Pluralist-Monism. Derived Category Theory as the Grammar of n-Awareness
aa r X i v : . [ m a t h . G M ] S e p Pluralist-Monism. Derived Category Theory as the Grammarof n -Awareness Shanna Dobson , and Robert Prentner Summary
In this paper, we develop a mathematical model of awareness based on the idea of plural-ity. Instead of positing a singular principle, telos, or essence as noumenon , we model itas plurality accessible through multiple forms of awareness (“ n -awareness”). In contrastto many other approaches, our model is committed to pluralist thinking. The noumenonis plural, and reality is neither reducible nor irreducible. Nothing dies out in meaningmaking. We begin by mathematizing the concept of awareness by appealing to the math-ematical formalism of higher category theory. The beauty of higher category theory liesin its universality. Pluralism is categorical. In particular, we model awareness using thetheories of derived categories and ( ∞ , n -declension) which could express n -awareness, accompaniedby a new temporal ontology (“ n -time”). Our framework allows us to revisit old problemsin the philosophy of time: how is change possible and what do we mean by simultaneityand coincidence? Another question which could be re-conceptualized in our model is oneof soteriology related to this pluralism: what is a self in this context? A new model of“personal identity over time” is thus introduced. Keywords : Awareness, mathematical model, derived categories, ( ∞ , n -morphisms, weak equivalence, homotopy theory, n -declension, n -time, simultaneity, selves 1 Introduction n -awareness Among contemporary substance metaphysicians, the dominant view is monism , that is,the belief that there exists only one kind of thing in the universe (one kind of “stuff”).Monism comes in different guises. The most widespread doctrine is physicalism [1], whichregards everything that exists as physical being. Notable alternatives are, for example, the“dual-aspect” monism of Baruch Spinoza [2] or the idealist monisms of Advaita Vedanta[3]. Monism could be differentiated from pluralism . The most well-known example beingdualism, for example the belief in physical and non-physical (“mental”) substances, oftenattributed to Ren Descartes [4]. Dualism lost much of its proponents in the course of thelast decades. But also other, perhaps more exotic forms of pluralism exist. For example,the “three-world” view of Karl Popper [5] or the more radical pluralism of Bruno Latour[6]. However, there are good reasons to find monisms (and here in particular physicalism) aswell as pluralism (and here in particular dualism) lacking: the former’s seeming inabilityto account for awareness and the latter’s seeming inability to account for the relationbetween different kinds of things, in particular the relation between awareness and theother substance(s). Terminologically consistent with traditional (critical) metaphysics, werefer to the being that is external to awareness as “noumenon” and adopt the quasi-Kantianview according to which a noumenon, while represented in awareness, exceeds any kind of“object-knowledge” acquirable via the senses but still needs to be posited as a “limitingconcept” [7](A253/B310). But rather than playing the “metaphysical game” of proposingor defending a particular metaphysical doctrine that makes statements about the natureof this noumenoun, we instead propose a mathematical model to represent awareness andthe relations between different forms of awareness. By this we hope to arrive at someinteresting conclusions that possibly relate back to the noumenon. We will find that ourmodel requires the noumenon to ground pluralism in awareness: While we stay silenton the ontology of this noumenon, it needs to be such that it affords an infinite varietyof non-equivalent representations. Hence we refer to it as “pluralist-monism”. This hasconsequences for an understanding of ourselves: who we are and what time is (for us).More specifically, we take “1-awareness” to represent what one would simply call aware-2ess. 1-awareness is, for example, the ability to concentrate on an exam or a zoom call,while being implicitly directed to the environment and to future events. We need not stopat 1-awareness. If our mathematical approach is correct, it is natural to assume “higherforms” of awareness. We refer to “2-awareness” as the ability to simultaneously sustainany combination of 1-awarenesses, such as in the following example: Imagine you are hav-ing lunch in your home on a Saturday at noon in the year 2020 while simultaneously youare a Cambridge Apostle having a discussion in the Moral Sciences Club on a Saturdayevening in 1888. Continuing, 3-awareness would be the ability to sustain any combinationof 2-awarenesses. For example, simulatenously being aware of simulatenously being awareof having lunch in your home on a Saturday at noon localized in the year 2020 while beinga Cambridge Apostle having a discussion in the Moral Sciences Club on a Saturday eveningin 1884, and of giving a thesis defense on perfectoid spaces on a Friday at 10am in 2024,while being at your desk and finishing the first chapter of said thesis later this evening...And so on, up to “ n -awareness”.How could such awareness be modeled mathematically, and given that it could, whatlanguage would be appropriate to express it? We use derived category theory and theconstruction of ( ∞ , n -category; second, as a linguisticmodel of n -declension for n -time. We focus on relations between multiple awarenesses thatgive rise to the notion of “time”. How to regard subjective experience against (the ontologyof) time is one of the most puzzling open questions in philosophy. And any mathematicaltreatment of awareness should be able to shed some light on its relation to time. We proposea mathematical model of awareness, together with a grammar for a rudimentary languagethat helps to express temporal experiences. We then have established the background torevisit some open problems in the philosophy of time and sketch a way to resolve themwithin the proposed model.An important consequence pertains to the problematic notion of personal identity overtime. Rather than thinking about “selves” as enduring entities wholly present at everymoment in time (i.e. like endurantists would), but also unlike the idea that we have(more or less well-defined) “temporal parts” (i.e. like perdurantists would), we presenta structural “bundle-like” theory of the self of weakly equivalent representations (acrosstimes). This raises some questions regarding the notion of simultaneity and coincidencewhich shall also be tentatively answered. 3 .2 Organization of the paper The outline of the paper is as follows: In section 2, we model awareness by higher categorytheory (the “category of n -awareness”, starting with n = 1). The objects of this categoryare representations of a noumenon; morphisms between such objects are maps betweensuch 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 ) to bea topological space of morphisms from X to Y in order to give us some more structure towork with.In line with pluralist ideas, we believe that different objects - through morphisms -affect each other (“they interact”). On a higher level ( k > k -awareness.Invertible morphisms imply a “mirroring” of representations in all other representationsit relates to – similar in spirit to the monadology of Leibniz [8]. This gives our model anotion of “self-reflexivity” on the level of the next higher category. Since k − k -category, the property of self-reflexivity is “lifted” to thenext higher level of awareness.However, having invertible morphisms implies strict structural equivalence. But thiswe feel cannot be quite right as this would destroy any plurality. The philosophical prob-lem underlying this dilemma is the following: If the universe evolves toward a system ofequivalent representations, how to prevent it eventually becoming “totalitarian” (at leastfrom a structural perspective), neglecting individuality?In order to guarantee both self-reflexivity and pluralism, we work in an enriched cat-egory called a “derived category” [9], which contains a model structure that allows fora new notion of equivalence, called a “weak equivalence”. We say that two objects areweakly equivalent if one can be “continuously deformed” into the other; this being a no-tion of homotopy theory, a branch of mathematics concerned with various deformationequivalences. Having merely weak equivalences means that having invertible morphismsbetween objects makes them the same only “up to” [31] homotopy equivalence (axioma-4ized in derived category theory ). Having invertible morphisms as weak equivalences givesa sense of pluralist-monism: Objects interact with each other and are weakly equivalentrepresentations, but still possess an irreducible type of individuality. Simply stated, weakequivalence guarantees plurality . We conclude by generalizing our setup to ( ∞ , n -declensionwhich follows from our mathematical model of n -awareness and its concomitant extensionto n -time. The grammar does not merely illustrate (or describe) the structure of awareness.By having a linguistic system to actually express experiences, this tells us much about theway how we concretely engage with the world (as individuals) – analogous to the idea ofa language game [12], but updated to a “language scheme”, inspired by the mathematicalmodel proposed.Further implications are discussed with respect to traditional problems of temporality. Time in our framework is modeled as nested hierarchies of “ k -times” ( k ≤ n ): 1-awarenessimplies a notion of 1-time, 2-awareness implies a notion of 2-time, ... , and n -awareness im-plies a notion of n -time. On our reading, temporality is just a consequence of a structure ofrepresentations. It is at most an ordering property. This is not an altogether new idea [10],but our mathematical model bears some natural consequences that haven’t, to our knowl-edge, been developed so far in the literature: because of our structural definition of n -time,there is no problem of temporal coincidences. A 2-awareness contains its 1-awarenesses,which are “simultaneously present” in 2-awareness. Events can be simultaneous to eachother indeed because simultaneity is a homotopy equivalence.Finally, we turn to a question deeply related to all soteriologies across all religions:what is a “self”? Without a clear notion of self, it becomes meaningless to ask any “mortalquestions” [11]. Prima facie , our structure sustains such a multiplicity without assumingpersonal identity over time. We replace the idea of personal identity over time with anupdated form of the “bundle theory of self” [13], based on a pluralist-monism and using Technically, we represent the noumenon as an abelian group [14] and form a complex [15] of these groups,which is informally a sequence of compositions. Then we “chain” the complexes together using a chain map.These complexes are the objects in the derived category. Thus we have upgraded the representation of thenoumenon from an object in Cat to an object in the derived category. The “bundle” refers to a collection of (weakly equivalent) awareness-objects which are related via
Hom ( X, Y ). But since this is assumed to bean enriched
Hom ( X, Y ) – the topologically structured collection of all morphisms from X to Y (that are only weakly equivalent ) – it is by definition a plurality. There is noconception of “enduring over time”, as if time were somehow exterior to experience; as iftime were constantly present allowing us to move through it (however that happens); oras if time were a well-ordering or counter. By contrast, we claim that time is a homotopicconcept, that there is simultaneity only up to homotopy; and that selves exists only up tohomotopy. The ( ∞ , A mathematical model should express this in a minimal and conceptually coherent way. Forthis, we appeal to category theory. (Higher) Category theory, while seemingly abstract,offers a flexible framework in which to define “relations”, arguably the basic structuralproperty of awareness [17, 18, 19, 20]. This is why we wish to model n -awareness, notwith respect to a substantial (e.g. temporal) ontology, but with respect to structure – as ifthe structure alone provided the ontology. We will use a specific approach, called derivedcategory theory and the theory of ( ∞ , 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 awareness. Awareness is sometimes treated as a “0-category”,e.g. functions between domains (or more generally: sets). One prominent approach alongthese lines is machine-state functionalism which identifies “mental states” with functional morphisms. 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. n -categories” to study awareness. A “1-category” contains objects and “1-morphisms” which are morphisms between these ob-jects. The objects of the 1-categories are possible 1-awarenesses – we call those objects“representations of the noumenon”. But unlike functionalism, we do not wish to enter thedebate surrounding the question what these things are (e.g. brain states [22] or funda-mental constituents of the universe [23, 24]), but only ask how they relate . Our hope isthat by establishing a general model to represent how awareness relates to itself, we canconjecture certain interesting properties about awareness and ontology. This method isdifferent from the way that the metaphysics of awareness is usually done. We do not startwith a “working definition” of what awareness is, but a give a model of what it does .The 1-category of awareness not only consists of experience-involving objects but alsoof the relations between them. This means that, for example, me being aware of thezoom call right now is contained (as representation) in a category made up of myriads of(potential) other experiences – sensing the joy in your voice, or being afraid of your dismis-sive reactions to what I want to say – but also my (implict) perceptions of environmentalgoing-ons, memories, and other (possibly explicit) background experience; and similar forthe examples of attending a meeting of the Cambridge apostles in 1888 (this experience isrelated to my potential experience of listening to Wittgenstein a few years later), or for theexperience of a thesis defense in 2024 (this experience is related to my actual experiencesof writing the thesis now), etc. See Fig. 1 (left).But one need not finish here. One could “increase the category number” to studypossible relations between relations. A “2-category” contains objects, 1-morphisms, and2-morphisms, which are morphisms between the 1-morphisms. In Fig. 1 (middle) we notonly have morphisms between objects but also morphisms between the morphisms. Thecollections of all morphisms from A to B form a set called the “homset”, Hom ( A, B ). In a2-category, each homset itself carries the structure of a category – a collection of objects andmorphism satisfying certain requirements – and thus morphisms between such homsets canbe regarded as morphisms between 1-categories. This higher dimensional structure allowsthe 2-category to represent two moments of awareness “at once”, represented structurally.See Fig. 1, (middle). Continuing further, a “3-category” contains objects, 1-morphisms,7 : φ $ , φ & . A f * * g B A f * * g g o γ o w B A f * * g h p γ n v B ❴ j t ❴ * C D g k k f s s C D g k k f s s C D g k k f s s Figure 1: A graphical overview for n -awareness, for n = 1 , ,
3. Left: 1-category consistingof objects
A, B, C, D (representations of the noumenon) and possible 1-morphisms, f i , g i between A and B (respectively between D and C ). Middle: 2-category which also includes2-morphisms, φ : f → f , γ : g → g between 1-morphisms (only two are shown). Right:3-category which also includes 3-morphisms between 2-morphisms (only one is shown).2-morphisms, and 3-morphisms between the 2-morphisms, where to 3-morphisms can beseen as relations between 2-categories. (Fig. 1, right).It follows that an n -category contains objects, 1-morphisms, 2-morphisms, , up to n -morphisms between the ( n − n -declension later on in the paper. Now that the basic idea is explained, we need a little more structure on our categories inorder to better capture the notion of awareness: • self-reflexivity At a higher level, we would like our morphisms to be invertible . That is, not onlydo we want a map from
Hom ( A, B ) to
Hom ( C, D ), we would like the map to bereversible too actio est reactio . An ( ∞ , k )-category is an infinity category in whichall morphisms higher than k are invertible. For our model, we chose to work in( ∞ ,
1) categories, where all k -morphisms (for k >
1) are invertible; this means that8he respective objects have the “same” structure (i.e. up to isomorphism). This alsomeans that any object 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 highercategory), it can be said that awareness is “self-reflexive”. We thus represent n -awareness as self-reflexive multiplicity, starting from a presentation of the originalnoumenon (i.e. the object of 1-awareness). • plurality, not mere multiplicity But such a total invertibility would (structurally) imply a strict equivalence betweeninstances of awareness (up to isomorphism). Phenomenologically, this would meanthat my awareness now is (in some sense) equivalent to my awareness yesterday, andeven worse: your awareness tomorrow is (in some sense) equivalent to my awarenesstwo days ago. This conflicts with our intuition that awareness is a unique experienceof a unique self. One could refine this strong notion of equivalence and develop theidea of a “weak equivalence” between objects.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. This is the reason we later work in the topoi setting , so that we canhave a space of maps between the objects we wish to study. We thus need to upgrade ourcategory to an enriched category which allows us to use homotopy theory to make theseideas more precise. We now state the most important concepts, the interested reader canfind a more in-depth discussion of the mathematical steps in the appendix:1.
Homotopy theory gives us a (relative) notion of equivalence that allows us to under-stand n -awareness as nested “simultaneous presence” of m moments of 1-awareness,thereby avoiding certain problems that are related to time. Homotopy theory, whichstudies deformation equivalences called homotopies, is defined as follows: Two con-tinuous functions from one topological space to another are called homotopic if onecan be continuously deformed into the other. Homotopy groups extend this notionto equivalences between topological spaces, recording information about the holes in Others have previously hypothesized, for slightly differrent reasons, that a mathematical model ofconsciousness should have at least the structure of a topological space [25, 26]. There is a remarkable freedom in reducing strong equivalences, such asthe claim that object A equals object B , to deformation equivalences, such as theclaim object A is “deformation equivalent” to object B because one can create thehomotopy map which makes these objects homotopy equivalent. We extrapolate thatto personal identity over time: You are not strictly “self-same”, with some substan-tialist notion of self. But, using our model of homotopy types, you could be homotopyequivalent to a “structural equivalent” of yourself. It is from this homotopy settingthat we derive the notion of a weak equivalence.2. Triangulated and derived categories.
Let A be a Grothendieck abelian category (e.g.,the category of abelian groups). We define K ( A ) to be the homotopy categoryof A whose objects are complexes [15] of objects of A and whose homomorphismsare chain maps modulo homotopy equivalence. The weak equivalences are quasi-isomorphisms defined as follows: A chain map f : X → Y is a quasi-isomorphism ifthe induced homomorphism on homology is an isomorphism for all integers n . K ( A )is endowed with the structure of a triangulated category . A triangulated category hasa translation functor and a class of exact triangles which generalize fiber sequencesand short exact sequences.Localization by quasi-isomorphisms preserves this triangulated structure. Bousfieldlocalization [29], particular to triangluated categories, allows us to make more mor-phisms count as weak equivalences and this is formally how we get from 1-awarenessto n -awareness. One of the axioms of a triangulated category states that given thediagram in Fig. 2, where A, B, C and
A, B, C form exact triangles, and the mor-phisms f and g are given such that the square ABBA commutes, then there existsa map C → C 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 transitiveand well-ordered by membership, where a set T is transitive if every element of T isa subset of T . Ordinals locate within a set as opposed to cardinality which referencesmerely size.The category of A represents all possible awarenesses (related by composition). There 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 . / / f (cid:15) (cid:15) B { { ✈✈✈✈✈✈✈✈ g (cid:15) (cid:15) C c c ❍❍❍❍❍❍❍❍ (cid:15) (cid:15) ✤✤✤✤✤✤ A ′ / / B ′ { { ✇✇✇✇✇✇✇✇ C ′ c c ●●●●●●●● Figure 2: A triangulated category composed of two exact triangles
A, B, C and
A, B, C and three commuting squares
ABBA , AC ′ C ′ A ′ , and BB ′ C ′ C , with a non-unique fill-in[44]. are an infinite number of 1-awarenesses in this category. Differentials define a 1-awareness relating with another 1-awareness. The composition of any two A s isthe zero map. This is the group law, the return to identity. Let B be the abeliangroup of 2-awarenesses. The differentials define relating with another 2-awareness.The diagram commuting means the 2-awareness is related with the 1-awareness.In essence, this diagram represents a higher-dimensional notion of commutativitythrough the map C → C ′ and the three squares commuting. By higher-dimensional,we mean the three commuting squares and the two exact triangles together form acone construction. There is a relation here between the higher commutativity andthe set-theoretic ordinal.3. Infinity-topoi . A topos (Greek for “place”) is a category which behaves like thecategory of sets but also contains a notion of localization. Topoi are modeled afterGrothendiecks notion of a sheaf on a site [45]. Formally, a topos is a categoryequivalent to the category of sheaves of sets on a site. A prototypical example of atopos is the category of sets, since it is the category of sheaves of sets on the onepoint space. A sheaf is a tool used to pack together local data on a topological space Informally, topoi are “nice” categories for doing geometry that act like models of intuitionist typetheory. They are abstract contexts “in which one can do mathematics independently of their interpretationas categories of spaces.” [9, 27]
11n ( ∞ , k -morphisms between ( k − Specifically an ( ∞ , ∞ , C which satisfies three conditions: C is presentable (withkappa filtered colimits), locally cartesian closed, and satisfies a descent condition(where an object in C is sent to the slice category C/u ). The objects in ( ∞ , ∞ , ∞ , ∞ , ∞ , We now turn to some important problems that are prompted by this treatment. Presently,we do not have a way to talk (or write) about n -awareness – our language is antiphrastical.But rather than declaring n -awareness to be merely an artifact of misguided language use(e.g. an illusionary result of playing a “language game la Wittgenstein [12]), we believethat it is important to actually find a (quasi-)linguistic system that could express it. Thisis due to the fact that having such a language is more than a mere “gloss” over the basic(abstract) structure of the world but an important precondition of (concretely) engagingwith it. The linguistic system we envision could be called a “language scheme” . Similaras metaphysics could be regarded a result of language (mis-)use, questions of ontologycould be rephrased in terms of the mathematical structure used in our model. We are“upgrading” Wittgensteins language games to language schemes which we regard as newmeta-language. In mathematics, “schemes” are generalizations of algebraic varieties [34].We use our model of n -declension and take as grammatical primitives commutative ringspectra. So, instead of (linguistic) meaning being derived from whatever game is at play, Higher category theory investigates the generalizations of ∞ -groupoids to directed spaces [41].
12e say meaning is derived from whatever scheme is at play. We now outline the basicgrammatical structure of this scheme.First, we propose a novel “grammar of multiplicity”. Modern English language containsthree grammatical cases (subjective, objective, and possessive) with different declensionsfor each case. We extend this grammatical structure by adding a number to the respectivepronoun, indicating the simultaneous presence of different “me’s” within an experienceof multiple awareness (Tab. 1). Formally, the resulting n -declension allows for all com-binations of n , n − . . . , 1-declensions to be present in a sentence. So for example, astatement compatible with double-awareness could contain a 1-I as well as a superposed2-I as subject. Analogously, n -conjugation could be defined. This is one way the grammarof sentences could sustain the presence of n -awareness.Second, we envision a novel temporal ontology. Spatio-temporally multiplied awarenessis obviously not reconcilable with theories which posit awareness to be bound to one par-ticular location in ( physical ) space and time. However, non-physicalist theories (outside ofspace-time) seem to be able to accommodate the concept of n -awareness. Many mysticsreported a similar kind of experience, in particular in the religious traditions from the Eastand West. If awareness is not bound to a single region in space and time, this suggests thatawareness cannot be understood as an emergent property of localized physical systems.We want an ontology that reconciles both the physicalist as well as the non-physicalistunderstanding. This calls for a novel ontology which could accommodate multiple formsof awareness throughout different points in time. Our model conceives of a temporalmultiplicity with a categorified model of n -time, evidenced in the proposed language schemeby way of n -inflection for n -conjugation. Modern English is spoken in local, linear time,yet it allows the inflectional change of verbs by way of conjugation. We extend the ideaof language spoken in linear time, conjugated over three tenses, to one of n -conjugationas follows: Instead of using only past, present, future, and their perfect correspondences,present-perfect, past-perfect, future-perfect, we allow 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 temporal experience, which has traditionallybeen expressed in terms of a “tensed” experienced time vis-a-vis an “untensed” physicalparameter time. 13able 1: Grammar of a language which could express n -awareness, extended from ordinary English case structure. n = 1 n = 2 n = 3 · · · n = k subjective I (we) → · · · k -I ( k -we) objective me (us) → · · · k -me ( k -us) dep. possessive my (our) → · · · k -my ( k -our) indep. possessive mine (ours) → · · · k -mine ( k -ours) .2 Revisiting problems of temporality Different temporal ontologies have been proposed throughout the ages, without coming todefinite conclusion. Rather than proposing yet another metaphysical framework, we wishto concentrate on particular problems that feature prominently in recent and not so recentdiscussions. We have posited a structural model of awareness using derived category theoryand hope that approaching these problems structurally will lead to a remedy.The problems we wish to focus are the following:1.
How is change possible?
Change is a manifest everyday experience. The idea that“change” does not really exist goes back to the works of Parmenides and the Eleaticschool of philosophy. While this philosophy has been influential up to this day, forexample, in the metaphysical thought of Martin Heidegger [46], the our common dayexperiences seems much better captured in the “everything flows” of Heraclitus [47].More specifically, in the philosophy of time the idea of a “tenseless” vs. a “tensed”time is a prominent distinction introduced by John McTaggert [48].The American philosopher David Lewis revived the problem of change for the philos-ophy of time in the problem of temporary intrinsics [49], 198f.:
For instance shape:when I sit, I have a bent shape; when I stand, I have a straightened shape. Bothshapes are temporary intrinsic properties; I have them only some of the time. Howis such change possible? I know of only three solutions.
According to David Lewis the problem of temporary intrinsics could be solved inthree ways. Either one acknowledges that there are no intrinsic properties, just“disguised relations”; or one believes that only those properties that exist at the present moment are real, whereas properties that an object seem to have had are,in some sense, fictional (this position is known as “presentism”); or one accepts thatobjects have genuine temporal parts (e.g. the me-yesterday, the me-now, and theme-tomorrow). The latter solution to the problem of temporary intrinsics has beendeemed the only viable solution to the problem of temporary intrinsics which is not“incredible” [49] and started the appreciation of “perdurance” theories in the modernphilosophy of time at the expense of so-called “endurance” theories that conceive ofpersisting wholes without temporal parts [50]. In addition to Lewis’ metaphysicalrejunevation, much support for a “perdurance-like” theory seems to come from sci-ence, in particular Einstein’s theory of relativity. Perdurance theory, so it is often15ut not uni-vocally believed, squares well with the believe that space-time forms afour-dimensional continuum as described by the special theory of relativity [50].2.
What is simultaneity?
The second problem worth mentioning in this respect is theproblem of simultaneity (coincidence). It seems that, when discussing n -awarenesswe believe in the simultaneous presences 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 clayhave temporal parts that overlap ). But it also seems to be in conflict with basicprinciples of physics understanding according to which there can be no “absolute”notion of simultaneity.However, note that the problem of simultaneity is mainly a (conceptual) “designissue” which stems from a linear notion of time where simultaneity is conceived interms of (“temporal”) coincidence, or, alternatively, from the treatment of time inthe framework of Minkowski space-time, the so called “fourth dimension”. We offera structural solution and define simultaneity by commutativity of diagrams of chaincomlexes in a homotopy category, and respectively by the “up to” notion.Commutativity classifies the equivalence of all possible ways to get to a destination.Take, for example, the chain map between complexes (see Fig. 4 in the appendix).The composition means that there are two ways to get to B n − , and there is no struc-tural difference in choosing one way over the other. There is no indicated startingpoint or canonical progression. Rather, we see all possible paths, and even infinitepaths are alluded to. The “up to” notion grants a universal equivalence which struc-turally corresponds to commutativity. For instance, making the statement that allmorphisms are equivalent “up to” homotopy means that they are equivalent withrespect to homotopy. There is no substantial way to distinguish one morphism overany other one. In a sense, commutativity in our diagrams is algebraically sustainingthe “up to” notion. Equivalence (and hence simultaneity) is never truly absolute.3. Temporal coincidence and synchronous reference?
While perdurantism claimsto solve problems of temporal coincidence (which are only problems if designed tobe so), it has its own problems when trying to account for the acts of synchronous Another argument in favor of perdurantism. n -awareness uses that of ( ∞ , X f / / g / / Y q / / q ′ ●●●●●●●●●●● Q u (cid:15) (cid:15) Q ′ 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) can be associated to a single and unique (up Isomorphisms are permutations of morphisms; the “up to” phrase exemplifies how distinct objects inthe same class can be considered equivalent under a particular condition [31]
17o 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.This also gives us a handle on the problem of simultaneity: Our view of simultaneousawareness is not aptly perdurantist, although superficially it seems a perdurantist represen-tation with the n -declension of 1-her, 2-hers, ... n -her, but these are relational propertiesand not necessarily temporal parts. It can be asked how different Mary-tomorrow is fromMary-today given that Mary-tomorrow has more morphisms? If Mary refers to the objectsof a category and the properties are its morphisms, then saying that Mary has “changed”is merely to say that Mary has added connections/morphisms. We say that Mary-today isthe same as Mary tomorrow “up to” isomorphism.Analogous to how perdurantists resolve the problem of coincidence by noting thattemporal parts can surely “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. are topological) could too be said to “overlap”. But this does not commit us toperdurantism as ontological position. The worry that our notion of “simultaneously beingaware” commits us to a strong notion of simultaneity which is in conflict with physics. Weinstead choose to model time in terms of 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 levelof time. 1-awareness corresponds to 1-time. 2-awareness corresponds to 2-time etc. There isonly a composition law not uniquely defined up to a homotopy of time. So simultaneity is ahomotopy equivalence and homotopy equivalences are neither perdurantist nor endurantist. In order to talk meaningfully about soteriology, we need a meaningful concept of self. Wedo have such a concept: the self as homset. To be self-reflexive would be to have invertiblemorphisms; offering a geometrical version of relationlism, rather than a set-theoretic one.Our model can be summarized as follows: We take 1-awareness as the presentation ofa (“pluralist-monist”) noumenon. To every 1-awareness we can associate (one or more) 1-morphisms which constitutes its “1-time”. 1-time is but the relation in which awarenessesperceived as present, past, future, possible or actual stand. Every 2-awareness has its18-time AND 1-time, since it contains 2-morphisms as well as 1-morphisms. Continuing,we have n -awareness containing n -time. This is new because current models of temporalontology still frame time as a static, 1-dimensional phenomena, static enough so that weclaim to have a “personal identity over time.” By contrast, we posit that this notion of“over time” is better conceptualized in terms of the relations that figure in n -categories.We extrapolate to personal identity over time. You are not self-reflexive with some illdefined notion of self. But, using our model of homotopy types, you could be homotopyequivalent to your other versions of yourself. It is from this homotopy setting that wederive the notion of a weak equivalence.The triangulated category (Fig. 2) is our structural representation of soteriology. It isa higher-dimensional notion of commutativity. Simply speaking, this means there is morethan one canonical way to reach divinity. Having such a higher notion of commutativityin a homotopy category means that there are many paths that are homotopy equivalentto the canonical path to divinity. As such, our soteriology is pluralistic. Soteriology is nota cardinal issue, it is an ordinal issue, having more to do with how close or far one is to(one’s own) divinity.This invites the following thought: Instead of taking n to infinity and reflecting, similaras in the Vedic tradition, the size (i.e. cardinality) of this infinity, one could look for thehighest ordinal of infinity, reflecting that awareness is not about size but about order -to be aware of the divine means being able to localize oneself within the (levels of the)infinite. Penitence is about navigating this distance. “Cardinality” is about size (e.g. thesize of a set); But “closeness is an ordinal concept. How to get there from many waysthats homotopy. Our work posits an ontology of plurality in three ways: structurally we begin with thenoumenon represented by the objects of 1-awareness and invertible n -morphisms whichrepresent a multiplicity. Second, our notion of n -time represents a temporal multiplicityand gives rise to a n -declension. Third and finally, our soteriology is pluralistic; our conceptof (“the one”) self is a homset.Pluralism is categorical. We are taking the monism of structure and making it plurasticby working in a derived setting appealing to the infinity topoi (with chain complexes with19ocalization giving more weak equivalences, cf. appendix). Similar ideas (in a less technicalsetting) can be found in the systems of Leibniz [8] or, perhaps, the Yogacara school of Bud-dhism [52]. It could be objected that our model really expresses a (self-reflexive) monism ,since we ground the model in a seemingly universal form of structural representation, thusmaking any soteriology which keeps with individuals difficult to sustain. We disagree. Ourmodel implies due to its notion of “weak equivalence” – a form of pluralism.( ∞ , logical statement could be internalized. ( ∞ , ∞ , except for those dependent on the law of the excluded middle and the axiom of choice [36]. eferences [1] Stoljar, D.. “Physicalism”, The Stanford Encyclopedia of Phi-losophy (Winter 2017 Edition) , edited by Zalta, E. N. https://plato.stanford.edu/archives/win2017/entries/physicalism/ (accessedAugust 7, 2020).[2] Spinoza, B..
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To later introduce the derived categorical setting, we will first explain the spirit of quasi-isomorphisms using the notion of quasi-categories formulated by Andre Joyal [39]. A quasi-isomorphism is a morphism A → B of chain complexes such that the induced morphisms H n ( A, · ) → H n ( B, · ) , H n ( A, · ) → H n ( B, . ) of homology groups are isomorphisms for all n . An important invariant of a mathematical space is encoded by its homology group -the number 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 thehomology 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 , { } otherwiseTake 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 , { } otherwiseQuasi categories are homotopoi [39] 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 onlyup to still higher order invertible morphisms. This means that all possible morphisms which respectively of cochain complexes. respectively of cohomology groups. n -awareness, where the n -awarenesses represent all possible morphisms relatedto each other by higher invertible morphisms. Homotopy theory gives us a (relative) notion of equivalence that allows us to understand n -awareness as “simulatenous presence” of n moments of 1-awareness, thereby avoidingcertain problems that a related to time. Homotopy theory, which studies deformationequivalences called homotopies, is defined as follows: Two continuous functions from onetopological space to another are called homotopic if one can be continuously deformedinto the other. Homotopy groups extend this notion to equivalences between topologicalspaces, recording information about the holes in each space. There is a remarkablefreedom in reducing strong equivalences, such as the claim that object A equals object B , to deformation equivalences, such as the claim object A is “deformation equivalent” toobject B because one can create the homotopy map which makes these objects homotopyequivalent. We extrapolate that to personal identity over time: You are not self-reflexive,with some substantialist notion of self. But, using our model of homotopy types, you couldbe homotopy equivalent to a “structural equivalent” of yourself. It is from this homotopysetting that we derive the notion of a weak equivalence.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 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 . ˜ f / / X ×{ } (cid:15) (cid:15) E π (cid:15) (cid:15) Y X ˜ f o o X × I f / / ˜ f = = B Y Ip O O O O A i O O f o o ˜ f ` ` Figure 3: 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 [42]. 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 [29], 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[40]. 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 3). 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.27e 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 [32]. The category of abelian groups is a prototyp-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. In particular, we take this group tomodel awareness, with an: • identity element, which serves as a basic notion of “self” • inverse which illustrates the “back-reaction” (reflectivity) for each element • associativity which defines an “order” of perception • a closure property which defines the “privacy” of awarenessGroups encode symmetries. But what is 1-awareness an symmetry of? We hypothesizethat it encodes a complexity class of Turing degree 0 [43].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 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.[30] For a more detailed exposition see the work of A. Caldararu [44]. . . A n − d A,n − o o f n − (cid:15) (cid:15) A nd A,n o o f n (cid:15) (cid:15) A n +1 d A,n +1 o o f n +1 (cid:15) (cid:15) . . . d A,n +2 o o . . . B n − d B,n − o o B nd B,n o o B n +1 d B,n +1 o o . . . d B,n +2 o o Figure 4: 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 − .two chain complexes, where d B , n ◦ f n = d A , n ◦ f n − . A chain map takes the form of thecommutative diagram in Fig. 4. ( f • ) ∗ : H • ( A • , d A, • ) → H • ( B • , d B, • ) on preserves cyclesand boundaries, so f induces a map on homology.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 objectsof A and whose homomorphisms are chain maps modulo homotopy equivalence. Thecategory K ( A ) has a model structure in which the cofibrations are the monomorphisms (categorical generalizations of injective functions) and the weak equivalences are the quasi-isomorphisms defined as follows: A chain map f : X → Y is a quasi-isomorphism if theinduced homomorphism on homology is an isomorphism for all integers n . K ( A ) is endowedwith the structure of a triangulated category . A triangulated category has a translationfunctor and a class of exact triangles which generalize fiber sequences and short exactsequences. Localization by quasi-isomorphisms preserves this triangulated structure.One of the axioms of a triangulated category states that given a diagram: A / / f (cid:15) (cid:15) B { { ✈✈✈✈✈✈✈✈ g (cid:15) (cid:15) C c c ❍❍❍❍❍❍❍❍ (cid:15) (cid:15) ✤✤✤✤✤✤ A ′ / / B ′ { { ✇✇✇✇✇✇✇✇ C ′ c c ●●●●●●●● Diagram of two exact triangles
A, B, C and
A, B, C and three commuting squares
ABBA , AC ′ C ′ A ′ ,and BB ′ C ′ C , with a non-unique fill-in [44]. A, B, C and
A, B, C form exact triangles, and the morphisms f and g are givensuch that the square ABBA commutes, then there exists a map C → C such that all thesquares commute, where this fill-in is not unique [44]. 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 .Through a localization process, called “Bousfield localization” [29], the derived category D ( A ) of the initial abelian category A is obtained by “pretending that quasi-isomorphismsin K ( A ) are isomorphisms. Specifically, the localization is constructed as follows: mor-phisms in D ( A ) between A · and B · will be ’roofs’ [44], with f, g morphisms in K ( A ) and f a quasi-isomorphism. This roof represents g ◦ f − .The category of A represents all possible awarenesses (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 ′′