Contents

foundations

category theory

topos theory

# Contents

## Idea

I am not a “Hegelian”. F. W. Lawvere 1

Aufhebung (sublation) is a central concept2 in the dialectical logic of the German philosopher G. W. F. Hegel. The German expression has several meanings for which tollere, elevare, conservare would be Latin equivalents.3

In his quest to 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 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 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 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 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 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): after a deduction of the categories of ‘being’, ‘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).

He goes on (p.113) 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 Versuch den Begriff der negativen Größen in die Weltweisheit einzuführen (1763) a distinction between contradictions and real oppositions4 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.5

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.6

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.

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.:

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, 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) 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 ordering. Let $L,R:N\to N$ be the two parallel functorseven’ and ‘odd’ defined by $L(n) \coloneqq 2n$ and $R(n) \coloneqq 2n+1$.

Both are 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 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,

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

3. it expresses the identity $L$ between $R$ 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

For convenience let us briefly recall the following

###### 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 when the reflection has furthermore a left adjoint.

If $l\dashv r\dashv i$ is an essential localization then $l$ is also full and faithful. If $\mathcal{B}$ is a topos, $\mathcal{A}$ is called an 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) that the essential subtoposes of a topos form a complete lattice. Therefore we say:

###### Definition

An essential subtopos of $\mathcal{B}$ is referred to as a 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).

###### Definition

Let $i,j$ be levels, def. , 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 short7) if $\bigcirc _j\Box_i=\Box _i$.

Finally a 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.

###### 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 $\bigcirc_j \Box_i X =\bigcirc_j X=\Box_i X =X$ 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).

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 $j$.

###### 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 categories or, more generally, over von Neumann regular categories (Lawvere 2002), it does. The Aufhebungs relation is also called the jump operator in Lawvere (2009).

###### Remark

Comparing with WdL under Lawvere’s translation and identifing the 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!

###### 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), 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 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).

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

#### 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 non being (the idempotent comonad constant on the initial object) and pure being (the idempotent comonad constant in the terminal object) whose Aufhebung is (at least in suitable cases, see below) the opposition of becoming

$\flat\dashv \sharp$

given by flat modality $\dashv$ sharp modality, between non-becoming vs. pure becoming (cf. Lawvere 1989a, 1989b, 1991a)8. This is what in SoL § 191 is called determinate being as it corresponds to localization at $\neg\neg$ and the double negation creates the determinateness of the ‘Etwasse’.

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 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 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

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, Shulman 15, section 3, see at pieces-to-points-transform – Relation to Aufhebung).

###### 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

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 $\emptyset \simeq \sharp \emptyset$.

###### 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).

###### Corollary

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

###### Proposition

Let $\mathcal{S}$ be a cohesive site (or ∞-cohesive site) and $\mathbf{H} = Sh(\mathcal{S})$ its 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. , the unity of opposites $(\emptyset \dashv \ast)$ which is becoming.

###### 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).

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. .

###### Proposition

Let $\mathcal{S}$ be a cohesive site (or ∞-cohesive site) and $\mathbf{H} = Sh(\mathcal{S})$ its 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. , of the duality of opposites of becoming $\emptyset \dashv \ast$ (“Dasein”).

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

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

###### Remark

As mentioned above, the Aufhebung of $\emptyset\dashv \ast$ is necessarily given by a 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 and hence happens to be a 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 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 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 that for cohesive sites over Set: $\mathbf{H}_{\neg\neg}=Set$ i.e. the double negation topos coincides with the base.

###### Example

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

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: 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 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).

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, 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:

###### Definition

Let $\mathcal{C}$ be a finitely complete category. An 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) 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).

The following is immediate:

###### 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. $\qed$

###### 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.

(…)

### Simplicial and cubical sets

(…) simplicial set (…)

simplicial skeleton$\dashv$ simplicial coskeleton

(…) cubical set (…)

## A guide to the literature

On the philosophical side, the lectures notes Koch (2009) that suggest the use of non-wellfounded set theory as interpretative tool may serve as a good general introduction to Hegel’s ideas on logic and metaphysics. Krahn (2014) 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) and Wolff (2010) the latter highlighting Hegel as ‘a philosopher of mathematics’ in this context.

On the mathematical side, the book by La Palme-Reyes-Zolfaghari (2004) 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) covers a good deal of ground yet stays accessible and is available online.

Lawvere introduced the Hegelian concepts in Lawvere (1989b). They get some attention in Lawvere (1991a,1992,1994a) with the second containing his ‘philosophical program’. By all means have a look at Lawvere (1996), this together with Lawvere (1989a,1999) exposes his ideas on homotopy theory. The work on graphic toposes (1989b,1991b,2002) concerns the Aufhebungs relation with the latter containing a discussion of the relevant concepts. Kelly-Lawvere (1989) 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) which is based on older phd-works by some of the authors.

Further results on essential localizations can be found in the papers by Borceux-Korotenski (1991), Johnstone (1996), Vitale (2001) and Lucyshyn-Wright (2011) or in SGA 4.

## References

• M.Artin, A.Grothendieck, J. L. Verdier (eds.), Théorie des Topos et Cohomologie Etale des Schémas - SGA 4, LNM 269 Springer Heidelberg 1972. (sec. IV 7.6., pp.414-416)

• J. C. Baez, M. Shulman, Lectures on n-categories and cohomology, pp.1-68 in J. C. Baez, P. May (eds.), Towards Higher Categories, Springer Heidelberg 2010. (preprint)

• F. Berto, Hegel’s Dialectic as a Semantic Theory: An Analytical Reading, European Journal of Philosophy 15 no.1 (2007) pp.19-39. (philpapers)

• F. Borceux, M. Korostenski, Open Localizations, JPAA 74 (1991) pp.229-238.

• Robert Brandom, Selbstbewusstsein und Selbstkonstitution , pp.46-77 in Halbig, Quante, Siep (eds.), Hegels Erbe , Suhrkamp Frankfurt am Main 2004.

• Oscar Daniel Brauer, Dialektik der Zeit - Untersuchungen zu Hegels Metaphysik der Weltgeschichte , frommann-holzboog Stuttgart 1982.

• J. Climent Vidal, J. Soliveres Tur, Functors of Lindenbaum-Tarski, Schematic Interpretations, and Adjoint Cylinders between Sentential Logics, Notre Dame Journal of Formal Logic 49 no.2 (2008) pp.185-202. (pdf) ‎

• H. F. Fulda, Aufheben, pp.318-320 in Ritter (ed.), Historisches Wörterbuch der Philosophie, Schwabe Basel 1971. (Heidi)

• M. Giovanelli, Trendelenburg and the Concept of Negation in Post-Kantian Philosophy, to appear in Munk (ed.), Proceedings of the Amsterdam 2010 Colloquium: Natur des Denkens und das Denken der Natur: Spinoza, Trendelenburg und H. Cohen. (draft)

• R. Goldblatt, Topoi - The Categorical Analysis of Logic, 2nd ed. North-Holland Amsterdam 1984. (Dover reprint New York 2006; project euclid)

• G. W. F. Hegel, Differenz des Fichte’schen und Schelling’schen Systems der Philosophie, pp.7-138 in Moldenhauer, Michel (eds.), Werke 2, Suhrkamp Frankfurt 1986[1801].

• G. W. F. Hegel, Begriffslehre für die Mittelklasse (1809/10), pp.139-162 in Moldenhauer, Michel (eds.), Werke 4, Suhrkamp Frankfurt 1986.

• G. W. F. Hegel, Wissenschaft der Logik I, Suhrkamp Frankfurt 1986[1812/13; revised 1831].

• P. Johnstone, Remarks on Quintessential and Persistent Localizations, TAC 2 no.8 (1996) pp.90-99. (pdf)

• G. M. Kelly, F. W. Lawvere, On the Complete Lattice of Essential Localizations, Bull. Soc. Math. de Belgique XLI (1989) pp.289-319.

• C. Kennett, E. Riehl, M. Roy, M. Zaks, Levels in the toposes of simplicial sets and cubical sets, JPAA 215 no.5 (2011) pp.949-961. (preprint)

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• R. Krahn, The Sublations of Dialectics: Hegel and the Logic of Aufhebung, PhD Guelf University Ontario 2014. (link)

• M. La Palme Reyes, G. E. Reyes, H. Zolfaghari, Generic Figures and their Glueings, Polimetrica Milano 2004.

• F. W. Lawvere, Qualitative Distinctions between some Toposes of Generalized Graphs, Cont. Math. 92 (1989) pp.261-299.

• F. W. Lawvere, Display of graphics and their applications, as exemplified by 2-categories and the Hegelian “taco”, Proceedings of the first international conference on algebraic methodology and software technology University of Iowa, May 22-24 1989, Iowa City, pp.51-74.

• F. W. Lawvere, Some Thoughts on the Future of Category Theory, pp.1-13 in LNM 1488 Springer Heidelberg 1991.

• F. W. Lawvere, More on Graphic Toposes, Cah. Top. Géom. Diff. Cat. XXXII no.1 (1991) pp.5-10. (pdf)

• F. W. Lawvere, Categories of Space and Quantity, pp.14-30 in: J. Echeverria et al (eds.), The Space of mathematics, de Gruyter Berlin 1992.

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• F. W. Lawvere, Unity and Identity of Opposites in Calculus and Physics, App. Cat. Struc 4 (1996) pp.167-174.

• F. W. Lawvere, Kinship and Mathematical Categories, pp.411-425 in: R. Jackendoff, P. Bloom, K. Wynn (eds), Language, Logic, and Concepts - Essays in Memory of John Macnamara, MIT Press 1999.

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• F. W. Lawvere, Linearization of graphic toposes via Coxeter groups, JPAA 168 (2002) pp.425-436.

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• F. W. Lawvere, Open Problems in Topos Theory, ms. (2009). (pdf)

• F. W. Lawvere, Combinatorial Topology, message to catlist November 2013. (link)

• F. W. Lawvere, M. Menni, Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness, TAC 30 no. 26 (2015) pp.909-932. (abstract)

• Mike Shulman, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (arXiv:1509.07584, doi:10.1017/S0960129517000147)

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1. (Lawvere 1989, p.74).

2. A pertinent passage is e.g. SoL §209.

3. As this polysemy is important for the concept and difficult to preserve in translation we prefer to use the German term in the following.

4. On the avatars of this synthetic negation from Kant through von Trendelenburg to Neo-Kantianism see Giovanelli (2015). 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) first published in 1981. The importance of this link has been stressed already in the 19th century by Karl Rosenkranz.

5. 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.

6. 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.

7. $\ll$ is called the way below relation in (KRRZ11).

8. 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.

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