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Aufhebung

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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 NN be the natural numbers {0,1,}\{0, 1,\dots\} viewed as a category via their usual ordering. Let L,R:NNL,R:N\to N be the two parallel functorseven’ and ‘odd’ defined by L(n)2nL(n) \coloneqq 2n and R(n)2n+1R(n) \coloneqq 2n+1.

Both are full and faithful, which means that they correspond to two subcategory inclusions and, accordingly, to two subcategories N evenN_{even} and N oddN_{odd}. We are now in situation where we have two subcategories that ‘oppose’ each other in that N evenN oddN_{even}\neq N_{odd} but are nevertheless ‘identical’ in that there is a bijection N evenN oddN_{even}\overset{\simeq}{\to} N_{odd}. Furthermore, both are ‘united’ as different parts in the encompassing NN whose overall structure they represent in that N evenNN oddN_{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 NNN\to N which with a clin d’oeil to C. S. Peirce's concept of thirdness we call TT , can encapsulate this bunch of relations in one sweep when it forms an adjoint triple LTRL\dashv T\dashv R with LL and RR:

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

  2. it expresses the opposition between LL and RR by an entailed adjunction LTRTL\circ T\dashv R\circ T,

  3. it expresses the identity LL between RR by the entailed equivalence TLTRT\circ L\simeq T\circ R .

In other words, TT unites, opposes and identifies LL and RR at the same time!

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

T(k)={k2kN even k12kN odd T(k)=\Bigg\{ \array{\frac{k}{2}\quad k\in N_{even} \\ \frac{k-1}{2} \quad k\in N_{odd}}

Whereas TLT\circ L and TRT\circ R are each the identity, the reverse compositions LTL\circ T and RTR\circ T yield an idempotent comonad sk:NNsk:N\to N and an idempotent monad cosk:NNcosk:N\to N, respectively, where sk(2n)=2nsk(2n)=2n and sk(2n+1)=2nsk(2n+1)=2n and cosk(2n)=2n+1cosk(2n)=2n+1 and cosk(2n+1)=2n+1cosk(2n+1)=2n+1: in new guises LL and RR resurface again but this time within an ‘opposition’ skcosksk\dashv cosk which expresses formally the ‘conflict’ between N evenN_{even} and N oddN_{odd}, even and odd, as well as their essential identity and unity.

Reflexivity vs. irreflexivity

Let us briefly consider a second example of a unity-and-identity of opposites:

The category Bin\mathbf{Bin} of sets equipped with a binary relation has objects pairs (X,ρ)(X,\rho) with XX a set and ρ\rho a binary relation on XX and morphims (X 1,ρ 1)(X 2,ρ 2)(X_1,\rho_1)\to (X_2,\rho_2) are functions f:X 1X 2f:X_1\to X_2 such that xρ 1yx\rho_1 y implies f(x)ρ 2f(y)f(x)\rho_2 f(y). Here we view a relation as set of ordered pairs (x 1,x 2)(x_1,x_2) of elements x 1,x 2Xx_1,x_2\in X. (,)(\empty,\empty) and ({*},(*,*))(\{\ast\},(\ast,\ast)) are the initial resp. the terminal object in Bin\mathbf{Bin}.

A relation ρ\rho on set XX is called reflexive (resp. irreflexive) when ρid X=id X\rho\cap id_X=id_X (resp. ρid X=\rho\cap id_X=\empty). In other words a reflexive relation contains (x,x)(x,x) for every xXx\in X. Let Refl\mathbf{Refl} and IrRefl\mathbf{IrRefl} be the respective full subcategories of Bin\mathbf{Bin}.

They are strongly opposed to each other in the sense that they have almost disjoint object classes, in fact the initial object of Bin\mathbf{Bin} (,)(\empty,\empty) is the only object that is reflexive and irreflexive. Furthermore, it is the only reflexive object (X,ρ)(X,\rho) at all that even admits a map (X,ρ)(Y,ι)(X,\rho)\to (Y,\iota) to some irreflexive object (Y,ι)(Y,\iota). This is because maps have to preserve the structure and (x,x)ρ(x,x)\in\rho requires (f(x),f(x))ι(f(x),f(x))\in\iota.

On the other hand, considering only reflexive objects among themselves in Refl\mathbf{Refl} the reflexive pairs in ρ\rho play no longer a role in determining whether a function f:XYf:X\to Y is structure preserving or not, since the codomain is equipped with a reflexive relation as well so requirements like (f(x),f(x))ι(f(x),f(x))\in\iota are automatically met. In other words, a function f:XYf:X\to Y between two sets underlies a structure preserving map (X,ρ)(Y,ζ)(X,\rho) \to(Y,\zeta) in Refl\mathbf{Refl} exactly iff it underlies a structure preserving map (X,ρid X)(Y,ζid X)(X,\rho\setminus id_X)\to (Y,\zeta\setminus id_X) in IrRefl\mathbf{IrRefl}. In functorial language, the functor ReflIrRefl\mathbf{Refl}\to\mathbf{IrRefl} that maps (X,ρ)(X,\rho) to (X,ρid X)(X,\rho\setminus id_X) and lets the (underlying) functions of the morphisms unchanged is an isomorphim with inverse (Y,ι)(Y,ιid X)(Y,\iota)\mapsto (Y,\iota\cup id_X). This isomorphism expresses the identity of reflexivity and irreflexivity.

The language of adjunctions permits one to encapsulate all the three relations of unity, opposition and identity:

The inclusion i:ReflBini:\mathbf{Refl}\hookrightarrow\mathbf{Bin} is an essential localization:

dri:ReflBin.d\dashv r \dashv i:\mathbf{Refl}\hookrightarrow\mathbf{Bin}\quad .

Here rr is the reflexivization functor that maps (X,ρ)(X,\rho) to (X,ρid X)(X,\rho\cup id_X), whereas d:ReflBind:\mathbf{Refl}\hookrightarrow \mathbf{Bin} is the irreflexivization functor that maps (X,ρ)(X,\rho) to (X,ρid X)(X,\rho\setminus id_X). The corresponding identical opposite of Refl\mathbf{Refl} is, of course, the full subcategory IrRefl\mathbf{IrRefl} of sets equipped with an irreflexive relation.

Another interesting essential localisation of Bin\mathbf{Bin} is given by the inclusion i:1Bini:1\hookrightarrow\mathbf{Bin}\quad, where 11 denotes the category with just one identity morphism (and, accordingly, just one object \bullet) and \bullet is mapped to the terminal object ({*},(*,*))(\{\ast\},(\ast,\ast)). The left adjoint to ii forcibly maps everything (X,ρ)(X,\rho) to \bullet, which in turn is right adjoint to the essentiality that maps \bullet to the initial object (,)(\empty,\empty). Whence the corresponding adjoint modalities oppose then the endofunctor that maps everything to the initial object to the one that maps everything to the terminal object: 010\dashv 1\quad.

How can we throw some philosophical meat in this mathematical soup? In Brandom (2004, p.60) it is suggested that the concept of an equivalence relation, i.e. transitive, reflexive and symmetric relation, lends itself as a crude mathematical model of Hegel’s concept of recognition that is a key concept in the dynamical unfolding of selfconciousness in the corresponding chapter in the “Phänomenologie des Geistes” (1807). On this suggestion we would interpret (X,ρ)(X,\rho) as a society XX of individuals xXx\in X and ρ\rho as a partial recognition relation between its members, a society (S,ϵ)(S,\epsilon) could be called intersubjective when ϵ\epsilon is an equivalence relation because then every member is a self xϵxx\epsilon x, recognition is reciprocal and if xx recognize yy and yy recognizes zz then xx recognizes zz as well.7 Note that such an intersubjective society could still be unspiritual in the sense that it is partitioned into disjoint sets of equivalence classes whose members recognize each other but not the members of the other classes, an extreme case of this is an ‘individualistic’ or discrete society (X,id X)(X,id_X).

Incidentally, the full subcategory Ind\mathbf{Ind} of Bin\mathbf{Bin} with objects the discrete societies is equivalent to SetSet and is reflective and coflective:

ΠΔΓ:BinSet. \Pi\dashv\Delta\dashv\Gamma: \mathbf{Bin}\to Set \quad .

Here Π\Pi lumps together all individuals x 1,x 2,x_1,x_2,\dots that have some chain of ρ\rho links x iρx jx_i\rho x_j between them into a group of maximally interrelated individuals and then maps (X,ρ)(X,\rho) to the set of these groups. Δ\Delta maps a set XX of individuals to the individualistic society (X,id X)(X,id_X) whereas Γ\Gamma maps a society (X,ρ)(X,\rho) to the set {xX:xρx}\{x\in X: x\rho x\} of selves.

Interestingly, crude as this picture is we can nevertheless interpret our adjoint oppositions 010\dashv 1 and IrReflRefl\mathbf{IrRefl}\dashv \mathbf{Refl} in it. For instance, the terminal object ({*},(*,*))(\{\ast\},(\ast,\ast)) models pure selfconciousness and the corresponding adjoint modality 010\dashv 1 can interpreted as saying that pure ego is empty, indeed a Hegelian contention and one of his reason to replace the Fichtean Ego with ‘pure being’ (see below!) as the starting point of philosophy. The opposition IrReflRefl\mathbf{IrRefl}\dashv \mathbf{Refl} can be interpreted as saying that a society where everything is a self is equivalent to a society consisting purely of (irreflexive) things, that would suggest to reinterpret a society as a Lebenswelt i.e. as consisting of set of individuals together with a set of things - in particular, the contradiction IrReflRefl\mathbf{IrRefl}\dashv \mathbf{Refl} expresses that a purely anamistic Lebenswelt where everything is a self is contradictory, in other words selfconciousness requires not only other selves but also not-selves!

Indeed, Brandom remarks (p.61) that in this philosophical context “reflexivity can be neglected” in the definition of an equivalence relation in other words we care for transitive and symmetric relations - the so called partial equivalence relations (per). Their inclusion as full subcategory i:PerBini:\mathbf{Per}\hookrightarrow\mathbf{Bin} has a reflector:

si:PerBin.s\dashv i:\mathbf{Per}\hookrightarrow\mathbf{Bin}\quad .

Here ss maps (X,ρ)(X,\rho) to the smallest per ρ¯\overline{\rho} containing ρ\rho, its symmetric and transitive closure. Note that general societies (X,ρ)(X,\rho) in Per\mathbf{Per} correspond to Lebenswelten since they consist of a set of things XX some of which occur as ρ\rho-relata and on these subset dom(ρ)dom(\rho) the restriction of ρ\rho is indeed an equivalence aka recognition relation.

Now the gambit in Lawvere’s approach is that by using the concept of adjoint opposition and a suitable mathematical notion of Aufhebung we can try to reconstruct the dynamics of selfconciousness in Hegel from ‘pure selfconciousness’ (010\dashv 1) to ‘(society of) general reflexivity’ (IrReflRefl\mathbf{IrRefl}\dashv \mathbf{Refl}). The key observation of Lawvere is that the opposed subcategories with object (,)(\empty,\empty) and ({*},(*,*))(\{\ast\},(\ast,\ast)), respectively, at the level of pure ego, are subcategories of IrRefl\mathbf{IrRefl} and Refl\mathbf{Refl}, respectively, at the level of ‘general reflxivity’ with the crucial difference that now (,)(\empty,\empty) is also contained in the righthand modality Refl\mathbf{Refl} i.e. abstract pure ego (= the category 11) is split into the two concrete contradictions (,)(\empty,\empty) and ({*},(*,*))(\{\ast\},(\ast,\ast)) at 010\dashv 1 which are reunited in the concrete Refl\mathbf{Refl}: in a sense the contradiction of pure selfconciousness is resolved at the level of general reflexivity.

Hopefully the two examples have provided some intuition how a mediated adjunction ltrl\dashv t\dashv r between two fully faithful functors l,rl,r captures the idea of unity-and-identity-of-opposites (UIO). The concept of Aufhebung then intends to capture the passage from such an UIO to another UIO thereby creating a dynamics between logical concepts.

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 𝒜\mathcal{A} whose reflection preserves finite limits. The localization is called essential when the reflection has furthermore a left adjoint.

If lril\dashv r\dashv i is an essential localization then ll is also full and faithful. If \mathcal{B} is a topos, 𝒜\mathcal{A} is called an essential subtopos and we write i !i *i *i_!\dashv i^*\dashv i_* in this case and call i !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,i,j,\dots .

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

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

  • the i-sheaves : XX\in\mathcal{B} with iXX\bigcirc _i X\simeq X (following the terminology at Lawvere-Tierney operator);

  • the i-skeleta : XX\in\mathcal{B} with iXX\Box _i X\simeq X (following the example of simplicial skeleta discussed below).

Definition

(Lawvere 1989b)

Let i,ji,j be levels, def. 2, of a topos 𝒜\mathcal{A} we say that the level ii is lower than level jj, written

i j i j \array{ \Box_i &\prec & \Box_j \\ \bot && \bot \\ \bigcirc_i &\prec & \bigcirc_j }

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

Let iji\prec j, we say that the level jj resolves the opposite of level ii, written

i j i j \array{ \Box_i &\ll& \Box_j \\ \bot && \bot \\ \bigcirc_i &\ll& \bigcirc_j }

(or just iji\ll j for short8) if j i= i\bigcirc _j\Box_i=\Box _i.

Finally a level i¯\bar{i} is called the Aufhebung of level ii

i i¯ i 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 ii, i.e. iff ii¯i\ll\bar{i} and for any kk with iki\ll k then it holds that i¯k\bar{i}\leq k in the order relation (by subtopos inclusion) between levels.

Remark

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

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

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

Thinking of i i= i\bigcirc_i\Box_i=\bigcirc_i as an expression of the negation of i\Box_i by i\bigcirc_i one could think of j i= i\bigcirc_j \Box_i=\Box_i as dialogical refinement of the opposition through a negation of the negation: ‘Proponent’ i\bigcirc_i gets updated - sublated to j\bigcirc_j in order to absorb the ‘opponent’ i\Box_i.

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

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

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 Ω\Omega and product preserving components functor Π 0\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)9. 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 𝒜\mathcal{A} with initial and terminal objects, via the adjoints to 𝒜{*}\mathcal{A}\to \{*\} that map ** to 00 and 11. 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=10=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 jj 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, 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 (X)(X)(\flat X \simeq \emptyset) \Leftrightarrow (X \simeq \emptyset).

Proof

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

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

(X) (X) (X) (X) (X). \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 (X)(X)(\flat X \simeq \emptyset) \Leftrightarrow (X \simeq \emptyset). Then for all XX

(X) (X) (X) (X) (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 Set^{\to} with objects maps XYX\to Y between sets X,YX,Y, the initial object is \emptyset\to\emptyset and the respective adjoint modalities are given by (XY)=X1\sharp(X\to Y)=X\to 1 and (XY)=XidX\flat(X\to Y)=X\overset{id}{\to} X. Since not only ()=\flat(\emptyset\to\emptyset)=\emptyset\to\emptyset but also (Y)=\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 0\leq 0 with respect to the \flat-modal base (∞,1)-topos.

Proposition

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

Then in H\mathbf{H} we have \sharp \emptyset \simeq \emptyset, hence that ()(\flat \dashv \sharp) resolves, def. 3, 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 𝒮\mathcal{S} has a terminal object *\ast, it follows that for XHX\in \mathbf{H} any sheaf X:𝒮 opSetX \colon \mathcal{S}^{op}\to Set then

XX(*) \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 U𝒮U\in \mathcal{S} there exists a morphism i:*Ui \colon \ast \to U hence for every XHX\in \mathbf{H} and every UU there exists a morphism i *:X(U)Xi^\ast \colon X(U)\to \flat X. This means that if X\flat X \simeq \emptyset then X(U)X(U) \simeq \emptyset for all U𝒮U \in \mathcal{S} and hence XX\simeq \emptyset. From this the claim follows with prop. 1.

Proposition

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

Then in H\mathbf{H} we have Aufhebung, def. 3, of the duality of opposites of becoming *\emptyset \dashv \ast (“Dasein”).

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

By prop. 2 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 j\mathcal{E}_j. Since the double negation topology ¬¬\neg\neg is the unique largest dense topology it follows in general that ¬¬ j\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 j= ¬¬\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 j\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: H ¬¬=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. 3 and hence exhibiting Aufhebung of becoming include

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

id id ʃ inf inf ʃ * \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 id_\mathcal{E} yielding the trivial adjoint string id id id id_\mathcal{E}\dashv id_\mathcal{E}\dashv id_\mathcal{E} with no less trivial adjoint modalities id id 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 id 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 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 nn is n+1n+1. Of course, id 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 id_\mathcal{E} when the lattice of levels is finite e.g. in the Sierpinski topos Set Set^{\to} (see below).

Though id 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 id_\mathcal{E} is the Aufhebung of no other level than itself absolute and id 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 id_\mathcal{E} but this is not the case. In fact id id_\mathcal{E} is only an instance of a whole class of essential localizations enjoying this property:

Definition

Let 𝒞\mathcal{C} be a finitely complete category. An essential localization lri:𝒞l\dashv r\dashv i:\mathcal{L}\to\mathcal{C} is called quintessential if ll is naturally isomorphic to ii.

To say that lri:𝒞l\dashv r\dashv i:\mathcal{L}\to\mathcal{C} is a quintessential localization amounts to say in Lawvere’s terminology (2007) that i:𝒞i:\mathcal{L}\to\mathcal{C} exhibits 𝒞\mathcal{C} as a quality type over \mathcal{L} with rr providing the right adjoint to ili\simeq l (provided \mathcal{L}, 𝒞\mathcal{C} are extensive).

The following is immediate:

Proposition

Let lri:𝒞l\dashv r\dashv i:\mathcal{L}\to\mathcal{C} be a quintessential localization. Then the corresponding adjoint modality lrirl\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 𝒞\mathcal{C} with objects pairs (X,e)(X, e) where XX is a set and e=e 2e=e^2 an idempotent map XXX\to X. A morphism f:(X 1,e 1)(X 2,e 2)f:(X_1, e_1)\to (X_2, e_2) is a function f:X 1X 2f:X_1\to X_2 with fe 1=e 2ff\cdot e_1=e_2\cdot f. These equivariant morphisms are bound to preserve fixpoints: when e 1(x)=xe_1(x)=x then f(e 1(x))=f(x)=e 2(f(x))f(e_1(x))=f(x)=e_2(f(x)). Then the fixpoint set functor r:𝒞Setr:\mathcal{C}\to Set with r(X,e)={xX|e(x)=x}r(X, e)=\{x\in X | e(x)=x \} is left as well as right adjoint to i(X)=(X,id X)i(X)=(X, id_X) since an equivariant morphism f:(X,e)(Y,id Y)f:(X,e)\to (Y,id_Y) is uniquely determined by its restriction to the fixpoints of ee and its values are given by f(e(x))f(e(x)). The adjoint modality irir:𝒞𝒞i\cdot r\dashv i\cdot r:\mathcal{C}\to\mathcal{C} corresponding to iri:Set𝒞i\dashv r\dashv i:Set\to\mathcal{C} maps (X,e)(X,e) to (r(X),id r(X))(r(X),id_{r(X)}). The corresponding level sublates *\emptyset\dashv\ast as well as itself whereas id id_\mathcal{E} only sublates itself.

For further properties of quintessential localizations see at quality type.

The example of the Sierpinski topos

(…)

Simplicial and cubical sets

(…) simplicial set (…)

simplicial skeleton \dashv simplicial coskeleton

(…) cubical set (…)

(Kennett-Riehl-Roy-Zaks (2011))

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

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.


(full pdf by In Situ Art Society)


References

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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. For another example of a such crude but useful mathematical modelling intended to capture social phenomena see at kinship.

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

  9. Lawvere suggests more generally to read i i\Box_ i\dashv\bigcirc_i as an opposition non-F vs. pure-F where FF is a property descriptively appropriate for the level.

Revised on September 10, 2017 11:19:31 by Thomas Holder (2.247.252.164)