Perfectoid Diamonds and n-Awareness. A Meta-Model of Subjective Experience
PPerfectoid Diamonds and n -Awareness. A Meta-Model ofSubjective Experience. Shanna Dobson and Robert Prentner Abstract
In this paper, we propose a mathematical model of subjective experience in terms of classesof hierarchical geometries of representations (“ n -awareness”).We first outline a general framework by recalling concepts from higher category theory,homotopy theory, and the theory of ( ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , n -declension”) for a novel language to express n -awareness, accompanied by a new temporalscheme (“ n -time”). Our framework allows us to revisit old problems in the philosophyof time: how is change possible and what do we mean by simultaneity and coincidence?We also examine the notion of “self” within our framework. A new model of personalidentity is introduced which resembles a categorical version of the “bundle theory”: selvesare not substances in which properties inhere but (weakly) persistent moduli spaces in the1 a r X i v : . [ m a t h . G M ] F e b -theory of perfectoid diamonds. Keywords : Perfectoid spaces, representation, ( ∞ , n -time, simultaneity, selves, localization, Efimov K-theory. “Representationalism” in the philosophy of mind is the well-known view that regards ourthoughts to be representations of external things, perceiving a “mental image” ratherthan directly accessing their nature. We are most interested in representationalist theories of consciousness [6]. There are many variants of such theories, and we refrain of a moredetailed overview, let alone listing and comparing their relative merits and problems. How-ever, representationalism about consciousness comes with several well-known problems thathave to do with a perceived mismatch between subjective experiences and representations: Privacy refers to the inaccessibility of subjective experience “from the outside”. The“inside” of an experience is not directly accessible, but only reflected by the outside. Ametaphor that illustrates this is that of a (mineralogical) diamond. A diamond displaysits (invisible) impurities as (visible) colors. The following properties particularly applyto the experience of selves:
Self-containedness refers to the property that any possibleexperiential pattern is “always already” [7] prefigured within the totally of an experience. Self-reflexivity refers to the idea that all parts that make up an experience are mirrored intheir whole experiential context.We propose a concrete mathematical structure, the “perfectoid diamond” [3], as amodel of subjective experience and appeal to Efimov K-theory [4] to study the equivalencerelations across all such diamonds. This would eventually lead to a geometric unificationof models that study (all possible forms of) subjective experience, which is why we referto it as our “meta-model”, analogous to a grand unified theory of mathematical structures(the “Langlands-program” [14]).We will first outline a basic (“vanilla”) framework of ( ∞ , This property further carries the connotation of being a “substance” in the sense of an independentlyexisting entity (e.g. [8]). • Conjecture 1 . The ( ∞ , ∞ , • Conjecture 2 . Topological localization, in the sense of Grothendieck-Rezk-Lurie( ∞ , ∞ , • Conjecture 3 . The meta-model takes the form of Efimov K-theory of the ( ∞ , Y on the category of perfectoid spaces [9] of characteristic p , which is constructed as the quotient of a perfectoid space by a pro-´etale equivalencerelation. Specifically, this quotient lives in a category of sheaves on the site of perfectoidspaces with pro-´etale covers. Perfectoid spaces are analytic spaces over nonarchimedeanfields, variants of Huber space [10]. The pro-´etale topology is a topology on the categoryof perfectoid spaces. A pro-´etale equivalance relation is a way of gluing together perfectoidspaces under the pro-´etale topology. A sheaf is a tool used to pack together local dataon a topological space. Diamonds contain impurities which are profinitely many copiesof geometric points Spa ( C ) → D [12]. A profinite set is a compact totally disconnectedtopological space. , A geometric point
Spa ( C ) → D inside a diamond D is made “visible”by pulling it back through a quasi-pro-´etale cover X → D , which gives rise to profinitelymany copies of Spa ( C ).While this property of perfectoid diamonds is taken to reflect the privacy of experience,talking about equivalence relations of classes of diamonds (and hence properties of “selves”)needs more work. This is the scope of our three conjectures, focusing on ( ∞ , Let S be a scheme defined over a field k and equipped with a morphism to Spec ( k ). A geometric point α in S is a morphism from the spectrum Spec (¯ k ) to S where ¯ k is an algebraic closure/separable closure of k . A profinite set S is extremely disconnected if the closure of any open subset U → S is still open [11]. The connected components of a totally disconnected space are of the form
Spa ( K, K +) for a perfectoidaffinoid field (
K, K + ). For strictly totally disconnected spaces, one has in addition that K is algebraicallyclosed. ∞ , K Efimov ( Y (cid:5) S,E ) One of the authors has recently proposed to extend K-theory to the study of (categories of)perfectoid diamonds [1]. K-theory is defined on the category of small stable ∞ -categorieswhich are idempotent, complete and where morphisms are exact functors. A certain cate-gory of large compactly generated stable ∞ -categories is equivalent to this small category.The condition for the colimit-preserving functor to preserve compact objects is that themorphisms are colimit preserving functors whose right adjoint also preserves colimits.In Efimov K-theory the idea is to weaken the condition of being compactly generated tobeing ‘dualizable’ such that K-theory is still defined. A category C being dualizable impliesthat C fits into a localization sequence C → S → X with S and X compactly generated.Then the Efimov K-theory should be the fiber of the K-theory in the localization sequence.The technical details are the following: 4igure 2: Efimov K-theory of diamonds as meta-model (picture taken from[1]). Whilethe perfectoid diamond is conjectured to model subjective experience, Efimov K-theoryof diamonds describes the relations between classes of diamonds (as objects in an ( ∞ , fimov Main Theorem Our terminology and small exposition follows Hoyois [4]: • ∞ -categories are called categories. • Let P r denote the category of presentable categories and colimit-preserving functors. • Let P r dual ⊂ P r denote the subcategory of dualizable objects and right-adjointablemorphisms (with respect to the symmetric monoidal and 2-categorical structures of P r ). • Let P r cg ⊂ P r be the subcategory of compactly generated categories and compactfunctors. Compact functors are functors whose right adjoints preserve filtered colim-its. • Let P r (cid:63)St denote the corresponding full subcategories consisting of stable categories. • Definition.
A functor F : P r dualSt → T is called a localizing invariant if it preservesfinal objects and sends localization sequences to fiber sequences. • Definition.
Let C ∈ P r be stable and dualizable. The continuous K-theory of C isthe space K cont ( C ) = Ω K( Calk ( C ) ω ). Lemma . If C is compactly generated, then K cont ( C ) = K ( C w ). Proof . The localization sequence is Ind of the sequence C w (cid:44) → C → Calk( C ) w . SinceK( C ) = 0, the result follows from the localization theorem in K-theory. Theorem [Efimov] . Let T be a category. The functor F un ( P r dualSt , T ) −→ F un ( Cat idemSt , T ), F (cid:55)→ F ◦ Ind ,restricts to an isomorphism between the full subcategories of localizing invariants, withinverse F (cid:55)→ F cont . In particular, if C ∈
P r cgSt , then F cont ( C ) = F ( C ω ). Proof . Let
A → B → C be a localization sequence in P r dualSt . Then for large enough κ wehave an induced localization sequenceCalk κ ( A ) → Calk κ ( B ) → Calk κ ( C ). 6t follows that F cont is a localizing invariant. By Lemma we have F cont ◦ Ind (cid:39) F , functo-rially in F . To C ∈ P r dualSt we can associate the filtered diagram of localization sequences C →
Ind( C κ ) → Calk κ ( C ) for κ (cid:29)
0. This gives a functorial isomorphism F (cid:39) ( F ◦ Ind) cont .The main goal is to develop a Waldhausen S-construction to obtain the K-theory spec-trum on the category of diamonds, and extend this to the ( ∞ , ∞ , ∞ , ∞ , ∞ , C [63]. Topological localizations are appropriate to our constructionbecause, under this type of localization of passing to the full reflective sub-( ∞ , ∞ -categorical localization of the category of chain complexes atthe class of quasi-isomporhisms [42]. The derived categories D ( A ) of abelian categories A are an important class of examples of triangulated categories. They are homotopycategories of stable ( ∞ , A [40]:If we ignore higher morphisms, we can “flatten” any ( ∞ , C into a 1-categoryho(C) called its homotopy category. When C is also stable, a triangulated structure cap-tures the additional structure canonically existing on ho(C). This additional structuretakes the form of an invertible suspension functor and a collection of sequences called dis-tinguished triangles, which behave like shadows of homotopy (co)fibre sequences in stable( ∞ , D ´ et and the six operations The Grothendieck six operations formalism, also known as six functor formalism, is aformalization of aspects of the refinement of Poincar´e duality from ordinary cohomologyto abelian sheaf cohomology. This has been adapted by Scholze in the following way [58]:7
Terminology . Fix a prime p . Let X be an analytic adic space on which p is top-logically nilpotent. To X was associate an ’etale site X ´ et . Let Λ be a ring suchthat n Λ = 0 for some n prime to p . There exists a left-completed derived category D ´ et ( X, Λ) of ´etale sheaves of Λ-modules on X ´ et . Let P erf d be the category of per-fectoid spaces and
P erf be the subcategory of perfectoid spaces of characteristic p .Consider the v -toplogy on Perf . • Definition . A diamond is a pro-´etale sheaf D on P erf which can be written as thequotient
X/R of a perfectoid space X by a pro-´etale equivalence relation R ⊂ X × X .Scholze restricts to a special class of diamonds which are better behaved. Denote theunderlying topological space of a diamond Y as | Y | = | X | / | R | . • Definition . A diamond Y is spatial if it is quasicompact and quasiseparated (qcqs),and | Y | admits a basis for the topology given by | U | , where U ⊂ Y ranges overquasicompact open subdiamonds. More generally, Y is locally spatial if it admits anopen cover by spatial diamonds. • Definition [from Definition 1.7 in [58]]. Let X be a small v -stack, and considerthe site X v of all perfectoid spaces over X , with the v -topology. Define the fullsubcategory D ´ et ( X, Λ) ⊂ D ( X v , Λ) as consisting of all
A ∈ D ( X v , Λ such that for all(equivalently, one surjective) map f : Y → X from a locally spatial diamond Y , f ∗ A lies in ˆ D ( Y ´ et , Λ). D ´ et ( X, Λ) contains the following six operations, see also Fig, 2.1. Derived Tensor Product. - (cid:78) L Λ - : D ´ et ( X, Λ) × D ´ et ( X, Λ) → D ´ et ( X, Λ).2. Internal Hom. R H om Λ ( − , − ) : D ´ et ( X, Λ) op × D ´ et ( X, Λ) → D ´ et ( X, Λ).3. For any map f : Y → X of small v -stacks, a pullback functor f ∗ : D ´ et ( X, Λ) → D ´ et ( Y, Λ).4. For any map f : Y → X of small v -stacks, a pushforward functor R f ∗ : D ´ et ( Y, Λ) → D ´ et ( X, Λ). The v -topology, where a cover { f i : X i → X } consists of any maps X i → X such that for anyquasicompact open subset U ⊂ X , there are finitely many indices i and quasicompact open subsets U i ⊂ X i such that the U i jointly cover U [58]
8. For any map f : Y → X of small v-stacks that is compactifiable, representablein locally spatial diamonds, and with dim.trg f < ∞ functor R f ! : D ´ et ( Y, Λ) → D ´ et ( X, Λ).6. For any map f : Y → X of small v -stacks that is compactifiable, representable inlocally spatial diamonds, and with dim.trg f < ∞ , a functor R f ! : D ´ et ( X, Λ) → D ´ et ( Y, Λ).
The notion of “1-awareness” refers to a network of representations, for example, when con-centrating on an exam or a zoom call, while being implicitly directed to the environmentand to future events. If a category theoretical approach is correct, it is natural to also con-sider higher representational levels, based on 1-awareness. We thus refer to “2-awareness”as simultaneously sustaining any combination of 1-awarenesses, such as in the followingexample: Imagine you are having lunch in your home on a Saturday at noon in the year2021 while simultaneously you are a Cambridge Apostle having a discussion in the MoralSciences Club on a Saturday evening in 1888. Continuing, 3-awareness would be the sus-taining of any combination of 2-awarenesses. For example, simultaneously being aware ofsimultaneously being aware of having lunch in your home on a Saturday at noon localizedin the year 2021 while being a Cambridge Apostle having a discussion in the Moral SciencesClub on a Saturday evening in 1884, and of giving a thesis defense on perfectoid spaces ona Friday at 10am in 2024, while being at your desk and finishing the first chapter of saidthesis later this evening... And so on, up to “ n -awareness”.In section 2, we model this idea based on higher category theory (the “category of n -awareness”, starting with n = 1). The objects of “1-awareness” are representations;morphisms between such objects are maps between such representations.We then distinguish “higher” categories, based on the category of 1-awareness. Theset Hom ( X, Y ) is the set of all morphisms between objects X and Y in a category. Wewould further want Hom ( X, Y ) to be more than a set. We would like
Hom ( X, Y ) tobe a topological space of morphisms from X to Y . One can appeal to homotopy theoryto precisely study transformation and equivalences between such topologically structuredspaces. We conclude this section by generalizing our setup to ( ∞ , In order to describe this more precisely, we will, in Conjecture 1, conjecture that the complexities of n -awareness (with n → ∞ for ( ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , ∞ , n -declension”) of a new language,which follows from our mathematical model of n -awareness and its concomitant extensionto n -time. Such a linguistic system enables us to actually express experience, revealing theway we concretely engage with the world – and constantly make the error of suspectingsubstantial (metaphysical) entities where there are none, analogous to the conception of“language games” [15], updated to a “language scheme” inspired by the mathematicalmodel proposed.The fiction of substantial temporal parts is rejected. How to regard subjective ex-perience against (the ontology of) time is one of the most puzzling open questions inphilosophy. And any mathematical treatment of subjective experience should be able toshed some light on its relation to time. Time in our framework is modeled as hierarchy of“ k -times” ( k ≤ n ): 1-awareness implies a notion of 1-time, 2-awareness implies a notion of2-time, ... , and n -awareness implies a notion of n -time. On our reading, temporality isjust a consequence of the geometry of representation. It is at most an ordering property.This is not an altogether new idea [16], but our mathematical model bears some naturalconsequences that have not, to our knowledge, been developed so far in the literature: ourstructural definition of n -time puts a novel spin on the problem of temporal coincidencesand simultaneity. A 2-awareness contains its 1-awarenesses, which are thus “simultane- covering maps can be represented in terms of the pro-´etale topology on the category of perfectoid spaces. selfhood ”. One desideratum that any theory of selfhoodshould meet is that any self’s experience is self-reflexive, i.e. all moments (parts) of suchan experience are referring to the total experiential context of the self. This distinguishesthe experience of a self from experience “as such”. We account for this self-reflexivity bythe idea of having invertible morphisms between objects in higher categories, implying a“mirroring” of representations in all other representations – an “internalized monadology”in the spirit of Leibniz [17]. This attributes self-reflexivity on the level of the next highercategory. Since ( k − k -category, the propertyof self-reflexivity is “lifted” to the next higher level of awareness. However, having strictlyinvertible morphisms would imply structural equivalence. This would destroy any notionof a “true individual” on the level of selves. The philosophical problem underlying thisdilemma is the following: If the universe evolves toward a system of equivalent represen-tations, how to prevent it eventually becoming “totalitarian”, neglecting individuality?However, in the framework of homotopy theory, there is a sense in which such equivalencesonly hold “up to” [18] homotopy equivalence. So, rather than thinking about “selves” asenduring entities wholly present at every moment in time (i.e. like endurantists would),but also unlike the idea that we have (more or less well-defined) “temporal parts” (i.e. likeperdurantists would), we replace the idea of personal identity over time with an updatedform of the “bundle theory of self” [19], based on a notion of “self-reflexivity” that pre-serves the individuality of experience.Some additional mathematical definitions can be found in an appendix. We later, in Conjecture 2, conjecture to work with abelian groups [20] and to form a complex [21] ofthese groups, which is informally a sequence of compositions. Then we “chain” the complexes togetherusing a chain map. These complexes are the objects in the “derived category” [22]. Thus we have upgradedrepresentations from being objects in a general category to being objects in the derived category. The “bundle” refers to a collection of (weakly equivalent) awareness-objects which are related viamorphisms. The vanilla framework
A mathematical model should express things in a minimal and conceptually coherent way.(Higher) category theory, while seemingly abstract, offers a flexible framework in whichto define “relations”, arguably the basic structural property of subjective experience [23,24, 26, 27, 28]. Our meta-model (“ n -awareness”) should not be regarded as model of asubstantial (e.g. spatio-temporal) entity, but as relational object – as if the structure alonedictated its properties. In particular, the following bears some similarities to the work ofAndr´ee Ehresmann and coworkers [23, 29], however with the difference that the primitiveobjects are not physiological entities (e.g. neurons), but representations.In mathematics, category theory formalizes the notion of “structure” in a very generalway. A category is defined by its objects , A, B , and morphisms (structure-preserving maps)between those objects, A → B , satisfying certain requirements (composing associativelyand the existence of an identity). We propose to change the “category number” of math-ematical models to study subjective experience. Awareness is sometimes identified with a“0-category”, e.g. computable functions between domains. One prominent approach alongthese lines is machine-state functionalism [30] which identifies “mental states” with func-tional relations involving sensory inputs and behavioral outputs, representable as stringof numbers. Importantly, functionalism also assumes that mental states are individuatedwith respect to the relations they have to other mental states.Upgrading this idea further, we propose “ n -categories” to study subjective experience:A “1-category” contains objects and “1-morphisms” which are morphisms between theseobjects. For disambiguation, the objects of the categories could be referred to as “mo-ments of awareness”, defined as dependent parts within an experiential whole, followingan early idea of Edmund Husserl [31] – in the case of 1-awareness we call those objects“representations” (Table 1).We do not wish to enter the debate surrounding the question what these represen-tations are, but we are primarily interested in how they relate . Our hope is that we can Two examples are the category of sets where this amounts to the fact that morphisms are functions,or the symmetric monoidal category where this condition means to satisfy hexagonal identity. Some popular proposals: functional states [32], patterns of brain states [33], “intentional” objects ofthought [34], “monads” [17], or fundamental constituents of the universe [35, 36, 37]. n -awareness”). n -awareness: a category of categories of... representations ↑ ...subjective experience 3-awareness: a category of categories of categories of representations ↓ does .1-awareness consists of various representations and the relations between them. Thismeans that, for example, me being aware of the zoom call right now is contained (asrepresentation) within a category populated by myriads of other (also potential) moments ofawareness – sensing the joy in your voice, or being afraid of your dismissive reactions to whatI want to say – but also my (implict) perceptions of environmental going-ons, memories, andother (possibly explicit) background experiences; and similar for the examples of attendinga meeting of the Cambridge apostles in 1888 (this experience is related to my potentialexperience of listening to Wittgenstein a few years later), or for the experience of a thesisdefense in 2024 (this experience is related to my actual experiences of writing the thesisnow), etc. See Fig. 3 (left).One could “increase the category number” to study possible relations between relations.A “2-category” contains objects, 1-morphisms, and 2-morphisms. In Fig. 3 (middle) we notonly have 1-morphisms between objects but also 2-morphisms between the 1-morphisms.The collection of all morphisms from A to B forms a set called the “homset”, Hom ( A, B ).In a 2-category, each homset itself carries the structure of a category – a collection ofobjects and morphism satisfying certain requirements – and thus morphisms between suchhomsets can be regarded as morphisms between 1-categories. This higher dimensionalstructure allows the 2-category to sustain two moments of awareness “at once”. See Fig. 3,(middle). Continuing further, a “3-category” contains objects, 1-morphisms, 2-morphisms,13 (cid:58) φ (cid:36) (cid:44) (cid:48) (cid:56) φ (cid:38) (cid:46) A f (cid:42) (cid:42) g (cid:52) (cid:52) B A f (cid:42) (cid:42) g (cid:52) (cid:52) (cid:103) (cid:111) γ (cid:111) (cid:119) B A f (cid:42) (cid:42) g (cid:52) (cid:52) (cid:104) (cid:112) γ (cid:110) (cid:118) B (cid:106) (cid:116) (cid:42) (cid:52) C D g (cid:107) (cid:107) f (cid:115) (cid:115) C D g (cid:107) (cid:107) f (cid:115) (cid:115) C D g (cid:107) (cid:107) f (cid:115) (cid:115) Figure 3: A graphical overview for n -awareness, for n = 1 , ,
3. Left: 1-category consistingof objects
A, B, C, D (representations) and possible 1-morphisms, f i , g i between A and B (respectively between D and C ). Middle: 2-category which also includes 2-morphisms, φ : f g → f g and γ : f g → g f . Right: 3-category which also includes 3-morphisms between2-morphisms.and 3-morphisms between the 2-morphisms, where 3-morphisms can be seen as relationsbetween 2-categories (or the respective moments). (Fig. 3, right).It follows that an n -category contains objects, 1-morphisms, 2-morphisms, . . . , up to n -morphisms between the ( n − Instead of having just a set of morphisms from one object to another (i.e. the homset),we want our categories to have the structure of a topological space of morphisms from oneobject to another , so that we can have a space of maps between the objects we wishto study. This also allows us to speak about transformations and equivalences betweensuch spaces, and thus between moments of awareness. We appeal to homotopy theory tomake these ideas more precise. This will allow us to understand n -awareness as (nested)“simultaneous presence” of m moments of 1-awareness, thereby avoiding certain problemsthat are related to time. Homotopy theory, which studies deformation equivalences calledhomotopies, is defined as follows: Two continuous functions from one topological space to Others have previously hypothesized, for slightly differrent reasons, that a mathematical model ofconsciousness should have at least the structure of a topological space [25, 27].
Homotopygroups extend this notion to equivalences between topological spaces, to therein classifytopological spaces. There is a remarkable freedom in reducing strong equivalences, suchas the claim that object A equals object B , to deformation equivalences, such as theclaim that object A is “deformation equivalent” to object B because one can create thehomotopy map which makes these objects homotopy equivalent. This allows us to speakof two experiences as being equivalent “up to” homotopy. A topos (Greek for “place”) is a category which behaves like the category of sets but alsocontains a notion of localization. Morphisms between objects of n -awareness represent theway how different moments of awareness are related to each other. In case morphisms are invertible , a mapping from Hom ( A, B ) to
Hom ( C, D ) implies the existence of a mappingfrom
Hom ( C, D ) to
Hom ( A, B ). An ( ∞ , k )-category is an infinity category ( n → ∞ ) inwhich all morphisms higher than k are invertible. For our model, we chose to work in( ∞ , k -morphisms (for k >
1) are invertible; this means that therespective moments of awareness could be transformed into an equivalent structure. Thisalso means that any moment of k -awareness “reflects” all the other objects it is related to,and looking at the whole category (or at the respective object in the next higher category),it can be said that n -awareness is “self-reflexive”.An ( ∞ , Theobjects in ( ∞ , ∞ , ∞ , ∞ , ∞ , C which satisfies three conditions: C is We say that two topological spaces, X and Y , are of the same homotopy type or are homotopy equivalentif we can find continuous maps f : X → Y and g : Y → X such that g ◦ f is homotopic to the identity map idX and f ◦ g is homotopic to idY . Topoi are modeled after Grothendieck’s notion of a sheaf on a site [38]. A prototypical example of atopos is the category of sets, since it is the category of sheaves of sets on the one point space. Informally,topoi are “nice” categories for doing geometry that act like models of intuitionist type theory. They areabstract contexts “in which one can do mathematics independently of their interpretation as categories ofspaces” [22, 63]. In this context, higher category theory investigates the generalizations of ∞ -groupoids to directedspaces [41]. C is sent to the slice category C/u ). We propose theseconditions to reflect the property of subjective experience to be “self-contained”. Allconceivable relations are built inside the framework.The ( ∞ , Y (cid:5) S,E = S × ( Spa O E ) (cid:5) ,the diamond Fargues-Fontaine curve of the geometrization of the local Langlands corre-spondence [14]. K Efimov ( Y (cid:5) S,E ) ∞ -categories+ invertible k -morphisms ( k > (cid:39) (cid:39) (cid:77) (cid:77) homotopy theory (cid:120) (cid:120) ( ∞ , − topoi (cid:83) (cid:83) Figure 4:
Gestell of a meta-model. The theory of ( ∞ , ∞ , ∞ , In general, topoi are models of internal logic, which means that almost any [63] logical statement couldbe internalized. ( ∞ , Arriving at the meta-model: Three conjectures
The vanilla framework of ( ∞ , Conjecture 1 . The ( ∞ , ∞ , Remark 1 . Recall, the category of sheaves on a (small) site is a Grothendieck topos.
Remark 2 . Lurie discusses the structure needed for this construction. Recall the fol-lowing:
Definition [see also Definition 6.2.2.6 in [66]]: An ( ∞ , ∞ , ∞ , Sh ( C ) L ← (cid:44) → P Sh ( C ) of an ( ∞ , ∞ , • L is a topological localization. •
2. There is the structure of an ( ∞ , C such that the objects of Sh ( C ) areprecisely those ( ∞ , A that are local objects with respect to the coveringmonomorphisms p : U → j ( c ) in P Sh ( C ) in that A ( c ) (cid:39) P Sh ( j ( c ) , A ) PSh(p,A) −−−−−−→
P Sh ( U, A ) is an ( ∞ , ∞ Grpd. •
3. The ( ∞ , ∞ , Remark 3 . This conjecture requires enlarging the class of higher pro-´etale morphisms ofdiamonds, restricting to locally spatial diamonds, and is motivated in [3]:Theorem 9.1.3 ([58, Propositions 9.3, 9.6, 9.7]). Descent along a pro-´etale cover X (cid:48) → X of a perfectoid space f : Y (cid:48) → X (cid:48) is effective in the following cases. •
1. If
X, X (cid:48) and
Y, Y (cid:48) are affinoid and X is totally disconnected. •
2. If f is separated and pro-´etale and X is strictly totally disconnected.17
3. If f is separated and ´etale. •
4. If f is finite ´etale.Moreover, the descended morphism has the same properties.The diamond properties of coherence (quasicompact, and quasiseparated), the pro-´etaletopology, and its presentability correlate with assumed properties of subjective experience.In a perfectoid diamond, privacy is mirrored in its interior “geometric points” being madevisible as profintely many covers. Geometric points have different appearances dependingon the “angle” they are viewed from, and complete information about these points cannotbe recovered. Informally, the structure of the diamond is defined topogically (by its pro-´etale topology). Self-containedness is a property inherited from the ( ∞ , ∞ ,
1) category of diamonds is a conjectured example.
Definition . Let
P erf d be the category of perfectoid spaces and
P erf be the subcategoryof perfectoid spaces of characteristic p . A diamond is a pro-´etale sheaf D on P erf whichcan be written as the quotient
X/R of a perfectoid space X by a pro-’etale equivalencerelation R ⊂ X × X .In essence, a diamond is a sheaf Y on the category Perf which is constructed as the quo-tient of a perfectoid space by a pro-´etale equivalence relation. More formally, this diamondquotient lives in a category of sheaves on the site of perfectoid spaces with pro-´etale covers.One example of a diamond is given by the sheaf SpdQ p that attaches to any perfectoidspace S of characteristic p the set of all untilts S Q p , where untilting , informally, isan operation that translates from characteristic p back to characteristic 0 [57]. The formaldefinition is: Definition . SpdQ p = Spa ( Q cyclp ) /Z × p . That is, SpdQ p is the coequalizer of Z × p × Spa ( Q cyclp ) (cid:91) ⇒ Spa ( Q cyclp ) (cid:91) , where one map is the projection and the other is the action;cf. [3].A second example is the moduli space of shtukas for ( G , b, { µ , ..., µ m ) } fibered over them-fold product SpaQ p × SpaQ p ... × m SpaQ p [3].18erfectoid spaces provide a framework for translating number-theoretic class field problemsfrom characteristic p to characteristic 0. The definition of perfectoid space is as follows. Definition . A perfectoid space is an adic space covered by affinoid adic spaces of theform
Spa ( R, R + ) where R is a perfectoid ring. Examples of perfectoid spaces, as detailed in [3], are the following: • The Lubin-Tate tower at infinite level: M LT, ∞ = ˜ U x × GL ( Q P ) GL ( Q P ) ∼ = (cid:116) Z ˜ U x [3]. • Perfectoid Shimura varieties: S K p ∼ lim ←− K p ( S K p K p (cid:78) E E p ) ad [57]. • Any completion of an arithmetically profinite extension (APF), in the sense of Fontaineand Wintenberger [59], is perfectoid. A nice source of APF extensions is p -divisibleformal group laws. • If K is a perfectoid field and K + ⊂ K is a ring of integral elements, then Spa ( K, K +)is a perfectoid space. • Zariski closed subsets of an affinoid perfectoid space support a unique perfectoidstructure. • It is mentioned that the nonarchimedean field Q p is not perfectoid as Q p does nothave a topologically nilpotent element ξ ∈ Z p whose p -th power divides p . • Definition.
A perfectoid space X is totally disconnected if it is quasicompact andquasiseparated (qcqs) and every open cover splits. A perfectoid space X is strictlytotally disconnected if it is qcqs and every ´etale cover splits. Any totally disconnectedperfectoid space is also affinoid.Why do we use adic spaces? Informally, adic spaces are versions of schemes associated tocertain topological rings and it is these structures that produce the nonarchimedean fieldswe need to represent the local/global nature of consciousness. Formally, an adic space is atriple ( X, O X , O + X ) where X is a topological space, O X is a sheaf of complete topologicalrings, and O + X is a subsheaf of O X that is locally of the form ( Spa ( A, A + ) , O A , O + A ).We assume that conscious agents operate via topological coverings, i.e. via establish-ing the right clustering of (data) points, rather than via calculating a particular function19ased on such a clustering. But in order to describe this process, we need a way tocapture the complexities of covering maps that preserves local homeomorphism and dif-feomorphism properties. ´Etale morphisms encapsulate the idea of maps which are localhomeomorphisms/local diffeomorphism. The pro-´etale site is the site of perfectoid spaceswith pro-´etale covers. Moreover, the language of sheaves provides a geometrization that can model the globalstorage of data and recollection of local data. Working over structures like
SpecR , thespectrum of a ring R , allows us to add additional structure to points, encoding geometricparameter spaces or moduli spaces. A more sophisticated mathematical structure allows usto model more complex behavior. Subjective experience seems to proceed from a local toglobal setting and back, mimicking the tilting and untilting operation of perfectoid spacesfrom characteristic 0 to characteristic p . Conjecture 2 . Topological localization, in the sense of Grothendieck-Rezk-Lurie ( ∞ , ∞ , Remark 4 . To construct a topology on the ( ∞ , ∞ , ∞ , ∞ , Definition [64]. The structure of an ( ∞ , ∞ , C is precisely thedata encoding an ( ∞ , ∞ , Sh ( C ) (cid:44) → P Sh ( C ) inside the ( ∞ , ∞ , C .Topological localizations are appropos to our diamond construction as objects and mor-phisms have reflections in the category, just as geometric points have reflections in theprofinitely many copies of Spa ( C ). Recall the definition of a reflective subcategory. Definition [65]. A reflective subcategory is a full subcategory C (cid:44) → D such that ob-jects d and morphisms f : d → d (cid:48) in D have “reflections” T d and
T f : T d → T d (cid:48) in C .Every object in D looks at its own reflection via a morphism d → T d and the reflection of The pro-´etale site tempers the “finiteness conditions” on the fibers of ´etale morphisms to pro-finitenessconditions, which are pro-´etale morphisms; cf. Remark 10.3.2 in [3].
20n object c ∈ C is equipped with an isomorphism T c (cid:39) c . The inclusion creates all limitsof D and C has all colimits which D admits. Remark 5 . Extending a perfectoid version of localization is essential to the constructionof the ( ∞ , D can be highly pathological and not even T . However, we will restrict toa special class of well-behaved diamonds, qcqs, given in [3]:Proposition 10.3.4. Let D be a quasicompact quasiseparated diamond. Then D is T .We previously appealed to the “up to” notion that comes naturally with homotopy theoryand to k -morphisms being invertible to mirror the self-reflexivity property. To make theidea of the associated equivalence (“equivalent up to homtopy”) more precise, we invokethe idea of a weak equivalence . We use derived category theory [22] to form a complex [21]of abelian groups, which is informally a sequence of compositions. Then we “chain” thecomplexes together using a chain map. Thus we have upgraded representations from beingobjects in a general category to being objects in the derived category. Weak equivalencescan also be referred to as “quasi-isomorphisms” in this context.Let A be a Grothendieck abelian category (e.g., the category of abelian groups). Wedefine K ( A ) to be the homotopy category of A whose objects are complexes of objects of A and whose homomorphisms are chain maps modulo homotopy equivalence. The weakequivalences are quasi-isomorphisms defined as follows: A chain map f : X → Y is aquasi-isomorphism if the induced homomorphism on homology is an isomorphism for allintegers n . K ( A ) is endowed with the structure of a triangulated category . A triangulatedcategory has a translation functor and a class of exact triangles which generalize fibersequences and short exact sequences. Localization by quasi-isomorphisms preserves thistriangulated structure. Bousfield localization [60] , particular to triangluated categories, An example from [3] details the quotient of the constant perfectoid space Z p over a perfectoid field bythe equivalence relation “congruence modulo Z ” which yields a diamond with underlying topological space Z p /Z . Through this process, the derived category D ( A ) of the initial abelian category A is obtained by “pre-tending” that quasi-isomorphisms in K ( A ) are isomorphisms. Specifically, the localization is constructedas follows: morphisms in D ( A ) between A · and B · will be ’roofs’ [61], with f, g morphisms in K ( A ) and f a quasi-isomorphism. This roof represents g ◦ f − . (cid:47) (cid:47) f (cid:15) (cid:15) B (cid:123) (cid:123) g (cid:15) (cid:15) C (cid:99) (cid:99) (cid:15) (cid:15) A (cid:48) (cid:47) (cid:47) B (cid:48) (cid:123) (cid:123) C (cid:48) (cid:99) (cid:99) Figure 5: A triangulated category composed of two exact triangles
A, B, C and A (cid:48) , B (cid:48) , C (cid:48) and three commuting squares ABB (cid:48) A (cid:48) , AC (cid:48) C (cid:48) A (cid:48) , and BB (cid:48) C (cid:48) C , with a non-unique fill-in[61].allows us to make more morphisms count as weak equivalences and this is formally howwe get from 1-awareness to n -awareness. One of the axioms of a triangulated categorystates that given the diagram in Fig. 5, where A, B, C and A (cid:48) , B (cid:48) , C (cid:48) form exact triangles,and the morphisms f and g are given such that the square ABB (cid:48) A (cid:48) commutes, then thereexists a map C → C (cid:48) such that all the squares commute. This triangulated category is acategorization of a set-theoretic ordinal . A set is an ordinal number if it is transitive andwell-ordered by membership, where a set T is transitive if every element of T is a subsetof T . Ordinals locate within a set as opposed to cardinality which references merely size. K ( A ) represents all possible moments of awareness (related by composition). Thereare an infinite number of 1-awarenesses in this category. Differentials define a 1-awarenessrelating with another 1-awareness. The composition of any two A ’s is the zero map. Thisis the group law, the return to identity. Let B be the abelian group of 2-awarenesses. Thedifferentials define relating with another 2-awareness. The diagram commuting means the2-awareness is related with the 1-awareness. In essence, this diagram represents a higher-dimensional notion of commutativity through the map C → C (cid:48) and the three squarescommuting. By higher-dimensional, we mean the three commuting squares and the twoexact triangles together form a cone construction. There is a relation here between thehigher commutativity and extending the pro-´etale morphisms of diamonds. Conjecture 3 . The meta-model takes the form of Efimov K-theory of the large sta-ble ( ∞ , Theorem [Efimov] [1].Specifically, we seek a diamond reformulation of the following Efimov theorem [1].
Theorem [Efimov*] . Let X be a locally compact Hausdorff topological space, C a stabledualizable presentable category, and R a sheaf of E -ring spectra on X . Suppose that Shv ( X ) is hypercomplete (i.e., X is a topological manifold). Let T be a stable compactlygenerated category and F : Cat idemSt → T a localizing invariant that preserves filtered col-imits. Then: F cont ( M od R ( Shv ( X, C ) (cid:39) Γ c ( X, F cont ( M od R ( C ))). Specifically, F cont ( Shv ( R n , C )) (cid:39) Ω n F cont ( C ). Proof . See [4], Theorem 15.
Diamond reformulation . The rough idea is the following:Conjecture [1]. Let S be a perfectoid space, D (cid:5) a stable dualizable presentable category,and R a p-adic shtuka on X . Let D be a stable compactly generated category and F :Cat idemSt → T a localizing invariant that preserves filtered colimits. Then: F cont ( Sht ( S n , C )) (cid:39) Ω n F cont ( D (cid:5) ) with S SpaQ p × SpaQ p . Presently, we do not have a way to talk (or write) about n -awareness – our language isantiphrastical. The Danish philosopher Søren Kierkegaard once remarked that piling up(empirical) knowledge in hope to get a glimpse of the nature of being commits one to “aperpetually self-repeating false sorites” [45]. One could interpret this as the problem ofarriving at knowledge about subjective experience by way of chaining together (empirical)propositions – a version of the explanatory gap argument [46]. An alternative, following23ierkegaard, would be to repeatedly live through one’s subjective experience, therebyexpressing its nature.Ontological statements could thereby be reconceived as being linguistic statements.We are “upgrading” Wittgenstein’s language games [15] to a “language scheme”. Thislanguage scheme follows from our meta-model. Instead of saying that meaning (reference)is a derivative from whatever game is at play, we say that meaning (reference) is a derivativefrom whatever scheme is at play. In view of Conjecture 1, the language scheme refers toa “profinite language” with expressions that consist of profinitely many words that are“glued together”. A basic example for language misuse is the stipulation of a substantive “self” thatendures through “time”, which might in actuality merely reflect the grammatical (andindexical) structure of our ordinary language. (Even though such a “self” does not referto an existing “thing out there”, it is highly useful, e.g., for communicating certain ideas,desires or intentions.) Another example would pertain to the idea of some “private internallanguage”. Actual languages depend on them being used by a community. The meaningof a word does not refer to an essentially private “thing”. Analogously, one could proceedin the case of a language scheme.First, we outline the basic grammatical structure of this scheme. Modern Englishlanguage contains three grammatical cases (subjective, objective, and possessive) withdifferent declensions for each case. We extend this grammatical structure by adding anumber to the respective pronoun, indicating the simultaneous presence of different “me’s”within an experience of multiple awareness (Tab. 2). Formally, the resulting “ n -declension”allows for all combinations of “ n , n − . . . , 1-declensions” to be present in a sentence. Sofor example, a statement expressing double-awareness could contain a “1-I” as well as asuperposed “2-I” as subjects. Analogously, “ n -conjugation” could be defined.Second, we envision a novel model of temporality. Spatio-temporally multiplied aware- What is the glossematics of such a profinite language? Recall that a profinite set S is called extremelydisconnected if the closure of any open subset U → S is still open. The philosopher and linguist RolandBarthes stated that “writing ceaselessly posits meaning ceaselessly to evaporate it, carrying out a systematicexemption of meaning.” [48] So, where is the meaning? We propose that word-meanings are a delicate thing:while we have profinitely many copies of words that are “glued together”, they never refer unambiguouslyto a single meaning “out there”. The glossemes correspond to the geometric impurities of the perfectoiddiamond, which is the glossematics. This is also true for “biological languages” implemented in a living “community” making up a body. n -awareness. Many mysticsin the religious traditions from the East and West reported a similar kind of experience.If awareness is not bound to a single region in space and time, this also suggests thatawareness cannot be understood as an emergent property of localized physical systems.Our model conceives of a temporal multiplicity with a categorified model of n -time,evidenced in the proposed language scheme by way of n -inflection for n -conjugation. Mod-ern English is spoken in local, linear time, yet it allows the inflectional change of verbsby way of conjugation. We extend the idea of language spoken in linear time, conjugatedover three tenses, to one of n -conjugation as follows: Instead of using only past, present,future, and their perfect correspondences, present-perfect, past-perfect, future-perfect, weallow for 1-past, 2-past, . . . , 1-present, 2-present, . . . , 2-present-perfect, 2-future-perfect, n -future-perfect etc, which is what we call n -time. This generalizes the discussion on tem-poral experience, which has traditionally been expressed in terms of a “tensed” experiencedtime vis-a-vis an “untensed” physical parameter time [49]. Different temporal ontologies have been proposed throughout the ages, without comingto definite conclusion. Rather than proposing yet another metaphysical framework, wewish to concentrate on particular problems that feature prominently in recent and not sorecent discussions. We hope that approaching these problems structurally, inspired by ourmeta-model, will lead to a remedy. The problems we wish to look at are the following:1.
How is change possible?
Change manifests itself every day. But what does itrefer to exactly? The idea that “change” does not really exist goes back to theworks of Parmenides and the Eleatic school of philosophy. While this philosophyhas been influential up to this day, for example, in the metaphysical thought ofMartin Heidegger [50], our everyday experience seems much better captured in the“everything flows” of Heraclitus [51].The American philosopher David Lewis revived the problem of change for the phi-losophy of time in the problem of temporary intrinsics [52]: An intrinsic property is a property an object has irrespective to the relations it bears to other objects. a b l e : G r a mm a r o f a l a n g u ag e s c h e m e f o r n - a w a r e n e ss , e x t e nd e d f r o m o r d i n a r y E n g li s h c a s e s t r u c t u r e . n = n = n = ··· n = k s ub j e c t i v e I ( w e ) → - I ( - w e ) - I ( - w e ) - I ( - w e ) ··· - I ( - w e ) - I ( - w e ) - I ( - w e ) - I ( - w e ) - I ( - w e ) ... k - I ( k - w e ) o b j e c t i v e m e ( u s ) → - m e ( - u s ) - m e ( - u s ) - m e ( - u s ) ··· - m e ( - u s ) - m e ( - u s ) - m e ( - u s ) - m e ( - u s ) - m e ( - u s ) ... k - m e ( k - u s ) d e p . p o ss e ss i v e m y ( o u r ) → - m y ( - o u r ) - m y ( - o u r ) - m y ( - o u r ) ··· - m y ( - o u r ) - m y ( - o u r ) - m y ( - o u r ) - m y ( - o u r ) - m y ( - o u r ) ... k - m y ( k - o u r ) i nd e p . p o ss e ss i v e m i n e ( o u r s ) → - m i n e ( - o u r s ) - m i n e ( - o u r s ) - m i n e ( - o u r s ) ··· - m i n e ( - o u r s ) - m i n e ( - o u r s ) - m i n e ( - o u r s ) - m i n e ( - o u r s ) - m i n e ( - o u r s ) ... k - m i n e ( k - o u r s ) are no intrinsic properties, just“disguised relations”; or one believes that only those properties that exist at the present moment are real, whereas the properties that an object seem to have hadpreviously are, in some sense, fictional (this position is known as “presentism”); orone accepts that objects have genuine temporal parts (e.g. the me-yesterday, theme-now, and the me-tomorrow). The latter solution to the problem of temporaryintrinsics has been deemed the only viable solution to the problem of temporaryintrinsics which is not “incredible” [52] and started the appreciation of “perdurance”theories in the modern philosophy of time at the expense of so-called “endurance”theories that conceive of persisting wholes without temporal parts [53]. In addition toLewis’ metaphysical rejunevation, much support for a “perdurance-like” theory seemsto come from science, in particular Einstein’s theory of relativity. Perdurance theory,so it is often but not always believed, squares well with the belief that space-timeforms a four-dimensional continuum as described by the special theory of relativity[53].2. What is simultaneity?
The second problem worth mentioning in this respect is theproblem of simultaneity (coincidence). It seems that, when discussing n -awarenesswe postulate the simultaneous presence of two experiences. This seems to violatethe basic intuition that no two objects could occupy the same place in time unlessthey are the same (or unless they share temporal parts: the statue and the clay havetemporal parts that overlap ). But it also seems to be in conflict with basic principlesof physics according to which there can be no “absolute” notion of simultaneity.3. Is synchronous reference possible?
While perdurantism claims to solve problemsof temporal coincidence (which are only problems if designed to be so), it has itsown problems when trying to account for the acts of synchronous and asynchronous
The converse would be a relational property: being an uncle is not a property that I have independent ofmy nephews. Another argument in favor of perdurantism. X f (cid:47) (cid:47) g (cid:47) (cid:47) Y q (cid:47) (cid:47) q (cid:48) (cid:35) (cid:35) Q u (cid:15) (cid:15) Q (cid:48) In category theory, a coequailizer refers to a single object (a “colimit”) associated to thedifferent morphisms f and g between objects X and Y , such that q ◦ f = q ◦ g . Furthermore,the objects Q is universal, meaning it is unique “up to” an isomorphism u . It follows that properties (i.e. morphisms within a category) are associated to a single and unique (upto isomorphism) object . Whereas perdurantism, translated into the language of categorytheory, is about change between such properties (i.e. the addition of new morphisms),endurantists refer to the unchanging (persisting) object defined by them. The diamond SpdQ p = Spa ( Q cyclp ) /Z × p is the coequalizer of Z × p × Spa ( Q cyclp ) (cid:91) ⇒ Spa ( Q cyclp ) (cid:91) , where onemap is the projection and the other is the action; cf. [3].Our solution superficially seems to correspond to a perdurantist representation withthe n -declension of 1-her, 2-hers, ... n -her, but these are relational properties that do28ot necessarily refer to temporal or intrinsic parts. It can be asked how different Mary-tomorrow is from Mary-today given that Mary-tomorrow has more morphisms? If Maryrefers to the objects of a category and the properties are its morphisms, then saying thatMary has “changed” is merely to say that Mary has added connections/morphisms. Else,we can say that Mary-today is the same as Mary tomorrow “up to” isomorphism.Analogous to how perdurantists resolve the problem of coincidence by noting thattemporal parts can indeed “overlap” without implying that the two objects that overlap areidentical, we note that categories (given they have at least some basic structural features:e.g. they are topological) could too be said to “overlap”. But this does not commit us toperdurantism as an ontological position.Note that the problem of simultaneity is mainly a (conceptual) “design issue” thatstems from a linear notion of time, where simultaneity is conceived in terms of (“temporal”)coincidence, or, alternatively, from the treatment of time in the framework of Minkowskispace-time (often called a “fourth dimension”). We instead choose to model time in termsof an equivalence relation using homotopy theory – “time” is not an absolute (ontological)notion, but instead refers to a relative (epistemic) “ordering” scheme of experiences. Thus,to every level of awareness, there corresponds a level of time. 1-awareness corresponds to1-time; 2-awareness corresponds to 2-time; etc.We offer a structural notion of “simultaneity of experience” by identifying it with thecommutativity of diagrams in a homotopy category, and respectively by the “up to” notion.Commutativity classifies the equivalence of all possible ways to get to a destination. Thereis no (structural) difference in choosing one way over the other. There is no indicatedstarting point or canonical progression. Rather, all possible paths are “revealed at once”(and even infinite paths might be alluded to). The “up to” notion grants a relative notion ofequivalence that corresponds to commutativity. Making the statement that all morphismsare equivalent “up to” homotopy means that they are equivalent with respect to homotopy.There is no substantial way to distinguish one morphism over any other. In a sense,commutativity is our way of geometrizing the “up to” notion. Equivalence and hencesimultaneity is never truly absolute. Simultaneity, on our view, expresses a homotopyequivalence and homotopy equivalences are neither perdurantist nor endurantist.Moreover, our meta-model provides a structural framework for (a)synchronous refer- Take, for example, the chain map between complexes: Fig. 7, in the appendix.
Spa ( C ). The asynchronicity comes from theprofinite condition. There is no canonical reference frame because of the existence of manysuch quasi-pro-´etale covers, the pullback of which gives the profinitely many copies. Eventsand their seeming synchrocity lie entirely in the quasi-pro-´etale covers. Two properties typically associated with the notion of self are, first, that the consciousexperience of a self is fundamentally self-reflexive: each moment of an experience alwaysrefers to the experiential context as a whole: Experience is embedded into a self thathas these experiences. If a self is nothing but a “bundle” of such experiences (or moretechnically: a weakly persistent moduli space [55] of relations between them), any momentof this experience refers to the whole bundle. This is not necessarily the case when wetalk about “minimal experiences as such” [56], but it should be the case when experienceis associated with a single self that has them. Second, we are interested in a modelthat acknowledges plurality and individuality, something that seems at odds with the ideaof a metaphysical monism that postulates all things to “be of the same kind” (at best,individual selves were instances of a universal natural process).If we work in an ( ∞ , your awareness tomorrow is (in some sense)equivalent to my awareness two days ago. This conflicts with our intuition that our ex-periences (across individuals but also across times) are unique . The notion of equivalence“up to” homotopy affords to deflate this strong notion of equivalence.You are not strictly the same “you”, with some substantialist notion of self in the Due to this mathematically “holographic” structure of the diamond, the idea is to reconstruct theAdS/CFT holographic principle [73] using diamonds, and the six operations in Scholze’s ´etale cohomologyof diamonds [58]. The diamonds represent the conformal field theory. The six operations reconstruct ananalytic version of what is Anti de Sitter space. Whether there truly is such an “experience as such” outside the context of a self, is still an open debate. Considering the results of subsection 4.2, there is no conception of a self “enduring overtime”, as if time were somehow exterior to experience; as if time were a constantly presentfluid through which selves move (however that happens); or as if time were a simple counterof experiences.
In section 2, we outlined a framework based on categories of representations and looked athierarchies that arise from the category theoretic treatment, starting with the morphismsdefining “1-awareness” up to (invertible) “‘ n -morphisms”. We emphasized the topologicalnature of the encountered relations between moments of awareness, and we postulatedto model this with homotopy theory and the theory of ( ∞ , • Conjecture 1 . The ( ∞ , ∞ , • Conjecture 2 . Topological localization, in the sense of Grothendieck-Rezk-Lurie( ∞ , ∞ , • Conjecture 3 . The meta-model takes the form of Efimov K-theory of the ( ∞ , Informally, this structural equivalence uses what one of the authors calls “perfectoid entanglement”.“Perfectoid entanglement entropy” is proposed as a measure of the degree of equivalence (cf. [5]).
31f “ n -declension” and “ n -time”), revisited some traditional problems in the philosophy oftime, and eventually investigated a notion of a self as a “bundle” of experiences. Thesediscussions had the purpose of, first, illustrating the framework in terms of language andgeneralizing a point made previously by various philosophers: how the intricacies of ourlanguage wrongly suggest to assume substantive entities (a disembodied “self”; “mean-ings” out there), where there in fact are none. With reference to the perfectoid diamondmodel, we proposed a “pro-finite” language with expressions that consists of profinitelymany words that are “glued together” without ever carrying some unambigious reference.Second, and related, many problems in the philosophy of time are artifacts of such aconfused language use. The remedy to these problems might not be found in a metaphys-ical solution (e.g. “perdurantism”), but in a re-conceptualization of problematic termssuch as “simultaneity”. We moreover hinted at a natural resolution of the problem of(a)synchronous reference within our meta-model.Third, the notion of a “self” could be understood in terms of a bundle of experiencesthat is “self-reflexive” and possess a form of “individuality”. The notion of “structuralequivalences” between classes of experiences is made precise in the Efimov K-theory ofperfectoid diamonds. Note that in each case certain supposedly “real” entities (linguisticmeanings, change, selves) were reduced to structuralist notions.We consider our model to reflect a new movement in mathematics which seeks to makefunctional analysis a branch of commutative algebra and therefore provide a geometrizationof analytical techniques. We conceive of replacing our model’s foundation of topologicalspaces with that of condensed mathematics [62]. The study of subjective experience isproposed to be a geometrical, rather than a computational, project.32 eferences [1] Dobson, S. Efimov K-theory of Diamonds , in preparation.[2] ncatlab authors. K-theory. Available online: https://ncatlab.org/nlab/show/K-theory (accessed February 10, 2021).[3] Scholze, P and Weinstein, J.
Berkeley Lectures on P-adic Geometry , Princeton Univer-sity Press, Annals of Mathematics Studies Number 207.[4] Hoyois, M.,
K-theory of Dualizable Categories (after rA. Efimov)
Perfectoid Quantum Physics and Diamond Nonlocality , in preparation.[6] Tye, M. ”Representational Theories of Consciousness”, in
The Oxford Handboook ofPhilospohy of Mind , edited byMcLaughlin, B.P., Beckermann, A., and Walter, S., OxfordUniversity Press, 2009, pp. 253–267.[7] Heidegger, M.
Sein und Zeit , Niemeyer, 1927.[8] Spinoza, B.
Ethica, ordine geometrico demonstrata , 1677.[9] ncatlab authors. Perfectoid Space. Available online: https://ncatlab.org/nlab/show/perfectoid+space (accessed January 29, 2021).[10] ncatlab authors. Huber Space. Available online: https://ncatlab.org/nlab/show/Huber+space (accessed January 29, 2021).[11] ncatlab authors. Profinite Space. Available online: https://ncatlab.org/nlab/show/profinite+space (accessed January 29, 2021).[12] ncatlab authors. Geometric Point. Available online: https://ncatlab.org/nlab/show/geometric+point (accessed January 29, 2021).[13] ncatlab authors. Grothendieck Topology. Available online: https://ncatlab.org/nlab/show/Grothendieck+topology (accessed January 29, 2021).[14] Fargues, L.,
Geometrization of Local Correspondence, an Overview , arXiv:1602.00999[math.NT] (accesseed February 10, 2021).3315] Wittgenstein, L.
Philosophical Investigations , Macmillan Publishing Company, 1953.[16] Huggett, N, and Hoefer, C. “Absolute and Relational Theories of Space andMotion”, in:
The Stanford Encyclopedia of Philosophy (Spring 2018 Edition) ,edited by Zalta, EN. https://plato.stanford.edu/archives/spr2018/entries/spacetime-theories/ (accessed January 29, 2021).[17] Leibniz, GW.
La Monadologie , 1714.[18] Wikipedia authors. Up to. Available online: https://en.wikipedia.org/wiki/Up_to (accessed January 29, 2021).[19] Hume, D.
A Treatise of Human Nature , 1740.[20] Wikipedia authors. Abelian group. Available online: https://en.wikipedia.org/wiki/Abelian_group (accessed January 29, 2021).[21] ncatlab authors. Complex. Available online: https://ncatlab.org/nlab/show/complex (accessed January 29, 2021).[22] Artin, M, Grothendieck, A, and Verdier, J-L.
Th´eorie des topos et cohomologie ´etaledes sch´emas, S´eminaire de G´eom´etrie Alg´ebraique du Bois-Marie 1963-64 , dirig´e par M.Artin, A. Grothendieck, J.-L. Verdier, SLN 269, 270, 305, Springer-Verlag, 1972, 1973.[23] Ehresmann AC, and Gomez-Ramirez. J. Conciliating neuroscience and phenomenologyvia category theory.
Prog. Biophys. , , 347-359.[24] Tsuchiya, N, Taguchi, S, and Saigo, H. Using category theory to access the relationbetween consciousness and the integrated information theory, Neuroscience Research , 107, 1–7.[25] Prentner, R. Consciousness and Topologically Structured Phenomenal Spaces,
Con-sciousness & Cognition , , 25–38.[26] Northoff, G, Tsuchiya, N, and Saigo, H. Mathematics and the Brain: A CategoryTheoretical Approach to Go Beyond the Neural Correlates of Consciousness. Entropy , 21(12), 1234–21.[27] Kleiner, J. Mathematical Models of Consciousness,
Entropy , , 609.3428] Signorelli CM, Wang, Q, and Kahn, I. A Compositional Model of Consciousness basedon Consciousness-Only, 2020. https://arxiv.org/abs/2007.16138 (accessed January29, 2021).[29] Ehresmann AC, and Vanbremeersch, J-P. Memory Evolutive Systems: Emergence,Hierarchy, Cognition.
Elsevier, 2007.[30] Putnam, H. “Psychological Predicates”, in
Art, Mind, and Religion , edited by Capi-tan, WH, and Merrill, DD. University of Pittsburgh Press, 1967, pp. 37–48.[31] Husserl E.
Logische Untersuchungen , Husserliana XIII, 1975.[32] Block, N. What is functionalism?
The Encyclopedia of Philosophy Supplement , 1996. (accessed Jan 26, 2021).[33] Wu, W. ”The Neuroscience of Consciousness”, in
The Stanford Encyclopedia ofPhilosophy (Winter 2018 Edition) , edited by Zalta, EN. https://plato.stanford.edu/archives/win2018/entries/consciousness-neuroscience/ (accessed January29, 2021).[34] Crane, T.
The Objects of Thought , Oxford University Press, 2013.[35] Hoffman, DD. Conscious realism and the mind-body problem.
Mind and Matter , , 87–121.[36] Hoffman, DD, and Prakash, C. Objects of Consciousness, Frontiers in Psychology , 5:577.[37] Goff, P.
Consciousness and Fundamental Reality , Oxford University Press, 2017.[38] Illusie, L. What is a topos?
Notices of the AMS , Vol 51, Number 5.[39] ncatlab authors. Topos. https://ncatlab.org/nlab/show/topos (accessed January29, 2021).[40] ncatlab authors. Triangulated Category. https://ncatlab.org/nlab/show/triangulated+category (accessed January 29, 2021).3541] ncatlab authors. Higher Category Theory. https://ncatlab.org/nlab/show/higher+category+theory (accessed January 29, 2021).[42] ncatlab authors. Derived Category. https://ncatlab.org/nlab/show/derived+category (accessed January 29, 2021).[43] ncatlab authors. Grothendieck Topology. https://ncatlab.org/nlab/show/Grothendieck+topology (accessed January 29, 2021).[44] ncatlab authors. (infinity,1)-topos. https://ncatlab.org/nlab/show/(infinity,1)-topos (accessed January 29, 2021).[45] Kierkegaard, S. Journal entry from August 17, 1838.[46] Levine, J. Materialism and Qualia: The Explanatory Gap.
Pacific Philosophical Quar-terly , 64, 354–361.[47] ncatlab authors. Descent. https://ncatlab.org/nlab/show/descent (accessed Jan-uary 29, 2021).[48] Barthes, R. The Death of the Author, in:
Image, Music, Text , Fontana Press, 1977.[49] McTaggart, JE. The Unreality of Time,
Mind , 1908.[50] Heidegger, M.
Parmenides, lecture in the spring semester 1942/43 , Vittorio Kloster-mann, 2018.[51] Diels, H, and Kranz, W.
Die Fragmente der Vorsokratiker , Weidmann, 1952.[52] Lewis, DK.
On the Plurality of Worlds , Blackwell, 1986.[53] Hawley, K. “Temporal Parts”, in
The Stanford Encyclopedia of Philosophy (Sum-mer 2020 Edition) , edited by Zalta, EN. https://plato.stanford.edu/archives/sum2020/entries/temporal-parts/ . (accessed January 29, 2021)[54] Russell, B.
The Analysis of Mind , G. Allen & Unwin Limited: London, 1921.[55] Wikipedia authors. Moduli space. https://en.wikipedia.org/wiki/Moduli_space (last accessed January 29, 2021) 3656] Metzinger, T. Minimal phenomenal experience. Meditation, tonic alertness, and thephenomenology of “pure” consciousness.
Philosophy and the Mind Sciences , 1(1),7.[57] Scholze, P. Perfectoid Spaces,
Publ. Math. IHES https://arxiv.org/abs/1709.07343 (accessed January 29, 2021).[59] Fontaine, J-M. and Wintenberger, J-P. Extensions alg´ebrique et corps des normesdes extensions APF des corps locaux,
C. R. Acad. Sci. Paris S´er. A–B https://ncatlab.org/nlab/show/Bousfield+localization+of+model+categories (accessed January29, 2021).[61] Caldararu, A.
Derived Categories of Sheaves: a Skimming , 2005. https://arxiv.org/pdf/math/0501094.pdf (accessed January 29, 2021).[62] ncatlab authors. Condensed Mathematics. Available online: https://ncatlab.org/nlab/show/condensed+mathematics(accessedJanuary29,2021) .[63] ncatlab authors. Topological localization. Available online: https://ncatlab.org/nlab/show/topological+localization(accessedFebruary2,2021) .[64] ncatlab authors. (Infinity,1)-Site. Available online: https://ncatlab.org/nlab/show/%28infinity%2C1%29-site(accessedFebruary2,2021) .[65] ncatlab authors. Reflective Subcateory. Available online: https://ncatlab.org/nlab/show/reflective+subcategory(accessedFebruary2,2021) .[66] Lurie, J.
Higher Topos Theory , Annals of Mathematics Studies 170, Princeton Uni-versity Press 2009.[67] Joyal, A. Notes on Quasi-Categories, 2008 https://web.math.rochester.edu/people/faculty/doug/otherpapers/Joyal-QC-Notes.pdf (accessed January 29,2021). 3768] ncatlab authors. Kan Complex. https://ncatlab.org/nlab/show/Kan+complex (ac-cessed January 29, 2021).[69] ncatlab authors. Homotopy Lifting Property. https://ncatlab.org/nlab/show/homotopy+lifting+property (accessed January 29, 2021).[70] ncatlab authors. Model category. https://ncatlab.org/nlab/show/model+category (accessed January 29, 2021).[71] Grothendieck, A., and Dieudonn´e, J. ´el´ements de g´eom´etrie alg´ebrique.
Publicationsmath´ematiques de l’IHES: Paris, 1960-1967.[72] Wikipedia authors. Turing degree. https://en.wikipedia.org/wiki/Turing_degree (accessed January 29, 2021).[73] Maldacena, J.,
The Large N limit of superconformal field theories and supergravity , Ad-vances in Theoretical and Mathematical Physics. 2 (4): 231–252. arXiv:hep-th/9711200.Bibcode:1998AdTMP...2..231M. doi:10.4310/ATMP.1998.V2.N2.A1.38 ppendix
An important invariant of a mathematical space is encoded by its homology group - thenumber of holes in that space. As such, they provide a means to compare spaces. For X a topological space, a set of topological invariants H ( X ), H ( X ),..., called the ho-mology groups of X , represent the homology of X . The number of k -dimensional holesin X is encoded by the k th Homology group H k ( X ). For instance, H ( X ) encodes the“path connected” components of X , where a (0-dimensional) hole encodes if the space isdisconnected.As an example, let us examine the homology groups of S , the 1-dimensional sphere(which is really just a circle). Take X to be S . X is connected and has one 1-dimensionalhole and no other holes for k >
1. The homology groups of X take the form: H k ( S ) = Z k = 0 , { } otherwise (1)Take X to be S , the 2-dimensional sphere (which is just the surface of a ball). S isconnected and has just one 2-dimensional hole. The homology groups of X are representedas: H k ( S ) = Z k = 0 , { } otherwise (2)To introduce the derived categorical setting, we will first explain the spirit of quasi-isomorphisms using the notion of quasi-categories formulated by Andre Joyal [67]. Aquasi-isomorphism is a morphism A → B of chain complexes such that the induced mor-phisms H n ( A, · ) → H n ( B, · ) , H n ( A, · ) → H n ( B, . ) of homology groups are isomorphismsfor all n .Quasi categories are homotopoi [67] which possess rich general structures and do notnecessarily have a uniquely defined composition of morphisms. Quasi-categories are likeordinary categories in that they are certain simplicial sets which contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). Unlikecategories, however, morphisms can be composed, but the composition is well-defined only Respectively of cochain complexes. Respectively of cohomology groups.
39p to still higher order invertible morphisms. This means that all possible morphisms whichserve as the composition of two 1-morphisms are related to each other by 2-morphismscalled 2-simplices, which resemble homotopies. It turns out that every Kan Complex [68]is a quasi-category. So from this Kan Complex we build the notion of the chain complex,which we will use in the next section.We mention two reasons that we work in the derived categorical setting. One reasonis that knowing the homology of a space does not give complete information about itshomotopy type. This is seen by that fact that there exist topological spaces X and Y such that H i ( X ) is isomorphic to H i ( Y ) for every i , but X is not homotopy equivalent to Y . The derived category remembers the entire complex, which is crucial to our model of n -awareness, and the consequent model structure gives us nice classes of morphisms whichaxiomatize homotopy theory. Another reason is that the derived category setting allowsus to localize in the category setting. Localization is a formal process of adding inverses toa space. A category can be localized by formally inverting certain morphisms, such as theweak equivalences in the homotopy category of a model category. We use a special case oflocalization called Bousfield localization [60], which assigns a new model category structurewith more weak equivalences to a given model category structure. So Bousfield localizationallows us to make more morphisms count as weak equivalences and this is formally how weget from 1-awareness to n-awareness.To axiomatize homotopy theory, we use the construction of a Quillen model structure[70]. A model structure on a category consists of three classes of morphisms: weak equiva-lences, fibrations, and cofibrations. Weak equivalences are quasi-isomorphisms, maps whichinduce isomorphisms in homology. Cofibrations are maps that are monomorphisms thatsatisfy the homotopy extension property. Fibrations are maps that are epimorphisms thatsatisfy homotopy lifting property (Fig 6). In the derived setting, quasi isomorphisms areused as the class of weak equivalences, fibrations mimic surjections, and the cofibrationsmimic inclusions. From this model structure, we will define the notion of simultaneity.We let A be a Grothendieck abelian category, such as the category of abelian groups. The Grothendieck abelian category is an AB category with a generator. AB categoriesare AB categories (abelian categories possessing arbitrary coproducts) in which filteredcolimits of exact sequences are exact [71]. The category of abelian groups is a prototyp- Grothendieck worked on unifying various constructions in mathematics. For instance, the Grothendieckgroup construction is the most universal way of constructing an abelian group from a commutative monoid. ˜ f (cid:47) (cid:47) X ×{ } (cid:15) (cid:15) E π (cid:15) (cid:15) Y X ˜ f (cid:111) (cid:111) X × I f (cid:47) (cid:47) ˜ f (cid:61) (cid:61) B Y Ip (cid:79) (cid:79) (cid:79) (cid:79) A i (cid:79) (cid:79) f (cid:111) (cid:111) ˜ f (cid:96) (cid:96) Figure 6: Homotopy lifting and extension property for topological spaces E and B . Inthe leftmost figure, the homotopy lifting property allows homotopies in the space B to beuplifted to the space E for any homotopy f : X × [0 , → B and for any map ˜ f : X → E such that f = π ◦ ˜ f . A lifting ˜ f corresponds to a dotted arrow giving a commutativediagram. In the rightmost figure, the homotopy extension property extends certain homo-topies defined on a subspace to a larger space. The homotopy extension property is dualto the homotopy lifting property [69].ical example of a Grothendieck category, with generator the abelian group Z of integers.The category of abelian groups has as objects abelian groups and as morphisms grouphomomorphisms. We use Grothendieck categories because we need a category universallyenriched over abelian groups to model n-awareness. Groups encode symmetries. But whatis 1-awareness an symmetry of? We hypothesize that it encodes a complexity class ofTuring degree 0 [72].We then construct a new “derived category”, D ( A ), whose objects are complexes ofobjects of A and whose morphisms are chain maps. D ( A ) contains a model structure thatwill be our model of n -awareness. Firstly, we define a chain complex. A chain complex ( A • , d • ) is a sequence of abeliangroups ..., A , A , A , A , A , ... connected by homomorphisms (called boundary operatorsor differentials) d n : A n → A n − , such that the composition of any two consecutive maps isthe zero map. Explicitly, the differentials satisfy d n ◦ d n +1 = 0, or with indices suppressed, d = 0. A chain complex has the form: · · · d ←− A d ←− A d ←− A d ←− A d ←− A d ←− · · · Secondarily, a chain map f between two chain complexes ( A • , d • ) , ( B • , d • ) is a sequence f n of homomorphisms f n : A n → B n for each n that commutes with the differentials on the For a more detailed exposition see the work of A. Caldararu [61]. d B , n ◦ f n = d A , n ◦ f n − . A chain map takes the form of thecommutative diagram in Fig. 7. ( f • ) ∗ : H • ( A • , d A, • ) → H • ( B • , d B, • ) on preserves cyclesand boundaries, so f induces a map on homology. . . . A n − d A,n − (cid:111) (cid:111) f n − (cid:15) (cid:15) A nd A,n (cid:111) (cid:111) f n (cid:15) (cid:15) A n +1 d A,n +1 (cid:111) (cid:111) f n +1 (cid:15) (cid:15) . . . d A,n +2 (cid:111) (cid:111) . . . B n − d B,n − (cid:111) (cid:111) B nd B,n (cid:111) (cid:111) B n +1 d B,n +1 (cid:111) (cid:111) . . . d B,n +2 (cid:111) (cid:111) Figure 7: A commutative diagram of a chain map between two chain complexes( A • , d • ) , ( B • , d • ), with a sequnce of morphisms f n : A n → B n and differentials d B , n ◦ f n = d A , n ◦ f n −1