anafunction

# Anafunctions

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## Idea

An *anafunction* is a [[relation]] between two [[sets]] $A,B$ such that to each [[element]] of $A$ there corresponds exactly one element of $B$.  It can (usually) equivalently be defined as a [[span]] $A \leftarrow F \to B$ of functions whose first leg $F\to A$ is both [[injective function|injective]] and [[surjective function|surjective]].

Every [[function]] yields an anafunction, namely its [[graph of a function|graph]], and this embeds the set of functions from $A$ to $B$ into the set of anafunctions.  Conversely, in ordinary mathematics every anafunction is the graph of some function, so these two sets are isomorphic (and indeed, in some foundations such as [[material set theory]], *equal*).  Thus in such cases the notion of anafunction is unneeded.

However, in more exotic contexts where the [[function comprehension|function comprehension principle]] (a.k.a. the "axiom of unique choice") fails, such as the [[internal logic]] of a [[quasitopos]] or a [[tripos]], it may be necessary to distinguish anafunctions from functions.  Indeed, the fact that every anafunction is a function, or equivalently that every injective and surjective function is an [[isomorphism]], is essentially the definition of the function comprehension principle.

Anafunctions are a [[decategorification]] of the notion of [[anafunctor]], and take their name from the latter.  Traditionally they would be called "total functional relations".

## In homotopy type theory

In [[dependent type theory]] ([[homotopy type theory]]), an anafunction is a [[type family]] $R$ indexed by types $A$ and $B$ such that for each [[term|element]] $a \colon A$, the [[dependent sum type]] $\sum_{b \colon B} R(a, b)$ is a [[contractible type]]:
$$
  \mathrm{isAnafunction}(R)
  \;\coloneqq\;
  \prod_{a:A}
  \mathrm{isContr}
  \left(
    \sum_{b \colon B}
    R(a, b)
  \right)
  \,.
$$

Since the [[principle of unique choice]] holds in dependent type theory, it follows that given any anafunction $a\colon A, b \colon B \vdash R(a, b) \; \mathrm{type}$, there is a function $a \colon A \vdash f(a) \colon B$.

The [[categorical semantics]] of an anafunction in [[homotopy type theory]] is an [[infinity-anafunctor|$\infty$-anafunctor]], since the [[identity types]] between two elements of a type are not required to be [[mere propositions]].

Given any [[type family]] $R$ indexed by types $A$ and $B$, there is a type family $R^\op$ indexed by $B$ and $A$, defined by $R^\op(b, a) \coloneqq R(a, b)$ for all $a \con A$ and $b \colon B$. An anafunction $R$ is an [[equivalence of types]] if both $R$ and $R^\op$ are anafunctions.

## See also

* [[anafunctor]]
* [[infinity-anafunctor]]
* [[one-to-one correspondence]]
* [[equivalence of types]]
* [[dependent anafunction]]

## References

For anafunctions in [[foundations of mathematics]] without the [[principle of unique choice]] see:

* [[Mike Shulman]], *Mathematics without the principle of unique choice*. [[MathOverflow]]. June 5, 2018. [Web](https://mathoverflow.net/questions/302037/mathematics-without-the-principle-of-unique-choice).

[[!redirects anafunctions]]


anafunctor

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### Context
#### Category theory
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#### Homotopy theory
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# Anafunctors
* automatic table of contents goes here
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## Idea

An _anafunctor_ $F\colon C \to D$ is a generalized [[functor]].

A basic fact in ordinary [[category theory]] is that a [[functor]] $f\colon C \to D$ is an [[equivalence of categories]] -- in that there is a functor $g\colon D \to C$ and natural isomorphisms $f \circ g \simeq Id_D$ and $g \circ f \simeq Id_C$ -- if and only if it is [[essentially surjective functor|essentially surjective]] and [[full and faithful functor|fully faithful]].  However, the "if" part of this statement depends crucially on the [[axiom of choice]]: the functor $g$ is obtained by _choosing_  for each [[object]] $d \in D$ an object $c \in C$ such that $f(c) \simeq d$.  In fact, the statement that "every fully faithful and essentially surjective functor is an equivalence of categories" is equivalent to the axiom of choice.

The notion of *anafunctor* is a generalization of the usual notion of *functor*, which enables us to recover a version of this statement without the axiom of choice.  It was first studied in detail ([Makkai](#Makkai)) with [[foundations|foundational concerns]] in mind, although it also appears unnamed in ([Kelly](#Kelly)). Later, they were applied by [[Toby Bartels]] to [[internal category|internal categories]], where the axiom of choice is simply not an option.  These "internal anafunctors" actually turned out to be known already (at least up to equivalence) in some contexts, in particular as [[Hilsum-Skandalis morphism]]s between [[Lie groupoid]]s.

We present three motivations and applications of anafunctors:

* [Anafunctors as a tool for handling foundational issues](#FoundationalMotivation)

* [Anafunctors as morphisms in internal categories](#InternalCatMotivation)

* [Anafunctors as morphisms in a homotopy category](#HomotopicalMotivation)

### Foundational motivation {#FoundationalMotivation}

There is a sense in which the construction of inverse equivalences is not a "real" use of the axiom of choice, because the choice of $d$ is determined up to unique isomorphism.  Note that in ordinary set theory, the axiom of choice is not necessary to make choices that are uniquely defined; this is sometimes called the "axiom of non-choice" or the "function comprehension principle."  Since in category theory, an object can only be expected to be determined up to unique isomorphism, it is natural to regard the above statement as really being a "[[functor comprehension principle]]" or an "axiom of non-choice for categories."  The fact that the full axiom of choice is required to make such choices is then an artifact of the usual [[foundation]]al choice to define categories as having a [[set]] of objects.

In fact, however, we can recover the functor comprehension principle while maintaining the definition of categories in set theory if we modify the notion of *functor*.  This results in the notion of _anafunctor_, which is essentially "a functor which determines its values on objects only up to isomorphism."  In particular, an anafunctor is an [[equivalence of categories]] (in the sense of having an inverse *anafunctor*) precisely if it is essentially surjective and full and faithful.  From this point of view, an anafunctor is not necessarily a fundamental notion, but rather an artifact that makes it possible to approximate the "natural" theory of categories, which doesn't need choice but has a functor comprehension principle, while still working in a set-theoretic foundation lacking choice.

Every functor may be interpreted as an anafunctor; that every anafunctor is equivalent to a functor is equivalent to the [[axiom of choice]], in which case the inclusion of functors into anafunctors is in fact an [[equivalence of categories]]. But if you ignore functors and deal only with anafunctors (or saturated anafunctors), then the theory becomes entirely [[constructive mathematics|constructive]] (without using the axiom of choice or even [[excluded middle]]).  Thus, anafunctors (or even saturated anafunctors) are the correct notion to use if you are doing [[constructive mathematics]] and you still want to [[foundations|found]] mathematics on some sort of [[set theory]].

### Motivation from internal categories {#InternalCatMotivation}

Since questions concerning the axiom of choice tend to look a bit esoteric to those not actively interested in questions of [[foundations]], it is helpful (and useful!) to think of this more generally in terms of [[internal category]] theory, where the concept is of independent use, and in fact well known by other names than "anafunctor".

Consider some ambient category $\mathcal{E}$ [[internalization|internal]] to which we want to do category theory. A good example to keep in mind is the category [[Top]] of [[topological space]]s.  We observe that "the axiom of choice fails in [[Top]]", but that this is a very non-esoteric and obvious statement: it just means that not every continuous [[epimorphism]] $P \to X$ between topological spaces has a _continuous_ [[section]].

Then it may easily happen that an internal functor $f\colon C \to D$ between [[internal category|internal categories]] in $\mathcal{E}$ (for instance between topological categories) is fully faithful, in the "internal" sense that
$$\array{C_1 & \overset{}{\to} & D_1\\
  \downarrow && \downarrow\\
  C_0 \times C_0 & \underset{}{\to} & D_0\times D_0}$$
is a [[pullback]], and essentially surjective, in the "internal" sense that $C_0 \times_{D_0} Iso(D) \to D_0$ is surjective (or even a quotient map, i.e. a [[regular epimorphism]]), but for which there still does not exist a weak inverse in $Cat(\mathcal{E})$.

For example, let $X$ be a topological space and consider the functor $C(U) \to X$, where $X$ is regarded as an internal category in $Top$ which is [[discrete category|discrete]] in the categorical sense, and $C(U)$ is the [[Cech nerve|Cech groupoid]] associated to an open [[cover]] $X = \bigcup_i U_i$ of $X$.  That means that the space of objects of $C(U)$ is $\coprod_i U_i$, and its space of morphisms is $\coprod_{i,j} (U_i\cap U_j)$.  Then the functor $C(U)\to X$ is fully faithful (essentially by definition of the morphisms in $C(U)$) and essentially surjective (since the $U_i$ are a cover of $X$) in the senses above.  However, in general it does not admit a weak inverse, since the continuous function $\coprod_i U_i \to X$ does not have a continuous section if $X$ is connected unless one of the $U_i$ is already equal to $X$.

Note that the foundational point of view also fits in this picture; we can simply take $\mathcal{E} = Set_{\not AC}$ to be the category [[Set]] of [[set]]s in a model of set theory that need not satisfy the [[axiom of choice]].  Here the same thing may happen: not every fully faithful and essentially surjective functor has a weak inverse.

### Homotopical motivation {#HomotopicalMotivation}

There is a standard way to deal with such situations where we are faced with a category -- here the category $Cat(\mathcal{E})$ of categories internal to $\mathcal{E}$ -- some of whose morphisms look like they ought to have inverses, but do not: we call these would-be invertible morphisms _weak equivalences_ such that our category becomes a [[category with weak equivalences]] or a [[homotopical category]]. Then we pass to the corresponding [[homotopy category]]: the universal "improvement" of our category such that all the would-be invertible morphisms do become invertible.

Here we take the weak equivalences in $Cat(\mathcal{E})$ to be the internal functors that are internally fully faithful and essentially surjective. It turns out that this choice of weak equivalences is particularly well-behaved in that it actually forms a [[calculus of fractions]]. Due to the early work on abstract [[homotopy theory]] by Gabriel and Zisman, there is simple explicit construction of the corresponding [[homotopy category]] $Ho(Cat(\mathcal{E}))$ in this case: the objects are the same as those of $Cat(E)$ -- hence [[internal category|categories internal to]] $\mathcal{E}$ for us -- and the morphisms $f\colon C \to D$ are [[span]]s of morphism in $Cat(\mathcal{E})$

$$
  \array{
    \hat C &\stackrel{\hat f}{\to}& D
    \\
    \downarrow^{\mathrlap{\in W}}
    \\
    C
  }
  \,,
$$

where the left leg is a weak equivalence, hence for us: where the left leg is an internal functor that is $k$-surjective for all $k$.  (This is the beginning of the construction of the [[Dwyer-Kan localization]] at our chosen weak equivalences.)

For the case $\mathcal{E} = Top$ such a span is a morphism out of a _Cech cover_. For instance for $C = X$ a topological space regarded as a topological category, for $G$ a [[topological group]] and $D = \mathbf{B}G$ its [[delooping]] one-object topological groupoid, such a span is a [[Cech cohomology|Cech cocycle]] on $X$ with values in $G$.

And finally: for the case that $\mathcal{E} = Set_{\not C}$ is the category of sets without the axiom of choice, such a span is an **anafunctor**: a functor $\hat C \to C$ that is surjective on objects and [[full and faithful functor|full and faithful]], together with a functor $\hat C \to D$ out of the "resolution" of $C$.

So one can understand ordinary anafunctors as follows:

1. first we consider that the [[axiom of choice]] may fail,
   which makes previously invertible functors  non-invertible;

1. then we universally _force_ the now non-invertible functors
   to become invertible after all,
   by throwing in formal inverses for them.

More generally, in any [[category of fibrant objects]] the morphisms in the [[homotopy category]] are represented by [[span]]s of the form

$$
  \array{
    \hat X &\to & Y
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
$$

with the left leg being an acyclic fibration. (This is a special case of the general statements of [[simplicial localization]]).

## Definitions

Given categories $C$ and $D$, an __anafunctor__ $F\colon C \to D$ may be rather slickly defined as a [[span]] of ordinary ([[strict functor|strict]]) [[functors]] $C \overset{\sigma}\leftarrow \overline{F} \overset{\tau}\rightarrow D$ (where $\overline{F}$ is some category), with the property that the functor $\sigma\colon {\overline{F}} \to C$ is both [[faithful functor|faithful]] and (strictly!) surjective on both objects and morphisms (therefore both [[full functor|full]] and [[essentially surjective functor|essentially surjective on objects]]).  It is also possible to define an anafunctor as a span in which $\sigma$ is merely a [[equivalence of categories|weak equivalence]] (that is, faithful, full, and essentially surjective on objects), although that is slightly more complicated to work with.

### Explicit set-theoretic definition {#SetDef}

In more explicit detail, an __anafunctor__ $F\colon C \to D$ consists of:

* a set ${|F|}$ of __specifications__ of $F$ (which corresponds to the set of objects of $\overline{F}$);
* maps $\sigma\colon {|F|} \to C$ and $\tau\colon {|F|} \to D$ (taking values in objects). Given $x\colon C$ and $y\colon D$, we say that $y$ is a __specified value__ of $F$ at $x$ if, for some $s\colon {|F|}$, $x = \sigma(s)$ and $y = \tau(s)$; in this case, $s$ __specifies__ $y$ as a value of $F$ at $x$, and we write $F_s(x) = y$.  That is,
  $$ F_s(x) \coloneqq \tau(s) .$$
  We say that $y$ is a __value__ of $F$ at $x$ if $y$ is isomorphic (in $D$) to some specified value of $F$ at $x$; we write $F(x) \cong y$. (There is no notion of *the* value of $F$ at $x$, except in the up-to-isomorphism sense of the [[generalised the]], and $F(x) = y$ is a meaningless statement.);
* for each $s, t\colon {|F|}$ and morphism $f\colon \sigma(s) \to \sigma(t)$ in $C$, a morphism
  $$ F_{s,t}(f)\colon F_s(x) \to F_t(y) $$
  in $D$, where $x \coloneqq \sigma(s)$ and $y \coloneqq \sigma(t)$. Similarly to the above, we can define whether a given morphism $g$ in $D$ is a __specified value__ of $F$ at a given morphism $f$ in $C$ or whether $g$ is (merely) a __value__ of $F$ at $f$. (Again, there is no notion of *the* value of $F$ at $f$.);
* $\sigma$ is a [[surjective function]]. Thus, $F$ has *some* value at any given object or morphism of $C$. (In the [[internalization|internalized]] case, this requirement can become quite complicated; for example, internal to [[Diff]], one requires a [[surjective submersion]].);
* $F$ preserves [[identity morphism|identities]]. That is, given $s\colon {|F|}$, the value of $F$ specified by $s$ and $s$ at the identity of $\sigma(s)$ is the identity of $\tau(s)$, or (in symbols) $F_{s,s}(\id_{\sigma(s)}) = \id_{\tau(s)}$, or (whenever this makes sense)
  $$ F_{s,s}(\id_x) = \id_{F_s(x)} ;$$
* $F$ preserves [[composition]]. That is, given $s, t, u\colon {|F|}$, $f\colon \sigma(s) \to \sigma(t)$, and $g\colon \sigma(t) \to \sigma(u)$,
  $$ F_{s,u}(f;g) = F_{s,t}(f);F_{t,u}(g) .$$
  (Here the semicolon indicates composition in the anti-Leibniz order.).

From the above explicit data, the category $\overline{F}$ is constructed as follows: the objects of $\overline{F}$ are the elements of ${|F|}$, while a morphism $s \to t$ in $\overline{F}$ is simply a morphism $\sigma(s) \to \sigma(t)$ in $C$.  Then $\sigma$ extends to a surjective faithful functor from $\overline{F}$ to $C$ (acting as the identity on morphisms), and $\tau$ extends to a functor from $\overline{F}$ to $D$ (mapping the morphism $f\colon s \to t$ in $\overline{F}$ to $F_{s,t}(f)\colon \tau(s) \to \tau(t)$ in $D$).

An anafunctor $F$ is __saturated__ if, whenever $F(x) \cong y$, $F_s(x) = y$ for some unique specification $s$, where the unicity of $s$ depends not only on $x$ and $y$ but also on how $y$ is a value of $F$ at $x$. To be precise: if $g\colon y' \to y$ is an isomorphism in $D$ and $F_{s'}(x) = y'$ for some specification $s'$, then there is a unique specification $s$ such that $F_{s',s}(\id_x) = g$ (where in particular, $\sigma(s) = x$ and $F_s(x) = y$). Every anafunctor $F\colon C \to D$ has a _saturation_ $\overline{F}$; $\overline{F}$ is a saturated anafunctor and $F \cong \overline{F}$ in the category of anafunctors from $C$ to $D$. In fact, the inclusion of the saturated anafunctors into the anafunctors (as a full subcategory) is an equivalence of categories (given fixed $C$ and $D$).

The usual notions of [[full functors]] and [[faithful functors]] can be generalized to anafunctors. An anafunctor $F$ is __full__ if the maps $F_{s, t}: Hom(\sigma(s), \sigma(t)) \to Hom(\tau(s), \tau(t))$ are all surjective, and it is __faithful__ if the maps are injective.

Anafunctors can be composed via pullback. Given anafunctors $F: C \to D$, $G: D \to E$, we can form the pullback
$$
\array{
  |GF|                & \to                 & |G|        & \overset{\tau_G}\to E\\
  \downarrow          &                     & \downarrow\mathrlap{\sigma_G}\\
  |F|                 & \underset{\tau_F}\to & D\\
  \mathllap{\sigma_F}\downarrow\\
  C
}
$$
More explicitly, the specifications are given by pairs $(s, t) \in |F| \times |G|$ such that $\tau_F(s) = \sigma_G(t)$, and $\sigma_{G F}(s, t) = \sigma_F(s)$, $\tau_{G F}(s, t) = \tau_G(t)$. For $(s, t), (s', t') \in |GF|$ and an arrow $f: \sigma_F(s) \to \sigma_F(s')$, we obtain an arrow $F_{s, s'}(f): \tau(s) \to \tau(s')$. This is also an arrow $\sigma_G(t) \to \sigma_G(t')$, so we can lift this map to $G_{t, t'}F_{s, s'}(f): \tau(t) \to \tau(t')$, and this completes the description of the anafunctor. The other axioms can be verified straightforwardly.

Categories, anafunctors, and a suitably defined notion of [[ananatural transformation]] between them form a [[bicategory]] $Cat_{ana}$; an internal [[equivalence]] in this 2-category is called an **anaequivalence**.  Every functor may be interpreted as an anafunctor, with ${|F|}$ always taken to be (the set of objects in) $C$ itself and $\sigma$ the [[identity functor]].  Indeed, there is a [[2-functor]] to $Cat_{ana}$ from the [[strict 2-category]] $Str Cat$ of categories, functors and natural transformations; this functor is an [[equivalence of categories|equivalence]] if and only if the [[axiom of choice]] holds.  Thus, most mathematicians will identify $Cat_{ana}$ and $Str Cat$ as simply [[Cat]], the $2$-category of categories; however, mathematicians who doubt the axiom of choice will distinguish them.  While anafunctors exist in any case, there is an ideological statement that may be implied by their use: that $Cat$ is *really* $Cat_{ana}$ rather than $Str Cat$.

In any case, (modulo "size issues" which one may want to impose) the inclusion of $Str Cat$ into $Cat_{ana}$ has a right adjoint, described using [[clique]]s. Accordingly, we can instead define anafunctors by means of clique categories, taking an anafunctor from $C$ into $D$ to be a genuine functor from $C$ into $Clique(D)$ (and the 2-category of anafunctors as the [[Kleisli category]] for the $Clique(-)$ 2-monad (in particular, natural transformations between anafunctors into $D$ are simply natural transformations of the corresponding genuine functors into $Clique(D)$)).

### Internal definition using covers

We generalise the slick definition of anafunctors as spans rather than the detailed definition involving specified values.

Let $S$ be a category containing a collection of morphisms called "covers" such that

* every [[isomorphism]] is a cover,
* covers are closed under composition,
* any [[pullback]] of a cover exists and is a cover ([[pullback stability]]),
* every cover is the [[quotient object]] of its [[kernel pair]], i.e. is an [[effective epimorphism]].  (Since all pullbacks of covers exist, this is equivalent to saying that every cover is a [[regular epimorphism]].)

Note that these are precisely the axioms saying that the singleton families $\{p\colon V\to U\}$ where $p$ is a cover form a [[subcanonical coverage|subcanonical]] [[Grothendieck pretopology]].  One important class of examples is when $S$ is a [[regular category]] and the covers are the [[regular epimorphisms]].  Another is when $S$ is the category of smooth manifolds and the covers are the surjective submersions.

In such a situation, if $C$ and $D$ are [[internal category|internal categories]] in $S$, we define an __anafunctor__ $C\to D$ to consist of a span $C\leftarrow F \to D$ of internal functors such that:

1. $F_0\to C_0$ (the map of objects) is a cover.

2. $F\to C$ is [[ff morphism|fully-faithful]], in the internal sense that the following is a pullback square:
$$\array{F_1 & \to & C_1 \\ \downarrow && \downarrow \\ F_0\times F_0&   \to & C_0\times C_0}$$

Note that assuming $F_0\to C_0$ is a cover, so is $F_0\times F_0\to C_0\times C_0$ (it is a composition of pullbacks of $F_0\to C_0$); thus the above pullback always exists.

By the remarks above, if $S$ is [[Set]] and "cover" means "[[surjection]]" (an example where the covers are the regular epimorphisms), then we recover the original external notion of ([[small category|small]]) anafunctor.  An anafunctor, defined in this way, is saturated just when the map $core(F) \to core(C\times D)$ of [[cores]] is an [[isofibration]], so we need an internal notion of core to define saturated anafunctors internally.  An anafunctor is an anaequivalence when $F\to D$ is fully faithful and [[essentially surjective functor|essentially surjective]], meaning the canonical map $F_0\times_{C_0}C_1 \to C_0$ has splits over a cover of $C_0$. For [[Lie groupoids]], these are the [[Morita equivalences]].

If $C\leftarrow F \to D$ and $C\leftarrow G \to D$ are internal anafunctors, we define an __[[ananatural transformation]]__ between them (or simply a _natural transformation_, given the context) to be a [[natural transformation]] between the two induced internal functors $F\times_C G \to D$.  We can then prove that internal categories, anafunctors, and natural transformations form a [[bicategory]].  (Interestingly, you may need the axiom of choice in the [[metalogic]] to conclude this, depending on whether there is a natural way to choose the necessary pullbacks; else you get an [[anabicategory]], in which the composition functors are anafunctors.)

The role of the assumptions about covers is:

* Identity maps must be covers in order to have identity anafunctors (and more generally, for every functor to give rise to an anafunctor).
* To compose anafunctors by pullback, the pullbacks of covers must exist and be covers, and covers must be closed under composition.
* To define $F \times_C G$ and obtain a notion of natural transformation, we again need covers to have pullbacks.
* To define composition of natural transformations between anafunctors, we need covers to be effective; see diagram (118) in [HGT1](#HGT1).

Note: in Section 1.1.5 of [HGT1](#HGT1), the following additional axiom was assumed on the class of covers:

* every [[congruence]] involving a cover has a quotient object which is a cover.

This is not needed for anafunctors but is used to relate descent to bundles (and then to $2$-bundles).

### As an operation on 2-categories

_This section is work in progress by [[David Roberts|me]]_

While internal anafunctors seem to require a lot of baggage, they can be defined very elegantly by working with the 2-category $Cat(S)$ (or some sub-2-category thereof) as a 2-category. One first makes the observation:

* functors internal to $S$ which are fully faithful and whose object component belongs to a singleton Grothendieck pretopology themselves form a (strict) singleton pretopology on $Cat(S)$. Thus one can consider anafunctors as spans in a 2-category where the source leg belongs to a strict, subcanonical singleton Grothendieck pretopology, all of whose covers are [[ff morphism|ff]].

### Homotopy-theoretic interpretation

Observe that the surjective-on-objects equivalences are precisely the [[model category|acyclic fibrations]] for the [[canonical model structure]] on [[Cat]].  Therefore, anafunctors can be identified with the "one-step generalized morphisms" in $Cat$ whose first leg is not just a [[weak equivalence]] but an acyclic fibration.  However, it appears that the canonical model structure on Cat only exists (with its weak equivalences being the fully faithful and essentially surjective maps) under the assumption of some choice---though full AC is not needed, [[COSHEP]] suffices.

More generally, it is proven in [EKV](#EKV) that if $S$ has a Grothendieck coverage, then under suitable additional conditions on $S$ (and, of course, the axiom of choice assumed external to $S$), there is a [[model category|model structure]] on the category $Cat(S)$ of internal categories in $S$ relative to that coverage.  The internal anafunctors relative to the given coverage, as defined above, can then once again be identified with the spans whose first leg is an acyclic fibration.

Since all objects in the canonical model structure on Cat are fibrant, according to Kenneth Brown's theorem in [[homotopical cohomology theory]] it follows that one-step generalized morphisms already realize the full localization, i.e. they represent all morphisms in the [[homotopy category]] $Ho(Cat)$.

If we specialize to groupoids, with their canonical model structure by [[Ronnie Brown|Brown]]--[[Marek Golasiński|Golasiński]], then by the general idea of [[homotopical cohomology theory]]
this means that anafunctors between groupoids represent [[nonabelian cocycle]]s on groupoids with values in groupoids. By the notion of [[descent|codescent]] such homotopical cocycles are related to [[descent|descent data]] that enters the definition of [[sheaf|sheaves]] and [[stack]]s.

## Examples

We will use the [explicit set-theoretic definition](/nlab/show/anafunctor#SetDef) in this section.

### Product anafunctor
Given a category $C$ with [[cartesian product|binary products]], we can form a product functor $P: C \times C \to C$ that sends a pair of objects $(A, B)$ to a product of them. This requires picking a product for each pair, and hence requires the [[axiom of choice]].

However, we can form the product anafunctor without using choice. The specifications are given by $|P| = \text{product diagrams in }\;C$, and the maps $\sigma: |P| \to C \times C$ and $\tau: |P| \to C$ are given by
$$
  \sigma(A \leftarrow D \rightarrow B) = (A, B),\;\; \tau(A \leftarrow D \rightarrow B) = D.
$$
Given product diagrams $A \leftarrow D \rightarrow B$ and $A' \leftarrow D' \rightarrow B'$, and a map $(f, g): (A, B) \to (A', C')$ in $C \times C$, we obtain $P(f, g): D \to D'$ by the universal property of the product. The compatibility conditions are easy to check.

### Anafunctor from functor
Suppose we have a usual functor $F: C \to D$. Then we can obtain an anafunctor as the span
$$
\array{
  C & \overset{F}\to & D\\
  \mathllap{id_C} \downarrow \\
  C
}
$$
The composition of anafunctors agree with the composition of functors.

### Inverses of anafunctors
Given an anafunctor $F: C \to D$ with $\sigma: |F| \to C$ and $\tau: |F| \to D$, if $F$ (ie. $\tau$) is essentially surjective, then its saturation is strictly surjective.

Then given a saturated full and faithful essentially surjective anafunctor, we can obtain an inverse anafunctor $F^{-1}: D \to C$ by swapping $\tau$ and $\sigma$ around. The conditions of being full and faithful and essentially surjective guarantees the axioms are still satisfied.

## Questions of size {#SizeQuestions}

Even if $C$ and $D$ are [[small categories]], then the category $Ana(C,D)$ of anafunctors from $C$ to $D$ is not necessarily even essentially small, and thus the 2-category $Cat_{ana}$ of categories and anafunctors is not [[cartesian closed category|cartesian closed]].  Some models in which this fails to be true are sketched in [this MO discussion](#HenryMO).

This is true, however, under the assumption of [[COSHEP]], since in that case (as above) anafunctors represent maps in $Ho(Cat)$, which is locally small by general model category theory.  More specifically, under COSHEP every anafunctor $C\leftarrow F \to D$ is equivalent to one where the set of objects of $F$ is $C_0'$, where $C_0'\to C_0$ is a projective cover of the set $C_0$ of objects of $C$.

COSHEP is actually stronger than necessary for this; all that is really needed is [[WISC]], i.e. for any set $X$, the full subcategory of $Set/X$ consisting of surjections has a [[weakly initial set]].  For in that case, any anafunctor $C\leftarrow F \to D$ is equivalent to one where the set of objects of $F$, equipped with its surjection to $C_0$, belongs to the weakly initial set.  Note that COSHEP implies WISC, as do the [[axiom of multiple choice]] and the axiom of [[small violations of choice]] (SVC).

In [Makkai's paper](#Makkai), he proves that $Ana(C,D)$ is essentially small under the assumption of his [[small cardinality selection axiom]] (SCSA), which also follows from SVC.  Although SCSA and WISC carry the same feel that "choice is violated only in a small way," Makkai's proof from SCSA is an "injective" approach, in that the set of possibilities for the objects of $F$ is constructed mainly from $D$, rather than purely from $C$ as in the "projective" approach above using COSHEP or WISC.

Another axiom ensuring that $Ana(C,D)$ is essentially small is the [[axiom of stack completions]] (ASC), since if $D\to \hat{D}$ is an intrinsic stack completion we have $Ana(C,D) \simeq Fun(C,\hat{D})$.

In particular, the bicategory of categories and anafunctors is locally essentially small and cartesian closed in the internal logic of any [[Grothendieck topos]], because the latter satisfies WISC, SCSA, and ASC.

##Anafunctors in homotopy type theory

Anafunctors are unnecessary when using "saturated/univalent" categories in [[homotopy type theory]] (see Def. 9.1.3 of [[the HoTT book]], and Chap. 9 notes), because of their [[functor comprehension principle]]. An anafunctor is a span whose first leg is a surjective and fully faithful functor, but for saturated categories any such functor is an equivalence (in the strong sense of having an inverse), so any anafunctor is equivalent to a functor.

## Related concepts

### Anafunctors versus representable profunctors

A different way of describing "a functor whose values are determined only up to isomorphism" is with a representable [[profunctor]] (a.k.a. distributor).  A profunctor $A ⇸ B$ is a functor $F\colon A \to [B^{op},Set]$, and we call it *representable* if each [[presheaf]] $F(a)$ is [[representable functor|representable]].

In the presence of [[axiom of choice|AC]], one can then choose a representing object $F_a\in B$ for each $a$ and thereby define a functor $A\to B$, but without choice this is generally not possible.  However, one can define an anafunctor from $A$ to $B$, whose specifications at $a$ are "representations" of $F(a)$.

Conversely, given an anafunctor represented by a span $A \xleftarrow{g} P \xrightarrow{f} B$, one can define a profunctor $A ⇸ B$ as the composite $f_* \circ g^*$, where $f_*(p)(b) = B(b,f(p))$ and $g_*(a)(p) = A(g(p),a)$, and this profunctor will be representable.  This defines a bijective-on-objects [[equivalence of 2-categories]] between $Cat_{ana}$ and $Prof_{rep}$, the [[locally full sub-2-category]] of [[Prof]] determined by the representable profunctors.  (This appears to have been written down first [here](http://permalink.gmane.org/gmane.science.mathematics.categories/6485) by [[Jean Benabou]]).

Anafunctors and representable profunctors each have advantages.  For purposes which require only the 2-category $Cat_{ana}\simeq Prof_{rep}$, either one is of course sufficient.  For instance, a non-[[cleavage|cloven]] [[Grothendieck fibration]] can equally well be turned into a [[pseudofunctor]] valued in $Cat_{ana}$ or in $Prof_{rep}$.  However, in this case $Prof_{rep}$ is arguably more natural, since an arbitrary functor (not necessarily a fibration) $A \to B$ can be turned into a normal [[lax 2-functor]] $B^{op}\to Prof$, which is a pseudofunctor landing in $Prof_{rep}$ exactly when the given functor was a fibration.  (It is a pseudofunctor landing in $Prof_{corep}$ iff the functor was an opfibration, and it is a pseudofunctor iff the functor was [[exponentiable functor|exponentiable]].)  More generally, one good point about using representable profunctors is that they fit in immediately with the general notion of profunctor.

On the other hand, sometimes it requires a little contortion to put something in the form of a representable distributor.  For instance, if $A$ has binary [[products]], then there is obviously a product-assigning representable distributor $P\colon A \times A  ⇸ A$ defined by $P(a_1,a_2)(a) = Hom_A(a,a_1) \times Hom_A(a,a_2)$.  But if A has binary [[coproducts]], then in order to define a coproduct-assigning representable distributor $C\colon  A \times A  ⇸ A$, one needs to say something like

* $C(a_1,a_2)(a)=$ the set of triples $(a_3,p_1,p_2,f)$ where $p_i\colon a_i \to a_3$ are the injections into a coproduct and $f$ is a morphism $a \to a_3$, modulo an equivalence relation $(a_3,p_1,p_2,f) \sim (a_3',p_1',p_2',f')$ if there exists a (necessarily unique iso)morphism $g\colon a_3 \to a_3'$ commuting with all the structure maps.

(This is essentially making explicit the functor $Cat_{ana} \to Prof_{rep}$ defined above.)  Of course, $C$ is more easily defined as a *[[corepresentable functor|corepresentable]]* distributor.  But if we want to define a functor that involves both limits and colimits, like $(a,b,c) \mapsto a \times (b + c)$, then it is not "naturally" represented as either a representable or a corepresentable profunctor.  However, with anafunctors, all of these functors can be represented "naturally" in analogous ways.  In the first case, we consider the span $A\times A \leftarrow P \to A$, where $P$ is the category of binary product diagrams in $A$.  In the second case, we consider the span $A\times A \leftarrow C \to A$, where $C$ is the category of binary coproduct diagrams.  And in the third case, we consider the span $A\times A\times A \leftarrow D \to A$, where $D$ is the category of binary coproduct diagrams together with a product diagram one of whose factors is the vertex of the coproduct diagram.

Roughly speaking, anafunctors are formulated exactly in order to describe "functors defined up to isomorphism," while representable distributors describe "functors valued in representable presheaves."  "Objects defined up to isomorphism" and "representable presheaves" are *formally* equivalent (without invoking AC), but not every "naturally occurring" object-defined-up-to-isomorphism is "given in nature" by the presheaf it represents.  Some are given by the [[copresheaf]] they corepresent; others aren't given directly in either of those ways.  One can also think of an anafunctor as a particularly convenient "presentation" of a representable distributor.

A further reason that the notion of anafunctor is useful is that when working with [[internal categories]], the quotienting operations necessary to define the composite of internal profunctors may not exist, whereas internal anafunctors can always be composed.  Thus, when working with (for instance) [[smooth manifold|smooth]] categories or groupoids, profunctors are not so much an option, but anafunctors are well-behaved (see the papers by Bartels and Roberts referenced below).

## Generalizations

### Higher versions

see [[infinity-anafunctor]]

### Lower version

see [[anafunction]]

### Additive version

The notion of abelian [[butterfly]] introduced by Behrang Noohi [Weak maps of 2-groups](http://arxiv.org/abs/math.CT/0506313) is the additive version of the notion of (saturated) anafunctor: the equivalence between, on the one hand, internal groupoids and internal functors and, on the other hand, arrows and commutative squares in an abelian category extends to an equivalence between saturated anafunctors and butterflies.

## References

The term "anafunctor" was introduced by Michael Makkai in

* {#Makkai} [[Michael Makkai]], _Avoiding the axiom of choice in general category theory_, Journal of Pure and Applied Algebra **108** isse 2 (1996) pp 109-173, doi:[10.1016/0022-4049(95)00029-1](https://doi.org/10.1016/0022-4049%2895%2900029-1), ([author's page](http://www.math.mcgill.ca/makkai/anafun/))

motivated in part to complete the analogy prophase:anaphase::profunctor:??.

The concept also appears, unnamed, in the article

[[Max Kelly]], _Complete functors in homology I_ (1963)
{#Kelly}

on [[homological algebra]].

The popularity of the term was notably pushed by [[Toby Bartels]], who considered [[internalization]]s of Makkai's definition in

* [[Toby Bartels]], _Higher Gauge Theory I: 2-Bundles_  ([arXiv:math.CT/0410328](http://arxiv.org/abs/math.CT/0410328))
{#HGT1}

A development and exposition of the general setup taking Makkai's and Bartels' motivations and the theory of [[homotopical category|homotopical categories]] into account is

* {#Roberts12} [[David Roberts]], _Internal categories, anafunctors and localisations_, [[Theory and Applications of Categories]], Vol. 26, 2012, No. 29, pp 788-829, [journal version](http://www.tac.mta.ca/tac/volumes/26/29/26-29abs.html), [arXiv:1101.2363](http://arxiv.org/abs/1101.2363)

and a purely formal construction using 2-categories appears in the notes

* {#Roberts18} [[David Roberts]], _The construction of formal anafunctors_ (2018) arXiv:[1808.04552](https://arxiv.org/abs/1808.04552), doi:[10.25909/5b6cfd1a73e55](https://doi.org/10.25909/5b6cfd1a73e55)

See also

* [[Erik Palmgren]], _Locally cartesian closed categories without chosen constructions_, [TAC](http://www.tac.mta.ca/tac/volumes/20/1/20-01abs.html).

Since anafunctors are a special case of a more general concept, they, or the general theory applying to them, has been considered under different terms elsewhere.

The general question of [[model category]] structures on categories of [[internal category|internal categories]] is discussed in

* T. Everaert, R.W. Kieboom and T. Van der Linden , _Model structures for homotopy of internal categories_  TAC,  Vol. 15, CT2004, No. 3, pp 66-94. ([web](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html) ([pdf](http://www.tac.mta.ca/tac/volumes/15/3/15-03.pdf)))
{#EKV}

Closely related, still a bit more general, are the considerations in

* Jardine, _Cocycle categories_ K-theory 0782 ([web](http://www.math.uiuc.edu/K-theory/0782/)) ([pdf](http://www.math.uiuc.edu/K-theory/0782/coc-cat3.pdf))

Some models of set theory in which the bicategory of anafunctors fails to be small are sketched in the answers to

* {#HenryMO} Simon Henry, _Non smallness of the set of anafunctors without AC?_, URL (version: 2017-03-16): <https://mathoverflow.net/q/264585>

[[!redirects anafunctor]]
[[!redirects anafunctors]]


anafunctor category

Given [[category|categories]] $C$ and $D$, the _anafunctor category_ $D^C$ has [[anafunctor|anafunctors]] $F: C \to D$ as [[object|objects]] and [[ananatural transformation|ananatural transformations]] between these as [[morphism|morphisms]].

This is the appropriate notion of [[functor category]] to use in the absence of the [[axiom of choice]] (including many [[internal category|internal]] situations).

Functor categories serve as the [[hom-category|hom-categories]] in the [[anabicategory]] [[Cat]].


analogy

#Contents#
* table of contents
{:toc}

## Examples

* [[function field analogy]]

* [[MKR analogy]]

* [[rubber-sheet analogy of gravity]]

## Related concepts

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

## References

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

[[!redirects analogies]]

[[!redirects analog]]
[[!redirects analogs]]


analyser

Before the modern theory of [[operad]]s was (re)invented there was a variant introduced earlier in

* M. Lazard, _Lois de groupes et analyseurs_, Ann. École Norm. Sup. __72__ (1955), pp. 299–400.

under the name __analyser__ which is often even nowdays referred by its original French version *analyseur*.

There is a generalization motivated by some questions in noncommutative algebra, the pseudoanalyser:

* Ralf Holtkamp, _A pseudo-analyzer approach to formal group laws not of operad type_, Journal of Algebra __237__, 1, 2001, p. 382-405, [doi](http://dx.doi.org/10.1006/jabr.2000.8566)

[[!redirects analysers]]
[[!redirects analyseur]]


analysis

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#### Analysis
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#### Functional analysis
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[[!include functional analysis - contents]]
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#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[mathematics]], _analysis_ usually refers to any of a broad family of fields that deals with a general theory of _[[limit of a sequence|limits]]_ in the sense of [[convergence]] of [[sequences]] (or more generally of [[nets]]), particularly those fields that pursue developments that originated in "the [[calculus]]", i.e., the theory of [[differentiation]] ([[differential calculus]]) and [[integration]] ([[integral calculus]]) of [[real numbers|real]] and [[complex numbers|complex]]-valued [[functions]]. The classical foundation of this general subject is usually based on the idea that the [[real numbers|real number system]] is describable as a [[complete space|complete]] [[ordered field]], or more generally on the concept of [[metric spaces]]. Their [[distance]] functions allow to formalize concepts like [[continuous functions|continuity]] and [[convergence]] in terms of existence of sufficiently small [[open balls]]. Many concepts of this "[[epsilontic analysis]]" have equivalent formulations in terms of simple [[combinatorics]] of [[open subsets]] with respect to the [[metric topology]] of metric spaces, and this way the field of analysis has a large overlap with the field of _[[topology]]_, this is particularly true for [[functional analysis]] and the theory of [[topological vector spaces]].

Analysis can also refer to other responses to the problem of founding these developments, especially "[[infinitesimal analysis]]" which admits [[infinitesimal quantities]] not found in the classical real number system and which takes various forms, for example the [[nonstandard analysis]] first introduced by [[Abraham Robinson]], or "[[synthetic differential geometry|synthetic differential analysis]]" whose rigorous foundations were largely introduced by [[William Lawvere]] and other [[category theory|category theorists]] who, following the example of [[Alexander Grothendieck]], consider [[nilpotent infinitesimals]] (instead of invertible ones à la Robinson) as a basis for understanding [[differentiation]].

## Entries related to analysis

### On mainstream analysis

Some of the $n$lab entries related to __mathematical analysis__ include  [[metric space]], [[normed vector space]], [[metric topology]], [[sequence]], [[net]], [[convergence]],

[[functional analysis]], [[harmonic analysis]], [[complex analysis]], [[Weierstrass preparation theorem]], [[several complex variables]], [[Fourier transform]], [[Pontrjagin dual]], [[differential geometry]], [[Legendre polynomial]], [[dilogarithm]], [[Hilbert space]], [[Banach space]], [[Banach algebra]], [[topological vector space]], [[locally convex space]], [[operator algebras]], [[Gelfand spectrum]], [[measure space]], [[measurable function]], [[Lebesgue space]], [[Sobolev space]], [[bounded operator]], [[compact operator]], [[Fredholm operator]], [[distribution]] (generalized function), [[hyperfunction]].[[spectral theory]], [[integral]], [[integration]]...and a book entry [[Handbook of analysis and its foundations]]. Many of the basic notions used in analysis courses are described in $n$lab in the more general [[topology|topological]] context if they belong there, e.g. [[compact space]], [[continuous map]], [[compact-open topology]] and so on. Many of the aspects of [[analytic geometry]] are treated in terms of Riemann surfaces, [[monodromy]], [[local system]]s and so on.

### On foundations

Alternative [[foundations]], especially [[constructive mathematics|constructive]] and those using [[topos theory]], are of traditional interest to the [[category theory]] community. For example the [[synthetic differential geometry]] of Lawvere and Kock (more in next paragraph) and the [[nonstandard analysis]] of Robinson, and its variant, [[internal set]] theory of Nelson are some of the principal examples. See also [[Fermat theory]], [[natural numbers object]], [[infinitesimal number]] etc. Many statements are about the versions without the [[axiom of choice]] and so on; we like to state clean and minimal conditions when possible.

### On smoothness and generalized Lie theory

Various smoothness concepts in geometry, rarely studied in standard courses of analysis, but sometimes relevant, were studied to fair extent (and sometimes with innovations) in the $n$lab. These smoothness concepts are built using some primitive notions in rather generalized (often categorical) setups: [[Kähler differential]], [[differential form]], [[tangent space]], [[jet bundles]], resolution of diagonal, [[infinitesimal object]], [[microlinear space]], [[generalized smooth algebra]], [[tangent category]], [[cotangent complex]] as defining ingredients of various notions of smoothness and smooth spaces. Main framework to systematize in geometry similar notion studied in $n$lab is [[synthetic differential geometry]] but many other examples are also represented. Let us mention [[generalized smooth space]], [[stratifold]], [[Frölicher space]], and some graded and super analogues ([[supermanifold]], [[NQ-supermanifold]], [[integration over supermanifolds]]); some concepts of smoothness are rather algebraic, e.g. [[formal smoothness]] of [[Grothendieck]]; see also [[algebraic approaches to differential calculus]]. Special attention in $n$lab has been paid to smooth group like objects like [[Lie group]], [[Lie groupoid]] and their superanalogues and [[categorification]]s, as well as to their tangent structures like [[Lie algebroids]] and their interrelations ([[Lie theory]]: [[integration]], [[Lie integration]]).

### On geometric function theory and quantization

Some other entries are related to the conceptual and categorical understanding of Feynman [[path integral]], however so far from physical, conceptual and formal point of view only (and not of analytic theory). This is closely related to understanding various higher categorical spaces of [[sections]] in geometry and in study of sigma-models in physics. This is here called [[geometric function theory]] (cf. [[space and quantity]], [[geometric quantization]]...).

### On quantization and the geometry of differential operators

Very relevant for [[quantization]] is also the geometric study of differential operators (see [[D-geometry]], [[diffiety]]) and distributions (cf. [[microlocal analysis]]), by analysis of oscillating integrals ([[semiclassical approximation]]), [[symplectic geometry]] (esp. the geometry of [[lagrangian submanifold]]s which could often be viewed as quantum points) etc. Some of the topological properties of differential operators are studied in [[index theory]], where special role have so called [[Dirac operator]]s. Sometimes it is possible or even useful to avoid fine analysis by using the [[algebraic approaches to differential calculus]] and [[regular differential operator|differential operators]], what also makes possible some noncommutative analogues.

### On contructivism and computable analysis

* [[constructive analysis]], [[computable analysis]]

## Related concepts

* [[geometric analysis]]

* [[epsilontic analysis]]

* [[infinitesimal analysis]]

* [[matrix analysis]]

## References
 {#References}

### General
 {#ReferencesGeneral}

Textbooks accounts:

* [[Tom Apostol]], *Mathematical Analysis* (1973) [ISBN:0201002884, [pdf](http://webpages.iust.ac.ir/amtehrani/files/Addison%20Wesley%20-%20Mathematical%20Analysis%20_%20Apostol%20%285Th%20Ed%29%20%281981%29.pdf)]

* {#Rudin64} [[Walter Rudin]], _Principles of Mathematical Analysis_, McGraw-Hill (1964, 1976) [[pdf](https://web.math.ucsb.edu/~agboola/teaching/2021/winter/122A/rudin.pdf)]

* [[Eric Schechter]], _[[Handbook of Analysis and its Foundations]]_, Academic Press (1996) ([web](http://www.math.vanderbilt.edu/~schectex/ccc/))

Discussion of the history, amplifying its roots all the way back in [[Zeno's paradoxes of motion]] is in

* {#Boyer49} Carl Benjamin Boyer, _The history of the Calculus and its conceptual development_, Dover 1949

See also

* Wikipedia, _[Mathematical analysis](https://en.wikipedia.org/wiki/Mathematical_analysis)_

See also the references at _[[calculus]]_.

### Constructive analysis
 {#ReferencesConstructiveAnalysis}

The formulation of [[analysis]] in [[constructive mathematics]], hence [[constructive analysis]], was maybe initiated in

* {#Bishop} [[Errett Bishop]], _Foundations of constructive analysis._ McGraw-Hill, (1967)

together with the basic notion of [[Bishop set]]/[[setoid]]. Implementations of constructive [[real number]] analysis in [[type theory]] implemented in [[Coq]] are discussed in

* R. O'Connor, _A Monadic, Functional Implementation of Real Numbers_. MSCS, 17(1):129-159, 2007 ([arXiv:0605058](http://arxiv.org/abs/cs/0605058))

* R. O'Connor, _Certified exact transcendental real number computation in Coq_, In TPHOLs 2008, LNCS 5170, pages 246--261, 2008.

* R. O'Connor, _Incompleteness and Completeness: Formalizing Logic and Analysis in Type Theory_, PhD thesis, Radboud University Nijmegen, 2009.

* Robbert Krebbers, [[Bas Spitters]], _Type classes for efficient exact real arithmetic in Coq_ ([arXiv:1106.3448](http://arxiv.org/abs/1106.3448/))

On [[constructive analysis]] such as in [[univalent foundations]] ([[homotopy type theory]]), see the references [there](https://ncatlab.org/nlab/show/constructive+analysis#References), such as:

* {#Booij20} [[Auke Booij]], *Analysis in Univalent Type Theory* (2020) [[etheses:10411](http://etheses.bham.ac.uk/id/eprint/10411), [pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf), [[Booij-AnalysisInUF.pdf:file]]]

[[!redirects mathematical analysis]]

[[!redirects real analysis]]


analysis - contents

**[[analysis]]** ([[differential calculus|differential]]/[[integral calculus|integral]] [[calculus]], [[functional analysis]], [[topology]])

[[epsilontic analysis]]

[[infinitesimal analysis]]

[[computable analysis]]

_[[Introduction to Topology -- 1|Introduction]]_

## Basic concepts

[[triangle inequality]]

[[metric space]], [[normed vector space]]

[[open ball]], [[open subset]], [[neighbourhood]]

[[metric topology]]

[[sequence]], [[net]]

[[convergence]], [[limit of a sequence]]

[[compact space|compactness]], [[sequentially compact space|sequential compactness]]

[[differentiation]], [[integration]]

[[topological vector space]]

## Basic facts

[[continuous metric space valued function on compact metric space is uniformly continuous]]

...

## Theorems

[[intermediate value theorem]]

[[extreme value theorem]]

[[Heine-Borel theorem]]

...


Analysis Situs

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#### Algebraic topology
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This page is about the article

* [[Henri Poincaré]]:

  **Analysis Situs**

  Journal de l'École Polytechnique. (2). 1: 1–123

  (1895)

  original: [gallica:12148/bpt6k4337198/f7](https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7)

  English translation by [[John Stillwell]]: [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf), [[Stillwell_AnalysisSitus.pdf:file]]

which laid modern foundations of [[topology]] (such as introducing the notion and terminology of *[[homeomorphisms]]*) and introduced basic concepts of  [[algebraic topology]] (such as [[singular homology]]) and of [[homotopy theory]] (such as the [[fundamental group]]).

See also:

* Wikipedia, *Analysis Situs_(paper)*

* [[Peter Hilton]], *Subjective History of Homology and Homotopy Theory*, Mathematics Magazine **61** 5 (1988) 282-291 $[$[doi:10.2307/2689545](https://doi.org/10.2307/2689545)$]$

category: reference


analytic (∞,1)-functor

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{:toc}

## Idea

In the context of [[Goodwillie calculus]] an [[(∞,1)-functor]] is called _analytic_ if it behaves in [[analogy]] with an [[analytic function]] in that it its [[Goodwillie-Taylor tower]] converges to it.

## Definition

### (Co-)Cartesian cubical diagrams

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]].

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

An $n$-[[cube]] in $\mathcal{C}$, hence an [[(∞,1)-functor]] $\Box^n \longrightarrow \mathcal{C}$, is called _strongly homotopy co-cartesian_ or just _strongly co-cartesian_, if all its 2-dimensional square faces are [[homotopy pushout]] [[diagrams]] in $\mathcal{C}$.

=--

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

An $n$-[[cube]] in $\mathcal{D}$, hence an [[(∞,1)-functor]] $\Box^n \longrightarrow \mathcal{D}$, is called _homotopy cartesian_ or just _cartesian_, if its "first" object exhibits a [[homotopy limit]]-[[cone]] over the remaining objects.

=--

### Analytic functors

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

An  [[(∞,1)-functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is _stably $n$-excisive_ with constants $c$ and $\kappa$_ -- or _satisfies "condition $E_n(c,\kappa)$"_ -- if for every strongly co-Cartesian $(n)+1$-cube $X$ in $\mathcal{C}$, def. \ref{StronglyCoCartesian}, such that $X(\emptyset) \to X(s)$ is $k_s$-[[n-connected object of an (infinity,1)-topos|connective]] for $k_s \geq \kappa$ for all $s\in \{1,\cdots, n+1\}$, then  $F(X)$ is an $(n+1)$-cube in $\mathcal{D}$ such that the comparison map

$$
  F(\emptyset) \longrightarrow \underset{{K\,subcube\,of\,F(X)}\atop {K \neq \emptyset}}{\lim} F(K)
$$

(to the indicated [[homotopy limit]]) is $(-c + \sum_s k_s)$-[[n-connected object of an (infinity,1)-topos|connective]].

The functor $F$ is called _$\rho$-analytic_ if there is $q$ such that it satisfies the condition $E_n(n\rho - q,\rho + 1)$ for all $n$.

=--

(e.g. [Johnson 95, def. 1.1, def. 1.3](#Johnson95))

## Properties

### Convergence of the Goodwillie-Taylor tower

For $\rho$-analytic functors their [[Goodwillie-Taylor tower]] converges to them on $\rho$-[[n-connected object of an (infinity,1)-topos|connective objects]]. See there.

## Examples

### The identity functor on homotopy types

The identity $(\infty,1)$-functor on [[∞Grpd]] is 1-analytic, def. \ref{rhoAnalyticFunctor}. For $n = 2$ this is the statement of the [[Blakers-Massey theorem]], for $n \gt 2$ this is the statment of the [higher cubical BM-theorems](Blakers-Massey+theorem#HigherCubical).

(see e.g. [Munson-Volic 15, example 10.1.18](#MunsonVolic15))

## Related concepts

* [[excisive (∞,1)-functor]]

* [[polynomial (∞,1)-functor]]

## References

The concept is due to

* [[Tom Goodwillie]], _Calculus II: Analytic functors_, K-Theory  01/1991; 5(4):295-332. DOI: 10.1007/BF00535644

Review includes

* [[Tom Goodwillie]], section 3 of _The differential calculus of homotopy functors_, Proceedings of the International Congress of Mathematicians in Kyoto 1990, Vol. I, Math. Soc. Japan, 1991, pp. 621–6 ([article pdf](https://math.mit.edu/juvitop/old/notes/2009_Fall/goodwillie-icm.pdf), [full proceedings Vol I pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1990.2/ICM1990.2.ocr.pdf), [[GoodwillieICM1990.pdf:file]])

* {#Johnson95} Brenda Johnson, _The derivatives of homotopy theory_, Transactions of the AMS, Volume 347, 1995 ([pdf](http://www.ams.org/journals/tran/1995-347-04/S0002-9947-1995-1297532-6/S0002-9947-1995-1297532-6.pdf))

A textbook account is in

* {#MunsonVolic15} [[Brian Munson]], [[Ismar Volic]], _Cubical homotopy theory_, Cambridge University Press, 2015 [pdf](http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf)

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

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

[[!redirects analytic functor]]
[[!redirects analytic functors]]


analytic affine line

#Contents#
* table of contents
{:toc}

## Idea

The analytic affine line is a [[Berkovich analytic space]] that is the analog of the [[affine line]] in [[analytic geometry]].

It is the [[analytic spectrum]] of the [[polynomial]] ring over the given base [[field]].

## References

* [[Jérôme Poineau]], _An introduction to the Berkovich line over $\mathbb{Z}$_ ([pdf](http://www-irma.u-strasbg.fr/~poineau/Textes/LineoverZ.pdf))

For general references see at _[[analytic space]]_.

[[!redirects analytic affine lines]]

[[!redirects Berkovich line]]
[[!redirects Berkovich lines]]


analytic completion of a ring

[[!redirects analytic completion]]

#Contents#
* table of contents
{:toc}

## Idea

For $A$ a [[commutative ring]] and $p$ a [[prime number]], then there is the  [[formal completion|topological completion]] at $(p)$, the _[[p-completion]]_ $A_p^\wedge \coloneqq \underset{\leftarrow}{\lim}_n A/(p^n A)$. The _analytic completion_ is instead the [[quotient]]

$$
  A[ [ x ] ]/(x-p)A[ [ x ] ]
  \,.
$$

For $A= \mathbb{Z}$ the [[integers]] both constructions agree, up to [[isomorphism]], and yield the [[p-adic integers]]. In general though they are different.

## References

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _[[Homotopy limits, completions and localizations]]_, Lecture Notes in Mathematics, Vol 304, Springer 1972

* [[Charles Rezk]], _Analytic completion_ ([pdf](https://rezk.web.illinois.edu/analytic-paper.pdf))

[[!redirects analytic completions]]


analytic continuation

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###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[analysis]], _analytic continuation_ refers to the extension of the [[domain]] of an [[analytic function]].

In particular it often refers to the extension of an analytic function on the [[real line]] to a [[holomorphic function]] on the [[complex plane]].

## Related concepts

* [[analytically continued Chern-Simons theory]]

* [[zeta function]]

* [[zeta function regularization]]

* [[Wick rotation]], [[thermal quantum field theory]]

## References

See also

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

In [[p-adic geometry]] the need for analytic continuation motivates the [[G-topology]], see the introduction of

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

[[!redirects analytic continuations]]


analytic function

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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Analytic geometry
+--{: .hide}
[[!include analytic geometry -- contents]]
=--
=--
=--

# Analytic functions
* table of contents
{: toc}

## Idea

An _analytic function_ is a [[function]] that is locally given by a converging [[power series]].

## Definitions

Let $V$ and $W$ be [[complete space|complete Hausdorff]] [[topological vector spaces]], let $W$ be [[locally convex space|locally convex]], let $c$ be an [[element]] of $V$, and let $(a_0,a_1,a_2,\ldots)$ be an [[infinite sequence]] of [[homogeneous operator]]s from $V$ to $W$ with each $a_k$ of degree $k$.

Given an element $c$ of $V$, consider the [[infinite series]]

$$
  \sum_k a_k(x - c)
$$

(a [[power series]]).  Let $U$ be the [[interior]] of the set of $x$ such that this series converges in $W$; we call $U$ the __domain of convergence__ of the power series.  This series defines a [[function]] from $U$ to $W$; we are really interested in the case where $U$ is [[inhabited set|inhabited]], in which case it is a [[balanced neighbourhood]] of $c$ in $V$ (which is Proposition 5.3 of [Bochnak--Siciak](#BS)).

Let $D$ be any [[subset]] of $V$ and $f$ any [[continuous function]] from $D$ to $W$.  This function $f$ is __analytic__ if, for every $c \in D$, there is a power series as above with inhabited domain of convergence $U$ such that
$$ f(x) = \sum_k a_k(x - c) $$
for every $x$ in both $D$ and $U$.  (That $f$ is continuous follows automatically in many cases, including of course the finite-dimensional case.)

## Generalisation

The vector spaces $V$ and $W$ may be generalised to [[analytic manifold]]s and (more generally) [[analytic space]]s.  However, these are [[manifolds]] and [[varieties]] modelled on vector spaces using analytic [[transition functions]], so the notion of analytic function between vector spaces is most fundamental.

## Complex-analytic functions of one variable

If $W$ is a vector space over the [[complex numbers]], then we have this very nice theorem, due essentially to [[Édouard Goursat]]:

+-- {: .un_theorem}
###### Theorem
A function from $D \subseteq \mathbb{C}$ to $W$ is [[differentiable function|differentiable]] if and only if it is analytic.
=--

(Differentiability here is in the usual sense, that the difference quotient converges in $W$.)  See [[holomorphic function]] and [[Goursat theorem]].

## Related concepts

* [[analytic geometry]]

* [[holomorphic function]], [[meromorphic function]]

* [[smooth function]]

* [[analytic (∞,1)-functor]]

* [[HoTT book real numbers]]

## References

The theory of analytic function was constructed to some extent by

* M. Krasner (1940)

and in full generality by

* [[John Tate]] (1961)

Textbook accounts:

* {#GunningRossi} [[Robert C. Gunning]], [[Hugo Rossi]], *Analytic functions of several complex variables*, Prentice-Hall Inc., Englewood Cliffs (1965)

* {#BS} Jacek Bochnak and Józef Siciak, _Analytic functions in topological vector spaces_; Studia Mathematica 39 (1971);  ([pdf](http://matwbn.icm.edu.pl/ksiazki/sm/sm39/sm3916.pdf)).

* {#Schanuel} [[Stephen Schanuel]], _Continuous extrapolation to triangular matrices characterizes smooth functions_, J. Pure App. Alg. 24, Issue 1 (1982), 59–71. [web](http://www.sciencedirect.com/science/journal/00224049/24/1)

[[!redirects analytic function]]
[[!redirects analytic functions]]
[[!redirects analytic map]]
[[!redirects analytic maps]]
[[!redirects complex analytic]]


analytic geometry

> This entry is about [[geometry]] based on the study of _[[analytic functions]]_, hence about [[analytic varieties]].

> This is unrelated to "analytic geometry" in the sense of methods in the geometry of $n$-dimensional [[Euclidean space]] involving *[[coordinate]] calculations* (as opposed to [[synthetic geometry]]); which is usually combined with linear algebra taught in a geometric way. For this latter meaning see at _[[coordinate system]]_.

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

#Contents#
* table of contents
{:toc}

## Idea

In research mathematics, when one says _analytic geometry_, then "analytic" refers to [[analytic functions]] in the sense of [[Taylor expansion]] and by __analytic geometry__ one usually means the study of geometry of [[complex manifolds]]/[[complex analytic spaces]], as well as their analytic subsets, [[Stein domains]] and related notions. More generally one may replace the complex numbers by [[non-archimedean fields]] in which case one speaks of _[[rigid analytic geometry]]_.

Similarly to an [[algebraic variety]], an [[analytic variety]] is locally given as a [[zero locus]] of a [[finite set]] of [[analytic functions]], i.e. of [[holomorphic functions]] in complex analytic geometry.

A short survey can be found in a chapter of Dieudonne's _Panorama of pure mathematics_.

In addition to analytic geometry over complex numbers, there is also another formalism which allows for [[archimedean field|nonarchimedean]] [[ground fields]]. This is the subject of [[rigid analytic geometry]] or [[global analytic geometry]]. Similarly to [[schemes]], rigid analytic varieties are glued from [[Bercovich spectrum|Bercovich spectra]] of certain commutative Banach algebras, so-called [[affinoid]]s, in a certain [[Grothendieck topology]]. (See _[[analytic space]]_.) There are several variants of the formalism (e.g. due Huber). The subject is closely related to [[formal geometry]] and has its main applications in [[arithmetic geometry]] and [[representation theory]].

It is an open problem to find an appropriate analogue of rigid analytic geometry in [[noncommutative geometry]], which is supposed to play an important role in [[mirror symmetry]].

Local properties of analytic manifolds and spaces are studied in [[local analytic geometry]].

## Theorems

(...)

* [[Ostrowski's theorem]]

(...)
## Holomorphic functions of several complex variables

This section is about certain aspects of
[[holomorphic functions]] $\mathbb{C}^n \to \mathbb{C}$.

Currently it concentrates on aspects of relevance in the application to [[AQFT]], such as a version of the [edge-of-the-wedge] (http://en.wikipedia.org/wiki/Edge-of-the-wedge_theorem) theorem.

From the viewpoint of complex [[manifold]]s this is the _local_ theory that describes the situation in coordinate patches.

* [Wikipedia] (http://en.wikipedia.org/wiki/Several_complex_variables)

When we talk about holomorphic functions in the following and do not specify the domain, we will always assume that the domain is an open, simply connected subset of $\mathbb{C}^n$.

### relevance to quantum field theory (QFT)

In the [[AQFT]] formulation (actually the following description is the [[Heisenberg picture]] of quantum mechanics in a nutshell) [[selfadjoint operator]]s $A$ on a [[Hilbert space]] $\mathcal{H}$ are the [[observables]] of a physical system, while normed vectors $x, y \in \mathcal{H}$ represent the [[states]] the system can be in. The [[real number]] $\langle y, A x \rangle$ represents the [[probability]] that a system starting in state $x$ will be in state $y$ after a [[measurement]] of $A$.

In [[AQFT]] we often encounter a set of operators indexed by several complex variables $z = (z_1, z_2, ...)$ and try to deduce properties of the theory from the function $f(z) := \langle y, A(z)x \rangle$. In this way, the theory of [[holomorphic functions]] of several variables is promoted to an irreplaceable tool in quantum field theory.

### Hartogs' theorem

One striking difference of functions of several _real_ variables and several _complex_ variables is described by Hartogs' theorem on separate analyticity:

* Theorem (Hartogs): Let f be a $\mathbb{C}$-valued function defined in an open set $U \subset \mathbb{C}^n$. Suppose that f is analytic in each variable $z_j$ when the other coordinates $z_k$ are fixed. Then f is analytic as a function of all n
coordinates.

_remark_: There are no other assumptions about f necessary, it needs not to be continous or even measurable. Note that in the real case the property of being partially differentiable alone encodes nearly no information about a function.

Reference:

* Paul Garrett: [Hartogs theorem] (http://www.math.umn.edu/~garrett/m/complex/hartogs.pdf)

_relevance_: When reading [[AQFT]] literature you will often encounter the claim that given functions are holomorphic, Hartogs' theorem simplifies the task of checking these claims considerably, because you have to check the holomorphy in every single variable only.

### analogues from the one-dimensional theory
Some results remain true in the multi dimensional case.

* Identity theorem: If a holomorphic function is zero in a neighbourhood of a point, it is the zero function.

_remark_: As usual the domain is supposed to be an open, simply connected (not necessarily proper) subset of $\mathbb{C}^n$, which implies that the point of the precondition of the theorem is an interior point of the domain.

### domains of holomorphy
One of the most notably difference of the theory of _one_ complex variable and of _several_ complex variables is that the [riemann mapping theorem] (http://en.wikipedia.org/wiki/Riemann_mapping_theorem) fails in several complex variables, which is in a certain sense the reason why in several complex variables there are domains which can be enlarged such that _all_ holomorphic functions extend to the larger domain.

_handwaving_ why this is not possible in one dimension: According to the riemann mapping theorem every domain (open, simply connected proper subset of $\mathbb{C}$) is biholomorph equivalent to the open disk $E: = \{ z: |z| \lt 1 \}$, which means that the rings of holomorphic functions are isomorph, too. But the ring of holomorphic functions on E has to every point in the boundary of E a function that has a pole in this point, so that E cannot be enlarged in a way that all holomorphic functions are extentable. Therefore this applies to every domain.

Some domains in $\mathbb{C}^n$ _do_ have the property that they cannot be enlarged, and since this is an interesting property, the name **domain of holomorphy** was coined for these, and the question how they could be described was promoted to an interesting research topic in the beginning of the 20th century.

* [Wikipedia] (http://en.wikipedia.org/wiki/Domain_of_holomorphy)

#### edge of the wedge theorems
These theorems desribe situations where holomorphic functions defined on specific domains (the wedges) can be continued to holomorphic functions of larger domains.

* [Wikipedia] (http://en.wikipedia.org/wiki/Edge-of-the-wedge_theorem)

They are a valuable tool in [[AQFT]] (and were in fact discovered by one of the fathers of the theory, [Nikolay Bogolyubov] (http://en.wikipedia.org/wiki/Nikolay_Bogoliubov)).

We state here one version that will be of use to the nLab:

* theorem (edge of the wedge): Let $K := \{ z \in \mathbb{C}^n: |z| \le r \}$ be a ball in $\mathbb{C}^n$ and let $\mathcal{C} \subset \mathbb{R}^n$ be an open convex cone such that $\mathcal{C} \cap (- \mathcal{C}) \neq \emptyset $. We put $z = x + iy$ with $x, y \in \mathbb{R}^n$ and define an open domain G by $G:= \{ z = x + iy: z \in K, y \in \mathcal{C} \}$. Let $f$ be a holomorphic function in G and assume that
$$
        \lim_{y \to 0, y \in \mathcal{C}} f(x + iy)
$$
exists for all $x \in B_r \subset \mathbb{R}^n$, where $B_r$ is an open ball with radius r. (The limit may not depend on the specific sequence chosen). Then $f$ is holomorph extendable into an open region $G \cup G_0$ with
$$
        G_0 := \bigcup_{x \in B_r} \{ z: |z-x| \le \theta \cdot dist(x, \partial U) \}
$$
with $0 \lt \theta \lt 1$ a constant that is independent from $x, B_r$, and $f$.

_proof_: V.S.Vladimirov, "theory of functions of several complex variables" ([ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0125.31904&format=complete))

## Related concepts

* [[several complex variables]],

* [[rigid analytic geometry]], [[analytic spectrum]], [[analytic space]], [[Stein space]], [[Berkovich space]], [[additive analytic geometry]]

* [[G-topology]]

* [[Oka principle]]

* [[p-adic geometry]]

* [[B1-homotopy theory]]

* [[global analytic geometry]]

  * [[overconvergent global analytic geometry]]

## References
 {#References}

Course notes on ([[global analytic geometry|global]]) analytic geometry are in

* [[Frédéric Paugam]], _Global analytic geometry and the functional equation_ (2010) ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-global-analytic-geometry.pdf))

and for [[rigid analytic geometry]] in

* Kiran Sridhara Kedlaya, _Introduction to Rigid Analytic Geometry_ ([web](http://www-math.mit.edu/~kedlaya/18.727/notes.html))

* [[Brian Conrad]], _Several approaches to non-archimedean geometry_ ([pdf](http://math.stanford.edu/~conrad/papers/aws.pdf))

A gentle and modern introduction to [[complex manifolds]] that starts with an extensive exposition of the local theory is this:

* [[Daniel Huybrechts]], _Complex geometry. An introduction._ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1055.14001&format=complete))

* [[Hans Grauert]], Reinhold Remmert, _Theory of Stein spaces_, Grundlehren der Math. Wissenschaften __236__, Springer 1979, xxi+249 pp.; _Coherent analytic sheaves_, Grundlehren der Math. Wissenschaften __265__, Springer 1984. xviii+249 pp.; _Komplexe Räume_, Math. Ann. __136__, 1958, 245–318, [DOI](http://dx.doi.org/10.1007/BF01362011)

Discussion of [[Berkovich space]] analytic geometry as [[algebraic geometry]] in the general sense of [[Bertrand Toën]] and [[Gabriele Vezzosi]] is in

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

For more see the references at [[rigid analytic geometry]] and at [[analytic space]].

category: analysis, geometry


analytic geometry -- contents

**[[analytic geometry]] ([[complex analytic geometry|complex]], [[rigid analytic geometry|rigid]], [[global analytic geometry|global]])**

[[geometry]]+[[analysis]]/[[analytic number theory]]

## Basic concepts

[[analytic function]]

[[analytic space]], [[analytic variety]], [[Berkovich space]]

* [[polydisc]]

* [[affinoid algebra]], [[analytic spectrum]]

* [[complex analytic ∞-groupoid|analytic ∞-groupoid]]

[[analytification]]

## Theorems

[[GAGA]]


analytic geometry ingredients -- table

|  [[algebra|algebraic structure]] | [[group]] | [[ring]]  | [[field]] | [[vector space]]  |  [[associative algebra|algebra]] |
|---|-----------|-----------|-------------------|--------------|---|
| (submultiplicative) [[norm]]  | [[normed group]]  | [[normed ring]] | [[normed field]]  | [[normed vector space]] |  [[normed algebra]]  |
| multiplicative [[norm]] ([[absolute value]]/[[valuation]]) |  |   | [[valued field]] |  |  |
| [[complete space|completeness]]  | [[complete normed group]] | [[Banach ring]] | [[complete field]] | [[Banach vector space]] | [[Banach algebra]] |


analytic Langlands program

[[!redirects analytic langlands program]]

#Contents#
* table of contents
{:toc}

## Idea

The aim of analytic version of the [[Langlands program]] is to find a theory of analytic motives that gives a full correspondence between all (not only algebraic) automorphic representations and all analytic motives.

There are many interesting automorphic representation that have nothing very clear to do with algebraic varieties (Mass forms, etc...). They should however be related to natural analytic motivic coefficients.

## References

* [[Pavel Etingof]], [[Edward Frenkel]], [[David Kazhdan]], *A general framework for the analytic Langlands correspondence* [[arXiv:2311.03743](https://arxiv.org/abs/2311.03743)]

[[!redirects analytuc Langlands correspondence]]


analytic manifold

#Contents#
* table of contents
{:toc}

## Idea

An **analytic manifold** is a [[manifold]] with [[analytic function|analytic]] transition functions over some [[field]].  The most widely studied are real-analytic and complex analytic manifolds.

Analytic manifolds are studied in [[analytic geometry]].

## References

* L. Schwarz, _Lectures on complex analytic manifolds_ (1955) ([pdf](http://www.math.tifr.res.in/~publ/ln/tifr04.pdf))

[[!redirects analytic manifolds]]

[[!redirects complex analytic manifold]]


analytic Markov's principle

#Contents#
* table of contents
{:toc}

## Definition ##

The **analytic Markov's principle** states that the [[pseudo-order]] on the [[Dedekind real numbers]] is a [[stable relation]]: for all [[real numbers]] $r \in \mathbb{R}$ and $s \in \mathbb{R}$, $\neg \neg (r \lt s)$ implies $r \lt s$.

This is equivalent to the usual formulation of the analytic Markov's principle, which says that for all real numbers $x \in \mathbb{R}$, $\neg (x \leq 0)$ implies $0 \lt x$. For if we take $x = s - r$, this becomes $\neg (s - r \leq 0)$ implies $0 \lt s - r$, and by the order and arithmetic properties of the real numbers, this is equivalent to $\neg (s \leq r)$ implies $r \lt s$, which is the same as $\neg \neg (r \lt s)$ implies $r \lt s$.

Other equivalent statements include that the [[tight apartness relation]] on the Dedekind real numbers is a [[stable relation]].

The analytic Markov's principle makes sense for any [[ordered local ring|ordered]] [[local Artinian algebra|local Artinian $\mathbb{R}$-algebra]] as well, where the relation $\lt$ is in general only a [[strict weak order]] instead of a [[pseudo-order]], the [[preorder]] $\geq$ is not a [[partial order]], and the equivalence relation $a \approx b$ derived from the preorder holds if and only if $a - b$ is [[nilpotent]]. The quotient of the Weil $\mathbb{R}$-algebra by its [[nilradical]] is the Dedekind real numbers satisfying the analytic Markov's principle.

## See also ##

* [[Markov's principle]]

* [[axiom of real cohesion]]

* [[intermediate value theorem]]

## References ##

* [[Mike Shulman]], *Brouwer’s fixed-point theorem in real-cohesive homotopy type theory*, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 ([arXiv:1509.07584](https://arxiv.org/abs/1509.07584), [doi:10.1017/S0960129517000147](https://doi.org/10.1017/S0960129517000147))


analytic monad

## Idea

Analytic monads are [[monads]] on [[Set]] that correspond to [[operads]] in [[Set]].

## Definition

More precisely, an [[operad]] $O$ in [[Set]] induces a [[monad]] $T$
on [[Set]]:
$$T(S)=\coprod_{n\ge0} O_n \times_{\Sigma_n} S^n.$$

Such a monad $T$ is equipped with a canonical weakly [[cartesian natural transformation]] to the [[monad]] $Sym$ arising from the [[commutative operad]].

[[finitary monad|Finitary monads]] correspond to [[algebraic theories]], and the analytic monads correspond to algebraic theories that are "linear regular", that is to say, they can be presented using only equations where the same variables appear on both sides and exactly once on each side (see e.g. [SzawielZawadowski](#SzawielZawadowski)).

## Properties

A theorem of [Joyal](#Joyal) states that
there is a [[monoidal equivalence]]
between the [[monoidal category]] of [[endofunctors]] $Set\to Set$
that admits a weakly [[cartesian natural transformation]] to $Sym$
and the [[monoidal category]] of [[species]],
i.e., symmetric sequences in [[Set]] with the [[substitution product]].

In particular, the category of analytic [[monads]] on [[Set]]
is equivalent to the category of [[operads]] in [[Set]].

## The colored case

The correspondence carries over to [[colored operads]] (with a [[set]]
of colors $C$)
if we use the [[slice category]] $Set/C$ instead of [[Set]].

## The nonsymmetric case

A similar correspondence can be established for [[nonsymmetric operads]],
except that we must include the data of a _cartesian_ (not weakly cartesian) transformation to the [[monad]] of the [[associative operad]],
which is no longer unique.

See Example 4.2.14 in [Leinster's book](#Leinster).

## The homotopical case

The correspondence generalizes to [[(∞,1)-categories]], with
some statements becoming more elegant.
See [Gepner–Haugseng–Kock](#GHK).

## Related concepts

* [[species]]

* [[operad]]

* [[monad]]

## References

* {#Joyal} [[André Joyal]], _Foncteurs analytiques et espèces de structures_, Combinatoire énumérative (Montréal/Québec, 1985), Lecture Notes in Mathematics 1234 (1986), 126-159.  [doi](http://dx.doi.org/10.1007/bfb0072514).

* [[Mark Weber]], _Generic morphisms, parametric representations and weakly Cartesian monads_, Theory Appl. Categ. 13 (2004), 191–234.

* [[Tom Leinster]], _[[Higher Operads, Higher Categories]]_, London Mathematical Society Lecture Note Series 298 (2004), [doi](http://dx.doi.org/10.1017/cbo9780511525896).

* {#GHK} [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], _∞-Operads as Analytic Monads_, [arXiv:1712.06469](https://arxiv.org/abs/1712.06469).

* {#SzawielZawadowski} [[Stanislaw Szawiel]] and [[Marek Zawadowski]], _Theories of analytic monads_, Math. Struct. Comp. Sci. 24 (2014). [arxiv:1204.2703](https://arxiv.org/pdf/1204.2703.pdf)


analytic motives

Analytic motives give a common generalization of classical homotopy theory (Ayoub's analytic motives over C), algebraic motivic homotopy theory (strict analytic motives over Z with the trivial non-archimedean norm), and Ayoub's non-archimedean motives.

The basic idea is to extend motivic homotopy theory to a general base Banach ring, by replacing the affine line by the overconvergent unit disc.

Analytic stable homotopy theory gives a natural setting to define homotopy invariant cohomologies, that are particularly interesting when one works over an extension of the field of rational numbers. However, they seem to be less well adapted to the study of torsion phenomena in de Rham type cohomology theories over an integral base, because homotopy invariance is not available in this situation, and imposing it artificially kills the p-torsion information in caracteristic p.

References:

Joseph Ayoub: Betti realization of motives and motives for rigid analytic varieties.

[[Frederic Paugam]]: Overconvergent global analytic geometry (in preparation).


analytic motivic homotopy theory

#Contents#
* table of contents
{:toc}

## Idea

Analytic motivic homotopy theory is an analytic generalization of [[motivic homotopy theory]] that gives back classical homotopy theory when one works with analytic varieties over $\mathbb{C}$ and motivic homotopy theory when one works with strict analytic varieties over a ring equipped with its trivial norm.

The proper setting for analytic motivic homotopy theory of quasi-projective varieties seems to be the setting of [[logarithmic motivic homotopy theory]].

## Related subjects

[[motivic homotopy theory]]

[[logarithmic motivic homotopy theory]]

## References

[[Joseph Ayoub]] _La realisation de Betti et les six opérations_ (defines complex analytic motives and shows that they are equivalent to classical homotopy types)

[[Joseph Ayoub]] _Motives of rigid analytic varieties_ (defines rigid motives and uses them to study the functor of vanishing cycles on a characteristic $0$ "trait").

[[Frédéric Paugam]] _Overconvergent global analytic geometry_ (defines motives over an arbitrary base Banach ring)


analytic number theory

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#Contents#
* table of contents
{:toc}

## Idea

The branch of [[number theory]] that uses [[analysis]]. Related to [[global analytic geometry]] as [[number theory]] is to [[arithmetic geometry]].

## References

* Wikipedia, _[Analytic number theory](http://en.wikipedia.org/wiki/Analytic_number_theory)_


analytic philosophy

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#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Analytic philosophy_ is a school of [[philosophy]] emphasizing clarity of argument, [[formal logic]], and aiming for strong connections with natural [[sciences]] such as [[physics]].

> Analytic philosophy is characterized above all by the goal of clarity, the insistence on explicit argumentation in philosophy, and the demand that any view expressed be exposed to the rigours of critical evaluation and discussion by peers. (European Society for Analytic Philosophy, homepage of website <http://www.dif.unige.it/esap>; accessed 18 October 2011)

Analytic philosophy to some extent defined itself, via people like [[Bertrand Russell]] in a movement known as the "revolt again idealism", in opposition to forms of German idealism, notably the [[objective idealism]] of [[Georg Hegel]] as expressed in his _[[Science of Logic]]_. See at _[Perception of Hegel's Naturphilosophie](Georg+Hegel#PerceptionOfHegelsPhilosophy)_ for more on this.

Specifically, analytic philosophy aims to detect the underlying logical form of propositions and analyse the concepts they rely upon. Since the surface grammar of natural language is taken to be often misleading, this work of analysis is often done in terms of formal languages such as [[first-order logic]] or [[modal logic]].

Examples:

* Where it might appear that holding the belief that 'Unpunctuality is reprehensible' commits one to the existence of something denoted by 'unpunctuality', rewording the sentence, as Gilbert Ryle did, as 'Whoever is unpunctual deserves that other people should reprove him for being unpunctual' avoids this commitment. As a further step, the meaning of 'X deserves Y' would now be a candidate for analysis.

* [[Martin Heidegger|Heidegger]] in _What is Metaphysics?_ passes from his claims "What is to be investigated is being only and — nothing else; being alone and further — nothing; solely being, and beyond being — nothing" to a discussion of what he takes to be their subject by asking "What about this Nothing?". For Carnap this was just to be have misled by the grammar which appear to make 'Nothing' a subject, but is to be properly analyzed in terms of negation and universal quantification.

So convinced was Russell of the power of the then new first-order logic that he wrote:

>The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds
of problems may be capable of solution, and what kinds are beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies, but must command the assent of all who are competent to form an opinion. (Bertrand Russell, '[[Logic as the Essence of Philosophy]]',
1914)

See also [SEP: Conceptions of Analysis in Analytic Philosophy](http://plato.stanford.edu/entries/analysis/#6).

**Remark.** While indeed [[first-order logic]] has little to no resemblance to [[Science of Logic|Hegel's logic]], for its refinement by [[type theory]] in the guise of [[modal type theory]] the situation is quite different. [[William Lawvere]] has argued that key aspects of [[Science of Logic|Hegel's logic]] do have a useful formalization in [[modal type theory]] (Lawvere mostly considered this in the corresponding [[categorical semantics]]). Lawvere's formalization of Hegel's concepts of _[[unity of opposites]]_, _[[Aufhebung]]_, _[[category of being]]_ etc. lead to non-trivial [[theorems]] in the [[foundations of mathematics]] and for the [[geometry of physics]].

## References

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


analytic ring

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## Definition

\begin{definition}([ScholzeLCM](#ScholzeLCM), Def. 7.1)

In [[condensed mathematics]], a _pre-analytic ring_ is a [[condensed ring]] $\underline{A}$ together with a [[functor]] $S\mapsto\mathcal{A}[S]$ from [[extremally disconnected topological spaces]] to $\underline{\mathcal{A}}$-[[modules]] in [[condensed abelian groups]] taking finite [[disjoint unions]] to [[products]], and a [[natural transformation]] $S \to\mathcal{A}[S]$.
\end{definition}

\begin{definition}([ScholzeLCM](#ScholzeLCM), Def. 7.4)
An _analytic ring_ is a pre-analytic ring $\mathcal{A}$ such that for any [[chain complex]] $C$ of $\underline{\mathcal{A}}$-modules in [[condensed abelian groups]] such that all $C_{i}$ are [[direct sums]] of objects of the form $\mathcal{A}[T]$ for varying [[extremally disconnected topological space|extremally disconnected]] $T$, the map
$$R\underline{\Hom}_{\underline{\mathcal{A}}}(\mathcal{A}[S],C)\to R\underline{\Hom}_{\underline{\mathcal{A}}}(\mathcal{A}[S],C)$$
of [[condensed abelian groups]] is an [[isomorphism]] for all extremally disconnected $S$.

\end{definition}

## Examples

The following examples come from [ScholzeLCM](#ScholzeLCM), Examples 7.3, as examples of pre-analytic rings. They are shown to be analytic rings as well later on in the same reference.

* The analytic ring $\mathbb{Z}_{\square}$ is given by $\underline{\mathcal{A}}=\mathbb{Z}$ and $S\mapsto \mathbb{Z}[S]^{\square}$ (in the notation of [[solid abelian group]]).

* For $A$ a discrete ring, the analytic ring $(A,\mathbb{Z})_{\square}$ is given by $\underline{\mathcal{A}}=A$ as a condensed ring and $S\mapsto \mathbb{Z}_{\square}[S]\otimes_{\mathbb{Z}} A$.

* For $A$ a finitely generated $\mathbb{Z}$-algebra, the condensed ring $A_{\square}$ is given by $\underline{\mathcal{A}}=A$ as a condensed ring and $S\mapsto
  A_{\square}[S]
  \;\coloneqq\;
  \underset
    {\underset{i}{\leftarrow}}
    {\lim}
  \;
  A\big[S_{i}\big]
  \,
$.

## Related concepts

* [[condensed mathematics]]

* [[solid module]]

## References

* {#ScholzeLCM} [[Peter Scholze]], _Lectures on condensed mathematics_, [pdf](https://www.math.uni-bonn.de/people/scholze/Condensed.pdf)


analytic space

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# Analytic spaces
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## Idea

Analytic spaces are [[spaces]] that are locally modeled on [[Isbell duality|formal duals]] of sub-[[associative algebra|algebras]] of [[power series]] algebras on elements with certain [[convergence]] properties with respect to given [[seminorms]]. This is in contrast to ([[formal geometry|formal]]) [[algebraic spaces]] (([[formal scheme|formal]]) [[schemes]]) where no [[convergence]] properties are considered.

In [[complex number|complex]] [[analytic geometry]] analytic spaces -- [[complex analytic space]] -- are a vast generalization of complex [[analytic manifolds]] and are usually treated in the formalism of [[locally ringed spaces]]. In this case the _[[GAGA]]-principle_ closely relates [[complex analytic geometry]] with [[algebraic geometry]] over the [[complex numbers]].

In the case of non-archimedean ground field, the topology of the affine space is totally disconnected what requires different approach than, say, over complex numbers. This leads to several variants like [[rigid analytic geometry]], [[Berkovich space]]s. [[Huber's adic spaces]] and so on.

## Related entries

* [[analytic geometry]],

  * [[complex analytic space]], [[analytic variety]], [[Berkovich space]]

* [[Banach analytic space]]

* [[B1-homotopy theory]]

* [[perfectoid space]]

## References

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

Discussion for [[complex analytic spaces]] and [[Stein spaces]] is in

* [[Hans Grauert]], [[Reinhold Remmert]], _Theory of Stein spaces_, Grundlehren der Math. Wissenschaften __236__, Springer 1979, xxi+249 pp.; _Coherent analytic sheaves_, Grundlehren der Math. Wissenschaften __265__, Springer 1984. xviii+249 pp.; _Komplexe Räume_, Math. Ann. __136__, 1958, 245–318, [DOI](http://dx.doi.org/10.1007/BF01362011)

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analytic spectrum

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## Idea

The concept of _analytic spectrum_ is a realization of the concept of [[spectrum (geometry)]] in the context of [[non-archimedean analytic geometry]]. Given an [[affinoid algebra]] $A$ over a [[non-archimedean field]], then the concept of [[spectrum of a commutative ring]] whose points are [[prime ideals]]/[[maximal ideals]] does not produce a sensible space that admits [[analytic geometry]]. Rather, instead of regarding points of the spectrum as ring homomorphisms to $A \to \mathbb{R}$, the analytic spectrum instead takes points to be multiplicative [[seminorms]] ${\vert -\vert} \colon A \to \mathbb{R}_{\geq 0}$ bounded by the norm on the given field.

If $Spec_{an}(A)$ is the set of all such multiplicative seminorms, then for $x \in Spec_{an}(A)$ a point one writes the corresponding seminorm as ${\vert - \vert}_x$ and thinks of it as being the norm on the function algebra on $Spec(A)$ which is given by "evaluating functions at $x$ and then applying the field norm to that".

One turns this $Spec_{an}(A)$ into a [[topological space]] in the usual way by choosing the weakest topology such that under this assignment the original elements of $A$ become [[continuous function]] on $Spec_{an}(A)$.

Globalizing this analytic spectrum construction leads to the concept of [[Berkovich analytic space]].

## Definition

For $A$ a [[normed ring]], its _analytic spectrum_ or _Berkovich spectrum_ $Spec_an A$ is the set of all non-zero multiplicative [[seminorms]] on $A$, regarded as a [[topological space]] when equipped with the weakest topology such that all [[functions]]

$$
  Spec_{an} A \to \mathbb{R}_+
$$

of the form

$$
  x \mapsto {\vert x(a)\vert}
$$

for $a \in A$ are [[continuous function|continuous]].

If $A$ is equipped with the structure of a [[Banach ring]], one takes the _[[bounded map|bounded]]_ multiplicative seminorms.

So a point in the analytic spectrum of $A$ corresponds to a non-zero [[function]]

$$
  {\Vert -\Vert} : A \to \mathbb{R}
$$

to the [[real numbers]], such that for all $x, y \in A$

1. ${\Vert x \Vert} \geq 0$;

1. ${\Vert x y \Vert} = {\Vert x \Vert} {\Vert y \Vert}$;

1. ${\Vert x + y \Vert} \leq {\Vert x \Vert} + {\Vert y \Vert}$

and boundedness means that there exists $C \gt 0$ such that for all $x \in A$

$$
  {\Vert x\Vert} \leq C {\vert x \vert}_A
  \,.
$$

(e.g. [Berkovich 09, def. 1.2.3](#Berkovich09))

## Examples

### Affine line

For $k$ a [[field]] and $k[T]$ the [[polynomial ring]] over $k$ in one generator,

$$
  \mathbb{A}_k := Spec_{an} k[T]
$$

is the [[analytic affine line]] over $k$.

If $k = \mathbb{C}$, then $\mathbb{A}_k = \mathbb{C}$ is the ordinary [[complex plane]].

## Related concepts

* [[spectrum (geometry)]]

* [[analytic geometry]], [[analytic space]]

* [[p-adic geometry]]

## References

### Original

The notion originates in

* {#Berkovich90} [[Vladimir Berkovich]], _Spectral theory and analytic geometry over non-Archimedean fields_, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp.

### Expositions and Lecture notes

Introductory exposition of the Berkovich analytic spectrum includes

* [[Sarah  Brodsky]], _Non-archimedean geometry_, brief lecture notes, 2012 ([pdf](http://math.berkeley.edu/~sstich/MAT_274/Math_274_3_Feb_2012.pdf))

* [[Scott Carnahan]], _Berkovich spaces I_ ([web](http://sbseminar.wordpress.com/2007/09/18/berkovich-spaces-i/))

* {#Poineau07} [[Jérôme Poineau]], _Global analytic geometry_, pages 20-23 in EMS newsletter September 2007 ([pdf](http://www.ems-ph.org/journals/newsletter/pdf/2007-09-65.pdf))

* [[Frédéric Paugam]], section 2.1.4 of _Global analytic geometry and the functional equation_ (2010) ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-global-analytic-geometry.pdf))

* {#Berkovich09} [[Vladimir Berkovich]], section 1 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))

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