AUC

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#### Category theory
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\tableofcontents

## Idea

AUC (ACU in the French language) is a method of naming [[magmoidal categories]]:

* A stands for associative

* U stands for unital

* C stands for commutative

Thus an AU category would be a [[monoidal category]]. The AUC terminology is commonly used in [[rewriting]] theory.

## References

For the AUC terminology in [[category theory]], see:

* [[Hoàng Xuân Sính]], _Gr-catégories_, Ph.D. thesis (1973) [[web](http://w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/Kuenzer/sinh.html)]

* [[Neantro Saavedra-Rivano]], *[[Catégories Tannakiennes]]*, Bulletin de la Société Mathématique de France 100 (1972) 417-430

* [[John Baez]], *Who Invented Monoidal Categories?*, nCafé ([web](https://golem.ph.utexas.edu/category/2023/07/who_invented_monoidal_categori.html))

> "John Baez wrote:

> > Right, it’s an efficient way to name all the possibilities. What makes it look baroque is that nowadays we’ve realized that 99% of the time, we only need two of the possibilities.

> It turns out that in the area of CS called rewriting theory, all the possibilities show up. So they’ve stuck to the AUC naming scheme."
> - [[Jacques Carette]]

[[!redirects ACU]]


Aufhebung

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### Context
#### Foundations
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#### Category theory
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#### Topos Theory
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# Contents
* table of contents
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[[!redirects aufhebung]]
[[!redirects Aufhebungs relation]]
[[!redirects jump operator]]
[[!redirects sublation]]

## Idea

> I am not a "Hegelian". F. W. Lawvere [^LQ]

**Aufhebung** (sublation) is a central concept[^ref] in the [[dialectic|dialectical]] logic of the German philosopher [[Georg Hegel|G. W. F. Hegel]]. The German expression has several meanings for which _tollere, elevare, conservare_ would be Latin equivalents.[^fine]

[^ref]:  A pertinent passage is e.g. [SoL §209](Science+of+Logic#209).

[^fine]: As this polysemy is important for the concept and difficult to preserve in translation we prefer to use the German term in the following.

In his quest to [[axiom|axiomatize]] the concepts of [[space]] and [[cohesion]], [[F. W. Lawvere]], inspired by [[homotopy theory]] proposed a mathematical rendering of the _Aufhebungs_ relation within [[topos theory]] or [[category theory]] more generally. It is the mathematical concept that will constitute the primary subject in the following.

##_Aufhebung_ in Hegel's 'Wissenschaft der Logik'

Although the two volumes of _'[[Science of Logic|Wissenschaft der Logik]]'_  (1st ed. 1812-1816) can be considered as one of the main texts of [[Hegel]]'s philosophy they fell into disfavour in the second half of the 19th century and most of the 20th century, and accordingly received much less attention than the 'Phänomenologie des Geistes' or the 'Rechtsphilosophie'. They shared this fate with Hegelian philosophy as a whole which apart from the philological interest it generated, was continued only through the political wing of Lefthegelianism which in either its existentialist interpretation by A. Kojève or its Marxist interpretation by G. Lukács openly rejected the concept of objective dialectics in nature thereby cutting the social thought from its broad foundation in [[ontology]] and [[logic]], whereas the [[natural philosophy|natural philosophical]] tradition in the vein of F. Engels petrified to the doctrines of dialectical materialism.

The 'Wissenschaft der Logik' has to be viewed against the background of philosophy in the early 19th century: **Kant** had embarked on a project of 'refoundation', or rather demolition, of [[metaphysics]] from an epistemological perspective and this project had been pushed further by his followers especially Fichte in his _Wissensschaftslehre_. However critical these [[idealism|idealist]] systems had been to the claims of traditional metaphysics and epistemology they all left the traditional logic untouched and in this respect fell behind Leibniz. It is at this point where Hegel starts: he sets out to extend the critical examination of the foundations of knowledge to logic itself.

Heavily influenced by the transcendental deductions and the chapters on dialectical paralogisms in Kant's 'Kritik der reinen Vernunft' he intends to start from indeterminate, immediate [[being]] and justify the autonomous development of the system of categories. Here dialectics and Aufhebung enter the picture as Hegel conceives the categories not only as not given apriorily but as actually [[becoming]]: _logic ceases to be an inventory of categories but becomes a system of transformations of categories!_ (Had Eilenberg and MacLane in 1945 intended their terminological loans from philosophy as a kind of joke, Lawvere would 25 years later take this terminological proximity at face value.)

Hegel parts with the traditional conception mainly in two points: the foundations of his logic coalesce with [[ontology]] into an **objective logic** as the first part is titled (a 'logic of things' as [[Charles Sanders Peirce|C.S. Peirce]] would later put it), i.e. he rejects the subject as a possible ground for logic, and he reassesses the status of **negativity** or conflict-contradiction in logic. The cornerstone of the edifice is the anti-eleatic unity of [[being]] and [[nothing|nothingness]] in the idea of [[becoming]]. It is precisely this 'positively being negative' that finds its expression in the concept of 'Aufhebung'.

A key passage on Aufhebung in 'Wissenschaft der Logik' comes at the end of the first chapter ([I.1.1Cc, p.113](#WdL)): after a deduction of the categories of '[[being]]', '[[nothing|nothingness]]', and their unity in '[[becoming]]' Hegel determines **[[dialectics]]** as 'the higher rational movement... in which the precondition of the separatedness (of the seemingly separated) is lifted ( _sich aufhebt_ )' ([p.111](#WdL)).

He goes on ([p.113](#WdL)) to explicate _Aufheben_ as one of the most important concepts in all of philosophy that constantly recurs everywhere. The sublated - _das Aufgehobene_ is not nothing which is an _unmediated_, but is a mediated - _ein Vermitteltes_; it is nothing - _das Nichtseiende_, but as a result that originated from a being and therefore still carries with it the determination from which it derives. This is inspired by Spinoza's 'omnis determinatio est negatio': **Aufhebung** is the mode of this coexistence of negation-affirmation.

Hegel draws his logic from a rich tradition of dialectic going back to Plato in general and to its renewal in Fichte and Schelling's attempts to transcendental philosophy in the 1790s for which in turn Kant's attitude was of capital importance: the latter had established in a short paper _'[[Attempt to Introduce the Concept of Negative Quantities into Philosophy|Versuch den Begriff der negativen Größen in die Weltweisheit einzuführen]]'_ (1763) a distinction between contradictions and real oppositions[^trendel] anticipating the later analytic-synthetic division - here originates the term 'Aufhebung' and gets tied to synthesis of oppositions which is terminologically present in Schelling's use of the terms 'thesis', 'anti-thesis' and 'synthesis' for the dialectical triad.[^triad]

Secondly, Kant had arranged the table of categories in triadic fashion with the third terms _grosso modo_ being the synthesis of the preceding positive respectively negative terms and claimed to have demonstrated the completeness of the resulting table though his deduction was generally considered inconclusive. Starting with Reinhold the deduction of the categories soon became a main concern of transcendental philosophy. In particular, Fichte rejected the primacy of the judgement forms in Kant's approach i.e. the primacy of propositions in logic and gave direct derivations of the categories from the transcendental ego by dialectical steps. Schelling and Hegel followed Fichte in this shift.[^schluss]

Thirdly, Kant's view that reason gets necessarily entangled in the contradictions of _transcendental dialectic_ by its own nature was interpreted by Hegel as an indication of the positive role of contradictions as the driving force of thought.

[^triad]: Hegel apparently didn't use these terms though, probably through their use by F. Engels who was a student of Schelling, the terms stick to Hegelian dialectic today.

[^trendel]: On the avatars of this synthetic negation from Kant through von Trendelenburg to Neo-Kantianism see Giovanelli ([2015](#Giovanelli15)). The connection between Kant's approach to negative quantities and Hegel's concept of contradiction against the background of the mathematics of their time is developed in the groundbreaking study Wolff ([2010](#Wolff10)) first published in 1981. The importance of this link has been stressed already in the 19th century by [[Karl Rosenkranz]].

[^schluss]: Eventually both distanced themselves from the Fichtean ego as the starting point though. Whereas Hegel gives primacy to 'deduction' in logic (cf. [[absolute conclusion]] - _'Alles Vernünftige ist ein Schluss'_) the philosophy of the mature Schelling with its emphasis on the [[transcendental ideal]] can be partly be seen as a return to the classical Pre-Kantian primacy of 'concept' in logic.

Clearly, these remarks can not do justice to the richness and subtlety of Hegel's logic and should only serve as canvas against which to get a better grasp of Lawvere's conceptual translation. The points to keep in mind from this view are:

* Aufhebung is a _pervasive_ concept: although Lawvere proposes only mathematical definitions for two terms of Hegel's logic, namely _[[unity of opposites]]_ and _Aufhebung_, these are in fact the key terms and go already far in a reconstruction of the whole edifice!

* Aufhebung unites _determinateness_ with annihilation of being. These recur at the mathematical level as the correspondance between being-being a sheaf, annihilation -being the adjoint opposite of sheaf (anti-sheaf), and the determination of Aufhebung as being the least (=the Leibnizian best) level of being simultaneously a sheaf and an anti-sheaf.

This is the original text on _Aufhebung_ from [Hegel 1812, book 1, section 1, chapter 1, C, 3. ](Science+of+Logic#AufhebenDesWerdens):

> Aufheben und das Aufgehobene (das Ideelle) ist einer der wichtigsten Begriffe der Philosophie, eine Grundbestimmung, die schlechthin allenthalben wiederkehrt, deren Sinn bestimmt aufzufassen und besonders vom Nichts zu unterscheiden ist.—Was sich aufhebt, wird dadurch nicht zu Nichts. Nichts ist das Unmittelbare; ein Aufgehobenes dagegen ist ein Vermitteltes, es ist das Nichtseyende, aber als Resultat, das von einem Seyn ausgegangen ist; es hat daher die Bestimmtheit aus der es herkommt, noch an sich.

> Aufheben hat in der Sprache den gedoppelten Sinn, daß es so viel als aufbewahren, erhalten bedeutet, und zugleich so viel als aufhören lassen, ein Ende machen. Das Aufbewahren selbst schließt schon das Negative in sich, daß etwas seiner Unmittelbarkeit und damit einem den äußerlichen Einwirkungen offenen Daseyn entnommen wird, um es zu erhalten.—So ist das Aufgehobene ein zugleich Aufbewahrtes, das nur seine Unmittelbarkeit verloren hat, aber darum nicht vernichtet ist. —Die angegebenen zwei Bestimmungen des Aufhebens können lexikalisch als zwei Bedeutungen dieses Wortes aufgeführt werden. Auffallend müßte es aber dabei seyn, daß eine Sprache dazu gekommen ist, ein und dasselbe Wort für zwei entgegengesetzte Bestimmungen zu gebrauchen. Für das spekulative Denken ist es erfreulich, in der Sprache Wörter zu finden welche eine spekulative Bedeutung an ihnen selbst haben; die deutsche Sprache hat mehrere dergleichen. Der Doppelsinn des lateinischen: tollere (der durch den ciceronianischen Witz tollendum esse Octavium, berühmt geworden) geht nicht so weit, die affirmative Bestimmung geht nur bis zum Emporheben. Etwas ist nur insofern aufgehoben, als es in die Einheit mit seinem Entgegengesetzten getreten ist; in dieser nähern Bestimmung als ein reflektirtes kann es passend Moment genannt werden. Gewicht und Entfernung von einem Punkt heißen beim Hebel, dessen mechanische Momente, um der Dieselbigkeit ihrer Wirkung willen bei aller sonstigen Verschiedenheit eines Reellen, wie das ein Gewicht ist, und eines Ideellen, der bloßen räumlichen Bestimmung, der Linie; s. Encykl. der philos. Wissenschaft 3te Ausg. _ 261. Anm.—Noch öfter wird die Bemerkung sich aufdringen, daß die philosophische Kunstsprache für reflektirte Bestimmungen lateinische Ausdrücke gebraucht, entweder weil die Muttersprache keine Ausdrücke dafür hat, oder wenn sie deren hat, wie hier, weil ihr Ausdruck mehr an das Unmittelbare, die fremde Sprache aber mehr an das Reflektirte erinnert.

## Lawvere's path to _Aufhebung_

> In early 1985, while I was studying the foundations of homotopy theory, it occurred to me that the explicit use of a certain simple categorical structure might serve as a link between mathematics and philosophy. ([Lawvere 1996](#Law96), p.167)

## Extracting the rational kernel

So now let's get down to business and do some mathematics!

### The mathematics of Yin and Yang

In ([Lawvere 2000](#Law00)) a particularly simple example of the [[adjoint cylinder]] was suggested that we use here as a warm up. Note that the categories involved are not toposes and even lack a terminal object!

Let $N$ be the [[natural numbers]] $\{0, 1,\dots\}$ viewed as a [[category]] via their usual [[poset|ordering]]. Let $L,R:N\to N$ be the two parallel [[functors]] '_even_' and '_odd_' defined by $L(n) \coloneqq 2n$ and $R(n) \coloneqq 2n+1$.

Both are [[fully faithful functor|full and faithful]], which means that they correspond to two subcategory inclusions and, accordingly, to two subcategories $N_{even}$ and $N_{odd}$. We are now in situation where we have two subcategories that 'oppose' each other in that $N_{even}\neq N_{odd}$ but are nevertheless 'identical' in that there is a bijection $N_{even}\overset{\simeq}{\to} N_{odd}$. Furthermore, both are 'united' as different parts in the encompassing $N$ whose overall structure they represent in that $N_{even}\simeq N\simeq N_{odd}$ - that is somewhat unusual for what is to follow below where the opposing parts are seldom equivalent to the whole but they will always be a pair consisting of a reflective and a coreflective subcategory.

Now it was Lawvere's observation that a third functor $N\to N$ which with a clin d'oeil to [[Charles Sanders Peirce|C. S. Peirce's]] concept of _thirdness_ we call $T$ , can encapsulate this bunch of relations in one sweep when it forms an [[adjoint triple]] $L\dashv T\dashv R$ with $L$ and $R$:

1. The triple expresses the _unity_ by the idempotency of $(R\circ T)^2=R\circ T$ and $(L\circ T)^2=L\circ T$ typical for (co)reflective subcategories,

1. it expresses the _opposition_ between $L$ and $R$ by an entailed adjunction $L\circ T\dashv R\circ T$,

3. it expresses the _identity_ between $L$ and $R$ by the entailed equivalence $T\circ L\simeq T\circ R$ .

In other words, *$T$ unites, opposes and identifies $L$ and $R$ at the same time*!

For our simple [[poset]] example the [[adjunctions]] $L\dashv T$ and  $T\dashv R $ amount to $L(n)\leq m$ iff $n\leq T(m)$ and $n\leq R(m)$ iff $T(n)\leq m$. When $T$ exists it must satisfy $T\circ L \cong id\cong T\circ R$ which in our case just gives $T\circ L = id = T\circ R$. Spelled out this says $T(2n)=n$ and $T(2n+1)=n$ which indicates as definition for $T$:

$$
T(k)=\Bigg\{ \array{\frac{k}{2}\quad k\in N_{even} \\
                       \frac{k-1}{2} \quad k\in N_{odd}}
$$

Whereas $T\circ L$ and $T\circ R$ are each the identity, the reverse compositions $L\circ T$ and $R\circ T$ yield an [[idempotent comonad]] $sk:N\to N$ and an [[idempotent monad]] $cosk:N\to N$, respectively, where $sk(2n)=2n$ and $sk(2n+1)=2n$ and $cosk(2n)=2n+1$ and $cosk(2n+1)=2n+1$: in new guises $L$ and $R$ resurface again but this time within an 'opposition' $sk\dashv cosk$ which expresses formally the 'conflict' between $N_{even}$ and $N_{odd}$, even and odd, as well as their essential identity and unity.

### The mathematics of _Aufhebung_
 {#TheMathematicsOfAufhebung}

For convenience let us briefly recall the following

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

A [[localization of a category]] $\mathcal{B}$ with [[finite limits]] is a [[reflective subcategory]] $\mathcal{A}$ whose reflection preserves finite limits. The localization is called _[[essential geometric morphism|essential]]_ when the reflection has furthermore a [[left adjoint]].

=--

If $l\dashv r\dashv i$ is an essential localization then $l$ is also [[fully faithful functor|full and faithful]]. If $\mathcal{B}$ is a [[topos]], $\mathcal{A}$ is called an _[[essential geometric morphism|essential]] [[subtopos]]_ and we write $i_!\dashv i^*\dashv i_*$ in this case and call $i_!$ the _essentiality_.

It is a result in ([Kelly-Lawvere 89](#KL89)) that the [[essential geometric morphism|essential]] [[subtoposes]] of a topos form a [[complete lattice]]. Therefore we say:

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

An [[essential geometric morphism|essential]] [[subtopos]] of $\mathcal{B}$ is referred to as a _[[level of a topos|level]]_ of $\mathcal{B}$ and levels are denoted by small letters $i,j,\dots$ .

=--

An [[adjoint triple]] $i_!\dashv i^*\dashv i_*$ yields two [[adjoint modalities]] $\Box _i\dashv\bigcirc _i$ on $\mathcal{B}$, namely $\Box _i \coloneqq i_!i^*$ and $\bigcirc _i \coloneqq i_*i^*$.

The [[modalities]] yield notions of _[[modal types]]_, which may be called

* the _i-sheaves_ : $X\in\mathcal{B}$ with $\bigcirc _i X\simeq X$ (following the terminology at _[[Lawvere-Tierney operator]]_);

* the _i-skeleta_ : $X\in\mathcal{B}$ with $\Box _i X\simeq X$ (following the example of [[simplicial skeleta]] discussed [below](#SimplicialAndCubicalSets)).

+-- {: .num_defn #Aufhebung}
###### Definition
([Lawvere 1989b](#Law89b))

Let $i,j$ be [[level of a topos|levels]], def. \ref{Level}, of a topos $\mathcal{A}$ we say that the level $i$ is _lower_ than level $j$, written

$$
  \array{
    \Box_i &\prec & \Box_j
    \\
    \bot && \bot
    \\
    \bigcirc_i &\prec & \bigcirc_j
  }
$$

(or $i\prec j$ for short) when every i-sheaf ($\bigcirc_i$-[[modal type]]) is also a j-sheaf and every i-skeleton ($\Box_i$-[[modal type]]) is a j-skeleton. This is equivalent to say that both $\bigcirc_j \bigcirc_i =\bigcirc_i$ and $\Box_j \Box_i =\Box_i$.

Let $i\prec j$, we say that the level $j$ _resolves the opposite_ of level $i$, written

$$
  \array{
    \Box_i &\ll& \Box_j
    \\
    \bot && \bot
    \\
    \bigcirc_i &\ll& \bigcirc_j
  }
$$

(or just $i\ll j$ for short[^wayb]) if $\bigcirc _j\Box_i=\Box _i$.

Finally a [[level of a topos|level]] $\bar{i}$ is called the _Aufhebung_ of level $i$

$$
  \array{
    \Box_i &\ll& \Box_{\bar i}
    \\
    \bot &\searrow& \bot
    \\
    \bigcirc_i &\ll& \bigcirc_{\bar i}
  }
$$

iff it is a minimal level which resolves the opposites of level $i$, i.e. iff $i\ll\bar{i}$ and for any $k$ with $i\ll k$ then it holds that $\bar{i}\leq k$ in the order relation (by subtopos inclusion) between levels.

=--

[^wayb]: $\ll$ is called the _way below_ relation in ([KRRZ11](#KRRZ11)).

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

The condition $\bigcirc_j \Box_i=\Box_i$ amounts to saying that every $i$-skeleton is a $j$-sheaf:

Suppose the condition holds and $X$ is an $i$-skeleton ($\Box_i X= X$) then $\bigcirc_j X =\bigcirc_j \Box_i X =\Box_i X =X$ i.e. $X$ is a $j$-sheaf. Conversely, if every $i$-skeleton is a $j$-sheaf then, since by the idempotency of $\Box_i$ $i$-skeleta are precisely the objects of form $\Box_i X$ for some $X$, $\Box_i X$ is by assumption a $j$-sheaf and that's precisely what $\bigcirc_j \Box_i X=\Box_i X$ asserts.

The resolution condition $\bigcirc_j \Box_i=\Box_i$ ensures that $i$-skeleta are in the intersection of the $j$-skeleta and $j$-sheaves at the resolving level.

Thinking of $\bigcirc_i\Box_i=\bigcirc_i$ as an expression of the negation of $\Box_i$ by $\bigcirc_i$ one could think of $\bigcirc_j \Box_i=\Box_i$ as dialogical refinement of the opposition through a _negation of the negation_: 'Proponent' $\bigcirc_i$ gets updated - _sublated_ to $\bigcirc_j$ in order to absorb the 'opponent' $\Box_i$.

Note that $\bigcirc_j \Box_i=\Box_i$ does not imply $\Box_j \bigcirc_i =\bigcirc_i$ , e.g. in the [[Sierpinski topos]] $Set^\to$ the level $\emptyset\dashv\ast$ is resolved by $\nabla\dashv ʃ$ but $(\nabla\circ\ast)\neq\ast$ (see [below](#SierpExample)).

We say that $j$ _co-resolves_ $i$ if $\Box_j \bigcirc_i =\bigcirc_i$. If $j$ resolves and co-resolves $i$ we say that $j$ bi-resolves $i$. In the latter case, all $i$-sheaves and $i$-skeleta are simultaneously $j$-sheaves and $j$-skeleta at the higher level $j$.

=--

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

The Aufhebung of a level is the smallest level that resolves its opposites or contradictions. Such a level need not exist in general for every level but in certain cases like [[presheaf toposes]] over [[graphic category|graphic categories]] or, more generally,  over [[von Neumann regular categories]] ([Lawvere 2002](#Law02)), it does. The Aufhebungs relation is also called the _jump operator_ in [Lawvere (2009)](#Law09).

=--

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

Comparing with [[Science of Logic|WdL]] under Lawvere's translation and identifing the [[level of a topos|levels]]  with logical categories of thinking in the ordinary sense (thought determinations - _Gedankenbestimmungen_), one sees that $\emptyset\dashv\ast$ corresponds to Hegel's logical category of _indeterminate being_ whereas the higher levels correspond to logical  categories of _determinate being_ - _Bestimmtheit_.

Furthermore one sees that the subtoposes corresponding to the levels trace out as _mathematical_ categories the _logical_ categories of thought as envisioned by Hegel thereby corroborating the terminological choices of Eilenberg and Mac Lane made in their 1945 paper!

=--

+-- {: .num_remark}
###### Remark
We can use the definition to try to shed some light on the apparently rather odd contention by Hegel that the method of logic is analytic and synthetic at the same time:

> Der Gang oder die Methode des absoluten Wissens ist ebensosehr analytisch als synthetisch. Die Entwicklung dessen, was im Begriff enthalten ist, die Analysis, ist das Hervorgehen verschiedener Bestimmungen, die im Begriff enthalten sind, somit zugleich synthetisch.  ([Begriffslehre für die Mittelklasse (1809/10)](#Begriff), p.161)

We can think of the inclusion of the sheaf category of a lower level into the higher sheaf category as an _analytic_ relation between the concepts involved: when viewed as a relation between the [[geometric theories]] classified by the respective subtoposes an inclusion relation corresponds indeed to an unpacking of the richer theory of the smaller subtopos e.g. the subtopos corresponding to the theory of [[local ring|local rings]] is included in the topos corresponding to the theory of rings which on the conceptual side is spelled out as _a local ring is a ring_, or, the concept 'local ring' implies the concept 'ring'. So the passage from subtopos to including supratopos corresponds to an unfolding of the concepts implied in the subtopos concept.

This analytic procedure seems close to the 'analytical reading' of Hegel's dialectic as a refinement of meaning postulates proposed by F. Berto e.g. in ([Berto 2007](#Berto07)).

Whereas on the _synthetic_ side, by demanding _essentiality_ of the subtoposes we get at each level skeletal 'determinations' corresponding to features _not_ contained in the concept on the sheaf side which by the resolution condition nevertheless get synthesized into the sheaf side on the higher levels.
=--

## Examples

### Aufhebung of Becoming -- Determinate being
 {#AufhebungOfBecoming}

#### From Faust's study

In the context of a **[[category of being]]**, aka a (sufficiently) [[cohesive topos]], which has a connected [[subobject classifier]] $\Omega$ and product preserving components functor $\Pi _0$,  there is an opposition

$$
  \empty\dashv \ast
$$

between _[[nothing|non being]]_ (the [[idempotent comonad]] constant on the [[initial object]]) and _[[being|pure being]]_ (the idempotent comonad constant in the [[terminal object]]) whose Aufhebung is (at least in suitable cases, see [below](#ExamplesBecomingFormalization)) the [[unity of opposites|opposition]] of [[becoming]]

$$
  \flat\dashv \sharp
$$

given by [[flat modality]] $\dashv$ [[sharp modality]], between _non-becoming_ vs. _pure becoming_ (cf. Lawvere 1989a, 1989b, [1991a](#Law91a))[^Pure]. This is what in [SoL § 191](Science+of+Logic#DaseinUberhaupt) is called _[[determinate being]]_ as it corresponds to localization at $\neg\neg$ and the double negation creates the determinateness of the 'Etwasse'.

[^Pure]: Lawvere suggests more generally to read $\Box_ i\dashv\bigcirc_i$ as an opposition _non-F_ vs. _pure-F_ where $F$ is a property descriptively appropriate for the level.

In terms of [[topos theory]] the Aufhebungs-condition $\sharp \emptyset \simeq \emptyset$ says equivalently that the [[subtopos]] of $\sharp$-[[modal objects]] is a [[dense subtopos]].

This lowest essential subtopos arises more generally for categories $\mathcal{A}$ with [[initial object|initial]] and [[terminal objects]], via the adjoints to $\mathcal{A}\to \{*\}$ that map $*$ to $0$ and $1$. Especially, the imposition of conditions that ensure the existence of $\flat\dashv \sharp$ can be viewed as intended to provide a specific resolution of the 'identity' $0=1$, the indeterminate confluence of truth and falsity at the lowest level which [[syntax|syntactically]] corresponds to the inconsistent [[geometric theory]].

Following Lawvere's suggestive terminology and identifying a level with its sheaf part, we could somewhat more loosely say that [[becoming]] is the Aufhebung of the opposition between [[nothing]] and [[being]], or more shortly, that _becoming is the Aufhebung of being_.

The Aufhebungs relation expresses precisely that the (positive) sheaf part of the higher level $j$ subsumes (the opposition between) the skeleton and the sheaf part of the lower level in a universal way - it is the smallest context in which negative and positive poles of the lower level can positively coexist. To elaborate this intuition somewhat, it is the minimal way to turn the negative part into a positive part yet retaining the positivity of its positive opposite.

For more on the relevant _metaphysical_ modalities see at [[adjoint modality]].

#### Over cohesive sites
 {#ExamplesBecomingFormalization}

We discuss Aufhebung of [[becoming]] in the above sense in [[cohesive toposes]] ([[cohesive (∞,1)-toposes]]) with a [[cohesive site]] ([[∞-cohesive site]]) of definition.

(More general discussion is now also in [Lawvere-Menni 15, lemma 4.1, 4.2](#LawvereMenni15), [Shulman 15, section 3](#Shulman15), see at [pieces-to-points-transform -- Relation to Aufhebung](points-to-pieces+transform#RelationToAufhebung)).

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

Given a [[topos]] equipped with a [[level of a topos]] given by an [[adjoint modality]] $(\Box\dashv \bigcirc) \coloneqq (\flat \dashv \sharp)$, then the condition $\sharp \emptyset \simeq \emptyset$ is equivalent to $(\flat X \simeq \emptyset) \Leftrightarrow (X \simeq \emptyset)$.

=--

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

In a topos the [[initial object]] $\emptyset$ is a [[strict initial object]], and hence $(X \simeq \emptyset) \simeq (X \to \emptyset)$.

In one direction, assuming $\sharp \emptyset \simeq \emptyset$ then

$$
  \begin{aligned}
    (X \simeq \emptyset)
    & \simeq
    (X \to \emptyset)
    \\
    & \simeq (X \to \sharp \emptyset)
    \\
    & \simeq (\flat X \to \emptyset)
    \\
    & \simeq (\flat X \simeq \emptyset)
  \end{aligned}
  \,.
$$

Conversely, assume that $(\flat X \simeq \emptyset) \Leftrightarrow (X \simeq \emptyset)$. Then for all $X$

$$
  \begin{aligned}
    (X\to \emptyset)
    & \simeq
    (X\simeq \emptyset)
   \\
    & \simeq (\flat X \simeq \emptyset)
    \\
    & \simeq (\flat X \to \emptyset)
    \\
    & \simeq (X\to \sharp \emptyset)
  \end{aligned}
$$

and hence by the [[Yoneda lemma]] $\emptyset \simeq \sharp \emptyset$.

=--

+-- {: .num_example}
###### Example
In the [[Sierpinski topos]] $Set^{\to}$ with objects maps $X\to Y$ between sets $X,Y$, the initial object is $\emptyset\to\emptyset$ and the respective adjoint modalities are given by $\sharp(X\to Y)=X\to 1$ and $\flat(X\to Y)=X\overset{id}{\to} X$. Since not only $\flat(\emptyset\to\emptyset)=\emptyset\to\emptyset$ but also $\flat(\emptyset\to Y)=\emptyset\to\emptyset$,  we find that $\flat\dashv \sharp$ does not resolve $\emptyset\dashv\ast$ (we expand on this example [below](#SierpExample)).

=--

+-- {: .num_cor}
###### Corollary

An [[(∞,1)-topos]] with Aufhebung $(\flat \dashv \sharp)$ of [[being]] has [[homotopy dimension]] $\leq 0$ with respect to the $\flat$-[[modal types|modal]] [[base (∞,1)-topos]].

=--

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

Let $\mathcal{S}$ be a [[cohesive site]] (or [[∞-cohesive site]]) and $\mathbf{H} = Sh(\mathcal{S})$ its [[cohesive topos|cohesive]] [[sheaf topos]] (or $\mathbf{H} = Sh_\infty(S)$ its [[cohesive (∞,1)-topos]] ).

Then in $\mathbf{H}$ we have $\sharp \emptyset \simeq \emptyset$, hence that $(\flat \dashv \sharp)$ resolves, def. \ref{Aufhebung},  the [[unity of opposites]] $(\emptyset \dashv \ast)$ which is [[becoming]].

=--

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

The [[flat modality]] $\flat$ in this case is given by forming [[global sections]] and re-embedding the resulting [[set]] as a [[constant sheaf]].

Since by assumption $\mathcal{S}$ has a [[terminal object]] $\ast$, it follows that for $X\in \mathbf{H}$ any sheaf $X \colon \mathcal{S}^{op}\to Set$ then

$$
  \flat X \simeq X(\ast)
$$

(where we may leave the constant re-embedding implicit, due to it being [[fully faithful functor|fully faithful]]).

Moreover by assumption, for every object $U\in \mathcal{S}$ there exists a morphism $i \colon \ast \to U$ hence for every $X\in \mathbf{H}$ and every $U$ there exists a morphism $i^\ast \colon X(U)\to \flat X$. This means that if $\flat X \simeq \emptyset$ then $X(U) \simeq \emptyset$ for all $U \in \mathcal{S}$ and hence $X\simeq \emptyset$. From this the claim follows with prop. \ref{AufhebungOfBecomingMeansOnlyInitialObjectHasNoGlobalPoints}.

=--

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

Let $\mathcal{S}$ be a [[cohesive site]] (or [[∞-cohesive site]]) and $\mathbf{H} = Sh(\mathcal{S})$ its [[cohesive topos|cohesive]] [[sheaf topos]] with values in [[Set]] (or $\mathbf{H} = Sh_\infty(S)$ its [[cohesive (∞,1)-topos]] ).

Then in $\mathbf{H}$ we have Aufhebung, def. \ref{Aufhebung}, of the [[duality of opposites]] of [[becoming]] $\emptyset \dashv \ast$ ("[[Dasein]]").

$$
  \array{
     \flat &\dashv& \sharp
     \\
     \vee &\nearrow& \vee
     \\
     \emptyset &\dashv& \ast
  }
$$

=--

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

By prop. \ref{OverCohesiveSiteBecomingIsResolved} we have that $(\flat\dashv \sharp)$ resolves $(\emptyset \dashv \ast)$ and so it remains to see that it is the minimal [[level of a topos|level]] with this property. But the [[subtopos]] of [[sharp modality|sharp]]-[[modal types]] is $\simeq$ [[Set]] which is clearly a [[two-valued topos|two-valued]] [[Boolean topos]]. By [this proposition](subtopos#BooleantoposesAreAtoms) these are the [[atoms]] in the [[subtopos lattice]] hence are minimal as non-trivial subtoposes and hence also as non-trivial [[level of a topos|levels]].

=--

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

As mentioned above, the Aufhebung of $\emptyset\dashv \ast$ is necessarily given by a _[[dense subtopos|dense]]_ subtopos $\mathcal{E}_j$. Since the [[double negation topology]] $\neg\neg$ is the unique largest dense topology  it follows in general that $\mathcal{E}_{\neg\neg}\subseteq\mathcal{E}_j$ , in particular in the case that $\mathcal{E}_{\neg\neg}$ happens to be [[essential geometric morphism|essential]] and hence happens to be a [[level of a topos|level]], the minimality condition on the Aufhebung of the initial opposition means that $\mathcal{E}_j = \mathcal{E}_{\neg\neg}$ is, in particular, a [[Boolean topos]].

=--

A special case of this are toposes $\mathcal{E}$ such that $\mathcal{E}_{\neg\neg}$ is [[open subtopos|open]] whence essential in particular; these are called $\bot$-scattered toposes. For the record we state:

**Proposition.**
Let $\mathcal{E}$ be a $\bot$-[[scattered topos]]. The Aufhebung of $\emptyset\dashv\ast$ is given by $\mathcal{E}_{\neg\neg}$.  $\qed$

Another consequence is that the Aufhebung $\mathcal{E}_j$ of $\emptyset\dashv\ast$ is [[Boolean topos|Boolean]] precisely when $\mathcal{E}_{\neg\neg}$ is essential e.g. for Boolean $\mathcal{E}$ this happens trivially and accordingly the Aufhebung of $\emptyset\dashv \ast$ is $\mathcal{E}$ in this case.

It also follows from the [above proposition](#OverCohesiveSiteBecomingIsAufgehoben) that for [[cohesive sites]] over [[Set]]: $\mathbf{H}_{\neg\neg}=Set$ i.e. the double negation topos coincides  with the base.

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

Examples of $\infty$-toposes satisfying the assumptions of prop. \ref{OverCohesiveSiteBecomingIsAufgehoben} and hence exhibiting Aufhebung of becoming include

* [[∞-groupoids]]

* [[Euclidean-topological ∞-groupoids]]

* [[smooth ∞-groupoids]]

* [[formal smooth ∞-groupoids]]

* [[super ∞-groupoids]]

* [[smooth super ∞-groupoids]]

Here [[formal smooth ∞-groupoid]] has its [[cohesion]] further refined to [[differential cohesion]], yielding

$$
  \array{
     id & \dashv & id
     \\
     \vee && \vee
     \\
     \Re &\dashv& ʃ_{inf} &\dashv& \flat_{inf}
     \\
      && \vee && \vee
     \\
     && ʃ &\dashv& \flat &\dashv& \sharp
     \\
     && && \vee &\nearrow& \vee
     \\
     && && \emptyset &\dashv& \ast
  }
$$

=--

### Absolute identity and _Selbstaufhebung_

Given a topos $\mathcal{E}$ the highest level is always given by $id_\mathcal{E}$ yielding the trivial [[adjoint string]] $id_\mathcal{E}\dashv id_\mathcal{E}\dashv id_\mathcal{E}$ with no less trivial adjoint modalities $id_\mathcal{E}\dashv id_\mathcal{E}$. Obviously, the sheaves and skeleta for this level coincide. Since the corresponding subtopos is simply $\mathcal{E}$, this level _resolves every other level_ and suggests to view the ascension from $\emptyset\dashv\ast$ to $id_\mathcal{E}\dashv id_\mathcal{E}$ as a process of increasing stepwise the number of objects that are sheaves as well as skeleta at a given level. The definition of resolution ensures that a level inherits the objects in the intersection from lower levels but also that all non-sheaves from the lower levels will be henceforth in the intersection.

Note that though $id_\mathcal{E}$ resolves every level it need not be the Aufhebung of any strictly lower level. This situation occurs e.g. in the examples from combinatorial topology discussed [below](#SimplicialAndCubicalSets): Here the levels correspond to the geometrical dimension of the 'triangulated spaces' involved plus the highest level 'at infinity' and the Aufhebung is a simple numerical relation between finite levels e.g. in the topos of ball complexes [[Roy (1997)](#Roy97)] the Aufhebung of level $n$ is $n+1$. Of course, $id_\mathcal{E}$ is always its own Aufhebung and we see, incidentally, that a level might be the Aufhebung of more than one level, namely e.g. itself and perhaps several other levels - this might happen with $id_\mathcal{E}$ when the lattice of levels is finite e.g. in the [[Sierpinski topos]] $Set^{\to}$ (see [below](#SierpExample)).

Though $id_\mathcal{E}$ is trivial from a mathematical point of view, paradoxically it nevertheless captures on the philosophical side one of the most enigmatic concepts of the early _identity philosophy_ of Hegel and Schelling, namely the **absolute** as the identity of identity and non-identity:

>Das Absolute selbst aber ist darum die Identität der Identität und der Nichtidentität; Entgegensetzen und Einssein ist zugleich in ihm. (Hegel [1801](#Diff), p.96)

We might call a topos $\mathcal{E}$ with the property that $id_\mathcal{E}$ is the Aufhebung of no other level than itself _absolute_ and $id_\mathcal{E}$ the absolute level. Note that these toposes will occasionally have infinitely many levels and are then from the perspective of the mature Hegel vulnerable to the charge of being a _'bad infinity'_.

The discussion so far might suggest that being self-sublating is a property peculiar to $id_\mathcal{E}$ but this is not the case. In fact $id_\mathcal{E}$ is only an instance of a whole class of essential localizations enjoying this property:

+-- {: .num_defn #quintessentialloc}
###### Definition
Let $\mathcal{C}$ be a finitely complete category. An [[level|essential localization]] $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ is called **quintessential** if $l$ is naturally isomorphic to $i$.
=--

To say that $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ is a quintessential localization amounts to say in Lawvere's terminology ([2007](#07)) that $i:\mathcal{L}\to\mathcal{C}$ exhibits $\mathcal{C}$ as a [[quality type]] over $\mathcal{L}$ with $r$ providing the right adjoint to $i\simeq l$ (provided $\mathcal{L}$, $\mathcal{C}$ are [[extensive category|extensive]]).

The following is immediate:

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

Let $l\dashv r\dashv i:\mathcal{L}\to\mathcal{C}$ be a quintessential localization. Then the corresponding [[adjoint modality]] $l\cdot r\dashv i\cdot r$ coincides up to natural isomorphism and provides its own Aufhebung.  $\qed$
=--

+-- {: .num_example}
###### Example
A simple _example_ of a non-trivial quintessential localization is given by the category $\mathcal{C}$ with objects pairs $(X, e)$ where $X$ is a set and $e=e^2$ an idempotent map $X\to X$. A morphism $f:(X_1, e_1)\to (X_2, e_2)$ is a function $f:X_1\to X_2$ with $f\cdot e_1=e_2\cdot f$. These equivariant morphisms are bound to preserve fixpoints: when $e_1(x)=x$ then $f(e_1(x))=f(x)=e_2(f(x))$. Then the fixpoint set functor $r:\mathcal{C}\to Set$ with $r(X, e)=\{x\in X | e(x)=x \}$ is left as well as right adjoint to $i(X)=(X, id_X)$ since an equivariant morphism $f:(X,e)\to (Y,id_Y)$ is uniquely determined by its restriction to the fixpoints of $e$ and its values are given by $f(e(x))$. The [[adjoint modality]] $i\cdot r\dashv i\cdot r:\mathcal{C}\to\mathcal{C}$ corresponding to $i\dashv r\dashv i:Set\to\mathcal{C}$ maps $(X,e)$ to $(r(X),id_{r(X)})$. The corresponding level sublates $\emptyset\dashv\ast$ as well as itself whereas $id_\mathcal{E}$ only sublates itself.
=--

For further properties of quintessential localizations see at [[quality type]].

### The example of the Sierpinski topos{#SierpExample}

(...)

### Simplicial and cubical sets
 {#SimplicialAndCubicalSets}

(...) [[simplicial set]] (...)

[[simplicial skeleton]] $\dashv$ [[simplicial coskeleton]]

(...) [[cubical set]] (...)

([Kennett-Riehl-Roy-Zaks (2011)](#KRRZ11))

### An open problem: the presheaf topos over non-empty finite sets

...

## Related pages

* [[Hegelian taco]]

* [[Science of Logic]]

* [[adjoint modality]]

* [[adjoint logic]]

* [[adjoint triple]]

* [[level of a topos]]

* [[negative moment]]

* [[quality type]]

* [[graphic category]]

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

* [[construction in philosophy]]

* [[infinite judgement]]

* [[co-Heyting boundary]]

* [[homotopy dimension]]

## Links

* [Online Journal of the Brazilian Hegel Society](http://revista.hegelbrasil.org/english-version-introduction/)

## A guide to the literature

On the philosophical side, the lectures notes [Koch (2009)](#Koch09) that suggest the use of [[pure set|non-wellfounded set theory]] as interpretative tool may serve as a good general introduction to Hegel's ideas on logic and metaphysics. [Krahn (2014)](#Krahn14) considers Hegel's concept of Aufhebung in the context of postmodern thought. For lucid accounts of Hegel's concept of dialectics in general consult Brauer ([1982](#Brauer82)) and Wolff ([2010](#Wolff10)) the latter highlighting Hegel as 'a philosopher of mathematics' in this context. Wegerhoff ([2008](#Wegerhoff08)) offers an interesting structural account of the dialectics inspired by Dedekind's theory of natural numbers.

On the mathematical side, the book by [La Palme-Reyes-Zolfaghari (2004)](#RRZ04) provides a good general entry to the 'mathematics of Lawvere' from an elementary point of view and contains even a page on the adjoint cylinder. Goldblatt's book on [[topos theory]] ([Goldblatt 1984](#Goldblatt84)) covers a good deal of ground yet stays accessible and is available online.

Lawvere introduced the Hegelian concepts in Lawvere ([1989b](#Law89b)). They get some attention in Lawvere ([1991a](#Law91a),[1992](#Law92),[1994a](#Law94a)) with the second containing his 'philosophical program'. By all means have a look at [Lawvere (1996)](#Law96), this together with Lawvere ([1989a](#Law89a),[1999](#Law99)) exposes his ideas on homotopy theory. The  work on graphic toposes ([1989b](#Law89b),[1991b](#Law91b),[2002](#Law02)) concerns the _Aufhebungs_ relation with the latter containing a discussion of the relevant concepts. [Kelly-Lawvere (1989)](#KL89) provides the technical prerequisites on essential localizations for _Aufhebung_.

The known mathematical results on the Aufhebungs relation are contained in the paper by [Kennett-Riehl-Roy-Zaks (2011)](#KRRZ11) which is based on older phd-works by some of the authors (e.g. [Roy  (1997)](#Roy97)).

Further results on essential localizations can be found in the papers by [Borceux-Korotenski (1991)](#BK91), [Johnstone (1996)](#JS96), [Vitale (2001)](#VT01) and [Lucyshyn-Wright (2011)](#LW11) or in [SGA 4](#SGA4).

***

<img src="http://ncatlab.org/nlab/files/AufhebungDetail.jpg" width="900">
([full pdf](http://www.in-situ-art-society.com/docs/2015-09-18+19_aufhebung_flyer.pdf) by [In Situ Art Society](http://www.in-situ-art-society.com/aufhebung.html))

***

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* [[R. Street]], *The petit topos of globular sets*, JPAA **154** (2000) pp.299-315.

* {#VT01}[[Enrico Vitale|E. M. Vitale]], *Essential Localizations and Infinitary Exact Completions*, TAC **8** no.17 (2001) pp.465-480. ([pdf](http://www.tac.mta.ca/tac/volumes/8/n17/n17.pdf))

* {#Wegerhoff08} Tilman Wegerhoff, _Hegels Dialektik - Eine Theorie der positionalen Differenz_ , Vandenhoeck&Ruprecht Göttingen 2008.

* {#Wolff10} Michael Wolff, _Der Begriff des Widerspruchs - Eine Studie zur Dialektik Kants und Hegels_ , Frankfurt UP ²2010.

* {#WR93}[[Gavin C. Wraith]], *Using the Generic Interval*, Cah. Top. Géom. Diff. Cat. **XXXIV** 4 (1993) pp.259-266. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1993__34_4/CTGDC_1993__34_4_259_0/CTGDC_1993__34_4_259_0.pdf))

[^LQ]: ([Lawvere 1989](#Law89b), p.74).


augmentation

#Contents#
* table of contents
{:toc}

## Idea

An **augmentation** of a [[simplicial set]] or generally a [[simplicial object]] $S_\bullet$ is a [[homomorphism]] of simplicial objects to a simplicial object constant ([[discrete object|discrete]]) on an object $A$:

$$
  \epsilon \colon S_\bullet \to A
  \,.
$$

Equivalently this is an _[[augmented simplicial set|augmented simplicial object]]_, namely a [[diagram]] of the form

$$
  \array{
     \cdots S_2 \stackrel{\to}{\stackrel{\to}{\to}} S_1 \stackrel{\to}{\to} S_0 \stackrel{\epsilon_0}{\to} A
  }
$$

(showing here only the face maps).

Under the [[Dold-Kan correspondence]] this yields:

The **augmentation of a chain complex** $V_\bullet$ (in non-negative degree) is a [[chain map]]

$$
  \epsilon \colon V_\bullet \to A
  \,.
$$

If $V_\bullet$ and $A$ are equipped with [[algebra]]-[[structure]] ($V$ might be an [[augmented algebra]] over $A$), then the [[kernel]] of the augmentation map is called the  **[[augmentation ideal]]**.

## Related concepts

* [[reduced homology]]

* [[augmented algebra]], [[augmented A-infinity algebra]]

* [[augmentation ideal]]

[[!redirects augmentations]].


augmentation ideal

+-- {: .rightHandSide}
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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $R \hookrightarrow A$ an [[associative algebra]] over a [[ring]] $R$ equipped with the structure of an [[augmented algebra]] $\epsilon \colon A \to R$, the _augmentation ideal_ is the [[kernel]] of $\epsilon$.

Specifically for $G$ a [[group]], and $R[G]$ its [[group algebra]] over a [[ring]] $R$, the _augmentation ideal_ is the [[ideal]] in $R[G]$ which consists of those [[formal linear combinations]] over $R$ of elements in $G$ whose sum of [[coefficients]] vanishes in $R$.

## Examples

### For group algebras

Let $G$ be a [[discrete group]] and $R$ a [[ring]]. Write $R[G]$ for the [[group algebra]] of $G$ over $R$.

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

Write

$$
  \epsilon \colon \mathbb{Z}[G] \to \mathbb{Z}
$$

for the [[homomorphism]] of [[abelian groups]] which forms the sum of $R$-[[coefficients]] of the [[formal linear combinations]] that constitute the group ring

$$
  \epsilon \colon r \mapsto \sum_{g \in G} r_g
  \,.
$$

This is called the [[augmentation]] map. Its [[kernel]]

$$
  ker(\epsilon) \hookrightarrow \mathbb{Z}[G]
$$

is the **augmentation ideal** of $\mathbb{Z}[G]$. (It is often denoted by $I(G)$.

=--

## Properties

### General

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

The augmentation ideal is indeed a left and right [[ideal]] in $R[G]$.

=--

### For group algebras

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

The $R$-[[module]] underlying the augmentation ideal of a [[group algebra]] is a [[free module]], free on the set of elements

$$
  \{
    g - e | g \in G,\; g \neq e
  \}
$$

in $R[G]$.

=--

+-- {: .num_prop }
###### Proposition
(For the case $R= \mathbb{Z}$)

As a $\mathbb{Z}[G]$-module, considered with the same generators, the relations are generated by those of the form

$$g_1(g_2-e)= (g_1g_2-e)-(g_1-e).$$

=--

##Related entries
 * [[derived module]]

[[!redirects augmentation ideals]]


augmented A-infinity algebra

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#### Higher algebra
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[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _augmented $A_\infty$-algebra_ is the analogue in [[higher algebra]] of the notion of _[[augmented algebra]]_ in ordinary algebra: an [[A-∞ algebra]] euipped with a [[homomorphism]] to the base [[E-∞ ring]] (which might be a plain [[commutative ring]]).

## Definition
 {#Definition}

Let $R$ be an [[E-∞ ring]] and $A$ an [[A-∞ algebra]] over $R$.

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

An **augmentation** of $A$ is an $R$-[[A-∞ algebra]] [[homomorphism]]

$$
  \epsilon \colon A \to R
 \,.
$$

=--

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

In as far as one considers [[A-∞ algebras]] are presented by [[simplicial objects]] or similar, there might also be a (less intrinsic) notion of [[augmentation]] as in _[[augmented simplicial sets]]_. This is _not_ what the above defines.

=--

Fully generally, a definition of augmentation of [[∞-algebras over an (∞,1)-operad]] is in ([Lurie, def. 5.2.3.14](#Lurie)).

## Examples

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

An augmentation of an [[E-∞ ring]] $R$, being an [[E-∞ algebra]] over the [[sphere spectrum]] $\mathbb{S}$, is a homomorphism

$$
  \epsilon \colon R \to \mathbb{S}
$$

to the [[sphere spectrum]], regarded as an [[E-∞ ring]].

Forming [[augmentation ideals]] constitutes an [[equivalence of (∞,1)-categories]]

$$
  E_\infty Ring_{/\mathbb{S}}
  \stackrel{\simeq}{\longrightarrow}
  E_\infty Ring^{nu}
$$

of $\mathbb{S}$-augmented $E_\infty$-rings and [[nonunital E-∞ rings]] ([Lurie, prop. 5.2.3.15](#Lurie)).

=--

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

A [[bipermutative category]] $\mathcal{C}$ induces (as discussed there) an [[E-∞ ring]] $\vert \mathcal{C}\vert$. If $\mathcal{C}$ is equipped with a bi-[[monoidal functor]] $\mathcal{C} \to \mathcal{Z}$ then this induces an augmentation of $\vert \mathcal{C}\vert$ over $H \mathbb{Z}$, the [[Eilenberg-MacLane spectrum]] of the [[integers]].

=--

See for instance ([Arone-Lesh](#AroneLesh))

## Related concepts

* [[augmented algebra]], [[augmentation ideal]]

* [[augmentation]], [[augmented simplicial set]]

* [[augmented ∞-group]]

## References

For $A_\infty$-algebras in [[characteristic]] 0 (in chain complexes) augmentation appears for instance as def. 2.3.2.2 on p. 81 in

* Kenji Lefèvre-Hasegawa, _Sur les A-infini catégories_ ([arXiv:math/0310337](http://arxiv.org/abs/math/0310337))

augmentation of $\mathbb{F}_p$ [[E-∞ algebras]] is considered in definition 7.1 of

* [[Michael Mandell]], _$E_\infty$-Algebras and $p$-adic homotopy theory_ ([pdf](hopf.math.purdue.edu/Mandell/einffinal.pdf))

The following articles discuss (just) [[augmented ∞-groups]].

Augmentation (of [[∞-groups of units]] of [[E-∞ rings]]) over the [[sphere spectrum]]  appears in

* Steffen Sagave, _Spectra of units for periodic ring spectra_ ([arXiv:1111.6731](http://arxiv.org/abs/1111.6731))

Augmentation over the [[Eilenberg-MacLane spectrum]] $H\mathbb{Z}$ appears in

* Gregory Arone, Kathryn Lesh, _Augmented $\Gamma$-spaces, the stable rank filtration, and a $b u$-analogue of the Whitehead conjecture_ ([pdf](http://www.math.union.edu/~leshk/papers/FilteredGammaSpaces-revision.pdf))
 {#AroneLesh}

See also

* {#Fresse06} [[Benoit Fresse]], _The Bar Complex of an E-infinity Algebra_, Adv. Math.  223 (2010), pages 2049-2096 ([arXiv:math/0601085](http://arxiv.org/abs/math/0601085))

and

* {#Schwede01} [[Stefan Schwede]], section 7.8 of _Stable Homotopy of Algebraic Theories_, 2001 ([pdf](http://www.math.uni-bonn.de/people/schwede/stable.pdf))

with comments on the relation to [[nonunital algebras]].

Fully general discussion in [[higher algebra]] is in

* {#Lurie} [[Jacob Lurie]], section 5.2.3 in _[[Higher Algebra]]_

[[!redirects augmented A-infinity-algebras]]

[[!redirects augmented A-∞ algebra]]
[[!redirects augmented A-∞ algebras]]

[[!redirects augmented E-∞ algebra]]
[[!redirects augmented E-∞ algebras]]

[[!redirects augmented E-∞ ring]]
[[!redirects augmented E-∞ rings]]


augmented algebra

+-- {: .rightHandSide}
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#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $R$ a [[ring]], an [[associative algebra]] over $R$ is a [[ring]] $A$ equipped with a ring inclusion $R \hookrightarrow A$.

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

If the $R$-algebra $A$ is equipped with an $R$-algebra [[homomorphism]] the other way around,

$$
  \epsilon \colon A \to R
  \,,
$$

then it is called an _augmented $R$-algebra_.

=--

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

In [Cartan-Eilenberg](#CartanEilenberg) this is called a _supplemented algebra_.

=--

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

The [[kernel]] of $\epsilon$ is called the corresponding [[augmentation ideal]] in $A$.

=--

## Examples

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

An augmentation of a bare [[ring]] itself, being an [[associative algebra]] over the ring of [[integers]] $\mathbb{Z}$, is a ring homomorphism to the integers

$$
  \epsilon \colon R \to \mathbb{Z}
$$

=--

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

Every [[group algebra]] $R[G]$ is canonically augmented, the augmentation map being the operation that forms the sum of [[coefficients]] of the canonical basis elements.

=--

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

If $X$ is a variety over an algebraically closed field $k$ and $x\in X(k)$ is a closed point, then the local ring $\mathcal{O}_{X,x}$ naturally has the structure of an augmented $k$-algebra. The augmentation map $\mathcal{O}_{X,x}\rightarrow k$ is the evaluation map, and the augmentation ideal is the maximal ideal of $\mathcal{O}_{X,x}$.

=--

## Related concepts

* [[augmentation]]

* [[augmented A-∞ algebra]]

## References

* [[Henri Cartan]], [[Samuel Eilenberg]], _Homological algebra_
 {#CartanEilenberg}

[[!redirects augmented algebras]]
[[!redirects augmentation map]]
[[!redirects augmentation maps]]


augmented infinity-group

#Contents#
* table of contents
{:toc}

## Idea

An [[∞-group]] equipped with a map to a fixed base [[∞-group]], such as that underlying the [[sphere spectrum]] or an [[Eilenberg-MacLane spectrum]].

## Related concepts

* [[augmentation]]

* [[augmented algebra]], [[augmentation ideal]]

* [[augmented A-infinity algebra]]

## References

Discussion of [[∞-groups of units]] of [[E-∞ rings]] as [[abelian ∞-groups]] augmented over the [[sphere spectrum]] is in

* Steffen Sagave, _Spectra of units for periodic ring spectra_ ([arXiv:1111.6731](http://arxiv.org/abs/1111.6731))

Augmentation over the [[Eilenberg-MacLane spectrum]] $H\mathbb{Z}$ appears in

* Gregory Arone, Kathryn Lesh, _Augmented $\Gamma$-spaces, the stable rank filtration, and a $b u$-analogue of the Whitehead conjecture_ ([pdf](http://www.math.union.edu/~leshk/papers/FilteredGammaSpaces-revision.pdf))
 {#AroneLesh}

[[!redirects augmented infinity-groups]]

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


augmented simplicial set

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

# Augmented simplicial sets
* table of contents
{: toc}

## Idea

Where an ordinary [[simplicial set]] $X$ may be thought of a [[space]] made up of $n$-[[simplices]] $\sigma \in X_n$ for all $n \in \{0,1,2,3, \cdots \}$, an _augmented simplicial set_ in addition has a set $X_{-1}$ of "$(-1)$-simplices" such that each 0-simplex has a single $(-1)$-dimensional face and such that the $(-1)$-dimensional faces of the two faces of any $1$-simplex coincide.

Equivalently this may be thought of as the data that encodes a [[morphism]] $X \to const X_{-1}$ in [[sSet]] between ordinary simplicial sets from an _underlying_ simplicial set to a _constant_ simplicial set with $X_{-1}$ as its set of $k$-simplices for all $k$.

It is in this latter form that augmented simplicial sets maybe mostly arise in practice, whereas the former incarnation offers a more succinct way of thinking about them.

For instance a major source of augmented [[simplicial object]]s are given by [[colimit]]s or rather [[homotopy colimit]]s over simplicial diagrams in a [[model category]]: for $X\colon \Delta^{op} \to C$ a simplicial object in a category with colimits, its colimit [[cocone]] may be thought of as a morphism

$$
  X_\bullet \to const \lim_{\to} X_\bullet
$$

into the constant simplicial object.

## Definitions

Denote by $\Delta_+$ (also denoted $\Delta_a$) the [[augmented simplex category]], which may be defined as the [[full subcategory]] of [[Cat]] on [[free categories]] over finite and _possibly empty_ linear [[directed graphs]], which are

* $[-1] := \emptyset$;

* $[0] := (0)$;

* $[1] := (0 \to 1)$;

* $[2] := (0 \to 1 \to 2)$;

* and so on.

An **augmented simplicial set** $X$ is a [[presheaf]] on $\Delta_+$ and the [[category]] of augmented simplicial sets, [[ASSet|$sSet_+$]], is the presheaf category

$$
  [\Delta_+^{op}, Set]
  \,.
$$

There is a canonical inclusion $\Delta \hookrightarrow \Delta_+$ of the ordinary [[simplex category]] and that the [[restriction]] of an augmented simplicial set along this inclusion is a simplicial set. This gives a [[forgetful functor]]

$$
  U : sSet_+ \to sSet
  \,.
$$

A cartoon of an augmented simplicial set, showing just the face maps, looks like

$$
  X_\bullet =
  \left(
  \cdots
  X_2 \stackrel{\to}{\stackrel{\to}{\to}} X_1 \stackrel{\overset{d^0_1}{\to}}{\underset{d^0_0}{\to}} X_0 \stackrel{d^{-1}}{\to} X_{-1}
  \right)
$$

where the only new [[simplicial identity]] satisfied by the new face map in degree $-1$ is that it coequalizes the degree-$0$ face maps in that

$$
  d^{-1}\circ d^0_1 = d^{-1}\circ d^0_0
  \,.
$$

It follows that $d^{-1}$ coequalizes in fact every pair of composites of face maps $X_n \to X_0$, so that equivalently an augmented simplicial set $X_\bullet$ is a morphism of ordinary simplicial sets

$$
  U(X_\bullet) \to const X_{-1}
  \,.
$$

We say that that $U(X_\bullet)$ is _augmented_ over $X_{-1}$.

More explicitly, an __augmented simplicial set__ consists of

*  for each [[integer]] $n \geq -1$, a [[set]] $X_n$ (so an [[infinite sequence]] $(X_{-1},X_0,X_1,X_2,\ldots)$ of sets),
*  for each integer $n \geq -1$ and each [[natural number]] $m \leq n + 1$, a __face map__ $d^n_m\colon X_{n+1} \to X_n$,
*  for each integer $n \geq -1$ and each natural number $m \leq n$, a __degeneracy map__ $s^n_m\colon X_n \to X_{n+1}$,
*  satisfying the [[simplicial identities]].

Above, we use the traditional system of numbering for a [[simplicial set]].  However, part of the motivation behind augmented simplicial sets is that this is a more sensible numbering system:

*  for each [[natural number]] $n$, a [[set]] $X_n$ (so an [[infinite sequence]] $(X_0,X_1,X_2,\ldots)$ of sets),
*  for each natural number $n$ and each natural number $m \leq n$, a __face map__ $d^n_m\colon X_{n+1} \to X_n$,
*  for each natural number $n$ and each natural number $m \lt n$, a __degeneracy map__ $s^n_m\colon X_n \to X_{n+1}$,
*  satisfying the [[simplicial identities]].

While this numbering is very nice for augmented simplicial sets, it is not standard and is can be easily misunderstood, so we don\'t use it in this article.

## Properties

Anything that applies to [[simplicial sets]] should also apply to augmented simplicial sets, if one properly takes care of the [[negative thinking]] necessary to deal with $X_{-1}$.

Every augmented simplicial set has an underlying unaugmented simplicial set found by forgetting $X_{-1}$ (and $d^{-1}_0$).  Conversely, every unaugmented simplicial set gives rise to a [[free object|free]] augmented simplicial set by augmentation over $\pi_0(X)$ (the set of [[connected component]]s of $X$) and a [[cofree object|cofree]] augmented simplicial set by augmentation over the [[point]] (the [[singleton set]]).  This defines [[adjunctions]]:
$$
\vdash\mathclap{\underoverset{\textsize{\operatorname{AugSimpSet}}}{\textsize{\operatorname{SimpSet}}}{\begin{matrix}\begin{svg}
<svg width="47" height="66" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="12691">
 <g>
  <title>Layer 1</title>
  <path fill="none" stroke="#000000" d="m15.14844,3c-21,22.035261 -18,43.469551 0,62.5" id="svg_12691_1" marker-start="url(#se_marker_start_svg_12691_1)"/>
  <path fill="none" stroke="#000000" d="m31.500429,65.25c21.000002,-21.947319 18.000002,-43.295465 0,-62.25" id="svg_12691_2" marker-end="url(#se_marker_end_svg_12691_2)"/>
  <line fill="none" stroke="#000000" x1="23.148436" y1="0.500004" x2="23.148436" y2="63.504595" id="svg_12691_3" marker-end="url(#se_marker_end_svg_12691_3)"/>
 </g>
 <defs>
  <marker id="se_marker_start_svg_12691_1" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50">
   <path id="svg_12691_4" d="m0,50l100,40l-30,-40l30,-40l-100,40z" fill="#000000" stroke="#000000" stroke-width="10"/>
  </marker>
  <marker id="se_marker_end_svg_12691_2" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50">
   <path id="svg_12691_5" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#000000" stroke="#000000" stroke-width="10"/>
  </marker>
  <marker id="se_marker_end_svg_12691_3" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50">
   <path id="svg_12691_6" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#000000" stroke="#000000" stroke-width="10"/>
  </marker>
 </defs>
</svg>
\end{svg}\includegraphics[width=35]{vertarrows}\end{matrix}}}\vdash
$$

## Examples

* For $C$ a [[site]], $\{U_i \to X\}$ a [[covering]] family we have the [[Cech nerve]] [[simplicial presheaf]] $C(U) \in [C^{op}, sSet]$. This comes canonically equipped with a morphism of simplicial presheaves to the one represented by $X\colon C(U) \to X$. This is an augmented simplicial object in the category of presheaves $[C^{op}, Set]$. Morover, over each object $V \in C$ its component

  $$
    C(U)(V) \to X(V)
  $$

  is an augmented simplicial set.

  More generally this applies to [[hypercover]]s.

## Related concepts

* [[augmentation]], [[augmentation ideal]]

* [[augmented algebra]]

[[!redirects augmented simplicial set]]
[[!redirects augmented simplicial sets]]

[[!redirects augmented simplicial object]]
[[!redirects augmented simplicial objects]]


augmented Teichmüller space

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

#Contents#
* table of contents
{:toc}

## Idea

The _augmented Teichmüller space_ ([Bers 73](#Bers73)) is [[Teichmüller space]] with also the points for the degenerate [[nodal curve]] admitted.

Accordingly, the [[orbifold quotient]] of the augmented Teichmüller space by the [[mapping class group]] is equivalently the [[Deligne-Mumford compactification]] of the [[moduli space of curves]] over the complex numbers ([Hubbard-Koch 13, main theorem](#HubbardKoch13)).

## References

The concept was introduced in

* {#Bers73} [[Lipman Bers]], _Spaces of degenerating Riemann surfaces. Discontinuous groups and Riemann surfaces_, Ann. Math. Stud. 79, 43–55 (1973).

Discussion of [[orbifold]] structure is in

* {#HinichVaintrob07} [[Vladimir Hinich]], [[Arkady Vaintrob]], _Augmented Teichmuller Spaces and Orbifolds_ ([arXiv:0705.2859](http://arxiv.org/abs/0705.2859))

Discussion of the relation to the [[Deligne-Mumford compactification]] of the [[moduli stack of curves]] is in

* {#HubbardKoch13} [[John Hubbard]], [[Sarah Koch]], _An analytic construction of the Deligne-Mumford compactification of the moduli space of curves_ ([arXiv:1301.0062](http://arxiv.org/abs/1301.0062))

[[!redirects augmented Teichmuller space]]


augmented virtual double category

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[[!include 2-category theory - contents]]
=--
=--
=--

# Augmented virtual double categories
* table of contents
{: toc}

## Idea

An **augmented virtual double category** is a [[virtual double category]] enhanced with additional [[2-cells]] whose vertical [[target]] has length 0 (i.e. is a single object rather than a horizontal arrow).

In particular, there is always a "vertical 2-category" consisting of the objects, vertical arrows, and 2-cells whose vertical source and target are both length 0.  A virtual double category, by contrast, does not have a vertical 2-category unless it has all units.

## Examples

[[large category|Large]] (not necessarily [[locally small category|locally small]]) [[categories]], [[functors]], small-set-valued [[profunctors]], and transformations form an augmented virtual double category.  Note that only locally small categories have units therein, so the underlying virtual double category does not have enough data to reconstruct the 2-category of large categories, functors, and natural transformations; but the augmented virtual double category does.

## Applications

Augmented Virtual double categories are a natural context in which to compare [[proarrow equipments]] and [[Yoneda structures]], which are two different approaches to [[formal category theory]].  In both cases there is a notion of "[[profunctor]]" which are generally considered to be small-set-valued, but [[Yoneda structures]] require non-locally-small categories (the presheaf categories of non-small categories).  In this context the Yoneda embedding can be given a universal property relative to the horizontal arrows.  See [(Koudenburg)](#Koudenburg19).

## Related pages

* [[virtual double category]]
* [[virtual equipment]]
* [[Yoneda structure]]

## Reference

* {#Koudenburg15} [[Seerp Roald Koudenburg]], *A double-dimensional approach to formal category theory* ([arXiv:1511.04070](http://arxiv.org/abs/1511.04070)) 2015.

  > (motivated by [[formal category theory]])

* {#Koudenburg19} [[Seerp Roald Koudenburg]], *Augmented virtual double categories*, Theory and Applications of Categories, Vol. 35, 2020, No. 10, pp 261-325 ([arXiv:1910.11189](https://arxiv.org/abs/1910.11189), [tac:35-10](http://www.tac.mta.ca/tac/volumes/35/10/35-10abs.html))

  > (streamlined and expanded version of Sec. 1-3 of [Koudenburg 15](#Koudenburg15) )

[[!redirects hypervirtual double category]]
[[!redirects hypervirtual double categories]]
[[!redirects augmented virtual double categories]]


August Ferdinand Möbius

* [Wikipedia entry](https://en.wikipedia.org/wiki/August_Ferdinand_M%C3%B6bius)

## related $n$Lab entries

* [[projective space]]

* [[Möbius strip]]

* [[Möbius transformation]]

category: people

[[!redirects August Möbius]]
[[!redirects Möbius]]

[[!redirects August Ferdinand Moebius]]
[[!redirects August Moebius]]
[[!redirects Moebius]]

[[!redirects August Ferdinand Mobius]]
[[!redirects August Mobius]]
[[!redirects Mobius]]


Auguste Bravais

* [Wikipedia entry](https://en.wikipedia.org/wiki/Auguste_Bravais)

## Selected writings

On [[crystallography]] and introducing the notion now known as *[[Bravais lattice]]*:

* {#Bravais1850}  [[Auguste Bravais]], *Mémoire sur les Systèmes Formés par les Points Distribués Régulièrement sur un Plan ou dans L'espace*, J. Ecole Polytech. 19 (1850) 1 $[$[ark:12148/bpt6k96124j](https://gallica.bnf.fr/ark:/12148/bpt6k96124j)$]$

category: people


Auguste Sagnotti > history

see _[[Augusto Sagnotti]]_


Augustin-Louis Cauchy

[[!redirects Augustin Cauchy]]

__Augustin-Louis Cauchy__ was a pioneer in [[analysis]] and [[group theory]].  He wrote an influential 1821 textbook, _[[Cours d'Analyse]]_.

Cauchy described the basic concepts of [[differential calculus|differential]] and [[integral calculus|integral]] [[infinitesimal calculus|calculus]] in terms of [[convergence|limits]].  His conception of limits was based on [[infinitesimal]] [[variables]], which do not appear as such in modern [[mathematics]], although they have been variously identified with [[sequences]] (that converge to zero), [[ultrafilters]] (that converge to zero), hyperpoints (in the [[infinitesimal neighborhood|infinitesimal neighbourhood]] of zero) in the sense of [[nonstandard analysis]], etc.  (His infinitesimals were *not* [[nilpotent infinitesimal|nilpotent]].)

Cauchy\'s student [[Karl Weierstrass]] defined [[limit of a sequence|limits]] in terms of [[Richard Dedekind]]\'s static conceptions of *[[real numbers]]* and *[[functions]]*, thereby creating modern [[analysis]].

Cauchy is associated with:

* [[Cauchy sequences]]

*  [[Cauchy sequences]] (and thus [[Cauchy nets]], [[Cauchy filters]], and [[Cauchy spaces]], although Cauchy himself knew none of these)

*  [[Cauchy–Riemann equations]]

* [[Cauchy-Schwarz inequality]]

*  [[Cauchy integral]]

* [[Cauchy surface]]

* [[Cauchy real number]]

* [[Cauchy principal value]]

*  [[Cours d'Analyse]], a textbook on [[infinitesimal analysis]]/[[epsilontic analysis]]

* [[Cauchy's theorems]]

  *  [[Cauchy integral theorem]] and [[Cauchy integral formula]] (for [[contour integrals]] in [[complex analysis]])

  * [[Cauchy sum theorem]] ("[[Cauchy's mistake]]") from his 1821 textbook [[Cours d'Analyse]]

  * [[Cauchy group theorem]]

category: people

[[!redirects Cauchy]]
[[!redirects Augustin Cauchy]]
[[!redirects Augustin-Louis Cauchy]]
[[!redirects Augustin Louis Cauchy]]


Augusto Sagnotti

* [Wikipedia entry](https://en.wikipedia.org/wiki/Augusto_Sagnotti)

## Selected writings

Review of the [[Green-Schwarz mechanism]] with an eye towards [[KK-compactification]] to 6d (see also at [[D=6 N=(1,0) SCFT]]):

* [[Augusto Sagnotti]], _A Note on the Green - Schwarz Mechanism in Open - String Theories_, Phys. Lett. B294:196-203, 1992 ([arXiv:hep-th/9210127](https://arxiv.org/abs/hep-th/9210127))

On [[D=6 supergravity]]:

* [[Sergio Ferrara]], [[Fabio Riccioni]], [[Augusto Sagnotti]], _Tensor and Vector Multiplets in Six-Dimensional Supergravity_, Nucl. Phys. B519 (1998) 115-140 ([arXiv:hep-th/9711059](https://arxiv.org/abs/hep-th/9711059))

* [[Fabio Riccioni]], [[Augusto Sagnotti]], _Some Properties of Tensor Multiplets in Six-Dimensional Supergravity_, Nucl. Phys. Proc. Suppl. 67 (1998) 68-73 ([arXiv:hep-th/9711077](https://arxiv.org/abs/hep-th/9711077))

On [[orientifolds]] and [[RR-field tadpole cancellation]]:

* {#Sagnotti88} [[Augusto Sagnotti]], _Open strings and their symmetry groups_ in G. Mack et. al. (eds.) Cargese ’87, "Non-perturbative Quantum Field Theory,"  (Pergamon Press, 1988) p. 521 ([arXiv:hep-th/0208020](https://arxiv.org/abs/hep-th/0208020))

* Carlo Angelantonj, [[Augusto Sagnotti]], _Open Strings_, Phys. Rept. 371:1-150,2002; Erratum ibid. 376:339-405, 2003 ([arXiv:hep-th/0204089](https://arxiv.org/abs/hep-th/0204089))

On [[higher spin gauge theory]] as [[string field theory]] for vanishing [[string tension]]:

* {#SagnottiTsulaia03} [[Augusto Sagnotti]], M. Tsulaia, _On higher spins and the tensionless limit of String Theory_, Nucl. Phys. B682:83-116, 2004 ([arXiv:hep-th/0311257](http://arxiv.org/abs/hep-th/0311257))

* [[Auguste Sagnotti]], M. Taronna, _String Lessons for Higher-Spin Interactions_, Nucl. Phys. B842:299-361,2011 ([arXiv:1006.5242](http://arxiv.org/abs/1006.5242))

* {#Sagnotti11} [[Augusto Sagnotti]], _Notes on Strings and Higher Spins_ ([arXiv:1112.4285](http://arxiv.org/abs/1112.4285))

Discussion of the [[Velo-Zwanziger problem]] for the higher [[string]] excitations that appear in [[string theory]]:

* [[Massimo Porrati]], Rakibur Rahman, [[Augusto Sagnotti]], _String Theory and The Velo-Zwanziger Problem_, Nucl. Phys. B846:250-282, 2011 ([arXiv:1011.6411](http://arxiv.org/abs/1011.6411))

## Related $n$Lab entries

* [[orientifold]]

* [[RR-field tadpole cancellation]]

* [[5d supergravity]]

category: people


Augustus De Morgan

Augustus De Morgan is a 19th-century mathematician and logician most famous for the [[De Morgan laws]], even though he did not formulate them.  He did, however, contribute (along with [[George Boole]], [[Charles Peirce]], etc) to the work of making [[logic]] into a mathematical discipline.

## Quotations

'Imagine a person with a gift of ridicule.  \[He might say\] First that a negative quantity has no logarithm; secondly that a negative quantity has no square root; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter.'  (That is, $\ln(-1)$ and $\sqrt{-1}$ are both imaginary, but their quotient is $\pi$.)

'I end with a word on the new symbols which I have employed. Most writers on logic strongly object to all symbols.  \[...\]  I should advise the reader not to make up his mind on this point until he has well weighed two facts which nobody disputes, both separately and in connexion.  First, logic is the only science which has made no progress since the revival of letters; secondly, logic is the only science which has produced no growth of symbols.'

## Related entries

* [[De Morgan laws]]

* [[De Morgan duality]]

* [[De Morgan Heyting algebra]]

* [[De Morgan Heyting category]]

* [[De Morgan algebra]]

* [[De Morgan topos]]

category: people

[[!redirects Augustus De Morgan]]
[[!redirects Augustus de Morgan]]


Auke Booij

[[!redirects Auke Bart Booij]]
[[!redirects Auke B. Booij]]

## Selected writings

On [[parametricity]], [[type universe]]-[[automorphisms]] and [[excluded middle]]

* [[Auke Bart Booij]], [[Martín Hötzel Escardó]], [[Peter LeFanu Lumsdaine]], [[Michael Shulman]], *Parametricity, automorphisms of the universe, and excluded middle*, in 22nd International Conference on Types for Proofs and  Programs (TYPES 2016)  $[$[arxiv:1701.05617](https://arxiv.org/abs/1701.05617), [drops:2018/9855](https://drops.dagstuhl.de/opus/volltexte/2018/9855), [blog entry](https://homotopytypetheory.org/2017/01/26/parametricity-automorphisms-of-the-universe-and-excluded-middle/)$]$

On [[constructive analysis]] with [[real numbers]] in [[univalent foundations]] ([[homotopy type theory]] with the [[univalence axiom]]):

* {#Booij18} [[Auke Booij]], _Extensional constructive real analysis via locators_, Mathematical Structures in Computer Science **31** 1 (2021) 64-88  [[arXiv:1805.06781](https://arxiv.org/abs/1805.06781), [doi:10.1017/S0960129520000171](https://doi.org/10.1017/S0960129520000171)]

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

category: people


Aurelio Carboni

Aurelio Carboni was an Italian category theorist, who was based in Como. He died on 11 December 2012.

* [webpage](http://www.uninsubria.eu/research/physmath/cv_Carboni.htm)

## Selected writings

On [[extensive categories]] and [[distributive categories]]:

* {#CarboniLackWalters93} [[Aurelio Carboni]], [[Stephen Lack]], [[Bob Walters|R. F. C. Walters]], _Introduction to extensive and distributive categories_, JPAA **84** (1993) pp. 145-158 (doi:10.1016/0022-4049(93)90035-R)

On [[reflective factorization systems]]:

* {#CJKP} [[Aurelio Carboni]], [[George Janelidze]], [[Max Kelly]], [[Robert Paré]], _On localization and stabilization for factorization systems_, Appl. Categ. Structures 5 (1997), 1--58 ([doi:10.1023/A:1008620404444](https://doi.org/10.1023/A:1008620404444))

On [[regular and exact completions]] (and on [[free coproduct completions]], etc.):

* [[Aurelio Carboni]], [[Enrico Vitale]], *Regular and exact completions*, Journal of Pure and Applied Algebra **125** 1–3  (1998) 79-116 (doi:10.1016/S0022-4049(96)00115-6)

On [[Cartesian bicategories]]:

* [[Aurelio Carboni]], [[Bob Walters]], _Cartesian Bicategories, I_, Journal of Pure and Applied Algebra, Volume 49, Issues 1–2, November 1987,  (doi:10.1016/0022-4049(87)90121-6)

* [[Aurelio Carboni]], [[Max Kelly]], [[Bob Walters]], [[Richard Wood]], _Cartesian Bicategories II_, Theory and Applications of Categories, Vol. 19, 2008, No. 6, pp 93-124. ([arXiv:0708.1921](https://arxiv.org/abs/0708.1921), [tac:19-06](http://www.tac.mta.ca/tac/volumes/19/6/19-06abs.html))

category:people

[[!redirects A. Carboni]]


Ausdehnungslehre

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

This page collects material related to the book

* {#Grassmann44} [[Hermann Grassmann]],

  _Die Wissenschaft der extensiven Grössen oder die Ausdehnungslehre_

    _Erster Teil, die lineale Ausdehnungslehre_,

  1844

  ([pdf scan of original](http://www.uni-potsdam.de/u/philosophie/grassmann/Werke/Hermann/Ausdehnungslehre_1844.pdf), [Internet Archive copy](https://archive.org/details/dielinealeausde00grasgoog/page/n11))

which introduced for the first time basic concepts of what today is known as [[linear algebra]] (including [[affine spaces]] as [[torsors]] over [[vector spaces]]) and introduced in addition an _exterior product_ (§37, §55) on [[vectors]], forming what today is known as _[[exterior algebra]]_ or _[[Grassmann algebra]]_, hence in fact _[[superalgebra]]_ (see [below](#SupercommutativeSuperalgebra)).

Grassmann advertizes his work (p. xxv) as being the theory of _[[extensive quantity]]_. The modern way of speaking about this is that the elements of the [[exterior algebra]] he considered are [[differential forms]] on [[Euclidean space]].

Prominent followups are:

* [[Giuseppe Peano]], *Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva*, Fratelli Bocca Editori, Torino, 1888, pp. XI, 171 [[Mathematica Italiana opere 138](http://mathematica.sns.it/opere/138), [pdf](http://mathematica.sns.it/media/volumi/138/Calcolo%20geometrico%20secondo%20l'Ausdehnungslhere%20di%20H.%20Grassmann_bw.pdf)]

which introduces the modern concept of *[[vector spaces]]*.

and

* [[William Clifford]], *Applications of Grassmann's extensive algebra*, American Journal of Mathematics **1** 4 (1878) 350-358 [[doi:10.2307/2369379](https://doi.org/10.2307/2369379), [jstor:2369379](https://www.jstor.org/stable/2369379)]

which introduces the notion of [[Clifford algebra]].

#Contents#
* table of contents
{:toc}

## Content

### Supercommutative superalgebra
 {#SupercommutativeSuperalgebra}

{#GrassmannDefinesSuperalgebra} Here is Grassmann introducing the [[signs in supergeometry|sign rule]] of [[supercommutative superalgebra]]:

> from [Grassmann 1844, p. 61](#Grassmann44)

> from [Grassmann 1844, p. 84](#Grassmann44)

## Reception
 {#Reception}

The appreciation of Grassmann's ideas took a long time:

From [Drew Armstrong](https://plus.google.com/103061162497127117651/posts/f4HHK1XhUL1):

> Perhaps owing to the abstraction of his work, he was not recognized by the mathematical community and was more famous during his lifetime for his work in philology (including a translation of the Rig Veda). Despite his failure to achieve mathematical success, Grassmann never doubted that his work was significant. Here's a striking quote from the preface of the 1862 second edition of the _[[Ausdehnungslehre]]_:

>> {#CompletelyConvinced} I remain completely confident that the labour I have expended on the science presented here and which has demanded a significant part of my life as well as the most strenuous application of my powers, will not be lost. It is true that I am aware that the form which I have given the science is imperfect and must be imperfect. But I know and feel obliged to state (though I run the risk of seeming arrogant) that even if this work should again remain unused for another seventeen years or even longer, without entering into the actual development of science, still that time will come when it will be brought forth from the dust of oblivion and when ideas now dormant will bring forth fruit. I know that if I also fail to gather around me (as I have until now desired in vain) a circle of scholars, whom I could fructify with these ideas, and whom I could stimulate to develop and enrich them further, yet there will come a time when these ideas, perhaps in a new form, will arise anew and will enter into a living communication with contemporary developments. For truth is eternal and divine.

> (Translation from Michael Crowe, _A History of Vector Analysis_)

{#WhyNoOneHasTranslated} Here is R. W. Genese in 1893 ([Nature volume 48, page 517 ](https://www.nature.com/articles/048517a0)) lamenting the delay in its translation:

>  Sir Robert Ball asks why no one has translated the "[[Ausdehnungslehre]]" into English. The answer is as regretable as simple—it would not pay. The number of mathematicians who, after the severe courses of the universities, desire to extend their reading is very small. It is something that a respectable few seek to apply what they have already learnt. The first duty of those who direct the studies of the universities is to provide that students may leave in possession of all the best means of future investigation. That fifty years after publication the principles of the "[[Ausdehnungslehre]]" should find no place in English mathematical education is indeed astonishing. Half the time given to such a wearisome subject as Lunar Theory would place a student in possession of many of the delightful surprises of Grassmann's work, and set him thinking for himself. The "[[Ausdehnungslehre]]" has won the admiration of too many distinguished mathematicians to remain longer ignored. Clifford said of it: "I may, perhaps, be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science.” Useful or not, the work is “a thing of beauty," and no mathematician of taste should pass it by. It is possible, nay, even likely, that its principles may be taught more simply; but the work should be preserved as a classic.

Grassmann's ideas were finally studied in

* Hermann Hankel, _Vorlesungen über die Complexen Zahlen und ihre Functionen_, Leipzig: Leopold Voss, 1867.

His ideas were developed by [[William Kingdon Clifford]] who turned Grassmann algebras into _[[Clifford algebras]]_ in

* [[William Clifford]], _Applications of Grassmann's extensive algebra_, American Journal of Mathematics 1 (4): 350–358. (1878). doi:[10.2307/2369379](https://doi.org/10.2307/2369379).

Today [[Grassmann algebra]] is ubiquituous in [[differential geometry]] ([[de Rham algebra]], [[supergeometry]]) and [[superalgebra]]. In fact, Grassmann had invented the [[signs in supergeometry|sign rule]] of [[supercommutative superalgebra]], which was fully appreciated only in the second half of the 20th century.

This way Grassmann's prediction on the eventual impact of his work did become true. From [[Freeman Dyson]], _Missed opportunities_ ([doi:10.1090/S0002-9904-1972-12971-9](https://doi.org/10.1090/S0002-9904-1972-12971-9)):

> In  the  year  1844 two  remarkable  events  occurred,  the  publication  by   Hamilton of  his  discovery  of  [[quaternions]],  and  the  publication  by Grassmann of  his  "[[Ausdehnungslehre]]."  With  the  advantage  of  hindsight  we  can  see  that  Grassmann's  was  the  greater  contribution  to   mathematics,  containing  the  germ  of  many  of  the  concepts  of  modern algebra,  and  including  vector  analysis  as a  special  case. However, Grassmann  was an  obscure  high-school  teacher  in Stettin, while Hamilton  was the  world-famous  mathematician  whose  official  titles  occupy  six  lines  of  print  after  his name at the beginning  of his 1844 paper.  So it  is regrettable,  but  not surprising,  that  quaternions  were  hailed  as  a  great  discovery, while Grassmann  had  to  wait  23  years  before  his  work  received  any recognition  at  all  from  professional  mathematicians.  When  Grassmann's  work  finally became  known,  mathematicians  were  divided  into  quaternionists and antiquaternionists, and  were spending more energy in polemical arguments  for  and  against  quaternions  than  in  trying  to  understand how  Grassmann  and  Hamilton  might  be  fitted  together  into  a  larger scheme  of  things.

Grassmann also had a profound influence on the thought of [[Gottlob Frege]]. See Sec. 2.1.1 of

* Paolo Mancosu, _Abstraction and infinity_, Oxford University Press, 2017.

## Discussion

Discussion of the book includes

* {#Lawvere96} [[William Lawvere]], _Grassmann's Dialectics and Category Theory_, in _Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar_, Boston Studies in the Philosophy of Science Volume 187, 1996, pp 255-264, doi:[10.1007/978-94-015-8753-2_21](https://doi.org/10.1007/978-94-015-8753-2_21)

and the similar text

* {#Lawvere95} [[William Lawvere]], _A new branch of mathematics, "The Ausdehnungslehre of 1844," and other works. Open Court (1995), Translated by Lloyd C. Kannenberg, with foreword by Albert C. Lewis_, _Historia Mathematica Volume 32, Issue 1, February 2005, Pages 99–106_, doi:[10.1016/j.hm.2004.07.004](https://doi.org/10.1016/j.hm.2004.07.004)

which says at one point that full appreciation of the _Ausdehnungslehre_ requires concepts of [[category theory]]

> The modern conceptual apparatus, involving levels of structure, [[categories]] of [[morphisms]] preserving given [[structure]], [[forgetful functor|forgetful reduct functors]] between categories, the [[adjoints]] to such functors, etc., seems to be necessary for ordinary mortals to be able to find their way through the riches of Grassmann's geometry.

The first part of the introduction of the _Ausdehnungslehre_ is concerned with [[philosophy]], about which

> Grassmann insists that his reason for including it is an attempt to provide an orientation to help the student form for himself the proper estimation of the relation between general and particular at every stage of the learning process ([Lawvere 95](#Lawvere95)).

The second part of the introduction, titled _Survey of the general theory of forms_ considers key concepts of [[algebra]]. For instance it considers the [[associativity law]] and states its [[coherence law]] (§3).

Grassmann writes that he uses the term "form" in place of "quantity" (German: "Grösse") (Introduction A.3, §2). It is "forms" that his algebraic operations are defined on, and which are produced by these.

> The last half of that introduction is essentially one of the first expositions of the rudimentary principles of what today might be called [[universal algebra]]. The content of the first half, after considerable study of the compact formulations, appears to be a simple and clear natural scientist's version of the basic principles of dialectical materialism, as applied to the formal sciences. ([Lawvere 95](#Lawvere95))

Curiously, while Grassmann complains (on p. xv) about the "unclarity and arbitrariness" of [[Hegel]]'s school of philosophy ([[German idealism]], predominant in Germany at Grassmann's time), the introduction of the _Ausdehnungslehre_ has much the same sound as Hegel, notably it discusses "[[category (philosophy)|categories]]" such as _[[being]]_, _[[becoming]]_ (p. xxii), _[[concrete particular|particulars]]_ (p.xx) and the [[dialectic]] of [[unities of opposites|opposites]] such as _[[flat modality|discrete]] $\dashv$ [[sharp modality|continuous]]_ (p.xxii) and, notably, of _[[intensive and extensive quantity]]_ (p. xxiv-xxv), which Grassmann advertizes as the very topic of his mathematical theory. That of course is the difference to [[Hegel]], that unambiguous mathematical formalization of these otherwise vague concepts is provided (according to [Lawvere 95](#Lawvere95) Grassmannn's formalization of the pair _[[being]]_ and _[[becoming]]_ is via points and [[vectors]] in an [[affine space]]), and in this sense Grassmann is clearly a forerunner of Lawvere's various proposals for formalizing Hegel's [[objective logic]] in [[categorical logic]]/[[topos theory]] (as discussed at _[[Science of Logic]]_).

## Related entries

* [[Grassmann algebra]]

* [[supercommutative superalgebra]]

* [[Berezin integral]], [[integration over supermanifolds]]

## References

* Wikipedia, _[Grassmann -- Mathematician](http://en.wikipedia.org/wiki/Hermann_Grassmann#Mathematician)_

* {#Dyson72} [[Freeman Dyson]], _Missed opportunities_, Bulletin of the AMS, Volume 78, Number 5 (1972) pp 635-652, doi:[10.1090/S0002-9904-1972-12971-9](https://doi.org/10.1090/S0002-9904-1972-12971-9)

  > In  the  year  1844  two  remarkable  events  occurred,  the  publication  by   Hamilton of  his  discovery  of  [[quaternions]],  and  the  publication  by Grassmann of  his  "Ausdehnungslehre."  With  the  advantage  of  hindsight  we  can  see  that  Grassmann's  was  the  greater  contribution  to   mathematics,  containing  the  germ  of  many  of  the  concepts  of  modern algebra,  and  including  vector  analysis  as a  special  case. However, Grassmann  was an  obscure  high-school  teacher  in Stettin, while Hamilton  was the  world-famous  mathematician  whose  official  titles  occupy  six  lines  of  print  after  his name at the beginning  of his 1844 paper.  So it  is regrettable,  but  not surprising,  that  quaternions  were  hailed  as  a  great  discovery, while Grassmann  had  to  wait  23  years  before  his  work  received  any recognition  at  all  from  professional  mathematicians.  When  Grassmann's  work  finally became  known,  mathematicians  were  divided  into  quaternionists and antiquaternionists, and  were spending more energy in polemical arguments  for  and  against  quaternions  than  in  trying  to  understand how  Grassmann  and  Hamilton  might  be  fitted  together  into  a  larger scheme  of  things.

category: reference


Auslander-Buchsbaum formula

Let $R$ be a commutative Noetherian local ring. Then for any nonzero finitely generated $R$-module $M$ of finite [[projective dimension]] $pd_R(M)$ the following formula holds
$$
pd_R(M) + depth(M) = depth(R)
$$

As a corollary a commutative ring is regular iff it has a finite global dimension. This is also proved by Serre.

This in turn is the essence of the fact that for any [[regular scheme]], hence in particular for any [[smooth scheme]] over a field, the bounded [[derived category of coherent sheaves]] coincides with the [[full triangulated subcategory]] of [[perfect complex]]es. In other words, the [[triangulated category of singularities]] of a smooth scheme is trivial.

One should be warned that another result on regular rings is usually known as [[Auslander-Buchsbaum theorem]] (from a later article in 1959), namely that every regular local ring is a [[unique factorization domain]].

## Literature

* Maurice Auslander, [[David Buchsbaum|David A. Buchsbaum]], _Homological dimension in local rings_, Trans. Amer. Math. Soc. __85__ (2): 390--405 (1957) [doi](https://doi.org/10.2307/1992937) MR0086822
* stacks project 10.111 Auslander-Buchsbaum tag:[090U](https://stacks.math.columbia.edu/tag/090U)
* wikipedia:[Auslander-Buchsbaum_formula](https://en.wikipedia.org/wiki/Auslander%E2%80%93Buchsbaum_formula)

[[!redirects Auslander–Buchsbaum formula]]