is such that $|c_I|\to 0$ as $I\to \infty$.

There is a [[Gauss norm]] on such series $\|\sum_I c_I x^I \| = max\{|c_I|\}_I$. This is indeed a [[norm]] making $T_{n,K}$ into a Banach $K$-algebra of countable type.

An __affinoid algebra__ is any [[Banach algebra]] which can be represented in a form (Tate algebra)/(closed ideal).

The [[category]] of **$k$-[[affinoid spaces]]** is the [[opposite category]] of the category of $k$-affinoid algebras and bounded [[homomorphisms]] between them.

## Properties

A version of the [[Weierstrass preparation theorem]] in this context implies a version of the
[[Hilbert basis theorem]]: $T_{n,K}$ is a [[noetherian ring]]. Moreover $T_{n,K}$ is a [[unique factorization domain]] of [[Krull dimension]] $n$.

## References

Affinoid algebras were introduced in

* [[John Tate]], (1961)

A standard textbook account is

* {#BoschGuntzerRemmert84} S. Bosch, U. Güntzer, [[Reinhold Remmert]], part B of  _[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry_, 1984 ([pdf](http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf))

See the references at _[[analytic geometry]]_ for more details.

Discussion of affinoid algebras as a [[site]] for a more [[topos]]-theoretic formulation of of [[analytic geometry]] is in

* {#BenBassatKremnitzer13} [[Oren Ben-Bassat]], [[Kobi Kremnizer]], section 7 of _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))

See also

* {#Bambozzi14} [[Federico Bambozzi]], _On a generalization of affinoid varieties_ ([arXiv:1401.5702](http://arxiv.org/abs/1401.5702))

[[!redirects affinoid algebras]]


affinoid domain

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## Idea

In the context of [[non-archimedean analytic geometry]], affinoid domains are basic model [[spaces]]: a [[Berkovich analytic space]] is, in particular, a [[topological space]] equipped with an [[atlas]] by ([[analytic spectra]] underlying) affinoid domains.

## Definition

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

An **[[affinoid domain]]** in an [[affinoid space]] $X = Spec_{an} A$ is a [[closed subset]] $V \subset X$ such that there is a [[homomorphism]] of $k$-affinoid spaces

$$
  \phi : Spec_{an} A_V \to X
$$

for some $A_V$, whose [[image]] is $V$, and such that every other morphism of $k$-affinoid spaces into $X$ whose image is contained in $V$ uniquely factors through this morphism.

=--

([Berkovich 09, def. 2.2.1](#Berkovich09))

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###### Definition

A morphism $f\colon X\to Y$ of [[affinoid spaces]] is an _affionoid domain embedding_ if it induces an [[isomorphism]] of $X$ with an affinoid domain in $Y$

=--

([Berkovich 09, def. 2.2.7](#Berkovich09))

These are the "admissible morphisms" in the site of affinoid domains. (...)

## Properties

* [[Tate's acyclicity theorem]]

## Related concepts

* [[closed cover]]

* [[Berkovich space]]

## References

* {#Berkovich09} [[Vladimir Berkovich]], section 2.2 of _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

* [[Doosung Park]], _Affinoid domains_, lecture notes ([pdf](http://math.berkeley.edu/~sstich/MAT_274/03-07.pdf))

[[!redirects affinoid domains]]

[[!redirects affinoid domain embedding]]
[[!redirects affinoid domain embeddings]]


affinoid space

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## Definition

The [[category]] of **$k$-[[affinoid spaces]]** is the [[opposite category]] of the category of $k$-[[affinoid algebras]] and bounded [[homomorphisms]] between them.

[Berkovich 09, def. 2.1.11](#Berkovich09)

## Related concepts

* [[Berkovich space]]

## References

* {#Berkovich09} [[Vladimir Berkovich]], _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

[[!redirects affinoid spaces]]


Agda

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## Overview

A [[dependent type|dependently typed]] [[functional programming language]] with applications to [[certified programming]]. It is also  used as a [[proof assistant]].

Besides [[Coq]], Agda is one of the languages in which [[homotopy type theory]] has been implemented ([Brunerie](#Brunerie)).

Agda can be compiled to [[Haskell]], Epic or Javascript.

## Variants

### Cubical Agda
 {#CubicalAgda}

[Cubical Agda](https://agda.readthedocs.io/en/latest/language/cubical.html)  is a mode of Agda (turned on by the flag `--cubical`) that implements a type theory similar to CCHM (De Morgan) [[cubical type theory]], and thus a form of [[homotopy type theory]].

Its main difference from CCHM is that instead of an exotype of "cofibrant propositions" it uses the interval itself, replacing cofibrant propositions by statements of the form $r \equiv 1$ for some dimension expression $r$.  This change does not prevent the construction of a model for the theory in De Morgan [[cubical sets]], although it doesn't technically fall under the Orton-Pitts axioms since $I$ is not a subobject of $\Omega$, and no one has checked whether this model can be strengthened to a [[Quillen model category]].

More problematically, to support [[identity types]] a la Swan (which are distinct from both cubical "path types" and Martin-Lof "identity types" -- the latter sometimes called "jdentity types" to emphasize their definition relative to the J-eliminator) the type of cofibrant propositions must support a [[dominance]].  Cubical Agda thus assumes that $I$ supports a dominance, but this is not true in De Morgan cubical sets.  So the semantics of the entirety of Cubical Agda, with Swan identity types, is unclear.  (For this reason, the Cubical Agda library generally avoids using Swan identity types, although Cubical Agda supports them.)

Ordinary Martin-Löf [[identity types]] should, in principle, also be definable in Cubical Agda as an indexed inductive family, with computational behavior as usual for any inductive types in cubical type theory.  As of March 2021, however, there is a bug in Cubical Agda that prevents jdentity types from computing correctly.

### Guarded Cubical Agda

The guarded cubical variant extends cubical Agda to support [[guarded recursive]] definitions which can be used to formalize [[synthetic guarded domain theory]].

### Agda-flat
  {#AgdaFlat}

`Agda-flat` is a mode of Agda that implements a [[comonad|co-]][[idempotent monad|monadic]] [[modal operator]] $\flat$ ("flat", following the notation used in [[cohesive homotopy type theory]] as introduced in [[schreiber:dcct|dcct]] and type-theorertically developed in [Shulman 15](cohesive+homotopy+type+theory#Shulman15)).  This makes Agda model a [[modal type theory]] and hence a [[modal homotopy type theory]], such as used, for instance, in [[schreiber:thesis Wellen|Wellen 2017]].

See:

* [`Agda-flat` documentation](https://agda.readthedocs.io/en/v2.6.2.2/language/flat.html)

* [installation instructions](http://www.cl.cam.ac.uk/~amp12/agda/internal-universes/code/agda-flat/install.txt)

## Little-known features {#LittleKnownFeatures}

Listed here are some little-known or undocumented features of Agda that are sometimes useful.  Note that undocumented "features" may change without warning; this list is current as of September 2022, Agda v2.6.2.

* There are a lot of useful documented [keybindings](https://agda.readthedocs.io/en/v2.6.2.2/tools/emacs-mode.html#keybindings) that you may not be aware of.
  * This appears not to be documented in the manual (although it is in the Emacs docstring): When in a hole, the commands `C-c C-,` and `C-c C-.` can be prefixed with `C-u C-u` to normalize the type of the hole (and the term, in the second case) before displaying them.

* The manual doesn't document the customizable variables in the Emacs mode.  Of particular note are:
  * `agda-input-user-translations` allows you to add new bindings to the Unicode input mode
  * `agda2-highlight-level`, when set to `interactive`, uses highlighting to display realtime information about which terms  and subterms in the buffer Agda is currently typechecking.
  * `agda2-program-args` allows you to add command-line arguments to be used every time (e.g. the `-v` options below).  This in in addition to the arguments specified by a particular file in the `OPTIONS` line.

  To change the values of these variables, run `M-x customize-variable RET` and enter the variable name, change the value, and then "Set and Save".

* The command `-v` (verbose output) accepts various additional options that are, according to the developers, "documented by their implementation".  These include:
  * `rewriting.rewrite:50` --- displays information about attempted uses of rewrite rules.
  * `rewriting.match:60` --- displays information about attempted matches during rewriting.
  * `import.chase:2` --- when compiling imported files, displays a notification when each file is completed in addition to when it is started.

  Unfortunately, the output produced by these flags appears in the `*Agda debug*` buffer, which is not visible by default, rather than the standard `AgdaInfo` buffer.  To turn these flags on, you can add (for instance) `-v import.chase:2` to `agda2-program-args` via Customization, as above, or to the `OPTIONS` line of a particular file.

## Minimizing bugs

Since Agda has many experimental features under active development, bugs in these features are not uncommon.  When [reporting a bug](https://github.com/agda/agda/issues/new/choose), it is helpful to "minimize" it to make the shortest possible [MRE](https://en.wikipedia.org/wiki/Minimal_reproducible_example).  Some tips for doing this [from the developers](https://github.com/agda/agda/issues/6067#issuecomment-1236313145) include

* Enable `--type-in-type`, `--no-termination-check`, `--no-positivity-check`, `--no-projection-like`, `--no-fast-reduce`
* Disable unused flags
* Remove unused imports
* Copy definitions from imported modules (other than `Agda.Builtin` modules)
* Delete unused definitions
* Turn functions into postulates
* Inline functions
* Remove clauses from definitions
* Replace patterns by `_` if they are not used
* Replace types by Set
* Delete unused arguments to functions or constructors
* Remove dependencies from types
* Replace (sub)terms by `w/e` where `w/e : ∀ {ℓ} {A : Set ℓ} → A` is a postulate
* Make implicit arguments explicit
* Replace wildcards `_` in terms by their solution

## Related concepts

* [[1lab]]

[[!include proof assistants and formalization projects -- list]]

## References
 {#References}

### General

Agda landing page:

* [wiki.portal.chalmers.se/agda/pmwiki.php](http://wiki.portal.chalmers.se/agda/pmwiki.php)

Documentation:

* [agda.readthedocs.io](https://agda.readthedocs.io/en/v2.6.3)

Online Agda interface:

* [[Ingo Blechschmidt]]: [agdapad.quasicoherent.io](https://agdapad.quasicoherent.io)

Plain Agda originates with:

* [[Ulf Norell]], _Towards a practical programming language based on dependent type theory_, PhD thesis (2007) [[pdf](https://www.cse.chalmers.se/~ulfn/papers/thesis.pdf), [[Norell-PracticalDTT.pdf:file]]]

* [[Ulf Norell]], *Dependently Typed Programming in Agda*, in: *Advanced Functional Programming* AFP 2008, Lecture Notes in Computer Science **5832** (2009) 230-266 [[doi:10.1007/978-3-642-04652-0_5](https://doi.org/10.1007/978-3-642-04652-0_5), [pdf](https://www.cse.chalmers.se/~ulfn/papers/afp08/tutorial.pdf)]

Textbook account:

* [[Aaron Stump]], *Verified Functional Programming in Agda*, Association for Computing Machinery and Morgan & Claypool (2016) [[doi:10.1145/2841316](https://doi.org/10.1145/2841316), ISBN:978-1-970001-27-3]

[[Cubical Agda]] (an implementation of [[cubical type theory]], for [[univalence axiom|univalently]] computing [[homotopy type theory]]) originates with:

* [[Anders Mörtberg]], *Cubical Agda* (2018) [[blog post](https://homotopytypetheory.org/2018/12/06/cubical-agda)]

* {#VMA19} [[Andrea Vezzosi]], [[Anders Mörtberg]], [[Andreas Abel]], *Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types*, Proceedings of the ACM on Programming Languages **3** ICFP 87  (2019) 1–29 [[doi:10.1145/3341691](https://doi.org/10.1145/3341691), [pdf](https://www.cse.chalmers.se/~abela/icfp19.pdf)]

### Libraries

Libraries for/with [[homotopy type theory]]/[[univalent foundations of mathematics]]:

* original HoTT library: [github.com/hott/hott-agda](https://github.com/hott/hott-agda)

* [[UniMath project]]: [unimath.github.io/agda-unimath](https://unimath.github.io/agda-unimath)

* [[1lab]]

  > (motivated towards [[univalent categories]])

E.g. on [[group theory]] (cf. *[[Symmetry]]*):

* [[UniMath project]]: [agda-unimath/src/group-theory](https://github.com/UniMath/agda-unimath/tree/master/src/group-theory)

Implementation of [[Cauchy real numbers]] (in [[Errett Bishop|Bishop]]-style [[constructive analysis]]) in [[Agda]] (cf. *[[exact real computer arithmetic]]*):

* [[Martin Lundfall]], *Formalizing real numbers in Agda* (2015) [pdf, [[Lundfall-RealNumbersInAgda.pdf:file]], [github](https://github.com/MrChico/Reals-in-agda)]

* [[Zachary Murray]], *Constructive Analysis in the Agda Proof Assistant* [[arXiv:2205.08354](https://arxiv.org/abs/2205.08354), [github](https://github.com/z-murray/honours-project-constructive-analysis-in-agda)]

### Introductions
 {#Introductions}

#### Quick expositions
 {#Expositions}

* _['Hello World!' in Adga](http://progopedia.com/example/hello-world/251/)_

* [[Guillaume Brunerie]], _The Agda proof assistant_ (2012) [slides:[[Brunerie-AgdaProofAssistant.pdf:file]]]

* [[Ulf Norell]], *Programming in Agda*, presentation at *Oregon Programming Languages Summer School* (2014) [lecture 1: [video](https://www.youtube.com/watch?v=NrSW7YsneVg), 2: [video](https://www.youtube.com/watch?v=X0JWsoWTWnI), 3: [video](https://www.youtube.com/watch?v=HjpLZr_sirY), 4: [video](https://www.youtube.com/watch?v=Yw2VGwxn_g8)]

* [[Scott Fleischman]], *Agda from Nothing*, talk at λC (2016) [[video 1](https://www.youtube.com/watch?v=-i-QQ36Nfsk&list=PLCqzQM-xq02Uv4OmcckrFsIyZ20hWEeAx&index=7), [video 2](https://www.youtube.com/watch?v=XprGyVWXwks)]

* [[Philip Wadler]], *(Programming Languages) in Agda = Programming (Languages in Agda)*, talk at *Codegram* (Sep 2019) [[video](https://www.youtube.com/watch?v=R49VgxNLmsY&t=372s)]

* [[Fredrik Nordvall Forsberg]], *The basic syntax of Agda*, talk at CS410 Advanced Functional Programming (2021) [[video](https://www.youtube.com/watch?v=yreLHTXwkts)]

On [[cubical Agda]]:

* [[Andrea Vezzosi]], *Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types* (2019) [[video](https://www.youtube.com/watch?v=AZ8wMIar-_c)]

#### Tutorials & Lectures

General:

* *Learn you an Agda* (2011) [[web](http://learnyouanagda.liamoc.net/pages/introduction.html), [github](https://github.com/liamoc/learn-you-an-agda)]

* [[Dan Licata]], Ian Voysey, _[Programming and proving in Agda](http://www.cs.cmu.edu/~drl/teaching/oplss13/)_, course material (2013)

* [[Peter Selinger]], *Lectures on Agda* (2021) [[web](https://www.mathstat.dal.ca/~selinger/agda-lectures/)]

* [[Ingo Blechschmidt]], *Agda -- a beautiful proof assistant*, course at *Teoria dei Tipi*, Padova (Apr 2023) [[webpage](https://agdapad.quasicoherent.io/~Padova)]

With emphasis on implementing [[homotopy type theory]] and [[univalent foundations of mathematics]]:

* {#Brunerie} [[Guillaume Brunerie]], _Agda for homotopy type theory_ [[github](https://github.com/guillaumebrunerie/HoTT/tree/master/Agda/tutorial)]

* [[Martín Hötzel Escardó]], *Introduction to Univalent Foundations of Mathematics with Agda* (2019) [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580), [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/)]

category: software

[[!redirects agda]]

[[!redirects Cubical Agda]]
[[!redirects cubical Agda]]
[[!redirects cubical agda]]

[[!redirects Agda-flat]]


age

In [[model theory]], an age of a [[structure in model theory|structure]] $M$ is the set $Age(M)$ of all finite structures which are embeddable in $M$.

* related entries: [[amalgamation]]
* wikipedia: [age (model theory)](http://en.wikipedia.org/wiki/Age_%28model_theory%29)
* J D Brody, _Model theory of graphs_, thesis, [pdf](http://tarski.fandm.edu/brody/mainthesis.pdf)


Agnese Bissi

* [webpage](https://katalog.uu.se/profile/?id=N17-1951)

## Selected writings

On [[string scattering amplitudes]] via [[AdS/CFT]] and the [[conformal bootstrap]]:

* {#ABP18} [[Luis Alday]], [[Agnese Bissi]], [[Eric Perlmutter]], _Genus-One String Amplitudes from Conformal Field Theory_, JHEP06(2019) 010 ([arXiv:1809.10670](https://arxiv.org/abs/1809.10670))

category: people


Agnès Beaudry

* [Website](https://www.colorado.edu/math/agnes-beaudry)

## Selected writings

On [[chromatic homotopy theory]]:

* [[Tobias Barthel]], [[Agnès Beaudry]], *Chromatic structures in stable homotopy theory*, in [[Handbook of Homotopy Theory]], Chapman and Hall/CRC Press (2019) [[arXiv:1901.09004](https://arxiv.org/abs/1901.09004), [doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)]

On the [[Steenrod algebra]] and the [[Adams spectral sequence]] (with motivation from [[functorial quantum field theory]]):

* [[Agnès Beaudry]], [[Jonathan A. Campbell]], *A Guide for Computing Stable Homotopy Groups*, in *Topology and Quantum Theory in Interaction*, Contemporary Mathematics **718**, Amer. Math. Soc. (2018)  [[arXiv:1801.07530](https://arxiv.org/abs/1801.07530), [ams:conm/718](https://bookstore.ams.org/view?ProductCode=CONM/718)]

On the [[K-theory classification of topological phases of matter]] (and more general classification in [[Whitehead generalized cohomology theories]]):

* {#BHMP23} [[Agnès Beaudry]], Michael Hermele, Juan Moreno, [[Markus Pflaum]], Marvin Qi, Daniel Spiegel, *Homotopical Foundations of Parametrized Quantum Spin Systems* [[arXiv:2303.07431](https://arxiv.org/abs/2303.07431)]

category:people


AGT conjecture > history

see _[[AGT correspondence]]_.