infinitary tensor product

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#### Measure and probability theory
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#### Monoidal categories
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#### Limits and colimits
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## Idea

In a [[category]] with all [[products]], one can take finite products of objects, and hence one has a [[monoidal category]].
One can however also take _infinite products_. In some situations, such as in [[categorical probability]], one is interesting in taking a infinitary version of a tensor product.

In measure-theoretic [[probability theory]], such a construction is provided by [[Kolmogorov's extension theorem]], and the same idea can be used in general:

- In a category with products, an infinite product can be expressed as a [[cofiltered limit]] of finite products.
- Similarly, one can define an *infinitary tensor product* as a cofiltered limit, if it exists, of finite *tensor* products*.

## Definition

Let $(C,\otimes,I)$ be a [[symmetric monoidal category]] equipped with a distinguished map $del_X:X\to I$ for each object $I$.
(This happens for example if $C$ is [[cartesian monoidal category|cartesian]] [[semicartesian monoidal category|semicartesian]], or if it has a [[Markov category|Markov or copy-discard structure]].)

Let $J$ be a possibly infinite set, and let $(X_i)_{i\in J}$ be a $J$-indexed collection of objects of $C$.

For every finite subset $F\subseteq J$, denote by
$$
\bigotimes_{i\in F} X_i
$$
the tensor product of the family $(X_i)_{i\in F}$. (Note that $F$ is not ordered, but the ordering of the terms in the tensor product does not matter up to coherent isomorphism, since $C$ is symmetric monoidal.)

Consider now finite subsets $F\subseteq G$ of $J$.
We have a canonical "projection" morphism
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
sep=large,%
]
\bigotimes_{i\in G} X_i \ar{r}{\pi_{G,F}} & \bigotimes_{i\in F} X_i
\end{tikzcd}
defined as follows.
First of all, for $i\in G$, let
$$
Y_i = \begin{cases}
X_i & i\in F ; \\
I & i\notin F.
\end{cases}
$$
Similarly, let $e_i:X_i\to Y_i$ be
$$
e_i = \begin{cases}
id_{X_i} & i\in F ; \\
del_{X_i} & i\notin F.
\end{cases}
$$
The map $\pi_{G,F}$ is given by the tensor product of the $e_i$ as follows,
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
\bigotimes_{i\in G} X_i \ar{rr}{\otimes_{i\in G} e_i} && \bigotimes_{i\in G} Y_i \ar{r}{\cong} & \bigotimes_{i\in F} X_i
\end{tikzcd}
where the isomorphism on the right is given by the [[unitors]] of the monoidal category.

Let now $Fin(J)$ be the [[poset]] of finite subsets of $G$. The union of finite subsets is finite, so $Fin(J)$ is a [[directed set]], hence a [[filtered category]].
The functor $Fin(J)^op\to C$ mapping the inclusion $F\subseteq G$ to $\pi_{G,F}$ defined above is hence a [[cofiltered diagram]].

We say that an object is an **infintary tensor product of the family $(X_i)_{i\in J}$** in $C$, and denote it by $X_J$, if

* It is a [[cofiltered limit]] of the functor $Fin(J)^op\to C$ described above, and moreover
* For every object $A$, the functor $A\otimes-:C\to C$ preserves this limit.

## Examples

* In a [[cartesian monoidal category]], every infinite product, if it exists, is an infinitary tensor product.

* In a [[Markov category]], [[Markov category#kolmogorov_products|Kolmogorov products]] are particular infinitary products compatible with the copy-discard structure. In particular, the category [[BorelStoch]] has all countable infinitary tensor products.

* Reversing all arrows, the *infinite tensor product of rings* is defined as the [[filtered colimit]] of all the finitary tensor products. (The canonical arrows $I\to X$ are the units of the rings.)

## See also

* [[monoidal category]], [[cartesian monoidal category]]
* [[filtered category]], [[filtered limit]]
* [[Markov category]], [[Kolmogorov extension theorem]]

## References

* [[George Janelidze]] and [[Ross Street]], _Real sets_, Tbilisi Mathematical Journal, Vol. 10, No. 3, 2017, pp. 23-49. [[doi:10.1515/tmj-2017-0101](https://doi.org/10.1515/tmj-2017-0101)]

* {#fritzrischel} [[Tobias Fritz]] and Eigil Fjeldgren Rischel, _Infinite products and zero-one laws in categorical probability_, Compositionality 2(3) 2020. ([arXiv:1912.02769](http://arxiv.org/abs/1912.02769))

[[!redirects infinitary tensor products]]
[[!redirects infinite tensor product]]
[[!redirects infinite tensor products]]


infinite cardinal

A [[cardinal]] which does not correspond to a [[finite set]] is called an _infinite cardinal_.

## Related concepts

* [[cardinal arithmetic]], [[transfinite arithmetic]]

[[!redirects infinite cardinals]]


infinite complex projective G-space

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## Idea

For $G$ a suitable [[equivariant homotopy theory|equivariance group]], the _infinite projective G-space_ is the [[projective G-space]] corresponding to the "complete infinite $G$-representation", namely to a given [[G-universe]].

Over the [[ground field]] of [[complex numbers]] this is the generalization to [[topological G-spaces]]/$G$-[[equivariant homotopy theory]] of the infinite [[complex projective space]], hence the [[classifying space]] for [[complex line bundles]], now classifying [[equivariant bundle|equivariant]] [[complex line bundles]].

## Definition

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###### Definition
**([[classifying space|infinite]] [[complex projective space|complex]] [[projective G-space]])**

For $G$ an [[abelian group|abelian]] [[compact Lie group]], let

\[
  \label{GUniverseForInfiniteComplexProjectiveSpace}
  \mathcal{U}_G
  \;\coloneqq\;
  \underset{k \in \mathbb{N}}{\bigoplus}
  \underset{\mathbf{1}_V \in R(G)}{\bigoplus}
  \mathbf{1}_V
\]

be the [[G-universe]] being the infinite [[direct sum]] of all [[complex numbers|complex]] 1-[[dimension|dimensional]] [[linear representations]] of $G$, regarded as a [[topological G-space]] with [[topological space|toplogy]] the [[colimit]] of its [[finite dimensional vector space|finite-dimensional]] [[linear subspaces]].

Then the _[[infinite complex projective G-space]]_ is the [[colimit]]

$$
  P\big(
    \mathcal{U}_G
  \big)
  \;\coloneqq\;
  \underset{
    \underset{
      {
        V \subset \mathcal{U}_G
      }
      \atop
      {
         dim(V) \lt \infty
      }
    }{\longrightarrow}
  }{\lim}
  P\big( V \big)
$$

of the [[projective G-spaces]] for all the [[finite dimensional vector space|finite-dimensional]] $G$-[[linear representations]] inside the [[G-universe]] (eq:GUniverseForInfiniteComplexProjectiveSpace).

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(e.g. [Greenlees 01, Sec. 9.2](#Greenlees01))

## Related concepts

* [[equivariant complex oriented cohomology theory]]

## References

* {#Greenlees01} [[John Greenlees]], Section 9.A of: _Equivariant formal group laws and complex oriented cohomology theories_, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 ([euclid:hha/1139840255](https://projecteuclid.org/euclid.hha/1139840255))

[[!redirects infinite complex projective G-spaces]]

[[!redirects infinite projective G-space]]
[[!redirects infinite projective G-spaces]]


infinite decimal representation of a unit interval

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## Definition ##

Let $R$ be an [[Archimedean integral domain]] and let $[0, 1]_R$ be the [[unit interval]] in $R$.

The __infinite decimal representation__ of $[0, 1]_R$ is a [[function]] $\mathcal{I}:[0, 1]_R \to (\mathbb{N} \to [0, 9]_\mathbb{N})$ from the unit interval in $R$ to the [[type]] of [[sequences]] in the [[natural numbers]] that are [[bounded set|bounded]] below by $0$ and bounded above by $9$, such that $r$ is equal to the [[limit of a sequence|limit of the following sequence]]

$$\prod_{r:[0, 1]_R} r = \lim_{n \to \infty} \sum_{i=0}^{n} \frac{\mathcal{I}(r)_i}{10^i}$$

## Properties ##

* The infinite decimal representation of the unit interval in the [[rational numbers]] consist of all the eventually periodic sequences in the [[natural numbers]] that are bounded below by $0$ and bounded above by $9$.
  Every rational number in the unit interval could be represented by two natural numbers $m:\mathbb{N}$ and $n:\mathbb{N}$ such that $p:m \leq n$. Let us define the sequences $\delta:\mathbb{N} \to \mathbb{N}$ and $\rho:\mathbb{N} \to \mathbb{N}$ inductively as
$$\delta_0 \coloneqq m \div n$$
$$\rho_0 \coloneqq m\ \%\ n$$
$$\delta_{s(i)} \coloneqq (\rho_i \cdot 10) \div n$$
$$\rho_{s(i)} \coloneqq (\rho_i \cdot 10)\ \%\ n$$
The sequence $\delta$ is the infinite decimal representation of the rational number $m/n$.

* The infinite decimal representation of the unit interval in the [[decimal numbers]] consist of all the sequences such that there is a natural number $N$ such that for all decimal number $i$ and natural numbers $n \geq N$, $\mathcal{I}(i)(n) = 0$ or $\mathcal{I}(i)(n) = 9$.

## See also ##

* [[real numbers]]

* [[pre-algebra real numbers]]

* [[sequence]]

* [[limit of a sequence]]

* [[unit interval]]


infinite dimensional smooth manifold > history

< [[infinite dimensional smooth manifold]]


infinite judgement

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## Idea

> Walfisch ist keine Tür.[^ref]

[^ref]: _The whale is no door_. Hegel ([1808/09](#Logik), p.108)

An **infinite judgement**, also called a limitative or indeterminate judgement, is a type of [[judgment|judgement]] in traditional logic that differs from a positive judgement by containing a negation operator and from a negative judgement by negating only the predicate term.

Infinite judgements enjoy a rather controversial status in traditional logic but have gained importance by being elevated to third position among the qualities of judgement in the table of judgement forms in Kant's [[Critique of Pure Reason|_Kritik der reinen Vernunft_]] (1781).

## Preliminaries

Before diving into the various proposals concerning infinite judgement let us chart the territory roughly to get an idea what's at stake. A 'typical' example of an infinite judgement would be _'anima est non mortalis'_ which could be rendered as _'the soul is immortal'_ or _'the soul is not-mortal'_ , so in a first approximation we have to do with expressions containing negative or negated predicate terms and immediately the questions arises how one should treat lexicalized forms like _'immortal'_ vs. rule-produced forms like _'not-mortal'_ and in a wider sense how much grammar a logic concerned with infinite judgements should take into account here.

Next arises the question how _'anima est non mortalis'_ is related to the _negative judgement_ _'anima non est mortalis'_ as well as to the affirmative _'anima est mortalis'_. Intuitively, an infinite judgement ressembles the first by containing a negation and the latter by having a copula that is not modified by a negation, so in a sense an infinite judgement is both negative and positive at the same time.

Proceeding to get a first grip on the semantics of these judgement types let us consider two arbitrary terms A and B: The intended interpretation of an affirmative _A is B_ can be symbolically rendered as $A\cap B=A$, and $A\cap B=\emptyset$ and $A\cap\bar{B}=A$ for negative and infinite judgements, respectively. Obviously, these symbols invite a set-theoretic interpretation and picturing as Venn diagrams. In particular, the situation of the infinite judgement can be visualized by a textured plane containing an oval denoting $A$ and a disjoint oval carved out of the texture denoting $B$ - this situation ressembles the negative judgement in that $A$ and $B$ are disjoint but whereas the containment relation in the case of positive judgements is one between ovals in the infinite judgement the oval $A$ is contained in the infinite unbounded punctured plane!

The set-theoretic interpretation also suggests that negative and infinite judgements are indeed semantically the same! This equivalence underlies the wide-spread dismissal of infinite judgement in traditional or modern logic. Note that one could still accept the extensional equivalence of _A is not B_ and _A is not-B_ but attaching an intensional or procedural sense to the different judgements. Another option is of course to reject the set-theoretic equivalence and attach different interpretations to the negations and complementations involved.

Note that the set-theoretic picture suggests a first idea on the difference between _'immortal'_ and _'not-mortal'_ in that one could assume that the former is tacitly type-restricted to _animated_ things so that the extension of _'immortal'_ is not the whole plane with mortal things cut out but merely the sphere of animated things with mortal things cut out! This in turn then raises the question whether the infinite judgement _'The soul is not-mortal'_ should be pictured like this as well since the soul is tacitly equipped with a type of animated things to which _'not-mortal'_ could be coerced as a bounded predicate, in other words one could challenge the idea that 'The soul is not-mortal' is an infinite judgement at all!

To sum up: the situation leaves open a lot of options and the way to choose among these might ultimately rely on the things one wants to accomplish with the resulting notion of 'infinite judgement'.

## Kant on infinite judgements

Due to the influence of Kant's philosophy in the 19th century, one could easily get the impression that the concept of an _infinite judgement_ originated with Kant, however the concept had already a long history in traditional logic going back to the example of _non-man_ as an _indeterminate noun_ in chap. X of Aristotle's _Peri Hermenaias_. Within this larger context the most relevant author for Kant was probably J. H. Lambert who gave the _termini infiniti_, how he called them, extensive discussion in his _'Neues Organon'_ (1764) and _'Anlage zur Architectonic'_ (1771).

For Kant, the concept becomes important for the table of judgements in _'Kritik der reinen Vernunft'_ (1781) in the transcendental analytics (B 95ff, A 70ff). He gives there a primarily syntactic definition differentiating them from negative judgements by the position of the negation particle _non_ with respect to the copula: _anima non est mortalis_ vs. (inifinite:) _anima est non mortalis_. He further distinguishes the latter from _anima est immortalis_  , though this is only implicit in the passage by his avoidance of the German 'ist unsterblich' and his use of 'ist nichtsterblich' instead which due to German word order fusions with the negative 'ist nicht sterblich'. The _Vienna logic_ (see below) is slightly more outspoken on the triple distinction which in fact is often ignored by later commentators like e.g. H. Cohen (or, more recently, O. Höffe or M. Wolff) who even recommends the less marked 'unsterblich' instead of 'nichtsterblich' (Cohen [1907](#Cohen07), p.48; Höffe [2003](#Hoeffe03) p.126; cf. Jesiolkiewicz [n.d.](#Jesiol) p.2).

Now, what does the distinction between 'unsterblich' and 'nichtsterblich' amount to ? We would like to suggest to the difference between _determination_ and _limitation_: both are formally positive in the sense that they behave like positive judgements in syllogism but only 'unsterblich' narrows the 'sphere' of its subject of predication whereas 'nichtsterblich' only limits the sphere in putting the subject in an infinite and effectively indeterminate sphere _'welches eigentlich gar keine Sphäre ist'_ (Jäsche-Logik §22).

Why would it be important for transcendental logic to make a difference between infinite and positive judgements ? Presumably, due to the role of infinite judgements in metaphysical 'reasoning' where the formally positive judgements create the dialectical illusion of positive determinations of thought objects (KdrV I.1 §23 B148; see also [Höffe 2003](#Hoeffe03), p.126) whereas thought actually meanders in an infinite space without ever being able to give an object in intuition corresponding to the negatively determined concept of reasoning e.g. the determination of the soul in rational psychology as not-mortal, not-material, not-spatial etc.

The so called _Vienna logic_ contains a lengthy passage on infinite judgements:

>Alle bejahende Sätze zeigen ihre Bejahung durch die copula _est_ , welche copula das Verhältniß zweyer Begriffe anzeigt. Wenn die
copula _est_ simpliciter besteht, bedeutet sie die Verknüpfung zweyer
— wenn die copula _est_ mit dem _non_ afficirt ist: so bedeutet sie Opposition der beyden Begriffe, und zeigt an, daß der eine Begriff nicht zum andern gehört, oder nicht in der sphaera des andern enthalten sey. Z. B. _anima non est mortalis_ , hier stelle ich mir vor, daß die Sterblichkeit die Seele nicht mit in sich schließt. Sage ich aber : _anima est non mortalis_ : so sage ich nicht bloß, daß die Seele nichts sterbliches enthalte, sondern daß sie auch in der sphaera alles deßen, was nicht sterblich ist, enthalten sey. Es ist also hierbey etwas besonders gesagt, daß ich einen Begriff nähmlich nicht bloß von der sphaera eines andern Begriffes ausschließe, sondern auch den Begriff unter der ganzen übrigen sphaera denke, die nicht unter dem Begriffe, der ausgeschlossen ist, gehört. Ich sage eigentlich nicht : _est immortalis_ , sondern ich sage : unter allen Begriffen überhaupt, die außerhalb dem Begriff der Sterblichkeit gedacht werden mögen, kann die Seele gezählt werden. Und dieses macht eigentlich die unendlichen Urtheile aus. —

>Bejahung und Verneinung sind demnach qualitaeten im Urtheil. Ein verneinendes Urtheil ist nicht ein jedes Urtheil, das negativ ist, sondern ein solches verneinendes Urtheil, wo die negation die copula afficirt. Ein
solches Urtheil aber, wo sie nicht die copula afficirt, sondern das praedicat, wie bey einem unendlichen Urtheil geschieht, und die copula also ohne alle negation ist, ist demnach ein bejahendes Urtheil, folglich sind alle unendliche Urtheile bejahend, weil die negation nur das praedicat afficirt. Obgleich aber jedes unendliche Urtheil die Natur des Bejahenden hat: so ist doch immer eine Verneinung da, zwar nicht des Urtheiles, d. i. des Verhältnißes der Begriffe, sondern des praedicats.

>Das Verhältniß ist zwar daßelbe, wie bey einem bejahenden Urtheil, aber die negation ist doch immer da, und folglich sind sie vom bejahenden unterschieden. Diese Sache scheint in der Logic eine subtilitaet zu seyn. Aber in der Metaphysic wird es von Wichtigkeit, sie hier nicht übergangen zu haben. Denn da ist der Unterschied zwischen realitaet, negation und limitation größer. Bey den limitationen denk ich etwas positives, aber nicht bloß, sondern auch negatives, und es ist etwas eingeschränkt positives.- Sie heissen _judicia infinita_ , weil sie unbegränzt sind. Sie sagen nur immer, was nicht ist, und solcher praedicate kann ich unzählige machen, denn die sphaera der praedicate, die mit _non_ afficirt vom subjecte können gesagt werden, ist unendlich.

>Das Princip von allen möglichen praedicatis contrarie oppositis muß aus der Sache kommen. Dieses ist das princip der durchgängigen Bestimmung. Diese durchgängige Bestimmung eines Dinges aber ist unmöglich, weil eine unendliche Erkenntniß dazu gehöret, alle die praedicate aufzusuchen, die einem Dinge zukommen, und ich kann daher ins Unendliche fortgehen, und doch das Ding nicht durchgängig bestimmen. Z. B. die Seele ist körperlich, nicht körperlich. Die Seele ist sterblich, nicht sterblich. Im logischen Gebrauche können sie hier vor bejahende gelten. Denn jedes Ding ist durch Bestimmung von andern unterschieden. Alle andere Dinge mit non afficirt können davon gesagt werden. Z. B. das Merkmahl des Steines ist die Härte. Nun kann ich immer fortgehen bis ins Unendliche, und sagen: ein Stein ist nicht Metall, nicht Holz etc. Sag ich dadurch aber etwas Neues ? Denn was hilft es, daß ich weiß, daß alles Übrige ausser dem Begriff nicht Stein ist ? Die sphaera dieses alles Übrigen ist unendlich, und deshalb nennt man diese judicia infinita.

Kant ([1966](#KantXXIV2),pp.930-31).

The _Jäsche-Logik_ from 1800 points out that the semantic difference between infinite and negative judgments vanishes in common or "formal" logic where the [[excluded middle|tertium non datur]] is assumed:

>Nach dem Principium der Ausschließung jedes Dritten (exclusi tertii) ist die Sphäre eines Begriffs relativ auf eine andre entweder ausschließend oder einschließend. - Da nun die Logik bloß mit der Form des Urteils, nicht mit den Begriffen ihrem Inhalte nach, es zu tun hat: so ist die Unterscheidung der unendlichen von den negativen Urteilen nicht zu dieser Wissenschaft gehörig.

Kant ([1800](#Jaesche), p.535).

In that context even the difference between positive and negative judgment becomes purely syntactic since every predicate is now semantically a [[complement]]: _A is B_ becomes equivalent to _not(A is not-B)_. It seems likely though that Kant would not apply his terminology to compound judgments like the latter.

One suggestion to clarify the question of infinite judgement in Kant employing the language of modern logic would be to link it to the [[coherent logic|Morleyization]] of finitary first-order formulas:

This is a systematic way to replace formulas containing negation with formulas without negation by defining via extra symbols and axioms negative versions of all predicates e.g. for predicate $P(x)$, add $\bar{P} (x)$ together with stipulations that $P(x)\wedge \bar{P}(x)=\bot$ and $P(x)\vee \bar{P}(x)=\top$. One can then show that a first-order theory $\mathbb{T}$ and its Morleyization $\mathbb{T}'$ have exactly the same models over coherent [[Boolean category|Boolean categories]] (cf. [Johnstone 2002](#JT02), p.859).

Now one could try to adapt this procedure for pseudocomplements i.e. leaving out the stipulation that $P(x)\vee \bar{P}(x)=\bot$ and sticking in further axioms for $\bar{P}(x)$ in order that the $\bar{P}(x)$ are interpreted as pseudocomplements turning into complements when valued in Boolean categories whence the difference between negative and infinite judgements vanishes in that context. One could then define "formal" logic as this transcendental logic valued in classical models. Depending on how much logic one is willing to grant Kant in an attempt to completely formalize transcendental logic with a modern constructive logic one might even bypass any pseudo-Morleyization and formalize infinite judgments simply as intuitionistic negation applied to predicate terms. This would suggest a possibility to formalize the difference between not-mortal and immortal as that between an intuitionistic and a weaker constructive negation e.g. minimal negation.

## Fichte's take

In Fichte's _Wissenschaftslehre_ (1794) infinite judgements appear in a key passage on his dialectic in I.§3. He transforms Kant's triad of positive, negative and infinite judgments into _synthetic_ , _antithetic_ and _thetic_ judgement and says that the first two require each a 'double ground' - a _relational_ and a _differential_ ground: the relational ground giving the field where subject and predicate notion (can) concide whereas the differential ground gives the field where they (can) differ.[^def] The synthetic judgement then reflects on the first but abstracts from the second, whereas the antithetic judgement abstracts from the first and reflects on the second.

[^def]: In the background here lurks the classical theory of definition by _genus proximum_ and _differentia specifica_.

>Ein thetisches Urteil würde ein solches sein, in welchem etwas keinem andern gleich- und keinem andern entgegengesetzt, sondern bloss sich selbst gleich gesetzt würde: es könnte mithin gar keinen Beziehungs- oder Unterscheidungsgrund voraussetzen: sondern das Dritte, das es der logischen Form nach doch voraussetzen muß, wäre bloß eine Aufgabe für einen Grund. Das ursprüngliche höchste Urteil dieser Art ist das: Ich bin, in welchem vom Ich gar nichts ausgesagt wird, sondern die Stelle des Prädikats für die mögliche Bestimmung des Ich ins Unendliche leer gelassen wird. ... Kant und seine Nachfolger haben daher diese Urteile sehr richtig unendliche genannt, obgleich keiner, soviel mir bewusst ist, sie auf eine deutliche und bestimmte Art erklärt hat.

Fichte, _Grundlage der gesamten Wissenschaftslehre_ , ([1794](#FichteI),pp.310-12).

This passage was rather consequential. First, Fichte frees the concept from its narrow relation to predicate term negation and considers purely positive judgements like 'I am' and 'A is beautiful' as infinite. Secondly, he elevates the infinite judgements to 'logico-ethical' tasks of determination of propositional grounds. Thirdly, he draws them near to identity statements.

For Schelling the connection to aethetic judgement as well as the interpretation of the 'I am' as infinite will become of capital importance. In Hegel's _Logik_ the connection to positive and identity statements
will return. But before we come to the latter let's have a look at what Fichte said on infinite judgements twenty years later.

In his 1812 lectures on transcendental logic, Fichte has apparently changed his views dramatically. Infinite judgements have now their own section in lecture 28 ([Werke VI](#FichteVI), pp.392-94). Contrary to the text of 1794 where he followed Kant in considering them as _logically positive_ , he now stresses that in syllogisms they behave like negative judgements, claims that the former position is actually unintelligible and criticizes Kant for following Leibniz who in this case 'didn't know what he was talking about'.

## Hegel's take

For Hegel infinite judgements become relevant in the third book of his _Wissenschaft der Logik_ (1816) on the subjective logic subsuming the '_logic formerly so called_'. His views present a synthesis of Kant and Fichte's contribution.

In the transcendental analytics of _KdrV_ , Kant had started from the table of judgement forms where he assigned infinite judgements to the third place in the second column of 'qualities of judgement' (B95, A70) and then went on to 'derive' the table of categories from it where now the category of _'limit'_ takes the place of infinite judgements in the third row of the second column (B107, A81). He goes on (B110-111):

>Über diese Tafel der Kategorien lassen sich artige Betrachtungen anstellen, die vielleicht erhebliche Folgen in Ansehung der wissenschaftlichen Form aller Vernunfterkenntniss haben könnten ...2te Anmerk. daß allerwärts eine gleiche Zahl der Kategorien jeder Klasse, nämlich drei sind, welches eben sowohl zum nachdenken auffodert, da sonst alle Einteilung a priori durch Begriffe Dichotomie sein muß. dazu kommt aber noch, daß die dritte Kategorie allenthalben aus der Verbindung der zweiten mit der ersten Klasse entspringt ...die Einschränkung (ist) nichts anders als Realität mit Negation verbunden.

One of the 'considerable consequences' of these 'well behaved' considerations would be the systematic logic (1812-16) of Hegel who turns Kant _'from the head on his feet'_ : in [[Science of Logic|WdL]] (1812-16) the categories in the first book on being precede now the judgement forms in the third book. Infinite judgements get the role, in analogy to the synthesis of 'reality' and 'negation' in the category of 'limit', to synthesize positive and negative judgements and to lead over from qualitative to reflective judgements by [[Aufhebung|sublation]].

Hegel overlays the (reversed) Kantian parallelism between qualitative 'being' and qualitative judgement with a theory of judgement as _division_ which in the case of quality takes the form of a successive predication S-P where the subject in analogy to the 'punctuality' of qualitative being is taken as the individual I of which is sucessively predicated the general G, the particular P, and, finally the individual $I$: I-G (the individual is the general), I-P (the individual is not the general) and I-I (the individual is the individual). The last tautology according to him sublates itself since it contradicts the division necessary for judgement (cf. Werckmeister [2009](#Werck09), pp.11f).

Apparently Hegel is in trouble at this point since the succession of predication around the subject I requires the tautology I-I as last step, on the other, the synthesis of positive and negative judgement suggests infinite judgements as last step and, finally, for the logic to move on he has to find a contradiction in both judgement forms!

Here Fichte's suggestions come to rescue who had already proposed a link between 'identity' statements and infinite judgements as well as identified purely positive aesthetic judgements as infinite (presumably not without input from Kant's _'Kritik der Urteilskraft'_). Hegel pushed by the necessity to find contradictory judgement forms goes a step further: for one he admits tautologies _tout court_ as infinite and, secondly, for the traditional infinite judgements he begins to stress their abolishment of the common sphere, e.g. the whale not-being-a-door, bringing the propositions on the verge of  absurdity. That is, in fact, he seems to work at the same time with Kants syntactic notion with, additionally, the nebulosity of the semantic sphere emphasized and a more intuitive notion of 'infinite judgement' influenced by Fichte where a judgement 'Peter is Peter' 'predicates' an infinity of predicates of Peter, unfortunately, without telling us what they are besides that Peter has them.

It goes without saying that all this, and especially the counting in of the tautologies which is mainly owed to the succession pattern of I-G, I-P, I-I, is a radical step that poses serious questions concerning its viability and was severely critized by [[J.M. Ellis McTaggart|J. M. Ellis McTaggart]] ([1910](#McT10), pp.202-205).

### A deontic aside

True to his conviction of the identity between thought and object Hegel uses in his writings on logic the _sphere of right_ to illustrate infinite judgements in action with crimes as actions that negate law totally e.g. theft vs. a repossession law suit where the latter corresponds to normal negation denying a specific property right within the framework of law. Interestingly, these examples return in his **philosophy of right** to bring out the logical structure of right as well (e.g. in Hegel [1983](#Law), p.84; also cf. Mohr [2014](#Mohr97)).

This analogy between actions and judgements of quality takes the following form in the philosophy of right: the lawful action is one where the individual will (I) is determined by the general will G, i.e. the law, resulting in the positive judgement I-G, whereas the lawsuit refutes or negates such a 'judgement' $\neg$ I-G, here both opponents accept the law (serving as the relational ground in the sense of Fichte) but (one of) the individual wills falls short of full generality resulting in I-P.

Finally, the infinite judgements correspond to (1.) _fraud_ i.e. the _positive_ infinite judgement I-I where the indiviual will I in predicate position takes on the appearance of generality, and (2.) to _crime_ using force i.e. the _negative_ infinite judgement where the individual will posits itself entirely outside the sphere of the general will: I-$\neg$G. These action and judgement forms are self defeating in that they are in conflict with the basic tenets of actions or judgements, namely, that they are determinations of the subjects.

These still to be fleshed out suggestions of Hegel are interesting for two reasons:

For one, they stress the importance of the social sphere as a background to read the science of logic against in the sense that the systematic logic takes its specific form because that form permits to display (social) reality as rational thereby illuminating the many oddities of the logic as well as proposing a motivation for the shift of the young Hegel from interest in the good society to systematic philosophy.

On the other hand, they point to the possibility to decode a possible formalisation of Hegel's subjective logic along the [[Aufhebung|lines suggested by Lawvere]] for the objective logic as a new and very rich kind of _deontic logic_.

## More recent positions

B. Bolzano pointed out the imprecisions and contradictions of Kant's account of the qualities of judgment forms in his _Wissenschaftslehre_ ([1837](#BolzWLII)).

R. H. Lotze ([1874](#Lotze74)) in his influential logic assesses infinite judgements negatively calling them an 'obvious nonsense' - _offensichtliche Grillen_ and 'absurd'. He maintains that in actual reasoning they are always replaced by negative judgements.

G. Patzig ([1982](#Patzig82), pp.42f) calls them superfluous from the perspective of modern logic since depending on the definition of 'negation' they can either be subsumed under positive or negative judgements. He suggests that Kant was probably motivated by his triadic scheme since his revision of the traditional table of judgements and categories consisted mainly in the addition of third terms.[^uberweg]

[^uberweg]: Already like this: _'Zu der Dreitheilung hat sich Kant durch die Vorliebe für die schematische Regelmässigkeit seiner Kategorientafel verleiten lassen'_ , F. Ueberweg, _System der Logik_ , Marcus Bonn 1857 (p.154). This unfortunately misses the point that the particular triadic form of the table is indeed crucial for Kant since the completeness of his logic hinges on it and that Kant offers a derivation of the terms.

M. Wolff ([1995](#Wolff95), pp.157-58,290-92), extending a suggestion by C. S. Peirce (1903) who had tentatively hypothesized that Kant follows traditional logic in that existential presuppositions for the subject terms depend on the quality of judgement (positive judgements have existential import whereas negative judgements lack them), views the difference between negative and infinite judgement in the lack of existential import in the former. This squares with a remark by Kant that infinite judgements behave like positive judgements in syllogism and another remark that negative judgement serve only to 'prevent error' without extending knowledge. On the other hand, Kant clearly invokes purely formal thought objects to which the law of excluded middle applies as the context for infinite judgements complying with the property that infinite judgements appear in the context of 'general logic' which applies to possible objects i.e. a context to which the extension of the concepts involved are not yet relevant - note that _anima_ in his example does not correspond to an object that could possibly be given in intuition whence existence can not be predicated of it and that the avatar of infinite judgements in the table of categories _limit_ ranges among the class of 'mathematical' categories that concern objects of intuition but _not_ their existence (§11, B110, [KdrV](#Kdrv1), p.121). In particular, it would seem that for coherence Kant would either have to stick to a surface syntactic definition or propose an intensional conceptual characterization invoking relations between conceptual features along the lines that e.g. an affirmative judgement is one where the features of the predicate concept are contained in the features of the subject concept.

## The finite verdict

It should have become clear by now that the concept of infinite judgement can count itself as a _terminus infinitus_ given the universal disagreements and vague boundaries of its definition. Let us see if we can sort things out a bit here:

1. The passage from the Vienna logic backs the idea that infinite judgements were intended by Kant to play a role exclusively in the transcendental dialectic i.e. to uncover negative (metaphysical) judgements in positive (sheep's) clothing (cf. Höffe [2003](#Hoeffe03)).

2. On the one hand, Kant distinguishes 'non-mortalis' from 'immortalis', on the other hand, it seems that the passages were infinite reasoning occurs it is rejected actually in the 'immortalis'-form raising the puzzle whether the distinction does any work transcendentally after all. This gives some plausibility to the claim that, not unlike Hegel, Kant was rather driven by triadic patterning than by logical demand (cf. Patzig [1982](#Patzig82), pp.42).

3. Another peculiarity are the examples that Kant uses: a prototypical infinite judgement should be _'sella est non mortalis'_ (the chair is not-mortal) because this indeed brings out the contrast between _'non-mortalis'_ and _'immortalis'_ i.e. it is doubtful that _'anima est non mortalis'_ is actually an infinite judgement at all since it seems to be synonymuous with _'anima est immortalis'_.

4. Kant's syntactic definition also creates problems when one considers predicates like _'non-combatant'_ (Lotze) or _'non-conducting'_ since they are syntactically infinite but fail to produce the indeterminate infinite semantic sphere tacitly supposed by Kant. In particular for the latter, one sees that progress in electrodynamics turned this into a perfectly determinate predicate which then only applies to subjects equipped with an appropriate electron structure.

5. It is also not clear how Kant can prevent the truly infinite cases like _'sella est non mortalis'_ from coinciding (=being logically equivalent) with ordinary negative sentences (cf. Lotze [1874](#Lotze74)). This points to a further ambiguity in Kant, namely that although he works with semantic extension as a metaphor the real difference he targets seems to be rather _procedural_ i.e. the expression do not really differ in extension but in the way the extension is computed (cf. above the remark on Morleyization).

We would like to suggest now that by taking over ideas from Hegel the problems inherent in Kant (dis)appear under a new light. For one, infinite judgements should not be defined surface syntactically, moreover we should allow syntactically positive statements and insist on the disjointness of the spheres i.e. neither should _'anima est immortalis'_ count as infinite nor _'anima est non-mortalis'_. We exclude the latter because it is extensively equivalent to the former and reject the former because its negation _'anima non est immortalis'_ is equivalent to _'anima est mortalis'_ which is a perfectly normal positive proposition. However, the negation of _'sella est non mortalis'_ is the _absurd_ '_sella est mortalis'_ whereas _'sella est immortalis'_ as well as its negation is absurd since they violate the sphere restriction to animated things inherent in _'immortalis'_ or _'mortalis'_.

So we arrive at the hypothesis that _a judgement is infinite precisely when its negation is absurd_. This immediately admits trivial identity statements like 'the morning star is the morning star' as infinite while non-trivial statements like 'the morning star is the evening star' are ordinary positive judgements. With some generosity one might even waive Fichte's _'I am'_ as infinite since one might conceive of _'I am not'_ as absurd. Of course, this leaves open the questions (1.) of how to formalize such a logic of absurdity (for some related problems cf. Zeilberger [2008](#Zeilberger08)) and then (2.) how to fit the dialectics of judgements into such a putative formalisation i.e. the precise sense in which the infinite judgement becomes the unity of positive and negative judgements i..e. how to account for the triadic patterning that is crucial for the systematic completeness of both Kant's and Hegel's logic.

To sum up: it seems that the most promising direction to spell out a concept of infinite judgement in the context of foundations of logic is to use a structural definition as the synthesis of positive and negative judgements i.e. to take the claims of logical systematicity serious, and try to flesh this out materially within a logic of absurdity, absurdity understood here as _immediate_ falsity.

## Related entries

* [[transcendental ideal]]

* [[co-Heyting negation]]

* [[absolute conclusion]]

* [[construction in philosophy]]

* [[Critique of Pure Reason]]

* [[Attempt to Introduce the Concept of Negative Quantities into Philosophy|Concept of Negative Quantities]]

* [[Aufhebung]]

* [[Science of Logic]]

* [[syllogism]]

* [[judgment|judgement]]

## Link

* SEP _Kant's Theory of Judgment_ ([link](http://plato.stanford.edu/entries/kant-judgment/))

## References

* {#AL11} A. Achourioti, M. Van Lambalgen, _A Formalisation of Kant's Transcendental Logic_ , Review of Symbolic Logic **4** no.2 (2011) pp.254-289.

* {#BolzWLII}[[Bernard Bolzano|B. Bolzano]], _Wissenschaftslehre II_ , von Seidel Sulzbach 1837. (§ 189 pp.266ff, [link](http://dml.cz/dmlcz/400486))

* {#Cohen07} [[Hermann Cohen|H. Cohen]], _Kommentar zu Immanuel Kants Kritik der reinen Vernunft_ , Dürr Leipzig 1907.

* {#McT10} [[J.M. Ellis McTaggart|J. M. Ellis McTaggart]], _A Commentary on Hegel's Logic_ , Russell&Russell New York 1964[1910].

* {#Law} [[Georg Hegel|G. W. F. Hegel]], _Philosophie des Rechts - Die Vorlesung von 1819/20 in einer Nachschrift_ ,  Suhrkamp Frankfurt a. M. 1983.

* {#Logik} [[Georg Hegel|G. W. F. Hegel]], _Logik für die Mittelklasse (1808/09)_ , pp.86-110 in Moldenhauer, Michel (eds.), _Werke 4_ , Suhrkamp Frankfurt a. M. 1986.

* [[Georg Hegel|G. W. F. Hegel]], _Phänomenologie des Geistes_ , Suhrkamp Frankfurt a. M. 1986[1807]. (pp.260-62)

* [[Georg Hegel|G. W. F. Hegel]], _Wissenschaft der Logik II_ , Suhrkamp Frankfurt a. M. 1986[1816]. (pp.324ff)

* [[Georg Hegel|G. W. F. Hegel]], _Enzyklopädie der philosophischen Wissenschaften I_ , Suhrkamp Frankfurt a. M. 1986[1830]. (§173, pp.324ff)

* {#Hoeffe03} O. Höffe, _Kants Kritik der reinen Vernunft_ , Beck München 2003. (pp.126-27,148)

* F. Ishikawa, _Kants Denken von einem Dritten: Das Gerichtshof-Modell und das unendliche Urteil in der Antinomienlehre_ , Lang Frankfurt a. M. 1990.

* {#Jaesche} G. B. Jäsche (ed.), _Immanuel Kants Logik ein Handbuch zu Vorlesungen_ , pp.419-582 in Kant, _Schriften zur Metaphysik und Logik 2_ , Suhrkamp Frankfurt a. M. 1985[1800].

* {#Jesiol} J. Jesiolkiewicz, _Das unendliche Urteil "Seele ist nichtsterblich" bei Kant_ , ms. University of Munich not dated. ([pdf](http://www.philosophie.uni-muenchen.de/lehreinheiten/philosophie_1/betreuung/promotionen/jakub_jesiolkiewicz/skizze.pdf))

* K. Joël, _Das logische Recht der kantischen Tafel der Urteile_ , Kant- Studien **27** no.1/2 (1922) pp.298-327. ([philpapers](http://philpapers.org/rec/JOLDLR))

* {#JT02} [[Peter Johnstone|P. T. Johnstone]], _Sketches of an Elephant II_ , Oxford UP 2002. (D1.5.13-14, pp.858ff)

* {#KdrV1} I. Kant, _Kritik der reinen Vernunft 1_ , Suhrkamp Frankfurt a. M. 1985[1781, rev. 1787].

* {#KantXXIV2} I. Kant, _Akademie Ausgabe XXIV 2: Vorlesungen über Logik_ , de Gruyter Berlin 1966.

* {#RRZ94} M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _The non-Boolean logic of natural language negation_ , Phil. Math. **2** no.1 (1994) pp.45-68.

* M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Models for non-Boolean negation in natural languages based on aspect analysis_ , pp.241-260 in Gabbay, Wansing (eds.), _What is Negation?_, Kluwer Dordrecht 1999.

* {#Lotze74} R. H. Lotze, _Logica - Logik_ , bilingual ed. Bompiani Milano 2010[1874]. (§40, pp.214-16)

* A. Maier, _Kants Qualitätskategorien_ , Metzner Berlin 1930.

* {#FichteI} F. Medicus (ed.), _J. G. Fichte - Ausgewählte Werke in sechs Bänden I_ , Lambert Schneider Darmstadt 2013[1911].

* {#FichteVI} F. Medicus (ed.), _J. G. Fichte - Ausgewählte Werke in sechs Bänden VI_ , Lambert Schneider Darmstadt 2013[1911].

* A. Menne, _Das unendliche Urteil Kants_ , Philosophia Naturalis **19** (1982) pp.151-162. ([digizeit](http://www.digizeitschriften.de/dms/img/?PID=PPN510319696_0019%7Clog14))

* {#Mohr97}G. Mohr, _Unrecht und Strafe (§§ 82-1004)_ , pp.95-124 in Siep (ed.), _Klassiker Auslegen - G. W. F. Hegel: Grundlinien der Philosophie des Rechts_ , Akademie Verlag Berlin ³2014.

* {#Patzig82} G. Patzig, _Immanuel Kant: Wie sind synthetische Urteile apriori möglich?_ , pp.9-70 in Specht (ed.), Grundprobleme der großen Philosophen: Philosophie der Neuzeit II, Vandenhoeck Göttingen 1982. (pp.41-43)

* N. Stang, _Kant on Complete Determination and Infinite Judgement_ , Brit. J. Hist. Phil. **20** no. 6 (2012) pp.1117-1139.

* N. Tennant, _Negation, Absurdity, and Contrariety_ , pp.199-222 in Gabbay, Wansing (eds.), What is Negation?, Kluwer Dordrecht 1999. ([draft](http://u.osu.edu/tennant.9/files/2014/07/nac-2gav3mo.pdf))

* H. Wansing, _Negation_ , pp.415-436 in Goble (ed.), _The Blackwell Guide to Philosophical Logic_ , Blackwell Oxford 2001.

* {#Werck09} G. Werckmeister, _Hegels absoluter Schluss als logische Grundstruktur der Objektivität_ , PhD TU Kaiserslautern 2009. ([link](https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2160))

* G. Wohlfahrt, _Das unendliche Urteil. Zur Interpretation eines Kapitels aus Hegels 'Wissenschaft der Logik'_ , Z. Phil. Forschung **39** no.1 (1985) pp.85-100. ([digizeit](http://www.digizeitschriften.de/dms/img/?PID=PPN511864582_0039%7Clog12))

* {#Wolff95}M. Wolff, _Die Vollständigkeit der kantischen Urteilstafel_ , Klostermann Frankfurt a. M. 1995.

* {#Zeilberger08}[[Noam Zeilberger|N. Zeilberger]], _On the unity of duality_ , APAL **153** (2008) pp.66-96.

[[!redirects infinite judgements]]
[[!redirects infinite judgment]]
[[!redirects infinite judgments]]


infinite loop space

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## Idea

A [[topological space]] is a [[loop space]] if it has a [[delooping]]. It is an **infinite loop space** if this delooping has itself a delooping, and so on.

In [[homotopy theory]] infinite loop spaces with a choice of deloopings are equivalent to [[connective spectra]].

## Properties

### Algebraic characterization

Infinite loop spaces with a choice of deloopings are the grouplike [[E-∞ algebras]] in [[Top]] (grouplike [[E-∞ space|E-∞ spaces]]).

See for instance ([Adams, pretheorem 2.3.2](#Adams)) and the references listed there for traditional accounts. See ([Lurie, section 5.1.3](#Lurie)) for a modern formulation.

(Compare to how just [[loop spaces]] are the grouplike [[A-∞ algebras]], see _[[looping and delooping]]_.)

### Free infinite loop space

See at _[[free infinite loop space]]_.

## Related concepts

[[!include k-monoidal table]]

## References

* [[Ib Madsen]], [[Victor Snaith]], [[Jørgen Tornehave]], *Infinite loop maps in geometric topology*, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 81, Issue 3, (1977)([doi:10.1017/S0305004100053482](https://doi.org/10.1017/S0305004100053482))

* [[Peter May]], *Infinite loop space theory*, Bull. Amer. Math. Soc. **83** 4 (1977) 456-494  [[pdf](http://www.math.uchicago.edu/~may/PAPERS/18.pdf), [doi:10.1090/S0002-9904-1977-14318-8](https://doi.org/10.1090/S0002-9904-1977-14318-8)]

  _Infinite loop space theory revisited_ [[pdf](http://www.math.uchicago.edu/~may/PAPERS/28.pdf)]

* {#Adams} [[John Adams]], *Infinite loop spaces*, Hermann Weyl lectures at IAS, Annals of Mathematics Studies **90** Princeton University Press (1978) [[ISBN:9780691082066](https://press.princeton.edu/books/paperback/9780691082066/infinite-loop-spaces-am-90-volume-90), [doi:10.1515/9781400821259](https://doi.org/10.1515/9781400821259)]

* [[Peter May]], _The uniqueness of infinite loop space machines_, Topology, vol 17, pp. 205-224 (1978) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/22.pdf))

Theorem 5.2.6.15 of

* {#Lurie} [[Jacob Lurie]], _[[Higher Algebra]]_

[[!redirects infinite loop spaces]]

[[!redirects infinite-loop space]]
[[!redirects infinite-loop spaces]]


infinite loop space object

## Related concepts

[[!include k-monoidal table]]

[[!redirects connective spectrum object]]


infinite number

\tableofcontents

## Definition

In a [[strictly weakly ordered ring]] $R$, the natural numbers are a subset of $R$, because every strictly weakly ordered ring has [[characteristic zero]]. A number or element $a \in R$ is an **infinite number** or an **infinite element** if for all natural numbers $n \in \mathbb{N}$, $n \lt a$.

Examples of rings with infinite numbers include the [[hyperreal numbers]] and the [[surreal numbers]].

A strictly weakly ordered ring $R$ which satisfies the [[archimedean property]] has no infinite numbers.

## See also

* [[infinitesimal]]

[[!redirects infinite number]]
[[!redirects infinite numbers]]

[[!redirects infinite element]]
[[!redirects infinite elements]]


infinite product

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# Infinite products

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{: toc}

## Definition

An **infinite product** is a sequence of numbers (usually [[real number|real]] or [[complex number|complex]] $(a_k)_{k\in\mathbb{N}}$ written as $\prod_{k=0}^\infty a_k$.  Like an [[infinite series]], we are interested in knowing whether such a product [[convergence|converges]], and if so, what it converges to.

### As an operator

For any group or semigroup $G$, we can [[inductive definition|inductively define]] the finitary product $\mathbb{N}$-[[action]]

$$\product_{n=0}^{(-)} (-)(n) : (\mathbb{N} \to G) \times \mathbb{N} \to G$$

as

$$\product_{n=0}^{0} a(n) = a(0)$$

$$\product_{n=0}^{k+1} a(n) = a(k+1) \product_{n=0}^{k} a(n) $$

for all $k:\mathbb{N}$ $a:\mathbb{N} \to G$, such that [[currying]] the action results in the infinite product [[operator]]

$$\prod_{n=0}^\infty (-)(n) : (\mathbb{N} \to G) \to (\mathbb{N} \to G)$$

over the [[function algebra|function $G$-module]] $\mathbb{N} \to G$.

## Internalisation

In a [[cartesian closed category]] $C$, let $(G,\cdot)$ be an multiplicative [[magma object]], let $(N,0,s)$ be a [[natural numbers object]] in $C$, and let $a:N\to G$ be an __infinite sequence object__ of elements in $G$. Then there exists an infinite sequence object $b:N\to G$ called the __left infinite product object__ of $a$ inductively defined by $b(0) = a(0)$ and $b(s(n)) = b(n) \cdot a(s(n))$, and an infinite sequence object $c:N\to G$ called the __right infinite product object__ of $a$ inductively defined by $c(0) = a(0)$ and $c(s(n)) = a(s(n)) \cdot c(n)$. The element $b(n)$ is called the __left $n$-th partial product__ of the sequence $a$, and the element $c(n)$ is called the __right $n$-th partial product__ of the sequence $a$. If the magma object is [[commutative]] and [[associative]], then the left and right infinite product objects of $a$ are equal and just called an __infinite product object__ $d = \prod_{n:N} a(n)$ of $a$, where the element $d(n)$ is the __$n$-th partial product__.

## Convergence

A naive definition of [[convergence]], by [[analogy]] with the [[sum]] of a [[series]], would be that $\underoverset{k=0}{\infty}{\prod} a_k = \underset{N\to\infty}{\lim} \prod_{k=0}^N a_k$ if the latter [[limit of a sequence|limit]] exists.  However, this has the flaw that it could happen that this limit exists and yet $\underset{k\to\infty}{\lim} a_k$ might not, whereas we would like to be able to say that if $\prod_{k=0}^\infty a_k$ converges then $\underset{k\to\infty}{\lim} a_k = 1$ (by analogy with the fact that if $\sum_{k=0}^\infty a_k$ converges then $\underset{k\to\infty}{\lim} a_k = 0$).  This failure can happen for two reasons:

1. If some $a_k = 0$, then $\underset{N\to\infty}{\lim} \prod_{k=0}^N a_k = 0$ since the partial products are eventually $0$, regardless of the eventual behavior of the sequence $(a_k)$.

1. If ${\vert a_k\vert}\le M\lt 1$, then $\underset{N\to\infty}{\lim} \prod_{k=0}^N a_k = 0$, whereas the sequence $(a_k)$ might approach any limit of [[absolute value]] $\lt 1$ or have no limit at all.

To avoid these "pathological" situations, we make the following modified definition.

+-- {: .num_defn}
###### Definition
Suppose at most finitely many of the $a_k$ are zero.  We say that $\prod_{k=0}^\infty a_k$ **converges** if
$$ \underset{N\to\infty}{\lim} \prod_{k=1\quad a_k\neq 0}^\infty a_k $$
exists and is nonzero.  If this is the case, we say that
$$ \prod_{k=0}^\infty a_k =
\begin{cases}
  0 &\quad \text{if some }a_k = 0\\
  \text{the above limit} &\quad\text{otherwise}
\end{cases}
$$
=--

If the above limit equals 0, one sometimes says that $\prod_{k=0}^\infty a_k$ **diverges to 0**.

## Examples

* [[Euler product]]

## Related pages

* [[series]]
* [[analytic function]]

[[!redirects infinite products]]

[[!redirects multiplicative series]]


infinite set

# Infinite sets
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{: toc}

## Idea

A [[set]] is infinite if it is not [[finite set|finite]].

The existence of an infinite set is usually given by an [[axiom of infinity]].  The main example is the set of [[natural numbers]].

## Definitions

As you can see from [[finite set]], there are at least five definitions of that term, which are all equivalent given the [[axiom of choice]].  The [[negation]] of any of these gives a definition of infinite set.

However, the definition usually used in practice in [[constructive mathematics]] is this:

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

A set $S$ is __infinite__ if, given any [[natural number]] $n$ and a finite [[sequence]] $(x_1, \ldots, x_n)$ of elements of $S$, there exists an element $y$ of $S$ such that $y = x_i$ is always false.
=--

In other words, given any [[function]] $f$ from a Kuratowski-finite set to $S$, there exists an element of $S$ that is not in the [[image]] of $f$.  (Although because only the image matters, the definition would be equivalent if we required $S$ to be Bishop-finite, that is finite in the strictest sense.)  This is essentially a variation of [[Richard Dedekind]]\'s definition of a __Dedekind-infinite set__.

Note that you can make this definition work without previously assuming the existence of natural numbers, by using an infinity-free definition of Kuratowski-[[finite set]].

### Strongly infinite sets

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

A set $S$ with a [[tight apartness relation]] $\#$ is __strongly infinite__ if, given any [[natural number]] $n$ and a finite [[sequence]] $(x_1, \ldots, x_n)$ of elements of $S$, there exists an element $y$ of $S$ such that $y \# x_i$.
=--

## Remarks

Probably a lot to say about the relation between the various definitions of infinite set (the one above, the negations of the definitions of finite set, and others that might be studied).  In the meantime, try [the English Wikipedia](https://en.wikipedia.org/wiki/Dedekind-infinite_set).

## See also

* [[finite set]]

[[!redirects infinite set]]
[[!redirects infinite sets]]

[[!redirects Dedekind-infinite set]]
[[!redirects Dedekind-infinite sets]]


infinite-dimensional Chern-Simons theory

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## Idea

The generalization of [[Chern-Simons theory]] to [[manifolds]] of infinite [[dimension]].

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * [[4d Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

  * [[11d Chern-Simons theory]]

  * [[AKSZ sigma-models]]

  * [[string field theory]]

## References

* [[Sylvie Paycha]], [[Steven Rosenberg]], _Chern-Weil Constructions on $\Psi$DO Bundles_ ([arXiv:math/0301185](http://arxiv.org/abs/math/0301185))

* [[Steven Rosenberg]], Fabián Torres-Ardila, _Infinite Dimensional Chern-Simons Theory_ ([arXiv:math/0411161](http://arxiv.org/abs/math/0411161))

* Andrés Larrain-Hubach, [[Steven Rosenberg]], Simon Scott, Fabián Torres-Ardila, _Characteristic Classes and Zeroth Order Pseudodifferential Operators_ ([arXiv:1003.0067](http://arxiv.org/abs/1003.0067))

* Yoshiaki Maeda, [[Steven Rosenberg]], Fabián Torres-Ardila, _Riemannian geometry on loop spaces, part II: characteristic classes on $LM$_ ([pdf](http://math.bu.edu/people/sr/articles/CAGv2.pdf))

* [[Sylvie Paycha]], Simon Scott, _Chern-Weil forms associated with superconnections_ ([pdf](http://math.univ-bpclermont.fr/~paycha/articles/PaychaScottfinalproceedings.pdf))

* Raymond Vozzo, _Loop Groups and Characteristic Classes_ (2010) ([pdf](http://www.mpim-bonn.mpg.de/webfm_send/24))


infinite-dimensional Lie group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[group object]] in [[infinite-dimensional smooth manifolds]] is an _infinite-dimensional [[Lie group]]_.

If the underlying type of manifolds is [[Banach manifold]], then one speaks of _Banach-Lie groups_, etc.

## Examples

* [[U(ℋ)]] in its [[norm topology]] is a [[Banach Lie group]] (but not in its [[operator topology]])

* [[stable unitary group]]

* [[loop group]]

* [[diffeomorphism group]]

  * [[symplectomorphism group]], [[quantomorphism group]]

* [[Butcher group]]

## References

* [[Andreas Kriegl]], [[Peter Michor]], _Regular infinite dimensional Lie groups_, Journal of Lie Theory Volume 7 (1997) 61-99 ([pdf](http://www.heldermann-verlag.de/jlt/jlt07/MICHPL.PDF))

* Rudolf Schmid, _Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics_, Advances in Mathematical Physics Volume 2010, ([doi:10.1155/2010/280362](https://doi.org/10.1155/2010/280362), [pdf](http://www.emis.de/journals/HOA/AMP/Volume2010/280362.pdf))

  > (with an eye towards application in [[mathematical physics]])

* Josef Teichmann, _Innite dimensional Lie Theory from the point of view of Functional Analysis_ ([pdf](http://www.math.ethz.ch/~jteichma/diss.pdf))

  > (with an eye towards [[functional analysis]])

* [[Karl-Hermann Neeb]], _Monastir summer school: Infinite-dimensional Lie groups_, 2005 ([pdf](http://www.math.uni-hamburg.de/home/wockel/data/monastir.pdf), [[Neeb_MonastirLieGroups.pdf:file]])

* [[Karl-Hermann Neeb]], _Towards a Lie theory of locally convex groups_, Japanese Journal of Math. 1 (2006), 291-468 ([arXiv:1501.06269](https://arxiv.org/abs/1501.06269))

* [[Alexander Schmeding]],  Chaper 3 of: *An introduction to infinite-dimensional differential geometry*, Cambridge University Press ([arXiv:2112.08114](https://arxiv.org/abs/2112.08114))

[[!redirects infinite-dimensional Lie groups]]

[[!redirects Banach-Lie group]]
[[!redirects Banach-Lie groups]]

[[!redirects Banach Lie group]]
[[!redirects Banach Lie groups]]


infinite-dimensional manifold

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

# Contents
* table of contents
{: toc}

## Idea

The basic definition of a [[manifold]] (especially a [[smooth manifold]]) is as a space locally modeled on a finite-dimensional [[Cartesian space]]. This can be generalized to a notion of smooth manifolds locally modeled on infinite-dimensional [[topological vector spaces]]. Typical examples of these are [[mapping spaces]] between finite-dimensional manifolds, such as [[loop space]]s.

## Definitions

See specific versions:

* [[Hilbert manifold]]

* [[Banach manifold]]

* [[ILH manifold]]

* [[Fréchet manifold]]

* [[convenient manifold]]

## Classes of examples

* [[manifold structure of mapping spaces]]

* [[infinite-dimensional Lie group]]

## Properties: Embedding into convenient toposes

Various types of smooth manifolds embed into the
[[quasi-toposes]] of [[diffeological spaces]]
and hence the [[topos]] of [[smooth spaces]].
See there for more.

## Related concepts

* [[nonlinear functional]]

* [[orthogonal structure]]

* [[integration over infinite-dimensional manifolds]]

* [[pro-manifold]]

## References

### General

Textbook accounts:

* [[Alexander Schmeding]],  *An introduction to infinite-dimensional differential geometry*, Cambridge University Press ([arXiv:2112.08114](https://arxiv.org/abs/2112.08114))

in terms of [[Frölicher spaces]] and [[diffeological spaces]]:

* {#KM} [[Andreas Kriegl]], [[Peter Michor]]: _[[The convenient setting of global analysis]]_, AMS (1997)

Aspects of the [[homotopy theory]] of infinite-dimensional manifolds:

* {#Palais66} [[Richard S. Palais]], *Homotopy theory of infinite dimensional manifolds*, Topology **5** 1 (1966) 1-16 (doi:10.1016/0040-9383(66)90002-4)

### Integration

On [[integration over infinite-dimensional manifolds]] (for instance [[path integrals]]):

* [[Irving Segal]], _Algebraic integration theory_, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489  ([Euclid](http://projecteuclid.org/euclid.bams/1183526903))

* Hui-Hsiung Kuo, _Integration theory on infinite-dimensional manifolds_, Transactions of the American Mathematical Society Vol. 159, (Sep., 1971), pp. 57-78 ([JSTOR]( http://www.jstor.org/stable/1995998))

* David Shale, _Invariant integration over the infinite dimensional orthogonal group and related spaces_,  Transactions of the American Mathematical Society
Vol. 124, No. 1 (Jul., 1966), pp. 148-157 ([JSTOR]( http://www.jstor.org/stable/1994441))

[[!redirects infinite dimensional manifold]]
[[!redirects infinite dimensional manifolds]]
[[!redirects infinite-dimensional manifold]]
[[!redirects infinite-dimensional manifolds]]

[[!redirects infinite dimensional smooth manifold]]
[[!redirects infinite dimensional smooth manifolds]]
[[!redirects infinite-dimensional smooth manifold]]
[[!redirects infinite-dimensional smooth manifolds]]

[[!redirects infinite-dimensional differential geometry]]


infinite-dimensional sphere

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

There are several ways to make sense of the notion of the [[n-sphere]] in the limit (rather: [[colimit]]) that $n \to \infty$. Typically the resulting _infinite-dimensional sphere_ $S^\infty$ has the remarkable property that it is [[contractible topological space|contractible]] as a [[topological space]] -- or at least weakly contractible, in that its map to the [[point]] is a [[weak homotopy equivalence]].

### In homotopy theory

Due to their [[weak homotopy equivalence]] to points, in [[homotopy theory]] infinite-dimensional spheres provide nothing new in themselves, but as a source of big contractible spaces they serve as a starting point for making concrete models of [[classifying spaces]].

For example, the [[n-sphere|2k-sphere]] is the total space of the [[tautological line bundle|tautological]] [[circle group]]-[[principal bundle]] over [[complex projective space|complex projective k-space]], so that in the [[colimit]] $2k \to \infty$ the infinite-dimensional sphere emerges as a model for the [[universal principal bundle]] over the [[classifying space]] $B \mathrm{U}(1)$. This being contractible relates to important statements such as that [[zero-section into Thom space of universal line bundle is weak equivalence|the zero-section into the Thom space of the universal line bundle is a weak equivalence]].

## Realizations

### An an infinite spherical cell complex
 {#AsAnInfiniteSphericalCellComplex}

Realizing the [[n-sphere]] as a [[cell complex]] of sorts, such that $S^n \hookrightarrow S^{n+1}$ is a [[relative cell complex]]-inclusion for all $n \in \mathbb{N}$, the infinite-dimensional sphere may be taken to be the [[colimit]] over this sequence of inclusions. Since the [[n-sphere|n+1-sphere]] is [[n-connected topological space|n-connected]] it follows that $S^\infty$ is $\infty$-connected and hence weakly contractible:

Specifically in the [[category]] [[Top]] of [[topological spaces]], the [[n-sphere]] has a standard [[CW-complex]] structure with exactly 2-cells in each dimension, obtained [[induction|inductively]] by [[space attachment|attaching]] two $n$-dimensional [[hemispheres]] to the $(n-1)$-sphere regarded as the [[equator]] in the $n$-sphere.

Since forming [[homotopy groups]] $\pi_k(-)$ commutes with taking the colimit over these [[relative cell complex]] inclusions (by the [[skeleton]] filtration), and since the [[n-sphere]] has trivial homotopy groups in dimension $k \lt n$, it follows at once that the homotopy groups of the infinite-dimensional sphere all vanish, and hence that it is [[weak equivalence|weakly equivalent]] to the point in the [[classical model structure on topological spaces]]:

$$
  \begin{aligned}
    \pi_k
    \big(
      S^\infty
    \big)
    & =\;
    \pi_k
    \big(
      \underset{ \underset{n \in \mathbb{N}}{\longrightarrow} }{\lim}
      \;
      S^n
    \big)
    \\
    & = \;
    \underset{ \underset{n \in \mathbb{N}}{\longrightarrow} }{\lim}
    \;
    \pi_k
    \big(
      S^n
    \big)
    \\
    & = \;
    \underset{ \underset{n \in \mathbb{N}}{\longrightarrow} }{\lim}
    \;
    \underset{
      = \, \ast
    }{
    \underbrace{
    \pi_k
    \big(
      S^{n+k+1}
    \big)
    }
    }
    \\
    & \simeq
    \ast
  \end{aligned}
$$

### As the unit spheres in a topological vector space

One can also talk about a sphere in an arbitrary (possibly infinite-dimensional) [[normed vector space]] $V$:
$$ S(V) = \{ x: V \;\text{such that}\; {\|x\|} = 1 \} .$$

If a [[LCTVS|locally convex topological vector space]] admits a continuous linear injection into a [[normed vector space]], this can be used to define its sphere.  If not, one can still define the sphere as a *quotient* of the space of non-zero vectors under the scalar action of $(0,\infty)$.

Homotopy theorists define $S^\infty$ to be the sphere in the (incomplete) [[normed vector space]] (traditionally with the $l^2$ norm) of infinite [[sequences]] almost all of whose values are $0$, which is the [[directed colimit]] of the $S^n$:

$$
  S^{-1}
    \hookrightarrow
  S^0
    \hookrightarrow
  S^1
    \hookrightarrow
  S^2
    \hookrightarrow
  \cdots
    S^\infty
  \,.
$$

If the vector space is a [[shift space]], then contractibility is straightforward to prove.

+-- {: .num_theorem #theorem}
###### Theorem
Let $V$ be a [[shift space]] of some order.  Let $S V$ be its sphere (either via a norm or as the quotient of non-zero vectors).  Then $S V$ is contractible.
=--

+-- {: .proof}
###### Proof
Let $T \colon V \to V$ be a [[shift map]].  The idea is to homotop the sphere onto the image of $T$, and then down to a point.

It is simplest to start with the non-zero vectors, $V \setminus \{0\}$.  As $T$ is injective, it restricts to a map from this space to itself which commutes with the scalar action of $(0,\infty)$.  Define a homotopy $H \colon [0,1] \times (V \setminus
 \{0\}) \to V \setminus \{0\}$ by $H_t(v) = (1 - t)v + t T v$.  It is clear that, assuming it is well-defined, it is a homotopy from the identity to $T$.  To see that it is well-defined, we need to show that $H_t(v)$ is never zero.  The only place where it could be zero would be on an eigenvector of $T$, but as $T$ is a [[shift map]] then it has none.

As $T$ is a [[shift map]], it is not surjective and so we can pick some $v_0$ not in its image.  Then we define a homotopy $G \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $G_t(v) = (1 - t)T v + t v_0$.  As $v_0$ is not in the image of $T$, this is well-defined on $V \setminus \{0\}$.  Combining these two homotopies results in the desired contraction of $V \setminus \{0\}$.

If $V$ admits a suitable function defining a spherical subset (such as a norm) then we can modify the above to a contraction of the spherical subset simply by dividing out by this function.  If not, as the homotopies above all commute with the scalar action of $(0,\infty)$, they descend to the definition of the sphere as the quotient of $V \setminus \{0\}$.

=--

## Related concepts

* [[Kuiper's theorem]]

* [[n-connected object]]

[[!redirects infinite-dimensional spheres]]

[[!redirects infinite-dimensional sphere is contractible]]
[[!redirects infinite-dimensional sphere is weakly contractible]]

[[!redirects the infinite-dimensional sphere is contractible]]
[[!redirects the infinite-dimensional sphere is weakly contractible]]

[[!redirects infinite-dimensional spheres are contractible]]
[[!redirects infinite-dimensional spheres are weakly contractible]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Broadly speaking, _high-_ or _infinite-temperature thermal field theory_ is [[thermal field theory]] in or around the [[limit of a sequence|limiting]] case of very high or (practically) [[infinity|infinite]] [[temperature]] $T$.

This is relevant for example in the study of extreme [[phases of matter]] such as the [[quark-gluon plasma]] of [[QCD]] ([Blaizot-Iancu-Rebhan 03, section 2.2.4](#BlaizotIancuRebhan03), [Blaizot 04, around p. 17](#Blaizot04)).

More concretely, with the formulation of [[thermal field theory]] via [[Wick rotation]] as a [[Euclidean field theory]] on a [[Riemannian manifold]] $X_3 \times S^1_\beta$ whose compact/periodic Euclidean time runs along a [[circle]] $S^1_\beta$ of [[length|circumference]] the [[inverse temperature]] $\beta = 1/T$, the limit of infinite temperature corresponds to the [[limit of a sequence|limit]] $\beta \to 0$ in which this [[circle]] [[fiber]] shrinks away.

In terms of the thermal [[Euclidean field theory]] on $X_3 \times S^1_\beta$ therefore the infinite-temperature limit is given by [[Kaluza-Klein reduction|Kaluaza-Klein-type dimensional reduction]] along this circle [[fiber]] to a 3-dimensional [[Euclidean field theory]] on $X_3$ ([Ginsparg 80](#Ginsparg80), [Appelquist-Pisarski 81](#AppelquistPisarski81), [Nadkarni 83](#Nadkarni83), [Jourjine 84](#Jourjine84), [Nadkarni 88](#Nadkarni88)).

As usual with [[KK-reduction]], some care must be exercised to ensure that the compactified theory is itself still a [[local field theory]]. Depending on how exactly one proceeds this may be subtle ([Landsman 89](#Landsman89)), but there exist robust approaches ([Reisz 92](#Reisz92), [Kajantie-Laine-Rummukainen-Shaposhnikov 96](#KajantieLaineRummukainenShaposhnikov96)).

## References

The expansion of thermal field theory around the [[infinite-temperature thermal field theory|infinite-temperature-limit]] (i.e. around $\beta = 1/T = 0$, i.e. [[KK-reduction]] in compact/periodic Euclidean time) is discussed in

* {#Ginsparg80} [[Paul Ginsparg]], _First and second order phase transitions in gauge theories at finite temperature_, Nuclear Physics B Volume 170, Issue 3, 15 December 1980, Pages 388-408 (doi:10.1016/0550-3213(80)90418-6)

* {#AppelquistPisarski81} [[Thomas Appelquist]], [[Robert Pisarski]], _High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics_, Phys. Rev. D **23** 2305 (1981) [[doi:10.1103/PhysRevD.23.2305](https://doi.org/10.1103/PhysRevD.23.2305)]

* {#Nadkarni83} Sudhir Nadkarni, _Dimensional reduction in finite-temperature quantum chromodynamics_, Phys. Rev. D 27, 917 (1983) ([doi:10.1103/PhysRevD.27.917](https://doi.org/10.1103/PhysRevD.27.917))

* {#Nadkarni88} Sudhir Nadkarni, _Dimensional reduction in finite-temperature quantum chromodynamics. II_, Phys. Rev. D 38, 3287 (1988) ([doi:10.1103/PhysRevD.38.3287](https://doi.org/10.1103/PhysRevD.38.3287))

* {#Jourjine84} Alexander N Jourjine, _Quantum field theory in the infinite temperature limit_, Annals of Physics Volume 155, Issue 2, July 1984, Pages 305-332 (doi:10.1016/0003-4916(84)90003-4)

* {#Landsman89} [[Klaas Landsman]], _Limitations to dimensional reduction at high temperature_, Nuclear Physics B Volume 322, Issue 2, 14 August 1989, Pages 498-530 (doi:10.1016/0550-3213(89)90424-0)

* {#Reisz92} T. Reisz, _Realization of dimensional reduction at high temperature_, Z. Phys. C - Particles and Fields (1992) 53: 169 ([doi:10.1007/BF01483886](https://doi.org/10.1007/BF01483886))

* {#Braaten95} Eric Braaten, _Solution to the Perturbative Infrared Catastrophe of Hot Gauge Theories_, Phys. Rev. Lett. 74, 2164 (1995) ([doi:10.1103/PhysRevLett.74.2164](https://doi.org/10.1103/PhysRevLett.74.2164))

* {#KajantieLaineRummukainenShaposhnikov96} K. Kajantie, M. Laine, K. Rummukainen, M. Shaposhnikov, _Generic rules for high temperature dimensional reduction and their application to the standard model_, Nuclear Physics B Volume 458, Issues 1–2, 1 January 1996, Pages 90-136 (doi:10.1016/0550-3213(95)00549-8)

and specifically with an eye to discussion of the [[quark-gluon plasma]] in

* {#BlaizotIancuRebhan03} Jean-Paul Blaizot, Edmond Iancu, [[Anton Rebhan]], section 2.2.4 of _Thermodynamics of the high temperature quark gluon plasma_, Quark–Gluon Plasma 3, pp. 60-122 (2004) ([arXiv:hep-ph/0303185](https://arxiv.org/abs/hep-ph/0303185), [spire:615570](http://inspirehep.net/record/615570))

* {#Blaizot04} Jean-Paul Blaizot, around p. 17 of _Thermodynamics of the high temperature Quark-Gluon Plasma_, AIP Conf. Proc. 739, 63-96 (2004) ([doi:10.1063/1.1843592](https://doi.org/10.1063/1.1843592))

[[!redirects high-temperature thermal field theory]]


infinitesimal analysis

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

#Contents#
* table of contents
{:toc}

## Idea

Once _infinitesimal analysis_ was a synonym for "infinitesimal [[calculus]]", hence for the foundations of the theory of [[differentiation]] and [[integration]], but it has come to  refer to the formulation of [[analysis]] using explicit [[infinitesimal objects]], in contrast to [[epsilontic analysis]].

Formalisms that make infinitesimal analysis precise include

* [[synthetic differential geometry]] with [[nilpotent infinitesimals]]

* [[nonstandard analysis]] with non-nilpotent infinitesimals

## References

* [[Augustin Cauchy]], _[[Cours d'Analyse]]_ (1821)

category: reference

[[!redirects infinitesimal calculus]]


infinitesimal and local - table

**Examples of sequences of local structures**

| [[geometry]] | point |  first order [[infinitesimal object|infinitesimal]]  |  $\subset$ | [[formal geometry|formal]] = arbitrary order infinitesimal | $\subset$ |  local = [[stalk]]wise | $\subset$ | finite  |
|--|--|--|--|--|--|--|--|--|
|  |   |  | $\leftarrow$ **[[differentiation]]** |  | **[[integration]]** $\to$  |  |  | |
| [[smooth functions]] |  |  [[derivative]] | |  [[Taylor series]] | | [[germ]] | | [[smooth function]] |
|  [[curve]] (path) |  |  [[tangent vector]] | | [[jet]] | | [[germ]] of [[curve]] | | [[curve]] |
|  [[smooth space]] |  | [[infinitesimal neighbourhood]] |  |  [[formal neighbourhood]]  |  | [[germ of a space]]   |   | [[open neighbourhood]]  |
|  [[function algebra]] |  |  [[square-0 ring extension]] | |  [[nilpotent ring extension]]/[[formal completion]] | |  |  | [[ring extension]] |
|  [[arithmetic geometry]] | $\mathbb{F}_p$ [[finite field]]  |    |   | $\mathbb{Z}_p$ [[p-adic integers]]  |   |  $\mathbb{Z}_{(p)}$ [[localization of a ring|localization at (p)]] |  |  $\mathbb{Z}$ [[integers]]  |
| [[Lie theory]] | |  [[Lie algebra]] |  | [[formal group]] |  |  [[local Lie group]] | | [[Lie group]] |
|  [[symplectic geometry]] |  | [[Poisson manifold]] |  | [[formal deformation quantization]] | | local strict deformation quantization | | [[strict deformation quantization]] |


infinitesimal bialgebra

An __infinitesimal bialgebra__ (abbreviated $\epsilon$-bialgebra) over a commutative unital ring $k$ is a $k$-module $A$ which is equipped with the structures of an associative unital $k$-algebra $(A,m,\eta)$, coassociative counital $k$-coalgebra $(A,\Delta,\epsilon)$ and such that the following compatibility condition holds: the comultiplication $\Delta:A\to A\otimes A$ is a derivation, where $A\otimes A$ is equipped with the standard algebra and standard (external) bimodule structure, i.e. $\Delta(a b) = \Delta(a)(1\otimes b) + (a\otimes 1)\Delta(b)$.

* [[Marcelo Aguiar]], _On the associative analogues of Lie bialgebras_, [pdf](http://www.math.tamu.edu/~maguiar/analog.pdf), _Infinitesimal Hopf algebras_, [pdf](http://www.math.tamu.edu/~maguiar/cor.pdf)


infinitesimal braid relation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #InfinitesimalBraidLieAlgebra}
###### Definition
**([[infinitesimal braid Lie algebra]])**

For $n \in \mathbb{N}$ [[natural numbers]], let $F(\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}})$ be the [[free Lie algebra]] (over a given [[ground field]]) on [[generators and relations|generators]] $t_{i j}$ in degree with two indices ranging from 1 to $n$ and distinct.

The _infinitesimal braid relations_ in dimension $D = 2$ are the following [[generators and relations|relations]] on the underlying [[vector space]] of the [[free construction|free]] [[Lie algebra]] on [[generators and relations|generators]] $\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}}$

\[
  \label{InfinitesimalBriadRelations}
 \left.
 \begin{array}{lcll}
    R0: \;\; & t_{i j} - t_{j i} & = 0
    \\
    R1: \;\; & [t_{i j}, t_{k l}] & =  0
    \\
    R2: \;\; & [t_{i j}, t_{i k} + t_{k j}] & = 0
    &
    \mathrlap{\,}
  \end{array}
  \right\}
  \text{for pairwise distinct indices}
  \;\;
\]

This defines a quotient [[Lie algebra]] often denoted

$$
  \mathcal{L}_n
  \;\coloneqq\;
  F(\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}})
  /(R0, R1, R2)
 \,.
$$

=--

More generally, there is a version of this in any dimension $D \in \mathbb{N}$ with some extra signs thrown in.

This is originally due to [Kohno 87 (1.1.4)](#Kohno87), it
appears also in [Kohno 88](#Kohno88), [Bar-Natan 96, Fact 3](#BarNatan96), [Fadell-Husseini 01](#FadellHusseini01), [Cohen-Gitler 01, Section 3](#CohenGitler01).

Authors have used different equivalent versions of this presentation, see [Cohen-Gitler 02, p. 2](#CohenGitler02) for a clear statement.

## Properties

### Relation to horizontal chord diagrams

If $F(\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}})/R0$ is identified with the [[commutator]] Lie algebra of [[horizontal chord diagrams]] on $n$ strands, generated by

and forming an [[associative algebra]] under [[concatenation]] of diagrams along strands, as in

then the remaining infinitesimal braid relations (eq:InfinitesimalBriadRelations) are equivalently the following:

R1 is the the [[2T relation]]:

R2 is the [[4T relation]]:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

Hence:

+-- {: .num_prop #UniversalEnvelopingOfInfinitesimalBraidsIsHorizontalChords}
###### Proposition
**([[universal enveloping algebra]] of [[infinitesimal braid Lie algebra]] is [[horizontal chord diagrams]] [[quotient vector space|modulo]] [[2T relations|2T]]&[[4T relations|4T]])

The [[associative algebra]]

$$
  \Big(
  \mathcal{A}_n^{pb}
  \;\coloneqq\;
  Span
  \big(
    \mathcal{D}_n^{pb}
  \big)/(2T, 4T)
  ,
  \circ
  \Big)
$$

of [[horizontal chord diagrams]] on $n$ strands with product given by [[concatenation]] of strands ([this Def.](horizontal+chord+diagram#AlgebraHorizontalChordDiagrams)), [[quotient vector space|modulo]] the [[2T relations]] and [[4T relations]] ([this Def.](horizontal+chord+diagram#2TAnd4TRelations)) is [[isomorphism|isomorphic]] to the [[universal enveloping algebra]] of the [[infinitesimal braid Lie algebra]] (Def. \ref{InfinitesimalBraidLieAlgebra}):

$$
  \big(\mathcal{A}_n^{pb}, \circ\big)
  \;\simeq\;
  \mathcal{U}(\mathcal{L}_n(D))
  \,.
$$

=--

## Related concepts

* [[braid group]]

* [[Yang-Baxter equation]]

* [[weight systems are cohomology of loop space of configuration space]]

[[!include chord diagrams and weight systems -- table]]

## References

* {#Kohno87} [[Toshitake Kohno]], (1.1.4) in: _Monodromy representations of braid groups and Yang-Baxter equations_, Annales de l'Institut Fourier, Volume 37 (1987) no. 4, p. 139-160 ([doi:10.5802/aif.1114](https://doi.org/10.5802/aif.1114))

* {#Kohno88} [[Toshitake Kohno]], _Linear representations of braid groups and classical  Yang-Baxter equations_, in: [[Joan S. Birman]], [[Anatoly Libgober]] (eds.) _Braids_ Cont. Math. 78 (1988), 339-363 ([doi:10.1090/conm/078](http://dx.doi.org/10.1090/conm/078))

* {#BarNatan96} [[Dror Bar-Natan]], Fact 3 in: _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

* {#FadellHusseini01} [[Edward Fadell]], [[Sufian Husseini]], Theorem 2.2 in: _Geometry and topology of configuration spaces_, Springer Monographs in Mathematics (2001), [MR2002k:55038](http://www.ams.org/mathscinet-getitem?mr=2002k:55038), xvi+313

* {#CohenGitler01} [[Fred Cohen]], [[Samuel Gitler]], Section 3 in: _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Jaume Aguadé, Carles Broto, [[Carles Casacuberta]]  (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7))

* {#CohenGitler02} [[Fred Cohen]], [[Samuel Gitler]], p. 2 of: _On loop spaces of configuration spaces_, Trans. Amer. Math. Soc. __354__ (2002), no. 5, 1705–1748, ([jstor:2693715](https://www.jstor.org/stable/2693715), [MR2002m:55020](http://www.ams.org/mathscinet-getitem?mr=1881013))

[[!redirects infinitesimal braid relations]]

[[!redirects infinitesimal braid Lie algebra]]
[[!redirects infinitesimal braid Lie algebras]]

[[!redirects universal enveloping of infinitesimal braids is horizontal chord diagrams]]


> This entry is about a variant of the concept of [[cohesive (∞,1)-topos]]. The definition here expresses an intuition not unrelated to that at [[infinitesimally cohesive (∞,1)-presheaf on E-∞ rings]] but the definitions are unrelated and apply in somewhat disjoint contexts.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[cohesive (∞,1)-topos]] is _infinitesimal cohesive_ if all its objects behave as though built from [[infinitesimally thickened points|infinitesimally thickened]] [[geometrically discrete ∞-groupoids]] in that they all have "precisely one point in each cohesive piece".

(There is an evident version of an infinitesimally cohesive 1-topos. In ([Lawvere 07, def. 1](#Lawvere07)) such is referred to as a "[[quality type]]". A hint of this seems to be also in ([Lawvere 91, p. 9](#Lawvere91))).

Infinitesimal cohesion may also be defined relative to any [[(∞,1)-topos]].

## Definition

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

A [[cohesive (∞,1)-topos]] $\mathbf{H}$ with its [[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]] denoted $ʃ \dashv \flat \dashv \sharp$ is _infinitesimal cohesive_ if the canonical [points-to-pieces transform](cohesive+topos#CanonicalComparison) is an [[equivalence]]

$$
  \flat \stackrel{\simeq}{\longrightarrow} ʃ
$$

in which case all three [[modal operators]] collapse to a single "[[classical modality]]" $\natural$ of a [[bireflective subcategory]].

=--

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

The underlying [[adjoint triple]] $\Pi \dashv Disc \dashv \Gamma$ in the case of infinitesimal cohesion is an [[ambidextrous adjunction]]. Such a [[localization of a category|localization]] is called a "quintessential localization" by [Johnstone (1996)](#Johnstone96), see at *[[bireflective subcategory]]*.

=--

## Examples

Given an [[(∞,1)-site]] with a [[zero object]], then the [[(∞,1)-presheaf (∞,1)-topos]] over it is infinitesimally cohesive. This class of examples contains the following ones.

### Super $\infty$-groupoids

[[super ∞-groupoids]] are infinitesimally cohesive over [[geometrically discrete ∞-groupoids]], while [[smooth super ∞-groupoids]] are cohesive over [[super ∞-groupoids]] and [[differential cohesion|differentially cohesive]] over [[smooth ∞-groupoids]]

$$
  \array{
    cohesion &&
    SmoothSuper\infty Grpds
    &\stackrel{\overset{\Pi^s}{\longrightarrow}}{\stackrel{\overset{Disc^s}{\leftrightarrow}}{\stackrel{\overset{\Gamma^s}{\longrightarrow}}{\underset{coDisc^s}{\leftarrow}}}}&
   Super \infty Grpds
    \\
    &&{}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}}
    &&
    {}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}}
    \\
    cohesion &&
    Smooth \infty Grpds
    &\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftrightarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}&
    \infty Grpds
    \\
    \\
    && diff.\;cohesion && inf.\;cohesion
  }
$$

### Formal moduli problems/ strong homotopy Lie algebras
 {#FormalModuliProblems}

[[synthetic differential ∞-groupoids]] are cohesive over generalized [[formal moduli problems]]/[[L-∞ algebras]] (generalized meaning without the condition of vanishing on the point and of without the condition of being [[cohesive (∞,1)-presheaf on E-∞ rings|infinitesimally cohesive sheaves in Lurie's sense]]) which in turn are infinitesimally cohesive over [[geometrically discrete ∞-groupoids]].

$$
  \array{
    cohesion
    &&
    SynthDiff\infty Grpds
    &\stackrel{\overset{\Pi^i}{\longrightarrow}}{\stackrel{\overset{Disc^i}{\leftrightarrow}}{\stackrel{\overset{\Gamma^i}{\longrightarrow}}{\underset{coDisc^i}{\leftarrow}}}}&
   L_\infty Alg^{op}_{gen}
    \\
    &&{}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}}
    &&
    {}^{\mathllap{Disc_{inf}}}\uparrow \downarrow^{\mathrlap{\Pi_{inf}}}
    \\
    cohesion && Smooth \infty Grpds
    &\stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftrightarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}&
    \infty Grpds
    \\
     && diff.\;cohesion && inf.\;cohesion
  }
$$

See also at _[[differential cohesion and idelic structure]]_.

### Goodwillie-tangent cohesion
 {#GoodwillieTangentCohesion}

A [[tangent (∞,1)-topos]] $T \mathbf{H}$ is infinitesimally cohesive over $\mathbf{H}$:

$$
  \array{
    && Stab(\mathbf{H})
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
    &
    Stab(\infty Grpd) \simeq Spectra
    \\
    && \simeq && \simeq
    \\
    && T_\ast \mathbf{H} && T_\ast \infty Grpd
    \\
    && \downarrow^{\mathrlap{incl}} && \downarrow^{\mathrlap{incl}}
    \\
    \mathbf{H}
    &\stackrel{\overset{d}{\longrightarrow}}{\underset{\Omega^\infty \circ tot}{\leftarrow}}&
    T \mathbf{H}
    &
    \stackrel{\overset{L\Pi^{seq}}{\longrightarrow}}{\stackrel{\overset{Disc^{seq}}{\leftarrow}}{\stackrel{\overset{\Gamma^{seq}}{\longrightarrow}}{\underset{coDisc^{seq}}{\leftarrow}}}}
  &
  T \infty Grpd
  \\
  &&
  {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}}
  &&
  {}^{\mathllap{base}}\downarrow \uparrow^{\mathrlap{0}}
  \\
  &&
    \mathbf{H}
    &
    \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  &
  \infty Grpd
  }
  \,.
$$

## Related concepts

* [[bireflective subcategory]]

* [[classical modality]]

* [[linear homotopy type theory]]

* [[differential cohesion]]

* [[differential cohesion and idelic structure]]

## References

The notion of infinitesimal cohesion appears under the name "quality type" in def. 1 of

* {#Lawvere07} [[William Lawvere]], _Axiomatic cohesion_, Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

An earlier hint of the same notion seems to be that on the bottom of p. 9 in

* {#Lawvere91}  [[William Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_ (1991)

The above examples of infinitesimal cohesion appear in

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ (2013)

Localization by an [[ambidextrous adjunction]] is also discussed in

* {#Johnstone96} [[Peter Johnstone]], _Remarks on quintessential and persistent localizations_, Theory and Applications of Categories, Vol. 2, No. 8, 1996, pp. 90–99 ([TAC](http://www.tac.mta.ca/tac/volumes/1996/n8/2-08abs.html))

[[!redirects infinitesimal cohesive (infinity,1)-toposes]]
[[!redirects infinitesimal cohesive (infinity,1)-topoi]]

[[!redirects infinitesimal cohesion]]
[[!redirects infinitesimally cohesive]]

[[!redirects infinitesimal cohesive (∞,1)-topos]]
[[!redirects infinitesimal cohesive (∞,1)-toposes]]
[[!redirects infinitesimal cohesive (∞,1)-topoi]]


infinitesimal disk bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}

## Idea

In a context of [[differential cohesion]], with [[infinitesimal shape modality]] $\Im$, then for every [[object]] $X \in \mathbf{H}$ its _infinitesimal disk bundle_ $T_{inf}X$ is the [[bundle]] over $X$ whose [[fiber]] over a [[point]] is the *[[infinitesimal neighbourhood]]* of that point, fomalized as the [[homotopy fiber]] at this point of the [[unit of a monad|unit]] of the [[infinitesimal shape modality]] $\Im$ on $X$.

The collection of these fiber hence forms the [[homotopy fiber product|homotopy]] [[fiber product]]

$$
  \array{
    T_{inf} X &\stackrel{ev}{\longrightarrow}& X
    \\
    \downarrow^{\mathrlap{p}} && \downarrow
    \\
    X &\stackrel{i}{\longrightarrow}& \Im(X)
  }
$$

of the $X$ component $i \colon X \to \Im X$ of the [[unit of a monad|unit]]  of the [[infinitesimal shape modality]] $\Im$ with itself.

Conversely, by the [[pasting law]], the [[fibers]] of $p \colon T_{inf}X \to X$ over [[global points]] of $X$ are indeed the [[infinitesimal disks]] around these points.

Evidently $T_{inf}X$ is the first stage in the [[Cech nerve]] of $X \to \Im(X)$, hence the object of morphisms of the [[groupoid object]] corresponding to this [[effective epimorphism]]. By the discussion at _[Lie algebroid -- General abstract definition](Lie+infinity-algebroid#GeneralAbstractDefinition)_ this is an [[infinity-Lie algebroid]], namely the (possibly higher jet order) [[tangent Lie algebroid]] of $X$.

More generally, for $(E \to X) \in \mathbf{H}_{/X}$ a [[bundle]] over $X$, then $T_{inf}E \coloneqq T_{inf} X \times_{\Im X} E \simeq X\times_{\Im X} E$, sitting in the [[pasting]] composite of pullbacks

$$
  \array{
    T_{inf} E &\longrightarrow& E
    \\
    \downarrow && \downarrow
    \\
    T_{inf} X &\stackrel{ev}{\longrightarrow}& X
    \\
    \downarrow^{\mathrlap{p}} && \downarrow
    \\
    X &\stackrel{i}{\longrightarrow}& \Im(X)
  }
  \,.
$$

Stated more abstractly, this means that forming infinitesimal disk bundles is the [[monad]]

$$
  T_{inf} X \times_X (-)
  =
  i^\ast i_!
$$

induced by the [[adjoint triple]] of [[base change]] along $i$

$$
  (i_! \dashv i^\ast \dashv i_\ast)
  \;\colon\;
  \mathbf{H}_{/X}
   \stackrel{\overset{i_!}{\longrightarrow}}{\stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}}
  \mathbf{H}_{/\Im X}
  \,.
$$

## Properties

### Relation to the formal neighbourhood of the diagonal
 {#RelationToFormalNeighbourhoodOfTheDiagonal}

In the standard models of [[differential cohesion]] (such as for [[formal smooth infinity-groupoids]]), $\Im X$ is the standard [[de Rham stack]] of $X$ obtained by identifying infinitesimal neighbours, and so then $T_{\inf }X$ is the [[formal neighbourhood of the diagonal]] of $X$, in the traditional sense. Indeed, in these standard models $X \to \Im X$ is a [[1-epimorphism]], hence [[effective epimorphism|effective]], and so on 0-truncated $X$ the above pullback equivalently equibits the [[de Rham stack]] $\Im X$ for 0-truncated $X$ as the [[coequalizer]] of the two projections out of the formal neighbourhood of the diagoal, which is the traditional definition of $\Im X$.

### Relation to tangent complexes

The [[tangent complex]] of a [[derived stack|derived]] [[algebraic stack]] $X$ is equivalently the (sheaf of modules of) sections of the [[formal neighbourhood of the diagonal]] of $X$ ([Hennion 13](#Hennion13)). Hence by the [above](#RelationToFormalNeighbourhoodOfTheDiagonal) one may generally think of (sections of) $T_{inf}X$ as being the [[tangent complex]] of $X$.

### Relation to jet bundles
 {#MonadicityAdjointToJetBundles}

The infinitesimal disk bundle construction is the [[left adjoint|left]] [[adjoint monad]] to the [[jet comonad]]

$$
  T_{inf} \dashv Jet
  \,.
$$

The [[underlying]] [[adjoint functor|adjunction]] has been observed in [Kock 1980, prop. 2.2](#Kock80), in the context of [[synthetic differential geometry]] .

In terms of [[differential cohesion]] this pair of [[adjoint monads]] is seen in [Khavkine & Schreiber 2017, p. 23](#KhavkineSchreiber17) as the adjoint pair [induced](monad#RelationBetweenAdjunctionsAndMonads) by the [[base change]] [[adjoint triple]]:

$$
  (i_! \dashv i^\ast \dashv i_\ast)
  \;\colon\;
  \mathbf{H}_{/X}
    \stackrel{\overset{i_!}{\longrightarrow}}{\stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}}
  \mathbf{H}_{/\Im X}
  \,.
$$

### Relation to frame bundles

For $X$ a $V$-[[manifold]], then its infinitesimal disk bundle is a [[fiber bundle]] ([[fiber infinity-bundle]]) with typical fiber $\simeq \mathbb{D}^V_e$. This is the [[associated bundle]] ([[associated infinity-bundle]]) to the [[frame bundle]] $Fr(X) \to X$ (or more generally of the [[higher order frame bundle]] when $(\Re \dashv \Im)$ encodes higher order infinitesimal thickening).

See at _[differential cohesion -- frame bundles](differential+cohesive+%28infinity%2C1%29-topos#GLnTangentBundles)_.

## Related concepts

* [[frame bundle]]

* [[jet bundle]]

## References

Discussion in [[synthetic differential geometry]] is, under the name "bundles of $k$-monads", in

* {#Kock80} [[Anders Kock]], above prop. 2.2 of: _Formal manifolds and synthetic theory of jet bundles_, Cahiers de Topologie et Géométrie Différentielle Catégoriques **21** 3 (1980)  [[numdam:CTGDC_1980__21_3_227_0](http://www.numdam.org/item?id=CTGDC_1980__21_3_227_0)]

* {#Kock09} [[Anders Kock]], p. 39 of _Synthetic Geometry of Manifolds_, 2009 ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

Discussion in [[differential cohesion]] is in

* {#KhavkineSchreiber17} [[Igor Khavkine]], [[Urs Schreiber]], _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

and formalization in [[homotopy type theory]] in

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, PhD Thesis, 2017

Relation to the [[tangent complex]] is discussed in

* {#Hennion13}  [[Benjamin Hennion]], _Tangent Lie algebra of derived Artin stacks_ ([arXiv:1312.3167](http://arxiv.org/abs/1312.3167))
[[!redirects infinitesimal disk bundles]]