pro-object

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#### Category theory
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[[!include category theory - contents]]
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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\section{Idea}

A **pro-object** of a [[category]] $C$ is a "formal [[filtered category|cofiltered]] [[limit]]" of [[objects]] of $C$.

The category of pro-objects of $C$ is written $pro$-$C$. Such a category is sometimes called a __pro-category__, but notice that that is *not* the same thing as a pro-object in [[Cat]].

"Pro" is short for "projective"
([[projective limit]] is an older term for [[limit]]). It is in contrast to "ind" in the dual notion of [[ind-object]], standing for "inductive", (and corresponding to [[inductive limit]], the old term for [[colimit]]).  In some (often older) sources, the term 'projective system' is used more or less synonymously for pro-object.

The definition of arrows in the category of pro-objects in $\mathcal{C}$ is perhaps more intuitive in the dual case of [[ind-object|ind-objects]] (pro-objects in $C^{op}$), where it can be seen as stipulating that the objects of $C$ are [[finitely presentable object|finitely presentable]] in $ind$-$C$.

## Definition

### As formal cofiltered limits
  {#AsFormalCofilteredLimits}

Let $\mathcal{C}$ be a [[small category]]. We write

\[
  \label{YonedaEmbedding}
  \mathcal{C}^{op}
  \xhookrightarrow{\;\;\; y \;\;\;}
  Func(\mathcal{C}, Sets)
\]

for the [[Yoneda embedding]] of its [[opposite category]], which we will often regard through its [[opposite functor]]

\[
  \label{OppositeYonedaEmbedding}
  \mathcal{C}
  \xhookrightarrow{\;\;\; y^{op} \;\;\;}
  Func(\mathcal{C}, Sets)^{op}
  \,.
\]

This is still a [[full subcategory]]-inclusion, now of $\mathcal{C}$ itself.

\begin{definition}\label{CategoryOfProObjects}
**(category of pro-objects)** \linebreak

The *category of pro-objects of $\mathcal{C}$*, according to [Grothendieck 1960, Section 2](#Grothendieck1960), is the [[full subcategory]] of the [[functor category]] $Func(\mathcal{C}, Sets)^{op}$ (eq:OppositeYonedaEmbedding)

\[
  \label{ProCAsFullSubcategoryOfOppositeOfCategoryOfPresheaves}
  Pro(\mathcal{C})
  \xhookrightarrow{\;\;}
  Func(\mathcal{C}, Sets)^{op}
\]

on those functors which are [[cofiltered limits]] of [[representable functors]] under the opposite Yoneda embedding (eq:OppositeYonedaEmbedding), hence of the form

\[
  \label{FilteredLimitOfRepresentablesInOppositeOfPresheafCategory}
  \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{\lim}
  y^{op}(X_i)
  \;\;
  \coloneqq
  \;\;
  \underset{\longleftarrow}{\lim}
  \big(
    \mathcal{I}
    \xrightarrow{\;X_\bullet\;}
    \mathcal{C}
    \xrightarrow{\;y^{op}\;}
    Func(\mathcal{C}, Sets)^{op}
  \big)
  \,,
\]

for $\mathcal{I}$ a [[cofiltered category]]. These objects of $Pro(\mathcal{C})$ are also called the *pro-objects of $\mathcal{C}$*.
\end{definition}

\begin{remark}
Since $Func(\mathcal{C}, Sets)^{op}$ (eq:OppositeYonedaEmbedding) is the [[free completion]] of $\mathcal{C}$, we may think of $Pro(\mathcal{C})$ (Def. \ref{CategoryOfProObjects}) as the *free completion of $\mathcal{C}$ under [[cofiltered limits]].* See also at *[[pro-representable functor]]*.
\end{remark}

\begin{remark}\label{BetweenFilteredLimitsAndFilteredColimits}
Under the identification of the [[objects]] of a category with those of its [[opposite category]], the  pro-objects (eq:FilteredLimitOfRepresentablesInOppositeOfPresheafCategory) are equivalently [[filtered colimits]] of [[presheaves]]:

$$
  \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{\lim}
  y^{op}(X_i)
  \;\simeq\;
  \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim}
  y(X^{op}_i)
  \;\;
  \coloneqq
  \;\;
  \underset{\longrightarrow}{\lim}
  \big(
    \mathcal{I}^{op}
    \xrightarrow{\;X^{op}_\bullet\;}
    \mathcal{C}^{op}
    \xrightarrow{\;y\;}
    Func(\mathcal{C}, Sets)
  \big)
  \,,
$$

since [[generalized the|the]] [[limit]] of a [[functor]] is the [[colimit]] of its [[opposite functor]].

\end{remark}

\begin{remark}\label{ProObjectAsFormalDualOfIndObject}
The category $Pro(\mathcal{C})$ is the [[opposite category]] of that of [[ind-objects]] in the opposite catgegory of $\mathcal{C}$ (e.g. [Kashiwara-Schapira 06, p. 138](#KashiwaraSchapira06)):
$
  Pro(\mathcal{C})
    \simeq
  (Ind(\mathcal{C}^{op}))^{op}
  \,.
$

\end{remark}

\begin{lemma}\label{ProObjectsAsFunctors}
  The pro-objects regarded as functors (eq:FilteredLimitOfRepresentablesInOppositeOfPresheafCategory)
  are objectwise on $c \in \mathcal{C}$ given by
  $$
    \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim}
    y^{op}(X_i)
    \;\;
    \colon
    \;\;
    c
    \;\mapsto\;
    \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim}
    \mathcal{C}(X_i, c)
    \,.
  $$

\end{lemma}
\begin{proof}
This is the composite of the following sequence of [[natural bijections]]:
$$
  \begin{aligned}
  \Big(
  \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{\lim}
  y^{op}(X_i)
  \Big)
  (c)
  &
  \;\simeq\;
  \Big(
  \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim}
  y(X^{op}_i)
  \Big)
  (c)
  \\
  &
  \;\simeq\;
  \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim}
  \Big(
    \big(y(X^{op}_i)\big)
    (c)
  \Big)
  \\
  &
  \;\simeq\;
  \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim}
  \;
  Func(\mathcal{C},Sets)
  \big(
    y(c), \, y(X_i)
  \big)
  \\
  &
  \;\simeq\;
  \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim}
  \mathcal{C}^{op}(c, X_i)
  \\
  &
  \;\simeq\;
  \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{\lim}
  \mathcal{C}(X_i, c)
  \mathrlap{\,,}
  \end{aligned}
$$
where

* the first step is Remark \ref{BetweenFilteredLimitsAndFilteredColimits};

* the second step uses that [[colimits of presheaves are computed objectwise]];

* the third step is the [[Yoneda lemma]];

* the fourth step is the [[Yoneda embedding]] (eq:YonedaEmbedding);

* the fifth step is the definition of the [[opposite category]].

\end{proof}

\begin{lemma}\label{HomSetsOfProObjectsAsLimitsOfColimitsOfHomoSetsInC}
**([[hom-sets]] between [[pro-objects]] as limits of colimits of hom-sets in $\mathcal{C}$)** \linebreak
The [[hom-sets]] in $Pro(\mathcal{C})$ (Def. \ref{CategoryOfProObjects})
are in [[natural bijection]] with [[limits]] of [[colimits]] of [[hom-sets]] in $\mathcal{C}$, as follows:

$$
  Pro(\mathcal{C})
  \Big(
    \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim}
    y^{op}(X_i),
    \,
    \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim}
    y^{op}(Y_j),
  \Big)
  \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim}
  \Big(
    \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim}
    \mathcal{C}
    \big(
      X_i,
      \,
      Y_j
    \big)
  \Big)
$$

\end{lemma}

\begin{proof}
This is the composite of the following sequence of [[natural bijections]]:
$$
  \begin{aligned}
  Pro(\mathcal{C})
  \Big(
    \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim}
    y^{op}(X_i),
    \,
    \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim}
    y^{op}(Y_j),
  \Big)
  &
  \;\simeq\;
  Func(\mathcal{C},Sets)^{op}
  \Big(
    \underset{\underset{i \in \mathcal{I}}{\longleftarrow}}{lim}
    y^{op}(X_i),
    \,
    \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim}
    y^{op}(Y_j),
  \Big)
  \\
  &
  \;\simeq\;
  Func(\mathcal{C},Sets)
  \Big(
    \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{lim}
    y(Y^{op}_j),
    \,
    \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim}
    y(X^{op}_i),
  \Big)
  \\
  & \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim}
  \;
  Func(\mathcal{C},Sets)
  \Big(
    y(Y^{op}_j),
    \,
    \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim}
    y(X^{op}_i),
  \Big)
  \\
  & \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim}
  \Big(
    \big(
      \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim}
      y(X^{op}_i)
    \big)
    \big(
      y(Y^{op}_j)
    \big)
  \Big)
  \\
  & \;\simeq\;
  \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{lim}
  \Big(
    \underset{\underset{i \in \mathcal{I}}{\longrightarrow}}{lim}
    \mathcal{C}
    \big(
      X_i,
      \,
      Y_j
    \big)
  \Big)
  \end{aligned}
$$

Here

* the first step is the [[fully faithful functor|fully faithfulness]] of the defining [[full subcategory]]-inclusion (eq:ProCAsFullSubcategoryOfOppositeOfCategoryOfPresheaves);

* the second step is the definition of the [[opposite category]], using Remark \ref{BetweenFilteredLimitsAndFilteredColimits};

* the third step uses that [[hom-functors preserve limits]], which means that $\mathcal{K}^{op}  \xrightarrow{\;\mathcal{K}(-,k)\;} Sets$ takes *[[colimits]] in $\mathcal{K}$* to *limits in [[Sets]], here for $\mathcal{K} = Func(\mathcal{C},Sets)$;

* the fourth step is the [[Yoneda lemma]];

* the fifth step is Lemma \ref{ProObjectsAsFunctors}.

\end{proof}

\linebreak

### Via spans
 {#DefinitionViaSpans}

\begin{rmk} We can give an explicit description of the arrows of pro-$\mathcal{C}$ as follows. First, for pro-objects $F: \mathcal{D} \rightarrow \mathcal{C}$ and $G: \mathcal{E} \rightarrow \mathcal{C}$ and for any object $e$ of $\mathcal{E}$, we introduce a relation $\sim$ on arrows with target $G(e)$ which identifies an arrow $f: F(d) \rightarrow G(e)$ with an arrow $f': F(d') \rightarrow G(e)$ for objects $d$ and $d'$ of $\mathcal{D}$ and an object $e$ of $\mathcal{E}$, if there is an object $d''$ of $\mathcal{D}$, an arrow $g: d'' \rightarrow d$ of $\mathcal{D}$, and an arrow $g': d'' \rightarrow d'$ of $\mathcal{D}$, such that $f \circ F(g) = f' \circ F(g')$.

This relation $\sim$ is in fact an [[equivalence relation]]. Symmetry is obvious. Reflexivity is immediately demonstrated using the identity arrows of $\mathcal{D}$. Transitivity would not hold for an arbitrary category, but follows from the assumption that $\mathcal{D}$ is cofiltered. Indeed, suppose that we have a zig-zag in $\mathcal{D}$ as follows.

\begin{centre}
  \begin{tikzcd}
          & d'_{0} \ar[dl, "g_{0}", swap] \ar[dr, "g_{1}"] & & d'_{1} \ar[dl, "g_{2}", swap] \ar[dr, "g_{3}"] &       \\
    d_{0} & & d_{1} & & d_{2}
  \end{tikzcd}
\end{centre}

The fact that $\mathcal{D}$ is cofiltered ensures that there is an object $d''$ of $\mathcal{D}$ fitting into the following diagram.

\begin{centre}
  \begin{tikzcd}
          & & d'' \ar[dl, "g'_{0}", swap] \ar[dr, "g'_{1}"] & & \\
          & d'_{0} \ar[dl, "g_{0}", swap] \ar[dr, "g_{1}"] & & d'_{1} \ar[dl, "g_{2}", swap] \ar[dr, "g_{3}"] &       \\
    d_{0} & & d_{1} & & d_{2}
  \end{tikzcd}
\end{centre}

Suppose that we have arrows $f_{0} : F(d_{0}) \rightarrow G(e)$, $f_{1}: F(d_{1}) \rightarrow G(e)$, and $f_{2}: F(d_{2}) \rightarrow G(e)$ such that $g_{0}$ and $g_{1}$ exhibit that $f_{0} \sim f_{1}$, and such that $g_{2}$ and $g_{3}$ exhibit that $f_{1} \sim f_{2}$. Then

$$
\begin{aligned}
f_{0} \circ F(g_{0} \circ g'_{0}) &= f_{0} \circ F(g_{0}) \circ F(g'_{0}) \\
  &= f_{1} \circ F(g_{1}) \circ F(g'_{0}) \\
  &= f_{1} \circ F(g_{2}) \circ F(g'_{1}) \\
  &= f_{2} \circ F(g_{3}) \circ F(g'_{1}) \\
  &= f_{2} \circ F(g_{3} \circ g'_{1}).
\end{aligned}
$$

This exhibits that $f_{0} \sim f_{2}$, as required.

With this equivalence relation $\sim$ to hand, we can give our explicit description of the arrows of pro-$\mathcal{C}$: an arrow of pro-$\mathcal{C}$ from a pro-object $F: \mathcal{D} \rightarrow \mathcal{C}$ to a pro-object $G: \mathcal{E} \rightarrow \mathcal{C}$ can be taken to be a set $\left\{ f_{e} : F\left(d_{e}\right) \rightarrow G(e) \right\}$ of arrows of $\mathcal{C}$, one for every object $e$ of $\mathcal{E}$, such that, for every arrow $g: e \rightarrow e'$ of $E$, $G(g) \circ f_{e} \sim f_{e'}$.

In other words: a set $\left\{ f_{e} : F\left(d_{e}\right) \rightarrow G(e) \right\}$ of arrows of $\mathcal{C}$, one for every object $e$ of $\mathcal{E}$, such that, for every arrow $g: e \rightarrow e'$ of $E$, there is an object $d$ of $\mathcal{D}$, an arrow $g_{e} : d \rightarrow d_{e}$ of $\mathcal{D}$, and an arrow $g_{e'}: d \rightarrow d_{e'}$ of $\mathcal{D}$ such that $G(g) \circ f_{e} \circ F(g_{e}) = \circ f_{e'} \circ F(g_{e'})$.

Two such sets $\left\{ f_{e} \right\}_{e \in Ob(\mathcal{E})}$ and $\left\{ f'_{e} \right\}_{e \in Ob(\mathcal{E})}$ are equal, i.e. define the same arrow from $F$ to $G$, if $f_{e} \sim f'_{e}$ for every object $e$ of $\mathcal{E}$.

\end{rmk}

\section{Characterisations}

In some cases, pro-objects in a category $\mathcal{C}$ can be viewed as actual limits in a certain category. We prove here some results of this kind.

\begin{prpn} \label{PropositionCategoriesEquivalentToProC} Let $\mathcal{C}$ be a category, and let $\mathcal{A}$ be a category with [[cofiltered limit|cofiltered limits]]. Suppose that we have a [[fully faithful functor]] $i: \mathcal{C} \rightarrow \mathcal{A}$ which lands in [[compact object|cocompact objects]]. Then $lim_{\mathcal{A}} \circ (i \circ) : Pro(\mathcal{C}) \to \mathcal{A}$ is fully faithful, and hence defines an equivalence onto its image.

\end{prpn}

\begin{proof} Let $F:\mathcal{D} \to \mathcal{C}$ and $G:\mathcal{E} \to \mathcal{C}$ be pro-objects. We then have a sequence of (natural) bijections:

$Hom_{Pro(\mathcal{C})}(F,G)$

$= \underset{e:\mathcal{E}}{lim} \, \underset{d:\mathcal{D}}{colim} \, Hom_{\mathcal{C}}(F d, G e)$

$\cong \underset{e:\mathcal{E}}{lim} \, \underset{d:\mathcal{D}}{colim} \, Hom_{\mathcal{A}}(i (F d), i (G e))$

$\cong \underset{e:\mathcal{E}}{lim} \, Hom_{\mathcal{A}}(\underset{d:\mathcal{D}}{lim} (i \circ F) d, i (G e))$

$\cong Hom_{\mathcal{A}}(\underset{d:\mathcal{D}}{lim} (i \circ F) d, \underset{e:\mathcal{E}}{lim} (i \circ G) e)$

$= Hom_{\mathcal{A}}(lim (i \circ F), lim (i \circ G))$

With these bijections being by definition of pro-object morphisms, fully faithfulness, cocompactness of $i (G e)$, definition of a limit, and definition respectively.
\end{proof}

\begin{example} \label{ExampleProfiniteGroupsEquivalentToCertainTopologicalGroups} Let $\mathcal{C}$ be the category [[Grp]] of groups, and let $\mathcal{A}$ be the category $\mathsf{Top-Grp}$ of [[topological group|topological groups]]. The fully faithful functor $\mathsf{Set} \rightarrow \mathsf{Top}$ sending a set to the [[discrete topological space]] on this set gives rise to a fully faithful functor $\mathsf{Grp} \rightarrow \mathsf{Top-Grp}$. Then (as finite discrete spaces are cocompact) Proposition \ref{PropositionCategoriesEquivalentToProC} implies that the category pro-$\mathsf{FinGrp}$ of pro-objects in $\mathsf{FinGrp}$, that is to say of [[profinite group|profinite groups]], is equivalent to the full sub-category of topological groups whose objects are obtained as a cofiltered limit of finite groups (viewed as topological groups via the discrete topology). \end{example}

\begin{rmk} \label{RemarkProfiniteGroupsEquivalentToCertainTopologicalGroups} Though it is less well-known, one can in Example \ref{ExampleProfiniteGroupsEquivalentToCertainTopologicalGroups} evidently replace $\mathsf{Top}$ with any category $\mathcal{A}$ for which there is a fully faithful functor $\mathsf{Set} \rightarrow \mathcal{A}$ which preserves finite products and lands in cocompact objects. See [[discrete object]] for one general setting in which finite product preserving functors exist. \end{rmk}

\begin{rmk} Both Example \ref{ExampleProfiniteGroupsEquivalentToCertainTopologicalGroups} and Remark \ref{RemarkProfiniteGroupsEquivalentToCertainTopologicalGroups} generalise from $\mathsf{Grp}$ to any [[finite product theory]], that is to say to the category of models of a [[finite product sketch]]. They generalise further to any [[finite limit theory]], that is to say to the category of models of  a [[finite limit sketch]], if the functor $\mathsf{Set} \rightarrow \mathcal{A}$ moreover preserves finite limits. \end{rmk}

## Examples

* [[profinite set]], [[profinite space]]

* [[profinite group]], [[progroup]]

* [[pro-étale morphism of schemes]], [[pro-étale site]]

* Since every ([[dg-coalgebra|dg-]])[[coalgebra]] is the [[filtered colimit]] of its finite-dimensional subalgebras (see at _[coalgebra -- as filtered colimit](coalgebra#AsFilteredColimits)_), the [[linear dual]] of a (dg-)coalgebra is canonically a pro-object in finite dimensional ([[dg-algebra|dg-]])[[associative algebra|algebras]]. This plays a role for instance for constructing [[model structures for L-infinity algebras]], see [there](model+structure+for+L-infinity+algebras#OnProAlg).

* [[pro-category of towers]]

## Applications

### Etale homotopy theory.

Procategories were used by Artin and Mazur in their work on [[étale homotopy]] theory. They associated to a scheme a “pro-homotopy type”.  (This is discussed briefly at [[étale homotopy]].) The important thing to note is that this was a pro-object in the _homotopy category_ of simplicial sets, i.e., in the pro-homotopy category. Friedlander rigidified their construction to get an object in the pro-category of simplicial sets, and this opened the door to use of 'homotopy pro-categories'.

### Shape theory.

The form of [[shape theory]] developed by [[Sibe Mardešić]] and [[Jack Segal]], at about the same time as the work in algebraic geometry, again used an approach equivalent to that of the [[pro-homotopy theory|pro-homotopy category]]. Under the name “Čech homotopy theory”, the use more formally of the pro-homotopy category, was introduced by Porter.  Strong shape, developed by Edwards and Hastings, also by [[Tim Porter]] and then in further work by [[Sibe Mardešić]] and [[Jack Segal]], used various forms of rigidification to get to the pro-category of spaces, or of simplicial sets.  There methods of model category theory could be used.

## Related concepts

* [[ind-object]] / [[ind-object in an (∞,1)-category]]

* **pro-object** / [[pro-object in an (∞,1)-category]]

* [[pro-representable functor]]

* [[ind-pro-object]]

* [[pro-left adjoint]]

* [[pro-homotopy theory]], [[profinite completion of a group]]

* [[2-pro-object]]

## References

* [[Alexander Grothendieck]], Section A.2 of: [[FGA]] *Technique de descente et théorèmes d'existence en géométrie algébriques. II. Le théorème d'existence en théorie formelle des modules*, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195, 22 p.  ([numdam:SB_1958-1960__5__369_0](http://www.numdam.org/item/?id=SB_1958-1960__5__369_0), English translation: [pdf](https://labs.thosgood.com/builds/fga-3-II.pdf), [web version](https://thosgood.com/fga/book/FGA-3-II))

* (SGA4-1) [[Alexander Grothendieck]], [[Jean-Louis Verdier]], _Préfaisceaux_, Exp. 1 ([retyped pdf](http://www.math.polytechnique.fr/~orgogozo/SGA4/01/01.pdf)) in _Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos_, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 ([[SGA 4]]). Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics __269__, Springer 1972. [pdf of SGA 4, Tome 1](http://www.iecn.u-nancy.fr/~gaillapy/SGA/grothendieck_sga_4.1.pdf)

* {#ArtinMazur69} [[Michael Artin]], [[Barry Mazur]], appendix of _Étale homotopy theory_, Lecture Notes in Maths. 100, Springer-Verlag, Berlin 1969.

* {#KashiwaraSchapira06} [[Masaki Kashiwara]], and [[Pierre Schapira]], Section 6 of: _[[Categories and Sheaves]]_, Grundlehren der mathematischen Wissenschaften 332 (2006)

* [[Peter Johnstone]], Section VI.1 of: _[[Stone Spaces]]_

* {#Isaksen02} [[Dan Isaksen]], _Calculating limits and colimits in pro-categories_, Fund. Math. 175 (2002), no. 2, 175–194.

* {#Lurie} [[Jacob Lurie]], Section 6.1 in: _Ultracategories_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Conceptual.pdf))

* [[Jean-Louis Verdier]], _Equivalence essentielle des systèmes projectifs_, C. R.A.S. Paris261 (1965), 4950 - 4953.

* [[John Duskin]], _Pro-objects (after Verdier)_, Sém. Heidelberg- Strasbourg1966 -67, Exposé 6, I.R.M.A.Strasbourg.

* A. Deleanu, P. Hilton, Borsuk shape and Grothendieck categories of pro-objects, Math. Proc. Camb. Phil. Soc.79-3 (1976), 473-482 [MR400220](http://www.ams.org/mathscinet-getitem?mr=400220)

* Walter Tholen, _Pro-categories and multiadjoint functors_, Canadian J. Math. __36__:1 (1984) 144-155 [doi](https://doi.org/10.4153/CJM-1984-010-2)

See also discussion in [[shape theory]]:

* [[Jean-Marc Cordier]], and [[Tim Porter]],  _Shape Theory_, categorical methods of approximation, Dover (2008) (This is a reprint of the 1989 edition without amendments.)

* [[Sibe Mardesic|S. Mardešić]], [[Jack Segal|J. Segal]], _Shape theory_, North Holland 1982

[[!redirects pro-objects]]
[[!redirects pro object]]
[[!redirects pro-category]]


pro-object in an (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### For small (∞,1)-categories

For $C$ a [[small (∞,1)-category]] and $\kappa$ a [[regular cardinal]], the $(\infty,1)$-category of $\kappa-$**pro-objects** in $C$ is the [[opposite (∞,1)-category]] of [[ind-object in an (∞,1)-category|ind-objects]] in the opposite of $C$:

$$
  Pro_\kappa(C) := (Ind_\kappa(C^{op}))^{op}
  \,.
$$

For $\kappa = \omega$ we write just $Pro(C)$.

By the properties listed there, if $C$ has all $\kappa$-small [[(∞,1)-limit]]s then this is equivalent to

$$
  \cdots \simeq Lex_\kappa(C, \infty Grpd)^{op} \subset Func(C,\infty Grpd)^{op}
$$

the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that preserve these limits.

### For large (∞,1)-categories

Generalizing this definition:

\begin{definition}\label{CategoryOfProObjectsOfLargeAccessibleCategory}
If $\mathcal{C}$ is a possibly [[large category|non-small]] but [[accessible (∞,1)-category|accessible]] [[(infinity,1)-category|$(\infty,1)$-category]] with [[finite (infinity,1)-limits|finite $(\infty,1)$-limits]], we write :

$$
  Pro(\mathcal{C})
  \;\coloneqq\;
  AccLexFunc(C ,\, \infty Grpd)^{op}
$$
for the [[opposite (infinity,1)-category|opposite $\infty$-category]] of [[(infinity,1)-functors|$\infty$-functors]] $\mathcal{C} \to $ [[∞Grpd]] which are

1. [[left exact functor|left exact]]  in that they [[preserved limit|preserve]] [[finite (infinity,1)-limit|finite $(\infty,1)$-limits]];

1. [[accessible (∞,1)-functor|accessible]].

\end{definition}
([Lurie 2009, Def. 7.1.6.1](#Lurie09))
\begin{remark}
Yet more generally, if $C$ is just [[locally small]], then one can take $Pro(\mathcal{C})$ to be the $\infty$-category of [[small functors]] whose [[Grothendieck construction]] is [[cofiltered]].  Equivalently, $Pro(\mathca;{C})$ consists of the functors which are "*small* cofiltered limits of representables".
\end{remark}

\begin{remark}\label{PlainObjectsInLargeAccessibleCategoryAreProObjects}
For $\mathcal{C}$ a possibly [[large category|large]] but [[accessible (∞,1)-category|accessible]]  [[(infinity,1)-category|$(\infty,1)$-category]] which is [[tensoring|tensored]] over [[∞Grpd]], in that there is a [[natural equivalence|natural]] [[equivalence in an (∞,1)-category|equivalence]]
$$
  \infty Grpd
  \big(
    S
    ,\,
    \mathcal{C}(X,Y)
  \big)
  \;\;
  \simeq
  \;\;
  \infty Grpd
  \big(
    S \cdot X
    ,\,
    Y
  \big)
$$
then it is still a [[full sub-(infinity,1)-category|full sub-$(\infty,1)$-category]] of its pro-objects, in the sense of Def. \ref{CategoryOfProObjectsOfLargeAccessibleCategory}, via the usual [[(infinity,1)-Yoneda embedding|$(\infty,1)$-Yoneda embedding]]:
\[
  \label{PlainObjectsAmongProObjects}
  \array{
    \mathcal{C}
    &\xhookrightarrow{\phantom{---}}&
    Pro(\mathcal{C})
    \\
    c
      &\mapsto&
    \mathcal{C}(c,-)
  }
\]
This is because the above tensoring means that $\mathcal{C}(c,-)$ is a [[right adjoint|right]]$\,$[[adjoint (infinity,1)-functor|adjoint $(\infty,1)$-functor]] and [[right adjoints preserve limits|these preserve limits]] and are accessible (by [this Prop.](accessible+infinity1-functor#AdjointFunctorsAreAccessible))
\end{remark}

## Related concepts

* [[ind-object]] / [[ind-object in an (∞,1)-category]]

* [[pro-object]]

* [[shape of an (infinity,1)-topos|shape of an $\infty$-topos]]

## References

The large version is mentioned in:

* {#Lurie09} [[Jacob Lurie]], around def. 7.1.6.1 of: _[[Higher Topos Theory]]_, 2009

See also:

* [[Ilan Barnea]], [[Yonatan Harpaz]], [[Geoffroy Horel]], _Pro-categories in homotopy theory_ ([arXiv:1507.01564](https://arxiv.org/abs/1507.01564)).

[[!redirects pro-object in an (∞,1)-category]]

[[!redirects pro-objects in an (∞,1)-category]]

[[!redirects pro-object in an (infinity,1)-category]]
[[!redirects pro-objects in an (infinity,1)-category]]

[[!redirects pro-object in an infinity1-category]]


pro-ring

A [[pro-object]] in [[rings]].

## Related concepts

* [[formal scheme]]

* [[adic ring]]

[[!redirects pro-rings]]


pro-set

# Pro-sets
* table of contents
{: toc}

## Definition

A **pro-set** is a [[pro-object]] in the [[category]] [[Set]].

This should not be confused with [[proset]], an abbreviation of "preordered set."

## Pro-sets versus locales

### The classifying locale of a pro-set

Since $Pro(Set)$ is the [[free completion]] of [[Set]] under [[cofiltered limits]], any functor out of $Set$ into a category with cofiltered limits extends uniquely to a cofiltered-limit-preserving functor from $Pro(Set)$.  In particular, we can consider the functor $Set \to Loc$ from $Set$ to the category of [[locales]] which regards a set as a [[discrete space|discrete]] locale.

If $(S_i)_i$ is a pro-set, which we may WLOG assume to be indexed on a [[directed poset]], then the corresponding locale $\lim S_i$ is presented by the following [[posite]].  Its underlying [[poset]] is the [[category of elements]] of the [[diagram]] $(S_i)_i$, i.e. its elements are pairs $(i,x)$ with $x\in S_i$, and we have $(i,x)\le (j,y)$ if $i\le j$ and $s_{i j}(x)=y$, where $s_{i j} \colon S_i \to S_j$ is the transition map.  The covers in the posite are generated by $(i,x) \lhd s_{i j}^{-1}(x)$ for any $i,j,x$.

Thus, the open sets in the locale $\lim S_i$ are the "ideals" for this coverage, i.e. sets $A$ of pairs $(i,x)$ which are down-closed and such that if $(j,y)\in A$ for some $j$ and all $y\in s_{i j}^{-1}(x)$, then $(i,x)\in A$.

### The pro-set $\Pi_0$ of a locale

On the other hand, there is a naturally defined functor $\Pi_0\colon Loc \to Pro(Set)$ which sends a locale to its pro-set of [[connected components]].  The vertices of the cofiltered diagram defining $\Pi_0(X)$ are decompositions $X = \coprod_{i\in I} U_i$ of $X$ as a [[coproduct]] of [[open subsets]], and the corresponding set is the index set $I$.  A morphism $(U_i) \to (V_j)$, called a *refinement*, consists of a function $f:I\to J$ such that $U_i$ is contained in $V_{f(i)}$; the corresponding function is of course $f$.

This diagram is cofiltered:
1. It is nonempty, since $X$ is the 1-ary coproduct of itself.
1. Given decompositions $(U_i)$ and $(V_j)$, the decomposition $(U_i \cap V_j)_{i,j}$ refines both of them.
1. Given parallel refinements $f,g:(U_i)\to (V_j)$, for each $i$ we have $U_i \subseteq V_{f(i)} \cap V_{g(i)}$.  If we define $K = \{ i | f(i) = g(i) \}$ and $W_i = U_i$ for $i\in K$, then we have an obvious refinement $h\colon (W_k) \to (U_i)$ and $f h = g h$.  It remains to show that $(W_k)$ is actually a cover of $X$.

   Since $V_{j_1} \cap V_{j_2} = \bigcup \{ V_{j_1} | j_1 = j_2 \}$ (the latter being the [[join]] of a [[subsingleton]]), we have $U_i \subseteq \bigcup \{ U_i | f(i) = g(i) \}$ (another join of a subsingleton) and thus $U_i \subseteq \bigcup_{k\in K} W_k$.  Thus, since the $U_i$ cover $X$, so do the $W_k$.

The [[classical mathematics|classical mathematician]] may be forgiven for thinking this last argument to be more confusing than necessary, since classically, either $f(i)=g(i)$ (in which case $W_i = U_i$) or $f(i)\neq g(i)$ (in which case $U_i = \emptyset$).  Constructively, however, the more involved argument is required.

Note that if $X$ is [[locally connected space|locally connected]], then it has a "minimal" such decomposition, namely its decomposition into [[connected components]].  Thus, in this case $\Pi_0(X)$ is a mere set.

This is the [[(0,1)-topos]] version of the [[fundamental group of a topos]] or the [[fundamental ∞-groupoid of an (∞,1)-topos]].

+-- {: .un_theorem}
###### Theorem
The functor $\Pi_0\colon Loc \to Pro(Set)$ is left adjoint to $\lim\colon Pro(Set)\to Loc$.
=--
+-- {: .proof}
###### Proof
Since $\lim$ is given by regarding a pro-set as a diagram of discrete locales and taking its limit, it suffices to show that morphisms of pro-sets $\Pi_0(X) \to S$, for a set $S$, are equivalent to morphisms of locales $X \to S_{disc}$.  But a locale map $X \to S_{disc}$ is precisely a decomposition of $X$ into disjoint opens indexed by $S$, which exactly defines a map $\Pi_0(X) \to S$.
=--

If $X$ is an [[overt locale]], then every decomposition is refined by a decomposition into [[positive elements]], so we may as well consider only decompositions into positive opens.  If we do this, the cofiltered category indexing $\Pi_0$ becomes a [[codirected poset]], since (in the argument above) each $(U_i)$ is covered by $\{ U_i | f(i) = g(i) \}$, which must therefore be an [[inhabited set]] for all $i$, so that $f=g$.  Moreover, since in any refinement $f\colon (U_i) \to (V_j)$, each $V_j$ is covered by $\{ U_i | f(i)=j\}$, in this case the transition maps of the resulting pro-set are surjective.  However, for a non-overt locale (which, recall, cannot exist [[classical mathematics|classically]]), it seems that the pro-set $\Pi_0(X)$ need not be surjective in this sense.

### The classifying locale functor is not an embedding

It is well-known that when restricted to the subcategory $Pro(FinSet)$ of [[profinite sets]], the functor $\lim$ is [[fully faithful]] and in fact lands inside the subcategory of [[topological locales]], its image being the category of [[Stone spaces]].

It is also true that when lifted to [[progroups]], the functor $\lim\colon Pro(Grp) \to Grp(Loc)$ into localic groups is fully faithful when restricted to *strict* or *surjective* progroups (those whose transition maps are surjective).

However, in general the functor $\lim\colon Pro(Set) \to Loc$ is not an embedding.  For a counterexample, consider morphisms $S\to 2$, where $S$ is a pro-set and $2=\{\bot,\top\}$, regarded as a pro-set in the trivial way (and thus giving rise to a discrete locale).  A morphism of pro-sets $S\to 2$ is determined by a partition of some $S_i = S_i^\bot \sqcup S_i^\top$ (modulo a suitable equivalence relation as we change $i$).  But a morphism of locales $\lim S_i \to 2$ consists of two ideals $A^\bot$ and $A^\top$ which are disjoint and whose union *generates* the improper ideal (which consists of all pairs $(i,x)$).  A pro-set morphism $S\to 2$ induces a locale map $\lim S_i \to 2$ where $A^\bot$ and $A^\top$ are the ideals generated by $S_i^\bot$ and $S_i^\top$, but in general not every  morphism $\lim S_i \to 2$ is induced by one $S\to 2$.

Specifically, consider the following pro-set, which is indexed on the natural numbers with the inverse ordering:
$$ \cdots \to S_i \to \cdots \to S_2 \to S_1 \to S_0 $$
We define $S_i = (\mathbb{N} \times \{a,b\}) / {\sim_i}$, where $\sim_i$ is the equivalence relation generated by $(k,a) \sim_i (k,b)$ for $k \ge i$.  The transition maps are the obvious projections, which are surjective.  Define
$$ A^\bot = \{ (i,(k,a)) | k \lt i \} \quad\text{and}\quad A^\top = \{ (i,(k,b) | k \lt i \}.$$
Then $A^\bot \cup A^\top$ generates the improper ideal, since for any $i$ we have $\{ (i+1, (i,a)), (i+1,(i,b)) \} \subset A^\bot \cup A^\top$, which covers $(i,(i,?))$, which covers $(i-1,(i,?))$, and so on down to $(0,(i,?))$.  However, no $S_i$ can be partitioned as $S_i = S_i^\bot \sqcup S_i^\top$ in such a way that $S_i^\bot$ generates $A^\bot$ and $S_i^\top$ generates $A^\top$.  Thus, this defines a locale map $\lim S_i \to 2$ which does not arise from a pro-set morphism $S\to 2$.

[[!redirects pro-set]]
[[!redirects pro-sets]]
[[!redirects pro(Set)]]
[[!redirects pro (Set)]]
[[!redirects pro-Set]]
[[!redirects pro Set]]
[[!redirects Pro(Set)]]
[[!redirects Pro (Set)]]
[[!redirects Pro-Set]]
[[!redirects Pro Set]]


pro-space

# Pro-spaces
* table of contents
{: toc}

## Idea

Sometimes  systems of 'spaces' are used  to approximate a space which does not have 'nice' properties.  This can be a general compact space (as in [[shape theory]], or an object of interest from algebraic geometry such as a scheme or algebraic stack. The use of [[Čech methods]], for instance, allows for the extraction of invariants (and other information) from an approximating pro-space using  methods from adapted standard homotopy theory, where standard method cannot be directly applied to the original 'spatial object'.

##Definition

A **pro-space** is a [[pro-object]] in a category of [[topological spaces]].

In many applications, the spaces considered will have the [[homotopy type]] of [[CW-complexes]]. This is the case in [[strong shape theory]], but also in the important applications to [[algebraic geometry]].

Pro-[[simplicial sets]] are often called pro-spaces following the tradition in certain schools of [[homotopy theory]] of referring to simplicial sets as 'spaces'.

There are several useful [[model category]] structures on categories of pro-spaces.  These are described in Isaksen's paper (listed below).

## References

* [[D. C. Isaksen]], _A model structure on the category of pro-simplicial sets_,
Trans. Amer. Math. Soc., 353, (2001), 2805–2841

[[!redirects pro-space]]
[[!redirects pro-spaces]]
[[!redirects prospace]]
[[!redirects prospaces]]


pro-étale (infinity,1)-site

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

## Idea ##

The variant of [[étale (infinity,1)-site]] which is an [[(infinity,1)-category]] theoretic version of a [[pro-étale site]].

## See also ##

* [[condensed infinity-groupoid]]

* [[pro-étale site]]

* [[étale (infinity,1)-site]]


pro-étale morphism of schemes

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A morphism of [[affine schemes]] $Spec A \to Spec B$ is _pro-étale_ if it is a [[filtered limit]] of [[étale morphisms of schemes]] over $Spec B$, dually if $B \to A$ is an [[ind-object|ind-]][[étale algebra]].

Pro étale morphisms into some [[scheme]] $X$ form the [[pro-étale site]] of $X$.

## References

* [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))
 {#BhattScholze13}

## Related concepts

[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[pro-étale morphism of schemes|pro-étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]

[[!redirects pro-étale morphism]]
[[!redirects pro-étale morphisms]]

[[!redirects pro-étale morphisms of schemes]]

[[!redirects pro-etale morphism]]
[[!redirects pro-etale morphisms]]
[[!redirects pro-etale morphism of schemes]]
[[!redirects pro-etale morphisms of schemes]]

[[!redirects ind-étale morphism]]
[[!redirects ind-étale morphisms]]


pro-étale site

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

#Contents#
* table of contents
{:toc}

## Idea

The _pro-étale site_ is a variant of the [[étale site]] where the finiteness conditions on the [[fibers]] of [[étale morphisms of schemes|étale morphisms]] is relaxed to a
[[pro-object|pro-]]finiteness condition ([[pro-étale morphism of schemes|pro-étale morphisms]])

The [[sheaf topos]] over the pro-étale site might be called the _pro-[[étale topos]]_. Its [[abelian sheaf cohomology]] -- _pro-étale cohomology_ -- improves on that of the original [[étale topos]] in that it genuinely contains ([Bhatt-Scholze 13](#BhattScholze13)) the [[Weil cohomology theory]] called [[ℓ-adic cohomology]] (while over the [[étale site]] this is only given by an [[inverse limit]] of [[abelian sheaf cohomology]]).

## Definition

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

For $X$ a [[scheme]], its _pro-étale site_ $X_{proet}$ is the [[site]]
whose [[objects]] are [[pro-étale morphisms]] into $X$ and whose [[Grothendieck topology]] is that of the [[fpqc site]].

=--

## Properties

### In terms of weakly étale maps

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

A [[morphism]] $f \colon X \longrightarrow Y$ of [[schemes]]
is called _weakly étale_ if

1. $f$ is a [[flat morphism of schemes]];

1. its [[diagonal]] $X \longrightarrow X \times_Y X$ is also flat.

=--

([Bhatt-Scholze 13, def. 4.1.1](#BhattScholze13))

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

For $X$ a [[scheme]], its _pro-étale site_ $X_{proet}$ is the [[site]]
whose [[objects]] are [[weakly étale morphisms]] into $X$ (hence weakly étale $X$-schemes) and whose [[Grothendieck topology]] is that of the [[fpqc site]].

=--

([Bhatt-Scholze 13, def. 4.1.1](#BhattScholze13))

### Generation by w-contractible rings

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

A [[commutative ring]] $R$ is a **[[w-contractible ring]]** if every [[faithfully flat morphism|faithfully flat]] [[pro-étale morphism]] $Spec A \to Spec R$ has a [[section]].

=--

([Bhatt-Scholze 13, def. 2.4.1](#BhattScholze13))

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

For every [[commutative ring]] $R$, there is a a w-contractible $A$, def. \ref{wContractible},
equipped with a [[faithfully flat morphism|faithfully flat]] [[pro-étale morphism]] $Spec A \to Spec R$.

=--

([Bhatt-Scholze 13, lemma 2.4.9](#BhattScholze13))

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

So the [[full subcategory]] on the [[w-contractible rings]] forms a [[dense subsite]] of the pro-étale site, consisting of objects with pro-[[étale homotopy type]] a set.

=--

### Relation to the étale topos

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

Since every [[étale morphism of schemes]] is in particular a [[pro-étale morphism of schemes|pro-étale morphism]], there is induced a [[geometric morphism]]

$$
  \nu
  \;\colon\;
  Sh(X_{proet})
  \longrightarrow
  Sh(X_{et})
$$

from the [[pro-étale topos]] to the [[étale topos]] of any [[scheme]] $X$.

=--

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

$\nu$ is a [[surjective geometric morphism]] with [[full and faithful functor|fully faithful]] [[inverse image]]. Hence the ordinary [[étale topos]] is a [[coreflective subcategory|coreflection]] of the pro-étale topos.

=--

([Bhatt-Scholze 13, lemma 5.1.2](#BhattScholze13))

## Examples

Given a [[field]] $k$ with [[separable closure]] $\overline{k}$, then the pro-etale site of $Spec(\overline{k})$ is equivalently the category of [[profinite sets]]. The pro-etale site of $Spec(k)$ identifies with the category of profinite continuous $Gal(\overline k/k)$-sets. ([Bhatt-Scholze 13, example 4.1.10](#BhattScholze13)).

## Related concepts

* [[étale homotopy]]

* [[étale cohomology]]

* [[condensed set]]

* [[condensed cohesion]]

## References

The pro-étale topology was suggested by Scholze and then fully developed in

* {#BhattScholze13} [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ Astérisque No. 369 (2015), 99–201([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))

Some results used in the study of weakly étale maps appeared earlier in

* {#GabberRamero03} [[Ofer Gabber]] and Lorenzo Ramero, _Almost ring theory_, volume 1800 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2003. ([arXiv:math/0201175](http://arxiv.org/abs/math/0201175))

A textbook account is in

* {#StacksProjectProEtaleCohomology} [[The Stacks Project]], _[Pro-étale Cohomology](http://stacks.math.columbia.edu/tag/0965)_ ([pdf](http://stacks.math.columbia.edu/download/proetale.pdf))

Further developments include

* Takashi Suzuki, _Duality for local fields and sheaves on the category of fields_ ([arXiv:1310.4941](http://arxiv.org/abs/1310.4941))

[[!redirects pro-etale site]]

[[!redirects pro-étale sites]]
[[!redirects pro-etale sites]]

[[!redirects pro-étale cohomology]]
[[!redirects pro-etale cohomology]]

[[!redirects pro-étale topos]]
[[!redirects pro-étale toposes]]
[[!redirects pro-etale topos]]
[[!redirects pro-etale toposes]]

[[!redirects proetale site]]
[[!redirects proetale sites]]
[[!redirects proetale topos]]
[[!redirects proetale toposes]]
[[!redirects proetale topoi]]


proadjoint

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[functor]] $F: C\to D$ we say that $F$ __admits a proadjoint__ if the canonical extension $pro(F): pro(C)\to pro(D)$ of $F$ to the categories of [[pro-object]]s
has a [[left adjoint]] $G$. In other words, there is a functor $G: pro(D)\to pro(C)$ and a [[bijection]]

$$
pro(C)(GY',X) \cong pro(D)(Y',y(F)X)
$$

[[natural isomorphism|natural]] in $X$ and $Y'$, where $y:{C^{op}}\hookrightarrow pro(C)$ is the [[Yoneda embedding]] into the category of proobjects $pro(C)\subset (Set^{C})^{op}$. Equivalently, for every [[prorepresentable functor]] $X:D\to Set$, the functor $X\mapsto X\circ F$ is also prorepresentable.

## Examples

### Shape theory, étale homotopy and Galois theory
 {#EtaleHomotopy}

If an [[(∞,1)-topos]] $\mathbf{H}$ has a genuine [[left adjoint|left]] [[adjoint (∞,1)-functor]] $\Pi$ to its [[constant ∞-stack]] functor $\Delta$, then that may be interpreted as sending each object to its [[fundamental ∞-groupoid]], see at _[[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]]_.

In general such a left adjoint $\Pi$ does not exist, but the [[pro-left adjoint]] $\Pi_{pro}$ to $\Delta$ always exists. This hence produces a pro-version of the [[fundamental ∞-groupoid]]-construction known generally as the [[étale homotopy type]]. In [[algebraic geometry]] and [[arithmetic geometry]] this reproduces the content of [[Galois theory]] (the pro-etale fundamental group is the [[Galois group]]/[[algebraic fundamental group]])  while in [[topology]] this reproduces the concept of [[shape]]. (Whence the term _[[shape modality]]_ for $\Pi$).

## Related concepts

* [[pro-homotopy theory]]

## References

* [[J.-M. Cordier]], [[T. Porter]], _Shape theory : Categorical Methods of Approximation_, (sec. 2.3), Mathematics and its Applications, Ellis Horwood Ltd., March 1989, 207 pages.Dover addition (2008)  (Link to publishers [here](http://store.doverpublications.com/048646623x.html))

* {#Hoyois13} [[Marc Hoyois]], _A note on Étale homotopy_, 2013 ([pdf](http://math.northwestern.edu/~hoyois/papers/etalehomotopy.pdf))

[[!redirects proadjonts]]

[[!redirects pro-adjoint]]
[[!redirects pro-adjoints]]


Prob

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

#Contents#
* table of contents
{:toc}

## Idea

A [[function]] between [[measure space|measure-]] or [[probability spaces]] is called *measure-preserving* if its [[preimages]] preserve the measures of subsets.

Such measure-preserving functions are closed under [[composition]], hence they constitute the [[morphisms]] of a [[category]] whose [[objects]] are [[measure spaces]]. When restricted to [[objects]] that are [[probability spaces]], the category is usually denoted by *Prob*.

## Definition

Let $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$ be [[probability spaces]].

A function $f \colon X\to Y$ is called

* *[[measurable function|measurable]]* if for every $B\in\mathcal{B}$ we have $f^{-1}(B)\in\mathcal{A}$;

* *measure-preserving* if it is measurable, and moreover for every $B\in\mathcal{B}$,

   $$
     p\big(f^{-1}(B)\big) \,=\, q(B)
     \,.
   $$

The [[category]] *Prob* has

* as [[objects]], [[probability spaces]];

* as [[morphisms]], measure-preserving functions.

## Related concepts

* [[probability space]]

* [[Markov kernel]]

* [[measurable function]]

* [[category of couplings]]

* [[categorical probability]]

* [[entropy]]

category: category

[[!redirects measure-preserving map]]
[[!redirects measure-preserving maps]]
[[!redirects measure preserving map]]
[[!redirects measure preserving maps]]
[[!redirects measure-preserving function]]
[[!redirects measure-preserving functions]]
[[!redirects measure preserving function]]
[[!redirects measure preserving functions]]


probability amplitude

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

# Contents
* table of contents
{: toc}

## Definition

Where a [[probability density function]] (on a [[measure space]]) is a [[real numbers|real]]-valued [[function]] $\rho$ (satisfying certain conditions), a _probability amplitude_ is a [[complex numbers|complex]]-valued function $\psi$ such that its pointwise [[absolute value]] squared

$$
  \rho(x) \coloneqq \psi^\ast(x) \psi(x)
$$

is a probability density.

## In quantum physics

### General

Probability amplitudes appear as [[pure states]] of [[quantum mechanical systems]] in the form of _[[wave functions]]_ $\psi$ on the [[phase space]] of a corresponding [[classical mechanics|classical system]]. The [[Born rule]] of [[quantum physics]] says  that the probability density $\rho = \psi^\ast \psi$ describes the probability to find the physical system in a given classical state (in a given region of its phase space).

The fact that probability amplitudes are complex-valued means that under addition ("[[superposition]]") they exhibit [[quantum interference]] (the addition of their [[complex phases]]) in stark contrast to the addition of probability densities, for which this cannot happen.

For instance on some [[probability space]] $(X,\mu)$ there are the probability amplitudes $ \exp(i \pi/2) \mu$ and $\exp(-i \pi/2)\mu$ whose associated probability densities are both just $\mu$ itself again. But the sum of these two probability amplitudes vanishes, in contrast to the sum of their associated probability densities. This is known as "destructive" [[quantum interference]].

### Scattering amplitudes

In [[perturbative quantum field theory]] the key probability amplitudes considered are _[[scattering amplitudes]]_, which encode the probability for a given  configuration of field quanta to come in from the far past, interact with each other and hence "scatter off" each other, and then re-emerge as some other set of field quanta in the far future.

These amplitudes are collected in the _[[scattering matrix]]_ of the [[perturbative quantum field theory]].

## Related concepts

* [[quantum logic]]

* [[interpretation of quantum mechanics]]

[[!include states and observables -- content]]

[[!redirects probability amplitude]]
[[!redirects probability amplitudes]]


probability density

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

A __probability density__ is a [[density]] on some [[space]] which under [[integration]] yields a [[probability measure]].  Similarly, a __probability density function__ is a [[measurable function]] on some [[measure space]] that under integration yields a probability measure.

## Related concepts

* [[moment]], [[cumulant]]

* [[probability amplitude]]

## References

* Wikipedia, _[Probability density function](http://en.wikipedia.org/wiki/Probability_density_function)_

category: probability

[[!redirects probability density]]
[[!redirects probability densities]]
[[!redirects probability density function]]
[[!redirects probability density functions]]


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## Idea

A _probability distribution_ is a [[measure]] used in [[probability theory]] whose [[integral]] over some subspace of a [[measurable space]] is regarded as assigning a _probability_ for some event to take values in this subset.

Often a _[[probability density]]_.

## Definition

### On measurable spaces

A __probability distribution__ is a [[measure]] $\rho$ on a [[measurable space]] $X$ such that

* it is positive: $\forall U \subset X : \int_U d\rho \geq 0$;

* it is normalized: $\int_X d\rho = 1$.

### On $\sigma$-frames

In measure theory, a __probability measure__ or __probability distribution__ on a [[sigma-frame|$\sigma$-frame]] or more generally a [[sigma-complete lattice |$\sigma$-complete]] [[distributive lattice]] $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a [[probability valuation]] $\mu:L \to [0, 1]$ such that the elements are mutually disjoint and the probability valuation is denumerably/countably additive
$$\forall s\in L^\mathbb{N}. \forall m \in \mathbb{N}. \forall n \in \mathbb{N}. (m \neq n) \wedge (s(m) \wedge s(n) = \bot)$$
$$\forall s\in L^\mathbb{N}. \mu(\Vee_{n:\mathbb{N}} s(n)) = \sum_{n:\mathbb{N}} \mu(s(n))$$

## Properties

The collection of all probability distributions on a measurable space carries various [[metric]] structures that are studied in [[information geometry]]:

* [[information metric]]

* [[Wasserstein metric]]

## Examples

* [[Gaussian probability distribution]]

## Related concept

* [[probability space]]

* [[expectation value]]

* [[moment]]

* [[entropy]]

* [[stochastic map]]

* [[positive-operator valued probability measure]]

* [[density of a subset]]

* [[unit interval]]

* [[sigma-frame]]

* [[sigma-complete lattice]]

* [[distributive lattice]]

* [[probability valuation]]

* [[sigma-continuous probability valuation]]

## References ##

* Alex Simpson, [Measure, randomness and sublocales](https://www.sciencedirect.com/science/article/pii/S0168007211001874).

category: probability

[[!redirects probability distribution]]
[[!redirects probability distributions]]
[[!redirects probability measure]]
[[!redirects probability measures]]

[[!redirects statistical distribution]]
[[!redirects statistical distributions]]


probability space

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## Idea

In _[[probability theory]]_, a _probability space_ is a [[measure space]] $(X,\mu)$ whose [[measure]] $\mu$ is a [[probability distribution]]: its [[integral]] is $\int_X \mu = 1$ (e.g. [Dembo 12, 1.1](#Dembo12)).

One thinks of the elements $x\in X$ as possible configurations of a system subject to randomness, hence of $X$ as a space of "[[possible worlds]]" in the idealized situation under consideration, and for any [[subset]] $U \subset X$ one thinks of $\int_U \mu$ as the [[probability]] that the system is found in a configuration $x$ which lies in $U$.

Accordingly, a [[measurable function]] $f$ on a probability space has the interpretation of a _[[random variable]]_. Its [[integral]] $\langle f\rangle \coloneqq \int_X f \cdot\mu$ is its [[expectation value]].

## Categories of probability spaces

* The category [[Prob]] has probability spaces as objects, and measure-preserving maps as morphisms;

* The [[category of couplings]] has probability spaces as objects and [[transport plans]] as morphisms, or equivalently, equivalence classes of measure-preserving [[Markov kernels]].

## Related concepts

* [[random variable]]

* [[expectation value]], [[moment]]

* [[entropy]]

* [[quantum probability space]]

* [[Prob]], [[category of coupling]].

## References

The modern formal concept originates around

* {#Kolmogorov33} [[Andrey Kolmogorov]], _Grundbegriffe der Wahrscheinlichkeitsrechnung_, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Springer Berlin Heidelberg, 1933

Surveys and lecture notes include

* {#Dembo12} [[Amir Dembo]], _Probability theory_, 2012 ([pdf](http://statweb.stanford.edu/~adembo/stat-310a/lnotes.pdf))

category: probability

[[!redirects probability spaces]]


probability theory

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## Idea ##

Probability theory is concerned with mathematical models of phenomena that exhibit _randomness_, or more generally phenomena about which one has incomplete information.

Its central mathematical model is based mostly on [[measure theory]]. So from a pure mathematical viewpoint probability theory today could be characterized as the study of [[measurable space]]s with a finite volume normalized to $1$.

Broader perspectives may stress the relevance of other pure mathematical concepts for probability theory, or include aspects of the interpretation of mathematical results to [[phenomenology]], the latter part making naturally contact with the field of [[statistics]].

Notice that in this respect probability theory has a similar status as (other(?!)) theories of [[physics]]: there is a mathematical model (measure theory here as the model for probability theory, or for instance [[symplectic geometry]] as a model for [[classical mechanics]]) which can be studied all in itself, and then there is in addition a more or less concrete idea of how from that model one may deduce statements about the observable world (the average outcome of a dice role using probability theory, or the observability of the next solar eclipse using [[Hamiltonian mechanics]]). The step from the mathematical model to its use as a tool for making statements about the observable world is subtle, maybe a subject of [[philosophy]], but in any case outside of the realm of [[mathematics]]. In probability theory the meaning of this step is traditionally a cause of debate, with two antagonistic main schools of thought being the _frequentist interpretation_ and the _[[Bayesianism|Bayesian perspective]]_ on the nature of the relation of probability theory to the observable world.

## Theory

### Basic theory

[[random variable|Random variables]] are defined typically in terms of [[probability spaces]], cf. the basic entries on [[measure space]], [[probability space]], [[conditional expectation|conditional probability]]. The modern point of view  emphasises that many facts about [[random variables]] do not depend much on the choice of the probability spaces; the random variables are also often identified with their distributions.

Some argue that in the study of measure and probability, one should start not only with sigma algebra of measurable sets but also another of null sets. Somehow this is abstractly captured by the approach of commutative [[von Neumann]] algebras.

### Stochastic (random) processes

(...)

[[ergodic process]]

(...)

### Statistical Manifolds

Families of probability distributions often form statistical models, that is, submanifolds of the space of all probability measures on a sample space. Techniques from [[differential geometry]] may be applied in a theory known as [[information geometry]].

### Probability theory from the nPOV

We describe here some perspectives on (parts of) probability theory from the categorical point of view (see [[nPOV]]). This perspective mainly applies to the study of situations involving Markov kernels and Chapman-Kolmogorov property.

See also [[categorical approaches to probability theory]].

[[Prakash Panangaden]] in [Probabilistic Relations](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.4840) defines the [[category]] $SRel$ (stochastic [[relation]]s) to have as [[object]]s [[set]]s equipped with a $\sigma$[[sigma-field|-field]]. [[morphism|Morphisms]] are conditional probability densities or stochastic kernels. So, a morphism from $( X, \Sigma_X)$ to $( Y, \Sigma_Y)$ is a function $h: X \times \Sigma_Y \to [0, 1]$ such that

1. $\forall B \in \Sigma_Y . \lambda x \in X . h(x, B)$ is a bounded measurable function,
1. $\forall x \in X . \lambda B \in \Sigma_Y . h(x, B)$ is a subprobability measure on $\Sigma_Y$.

If $k$ is a morphism from $Y$ to $Z$, then $k \cdot h$ from $X$ to $Z$ is defined as $(k \cdot h)(x, C) = \int_Y  k(y, C)h(x, d y)$.

This is based on earlier work by Michele Giry, see [[Giry's monad]].

* Michèle Giry, _A categorical approach to probability theory_  Categorical aspects of topology and analysis (Ottawa, Ont., 1980), pp. 68–85, Lecture Notes in Math., 915, Springer.

Panangaden's definition differs from Giry's in the second clause where subprobability measures, rather than ordinary probability measures, are allowed.

Panangaden emphasises that the mechanism is similar to the way that the category of relations can be constructed from the [[power set]] [[functor]]. Just as the category of relations is the [[Kleisli category]] of the powerset functor over the category of sets [[Set]], $SRel$ is the Kleisli category of the functor over the category of [[measurable space]]s and [[measurable function]]s which sends a measurable space, $X$, to the measurable space of sub[[probability measure]]s on $X$. This functor gives rise to a [[monad]].

What is gained by the move from probability measures to subprobability measures? One motivation seems to be to model probabilistic processes from $X$ to a [[coproduct]] $X + Y$. This you can iterate to form a process which looks to see where in $Y$ you eventually end up. This relates to $SRel$ being traced.

There is a [[monad]] on $MeasureSpaces$, $1 + -: Meas \to Meas$. A probability measure on $1 + X$ is a subprobability measure on $X$. Panangaden's monad is a composite of Giry's and $1 + -$.

The opposite of the [[Kleisli category]] of [[Giry's monad]] has as morphisms $X \to Y$, linear maps from bounded functions on $X$ to bounded functions on $Y$, which send the characteristic function on $X$ to the characteristic function on $Y$.

For more details on Giry's monad and its variants see [[probability monad]].

### Generalizations

* [[quantum mechanics|Quantum mechanics]] studies [[complex number|complex]] probability amplitudes whose absolute square can be interpreted as the usual probability in the process of [[measurement]], i.e. quantum reduction. An alternative approach via [[Wigner's function]] has real, but possibly outside $[0,1]$, probabilities.

* Relatedly, noncommutative [[von Neumann algebras]] may be interpreted as a noncommutative measure theory, analogous to the role that [[C*-algebras]] play in [[noncommutative geometry]], see at _[[quantum probability]]_.

* The [[free probability]] theory of Voiculescu and others is another noncommutative generalization, with physical applications related to [[random matrix theory]].

## Applications

* [[statistical mechanics]]
* [[stochastic Loewner equation]] has relation to [[conformal field theory]] discovered by Schramm and collaborators including Fields medalist Werner.

## Related concepts

* [[probability space]], [[random variable]]

* [[expectation value]],

* [[almost surely]]

* [[Radon-Nikodym derivative]],

* [[cumulant]],

* [[ergodic theory]],

* [[statistics]],

* [[stochastic process]],

* [[Wiener measure]]

* [[Bayesian reasoning]]

* [[central limit theorem]]

* [[infra-Bayesianism]]

* [[imprecise probability]]

* [[free probability]]

* [[quantum probability]]

* [[synthetic probability theory]]

* [[statistical significance]]

* [[p-value]]

* [[Monte Carlo method]]

## References

### General

The modern formalization of probability theory in [[measure theory]] originates around

* {#Kolmogorov33} [[Andrey Kolmogorov]], _Grundbegriffe der Wahrscheinlichkeitsrechnung_, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Springer Berlin Heidelberg, 1933

Lecture notes include

* {#Grigoryan08} Alexander Grigoryan, _Measure theory and probability_, 2008 [pdf](https://www.math.uni-bielefeld.de/~grigor/mwlect.pdf)

* {#Tao10} [[Terence Tao]], _A review of probabiltiy theory_, 2010 ([web](https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/))

> just as the natural numbers can be defined abstractly without reference to any numeral system (e.g. by the Peano axioms), core concepts of probability theory, such as random variables, can also be defined abstractly, without explicit mention of a measure space; we will return to this point when we discuss free probability later in this course.

* [[Terence Tao]], _Free probability_, 2010 ([web](https://terrytao.wordpress.com/2010/02/10/245a-notes-5-free-probability/))

* {#Dembo12} [[Amir Dembo]], _Probability theory_, 2012 ([pdf](http://statweb.stanford.edu/~adembo/stat-310a/lnotes.pdf))

For references related to [[Giry's monad]] and variants see there.

Formulation in [[category theory]]:

* [[Tobias Fritz]], _A synthetic approach to Markov kernels, conditional independence, and theorems on sufficient statistics_, ([arXiv:1908.07021](https://arxiv.org/abs/1908.07021))

* [[Kirk Sturtz]], _Categorical probability theory_ ([arXiv:1406.6030](https://arxiv.org/abs/1406.6030))

Formulation in [[topos theory]]:

* [[Alex Simpson]], _Probability sheaves and the Giry monad_, 2017 ([pdf](https://coalg.org/mfps-calco2017/calco-papers/calco2017-1.pdf), [talk recording](https://youtu.be/IMGoluu1mgc))

For a more convenient setting for 'higher-order' probability theory, that is, one which admits higher-order functions, the following article uses the [[cartesian closed category]] of [[quasi-Borel space|quasi-Borel spaces]] rather than the category of measurable spaces:

* [[Chris Heunen]], Ohad Kammar, Sam Staton, Hongseok Yang, _A Convenient Category for Higher-Order Probability Theory_, ([arXiv:1701.02547](https://arxiv.org/abs/1701.02547))

For big picture in probability theory see answers to

* MathOverflow: [is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p](http://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p)

An instance of a "categorical thinking" (in a generalized sense) in solving probability problems is a solution to Buffon's noodle problem ([wikipedia](http://en.wikipedia.org/wiki/Buffon%27s_noodle)) discussed by
Tom Leinster at nCafe [here](http://golem.ph.utexas.edu/category/2010/03/a_perspective_on_higher_catego.html).

* Klain, [[Gian-Carlo Rota]], _Introduction to geometric probability_

* John C. Baez, Jacob D. Biamonte, _A course on
quantum techniques for stochastic mechanics_, [pdf](http://math.ucr.edu/home/baez/stoch_stable.pdf)

Discussion from a perspective of [[formal logic]]/[[type theory]] is in

* {#Toronto14} [[Neil Toronto]], _Useful Languages for Probabilistic Modeling and Inference_, PhD Thesis, 2014 ([pdf](http://cs.umd.edu/~ntoronto/papers/toronto-2014diss.pdf), [slides](http://cs.umd.edu/~ntoronto/papers/toronto-2014diss-slides.pdf))

[[Mikhail Gromov]] on possible generalizations/modifications of probability theory (especially probability theory seen as, fundamentally, a ""functor" from a "complex category" to a "simple category""), as well as applications of probability within and without pure mathematics:

* _Probability, symmetry, linearity. (six lectures)_. IHES, Nov 2014. ([videos](https://www.youtube.com/watch?v=aJAQVletzdY)) ([pdf](https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/probability-huge-Lecture-Nov-2014.pdf)).

### As Euclidean field theory
 {#AsEuclideanFieldTheories}

Probability regarded as [[Euclidean quantum field theory]]:

(...) see references at _[[conformal field theory]]_

In relation to the [[mass gap problem]] in [[lattice gauge theory]]:

* [[Sourav Chatterjee]], _Yang-Mills for probabilists_, in: _Probability and Analysis in Interacting Physical Systems_, PROMS **283** (2019) Springer ([arXiv:1803.01950](https://arxiv.org/abs/1803.01950), [doi:10.1007/978-3-030-15338-0](https://link.springer.com/book/10.1007/978-3-030-15338-0))

* {#Chatterjee20} [[Sourav Chatterjee]], _A probabilistic mechanism for quark confinement_, Comm. Math. Phys. 2020 ([arXiv:2006.16229](https://arxiv.org/abs/2006.16229))

category: probability
[[!redirects probability]]
[[!redirects probabilities]]


probability valuation

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## Definition ##

In measure theory, a __probability valuation__ on a [[lattice]] $(L, \leq, \bot, \vee, \top, \wedge)$ is a [[monotonic function]] $\mu:L \to [0, 1]$ such that $\mu(\bot) = 0$, $\mu(\top) = 1$, and the modularity condition is satisfied: for all elements $a, b \in L$,
$$\mu(a) + \mu(b) = \mu(a \wedge b) + \mu(a \vee b)$$

## See also ##

* [[unit interval]]

* [[valuation (measure theory)]]

* [[sigma-continuous probability valuation]]

## References ##

* Alex Simpson, [Measure, randomness and sublocales](https://www.sciencedirect.com/science/article/pii/S0168007211001874).


probe brane

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## Idea

In the [[string theory]] literature, by "probe branes" authors refer to the idea of "adding" to a configuration of [[black branes]] a small number of further branes whose ([[gravity|gravitational]], [[electromagnetism|electric]], and/or [[RR-field]]) [[backreaction]] on the given [[black brane]] [[background field|background]] is neglected. In this sense a probe brane "probes" the (forces exerted by a given) [[background field]] configuration without backreacting onto it (up to some approximation), whence the terminology.

This is, of course, the default perspective for [[Green-Schwarz sigma-model]]-branes in the given background, but authors tend to speak of "probe branes" when they think of these branes as actually being [[black branes]] themselves, just so light (i.e. of such small [[number]] compared to the number of background branes) that their backreaction may be neglected not in principle but for practical purposes.

One will not find this as a formal definition in the literature, but it transpires for instance in the following passages:

> [Polchinski & Strassler (2000), p. 18 ](#PolchinskiStrassler00): "The D3-brane charge of the probe is $n$ [...] This is assumed in this section to be small compared to $N$ so that the effect of the probe on the background can be ignored."

> [Karch & Randall (2001)](#KarchRandall01): "4. The Probe Picture [...] In this section we will show that when neglecting the backreaction of the D5 branes on the geometry [...] Of course this is just the probe approximation. Ultimately, one has to account for the back reaction."

> [Bozhilov (2003), p. 1](#Bozhilov03): "The probe branes approach for studying issues in the string/M-theory uses an approximation, in which one neglects the back-reaction of the branes on the background. In this sense, the probe branes are multidimensional dynamical systems, evolving in given, variable in general, external fields."

One particular use of (the idea of) probe branes is as [[flavor branes]] in backgrounds appearing in the [[AdS/QCD correspondence]], see the references [below](#ReferencesAsFlavorBranes).

## References

### General

Early appearance of the concept of "probe branes":

* {#PolchinskiStrassler00} [[Joseph Polchinski]], [[Matthew J. Strassler]], *The String Dual of a Confining Four-Dimensional Gauge Theory*, [[arXiv:hep-th/0003136](https://arxiv.org/abs/hep-th/0003136), [spire:524994](https://inspirehep.net/literature/524994)]

* {#KarchRandall01} [[Andreas Karch]], [[Lisa Randall]], Section 3 of: _Open and Closed String Interpretation of SUSY CFT's on Branes with Boundaries_, JHEP 0106:063 (2001) [[arXiv:hep-th/0105132](https://arxiv.org/abs/hep-th/0105132)]

Further discussion:

* {#Bozhilov03} [[Plamen L. Bozhilov]], *Probe Branes Dynamics: Exact Solutions in General Backgrounds*, Nucl. Phys. B **656** (2003) 199-225 [[arXiv:hep-th/0211181](https://arxiv.org/abs/hep-th/0211181), doi:10.1016/S0550-3213(03)00119-6]

### As flavor branes
 {#ReferencesAsFlavorBranes}

Interpretation of probe branes as [[flavor branes]] in the context of the [[AdS/QCD correspondence]]:

* {#KarchKatz02} [[Andreas Karch]], [[Emanuel Katz]],  _Adding flavor to AdS/CFT_, JHEP 0206:043, 2002 ([arxiv:hep-th/0205236](https://arxiv.org/abs/hep-th/0205236))

* [[Peter Ouyang]], _Holomorphic D7-Branes and Flavored $\mathcal{N}=1$ Gauge Theories_, Nucl. Phys. B699:207-225, 2004 ([arXiv:hep-th/0311084](https://arxiv.org/abs/hep-th/0311084))

* Thomas S. Levi, [[Peter Ouyang]], _Mesons and Flavor on the Conifold_, Phys. Rev. D76:105022, 2007 ([arXiv:hep-th/0506021](https://arxiv.org/abs/hep-th/0506021))

[[!redirects probe branes]]


Problems in the theory of automorphic forms -- 45 years later

This page records material related to the talk

* [[Robert Langlands]]

  _Problems in the theory of automorphic forms: 45 years later_

  at

  _[Symmetries and correspondences in number theory, geometry, algebra, physics: intra-disciplinary trends](https://www.maths.nottingham.ac.uk/personal/ibf/files/sc3.html)_

  Oxford, July 5 - July 8, 2014

  * [a message to Sarnak on general issues](https://publications.ias.edu/node/2605)

  * [review and discussion of further issues (pdf)](https://www.maths.nottingham.ac.uk/personal/ibf/files/RL-oxford.pdf)

  * [lecture video 1](http://video.ias.edu/files/ams/2014/LanglandsEdit.mp4), [video 2](http://video.ias.edu/files/ams/2014/LanglandsP1.mp4), [video 3](http://video.ias.edu/files/ams/2014/LanglandsP2.mp4).[^wontPlay]

  * technical remarks on the video material ([[Langlands-VideoForFosenko.pdf:file]])

  * "Video for Fosenko" ([[Langlands-VideoForFosenko.pdf:file]])

on the state of the art of the [[Langlands program]], including some cautionary remarks on the [[geometric Langlands correspondence]].

[^wontPlay]: Be aware that the [page](https://publications.ias.edu/node/2617) linking these video files says: *"Please note that the following videos might not play in Mozilla Firefox, but should in Google Chrome and possibly other browsers."* If your computer runs a recent version of Quicktime (e.g. 10.3) then copy the address from here and paste it into the location bar in Quicktime and it will work. This difficulty should be resolved with up-dates of Firefox and other browsers.

category: reference

[[!redirects LanglandsOxford2014]]


This page compiles pointers related to the book

* [[Samuel Eilenberg]], [[David K. Harrison]], [[Saunders MacLane]], [[Helmut Röhrl]] (eds.):

  **Proceedings of the **

  **Conference on Categorical Algebra**

  **--- La Jolla 1965**

  Springer (1966)

  [doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)

on [[categorical algebra]], nominally, but really about fundamental [[category theory]] as such, to a large extent.

## Contents

On axiomatization of the [[2-category]] [[Cat]] [[category of categories|of categories]] (cf. [[ETCC]]):

* {#Law66} [[William Lawvere]], *The Category of Categories as a Foundation for Mathematics*, pp. 1-20

On [[fibered categories]]:

* [[John W. Gray]], *Fibred and Cofibred Categories*, pp 21-83

On [[monadic cohomology]] of [[algebras]] (related to [[Hochschild cohomology]]):

* [[Michael Barr]], *Cohomology in tensored categories*, [[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]], Springer (1966) 344–354 [[doi:10.1007/978-3-642-99902-4_17](https://doi.org/10.1007/978-3-642-99902-4_17), [pdf](https://www.math.mcgill.ca/barr/papers/ctc.pdf), [[Barr-CohomologyTensored.pdf:file]]]

On [[monoidal categories]] and introducing [[closed categories]] as well as ([[strict 2-category|strict]]) [[2-categories]] (under the name "hypercategories"):

* {#EK65} [[Samuel Eilenberg]], [[G. Max Kelly]], *Closed Categories*, in:  [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

(...)

category: reference


process algebra

_your text_


product > history

> see _[[Cartesian product]]_


product category

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## Definition

For $C$ and $D$ two [[categories]], the **product category** $C \times D$ is the category whose

* [[object]]s are ordered [[pairs]] $(c,d)$ with $c$ an object of $C$ and $d$ an object of $D$: $Obj(C \times D) = Obj(C) \times Obj(D)$.

* [[morphism]]s are ordered pairs $((c \stackrel{f}{\to} c'),(d \stackrel{g}{\to} d'))$: $Mor(C \times D) = Mor(C) \times Mor(D)$

* composition of morphisms is defined componentwise by composition in $C$ and $D$.

This operation is the [[cartesian product]] in the 1-category [[Cat]].

## In homotopy type theory
 {#InHomotopyTypeTheory}

We discuss cartesian products for [[internal category in homotopy type theory|categories in homotopy type theory]].

Note: [UFP 2013](#UFP13) calls a [[internal category in homotopy type theory|category]] a "precategory" and a [[univalent category]] a "category", but here we shall refer to the standard terminology of "category" and "univalent category" respectively.

### Definition

For categories $A$ and $B$, their **product** $A \times B$ is a category with $(A\times B)_0\equiv A_0 \times B_0$ and

$$hom_{A \times B}\big((a,b),(a',b')\big)\equiv hom_A (a,a')\times hom_B (b,b')$$

Identities are defined by $1_{(a,b)}\equiv (1_a,1_b)$ and composition by

$$(g,g')(f,f') \equiv \big((gf),(g'f')\big)$$

### Properties
 {#InHoTTProperties}

\begin{lemma}\label{Lemma953}
**(Lemma 9.5.3 in [UFP13](#UFP13))**
\linebreak
For [[categories]] $A,B,C$, the following types are [[equivalent]]:

1. Functors $A \times B \to C$
1. Functors $A \to C^B$

\end{lemma}

\begin{proof}
Given $F : A \times B \to C$, for any $a:A$ we obviously have a functor $F_a : B \to C$. This gives a function $A_0 \to (C^B)_0$. Next, for any $f: hom_A(a,a')$, we have for any $b:B$ the morphisms $F_{(a,b),(a',b')}(f,1_b):F_a(b) \to F_{a'}(b)$.

These are the components of a [[natural transformation]] $F_a \to F_{a'}$. Functoriality in $a$ is easy to check, so we have a functor $\hat{F} : A \to C^B$.

Conversely, suppose given $G:A \to C^B$. Then for any $a:A$ and $b:B$ we have the object $G(a)(b):C$, giving a function $A_0 \times B_0 \to C_0$. And for $f:hom_A(a,a')$ and $g : hom_B(b,b')$, we have the morphism

$$G(a')_{b,b'}(g) \circ G_{a,a'}(f)_b= G_{a,a'}(f)_{b'} \circ G(a)_{b,b'}(g)$$

in $hom_C(G(a)(b),G(a')(b'))$. Functoriality is again easy to check, so we have a functor $\v{G} : A \times B \to C.$ Finally, it is also clear that these operations are inverses.
\end{proof}

## Related concepts

* [[enriched product category]]

* [[bifunctor]]

* [[extranatural transformation]]

* [[Yoneda lemma]]

## References ##

Textbook accounts:

* [[Saunders MacLane]], §II.3 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

Discussion in [[homotopy type theory]]:

* {#UFP13} [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

[[!redirects product categories]]

[[!redirects product of categories]]
[[!redirects products of categories]]

[[!redirects product groupoid]]
[[!redirects product groupoids]]

[[!redirects product of groupoids]]
[[!redirects products of groupoids]]


product extensionality

\tableofcontents

\section{Idea}

The counterpart to [[function extensionality]] but for [[products]]/[[pairs]] of [[elements]] $(a, b)$. Product extensionality characterizes the equality of elements in [[cartesian products]] in [[set theory]] and [[product types]] in [[type theory]].

\section{Definition}

\subsection{In set theory}

Product extensionality states that for all elements $a \in A$, $a' \in A$, $b \in B$, and $b' \in B$, $a = a'$ and $b = b'$ if and only if $(a, b) = (a', b')$. Product extensionality is sometimes assumed as an axiom in [[material set theory]] to make the [[pairing structure]] into an [[ordered pairing structure]] $(a, b)$.

\subsection{In type theory}

In [[intensional type theory]], [[equality]] is represented by the [[identity type]], and furthermore, there might be more than one element of the identity type in [[intensional type theory]], so a naive translation of product extensionality into intensional type theory doesn't result in the right statement.

Instead, product extensionality is the statement that given types $A$ and $B$, the [[binary function application to identities]] $\mathrm{apbinary}_{(-,-)}(a, a', b, b')$ of the function $a:A, b:B \vdash (a, b):A \times B$ defined in the [[introduction rule]] for the [[product type]] is an [[equivalence of types]] for all elements $a:A$, $a':A$ and $b:A$, $b':A$:
$$\mathrm{prodext}(a, a', b, b'):\mathrm{isEquiv}(\mathrm{apbinary}_{(-,-)}(a, a', b, b'))$$

\subsection{Rules for product extensionality}

Formation rules for the identity type of negative product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash x:A \times B \quad \Gamma \vdash y:A \times B}{\Gamma \vdash \mathrm{Id}_{A \times B}(x, y) \; \mathrm{type}}$$

Introduction rules for the identity type of negative product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash a:A  \quad \Gamma \vdash a':A \quad \Gamma \vdash b:B \quad \Gamma \vdash b':B}{\Gamma \vdash \mathrm{apbinary}_{(-,-)}(a, a', b, b'):\mathrm{Id}_{A}(a, a) \times \mathrm{Id}_{B}(b, b') \to \mathrm{Id}_{A \times B}((a, b), (a', b'))}$$

Elimination rules for the identity type of negative product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash x:A \times B \quad \Gamma \vdash y:A \times B}{\Gamma \vdash \mathrm{ap}_{\pi_1}(x, y):\mathrm{Id}_{A \times B}(x, y) \to \mathrm{Id}_{A}(\pi_1(x), \pi_1(y))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash x:A \times B \quad \Gamma \vdash y:A \times B}{\Gamma \vdash \mathrm{ap}_{\pi_2}(x, y):\mathrm{Id}_{A \times B}(x, y) \to \mathrm{Id}_{B}(\pi_2(x), \pi_2(y))}$$

Computation rules for the identity type of negative product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash x:A \times B \quad \Gamma \vdash y:A \times B}{\Gamma, p:\mathrm{Id}_{A}(\pi_1(x), \pi_1(y)), q:\mathrm{Id}_{B}(\pi_2(x), \pi_2(y)) \vdash \mathrm{ap}_{\pi_1}(\mathrm{apbinary}_{(-,-)}(\pi_1(x), \pi_2(x), \pi_1(y), \pi_2(y))(p, q)) \equiv p:\mathrm{Id}_{A}(\pi_1(x), \pi_1(y))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash x:A \times B \quad \Gamma \vdash y:A \times B}{\Gamma, p:\mathrm{Id}_{A}(\pi_1(x), \pi_1(y)), q:\mathrm{Id}_{B}(\pi_2(x), \pi_2(y)) \vdash \mathrm{ap}_{\pi_2}(\mathrm{apbinary}_{(-,-)}(\pi_1(x), \pi_2(x), \pi_1(y), \pi_2(y))(p, q)) \equiv q:\mathrm{Id}_{A}(\pi_2(x), \pi_2(y))}$$

Uniqueness rules for the identity type of negative product types:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma \vdash a:A  \quad \Gamma \vdash a':A \quad \Gamma \vdash b:B \quad \Gamma \vdash b':B}{\Gamma, r:\mathrm{Id}_{A \times B}((a, b), (a', b')) \vdash \mathrm{apbinary}_{(-,-)}(a, a', b, b')(\mathrm{ap}_{\pi_1}(r), \mathrm{ap}_{\pi_2}(r)) \equiv r:\mathrm{Id}_{A \times B}((a, b), (a', b'))}$$

These rules ensure that $\mathrm{apbinary}_{(-,-)}$ is a judgmental equivalence of types.

\subsection{Judgmental product extensionality}

One could replace the equivalence of types above with a [[judgmental equality]] of types, resulting in judgmental product extensionality, that for all types $A$ and $B$ and elements $a:A$, $a':A$, $b:B$, and $b':B$,
$$(a, b) =_{A \times B} (a', b') \equiv (a =_A a') \times (b =_B b')$$

Judgmental product extensionality holds in [[cubical type theory]] and [[higher observational type theory]].

\subsection{Relation to dependent product extensionality}

Product extensionality is equivalent to dependent product extensionality for dependent products indexed by the [[two-valued type]] $\mathbb{2}$. Indeed, given types $A$ and $B$, we can define a type family $C$ indexed by $\mathbb{2}$ as $C(0) \coloneqq A$ and $C(1) \coloneqq B$. Then $A \times B$ is the same as $\prod_{x:\mathbb{2}} C(x)$.

We define $c:\prod_{x:\mathbb{2}} C(x)$ and $c':\prod_{x:\mathbb{2}} C(x)$ by $c(0) \coloneqq a$, $c(1) \coloneqq b$, $c'(0) \coloneqq a'$, $c'(1) \coloneqq b'$.

There is a canonical function
$$\mathrm{idstodepprodid}(c, c'):(c(0) =_{C(0)} c'(0)) \times (c(1) =_{C(1)} c'(1)) \to (c =_{\prod_{x:\mathbb{2}} C(x)} c')$$
inductively defined by
$$\mathrm{idstodepprodid}(c, c)(\mathrm{refl}_{C(0)}(c(0)), \mathrm{refl}_{C(1)}(c(1))) \equiv \mathrm{refl}_{\prod_{x:\mathbb{2}} C(x)}(c):\Omega(\prod_{x:\mathbb{2}} C(x), c)$$
where $\Omega(A, a)$ is the [[loop space type]] $a =_A a$ of $A$ at $a:A$.

We could also make $(c(0) =_{C(0)} c'(0)) \times (c(1) =_{C(1)} c'(1))$ into a dependent product as well, as $\left(\prod_{x:\mathbb{2}} (c(x) =_{C(x)} c'(x))\right)$, making the above function

$$\mathrm{idstodepprodid}(c, c'):\left(\prod_{x:\mathbb{2}} (c(x) =_{C(x)} c'(x))\right) \to (c =_{\prod_{x:\mathbb{2}} C(x)} c')$$

Product extensionality is the statement that the function $\mathrm{idstodepprodid}(c, c')$ is an [[equivalence of types]] for all elements $c:\prod_{x:\mathbb{2}} C(x)$ and $c':\prod_{x:\mathbb{2}} C(x)$:
$$\mathrm{prodext}(c, c'):\mathrm{isEquiv}(\mathrm{idstodepprodid}(c, c'))$$

\section{See also}

* [[extensionality]]
* [[tuple extensionality]]

\section{References}

For judgmental product extensionality in higher observational type theory, see

* [[Mike Shulman]], *Towards a Third-Generation HOTT Part 1* ([slides](https://www.cmu.edu/dietrich/philosophy/hott/slides/shulman-2022-04-28.pdf), [video](https://www.youtube.com/watch?v=FrxkVzItMzA))

For a proof that the given function above between identity types is a [[quasi-inverse function]] in [[intensional type theory]], see section 2.6:

* *Homotopy Type Theory: Univalent Foundations of Mathematics*, The [[Univalent Foundations Project]], Institute for Advanced Study, 2013. ([web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf))


product formula for the sine function

\tableofcontents

## Statement

The infinite product formula for the [[sine]] function, due to Euler, is

$$\sin x = x \prod_{n=1}^\infty (1 - \frac{x^2}{\pi^2 n^2}).$$

Taking logarithms and then derivatives, this implies the formula

$$\pi \cot \pi x = \frac1{x} + \sum_{n=1}^\infty \left(\frac1{x+n} + \frac1{x-n} \right).$$

Conversely, assuming this cotangent identity, then integrating and exponentiating, we deduce that the product formula is correct up to a constant factor $C$, and we further deduce that $C = 1$ because $\underset{x \to 0}{\lim}\; \frac{\sin x}{x} = 1$. In this sense, we may say that the product formula is equivalent to this expansion formula for the [[cotangent]].

## Proof

Various proofs of the product formula are known, some relating the product formula to the theory of the Gamma function, some using complex analysis, among others. Below we prove the corresponding identity for the [[cotangent]] stated above, using an ingenious argument due to Herglotz.

+-- {: .num_theorem #OrderOfZero}
###### Lemma
$\underset{x \to 0}{\lim} \pi \cot \pi x - \frac1{x} = 0$.
=--

\begin{proof} In the right-hand side of the equation

$$\pi \cot \pi x - \frac1{x} = \frac{\pi x\cos \pi x - \sin \pi x}{x \sin{\pi x}},$$
the numerator has a zero of order $3$ at $x = 0$, and the denominator has a zero of order $2$ at $x = 0$, by MacLaurin expansions.
\end{proof}

+-- {: .num_theorem #DoubleAngle}
###### Lemma
Let $f(x) = \pi \cot \pi x$ and $g(x) = \frac1{x} + \sum_{n=1}^\infty \left(\frac1{x+n} + \frac1{x-n} \right)$. Then

$$f\left(\frac{x}{2}\right) + f\left(\frac{x+1}{2}\right) = 2f(x); \qquad g\left(\frac{x}{2}\right) + g\left(\frac{x+1}{2}\right) = 2g(x).$$
=--

\begin{proof} The first follows from $\cos\frac{\pi(x+1)}{2} = -\sin\frac{\pi x}{2}$, $\sin\frac{\pi(x+1)}{2} = \cos\frac{\pi x}{2}$, and double-angle formulas. To prove the second identity, put

$$g_N(x) = \frac1{x} + \sum_{n=1}^N \left(\frac1{x+n} + \frac1{x-n} \right)$$
and note that

$$g_N\left(\frac{x}{2}\right) + g_N\left(\frac{x+1}{2}\right) = 2g_{2N}(x) + \frac{2}{x + 2N + 1}.$$
Taking the limit as $N \to \infty$, the result follows.
\end{proof}

\begin{lemma} The function $h$, defined by $h(x) = f(x) - g(x)$ for $x \in \mathbb{R} \setminus \mathbb{Z}$ and $h(x) = 0$ for $x \in \mathbb{Z}$, is a continuous odd function of period $1$.
\end{lemma}

\begin{proof} The oddness and periodicity assertions are obvious, as is continuity at points $x \notin \mathbb{Z}$. Continuity at points $x \in \mathbb{Z}$ follows from Lemma \ref{OrderOfZero} and periodicity.
\end{proof}

Now we prove the cotangent identity, following the notations of the previous two lemmas:

\begin{proof} By continuity and periodicity, $h(x)$ attains a maximum value $M$, say at $x_0$. Since $h\left(\frac{x}{2}\right) + h\left(\frac{x+1}{2}\right) = 2h(x)$ by Lemma \ref{DoubleAngle}, $h\left(\frac{x_0}{2}\right) = M$ follows (else, $h\left(\frac{x_0}{2}\right) \lt M$ implies $h\left(\frac{x_0+1}{2}\right) \gt M$). By iterating, $h\left(\frac{x_0}{2^m}\right) = M$ for all $m$. By continuity,

$$M = \underset{m \to \infty}{\lim} h\left(\frac{x_0}{2^m}\right) = h(0) = 0.$$
Thus $h(x) \leq M = 0$ for all $x$. From $h(x) \leq 0$ and oddness we get that $h(x) = 0$ identically, and so $f(x) = g(x)$ identically. This completes the proof.
\end{proof}


product law

#Contents#
* table of contents
{:toc}

## Idea

The _product law_ or _product rule_ or _[[Leibniz]] rule_ of [[differentiation]] says that for $f,g : X \to \mathbb{R}$ two [[differentiable functions]] and $f g$ their (pointwise) product the [[derivative]] of their product is

$$
  d (f g) = (d f) g + f (d g)
  \,.
$$

Generalized to [[differential forms]] the product law says that $d$ is a [[derivation]] of degree +1 on the [[graded commutative algebra]] of differential forms:

$$
  d (f \wedge g) = (d f) \wedge g + (-1)^{\deg f} f \wedge (d g)
  \,
$$

if $f$ is homogeneous.

## Categorification

One categorified form of the product rule occurs in the theory of [[species]]. Let $V$ be a [[symmetric monoidal category]] with finite [[coproducts]] over which the tensor product [[preserved limit|distributes]], and let $FB$ be the [[core]] [[groupoid]] consisting of [[finite sets]] and [[bijections]]. Recall the [[Day convolution|convolution product]] for species $F, G: FB \to V$ is given by the formula

$$(F \otimes G)[S] = \sum_{S = T + U} F[T] \otimes G[U]$$

and the derivative of $F$ is given by the formula

$$F'[S] = F[S \sqcup \{\ast\}].$$

Then it is easy to check that the canonical map

$$F' \otimes G + F \otimes G' \to (F \otimes G)'$$

is an [[isomorphism]]. More generally, for any finite set $X$ one can define an $X^{th}$ derivative by the formula

$$F^{(X)}[S] = F[S + X]$$

and then one has a canonical isomorphism

$$(F \otimes G)^{(X)} \cong \sum_{X = Y + Z} F^{(Y)} \otimes G^{(Z)}$$

which is a [[categorification]] of the general product rule

$$(f \cdot g)^{(n)} = \sum_{n = j + k} \frac{n!}{j! k!} f^{(j)} \cdot g^{(k)}.$$

## Related concepts

* [[derivation]]

* [[multiplicative spectral sequence]]

[[!redirects product law]]
[[!redirects product rule]]


product manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given two [[manifolds]] $X$, $Y$ (e.g. [[topological manifolds]], [[differentiable manifolds]], [[smooth manifolds]], etc.) the _product manifold_ $X \times Y$ is the [[Cartesian product]] in the corresponding [[category]] of [[manifolds]]: its underlying [[topological space]] is the [[product topological space]] and its [[charts]] are the [[Cartesian product]] of the given charts of $X$ and $Y$.

## Examples

For [[Cartesian spaces]] we have

$\mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \simeq
\mathbb{R}^{n_1 + n_2}$

## Related concepts

* [[product topological space]]

[[!redirects product manifolds]]


product natural deduction - table

| | [[type theory]] | [[category theory]] |
|--|--|--|
|  | [[syntax]] | [[semantics]] |
|  |  [[natural deduction]] |  [[universal construction]] |
|  | **[[product type]]** | **[[product]]** |
| [[type formation]] |  $\frac{\vdash \;A \colon Type \;\;\;\;\; \vdash \;B \colon Type}{\vdash A \times B \colon Type}$ | $A,B \in \mathcal{C} \Rightarrow A \times B \in \mathcal{C}$ |
| [[term introduction]] | $\frac{\vdash\; a \colon A\;\;\;\;\; \vdash\; b \colon B}{ \vdash \; (a,b) \colon A \times B}$ | $\array{ && Q\\ & {}^{\mathllap{a}}\swarrow &\downarrow_{\mathrlap{(a,b)}}&  \searrow^{\mathrlap{b}}\\ A &&A \times B&& B }$  |
| [[term elimination]] | $\frac{\vdash\; t \colon A \times B}{\vdash\; p_1(t) \colon A} \;\;\;\;\;\frac{\vdash\; t \colon A \times B}{\vdash\; p_2(t) \colon B}$ | $\array{ && Q\\ &&\downarrow^{t} && \\ A &\stackrel{p_1}{\leftarrow}& A \times B &\stackrel{p_2}{\to}& B}$ |
| [[computation rule]] | $p_1(a,b) = a\;\;\; p_2(a,b) = b$ | $\array{ && Q \\ & {}^{\mathllap{a}}\swarrow &\downarrow_{(a,b)}& \searrow^{\mathrlap{b}}  \\ A &\stackrel{p_1}{\leftarrow}& A \times B& \stackrel{p_2}{\to} & B}$ |