Category Theory and Philosophy

Prolegomena to a future logico-mathematical metaphysics

The pure categories (captured by higher order categorical logic, etc.) must be unfolded and specified via schematism and regional ontologies. Such can only be geometric and geometro-modal-dynamic although in a suitable categorical framework.  On the other hand a formal axiomatic philosophy (a theory of 'manifolds') can be obtained via abstraction from certain geometro-modal-dynamic frameworks found in applications. By modal we mean everything pertaining to phase spaces, configuration spaces, moduli spaces, the only right way to treat modality.  These geometro-modal-dynamic frameworks are not only found in mathematical physics or mathematical biology but in the study of concepts, in semantics, in mathematical linguistics. The question is, can we follow Husserl and Gödel this way to study all possible worlds to all possible minds ?

Lawvere's theory of smooth toposes and their use for formulating differential geometry and differential equations can be seen as an unveiling of fundamental synthetic a priori categories and principles. The adequacy and conformity to scientific experience (in the spirit of critical idealism) is a vital dimension in the philosophical deployment of category theory.

The sequence seems to be : topos $->$ ringed topos $->$ lined topos $->$  smooth topos.  Our task is to elucidate the phenomenological and categorical (in the philosophical sense) meaning of the concept of topos.  Also to understand why the internal version of the concepts of commutative ring, commutative algebra, linear map (cf. Weil algebras and the Kock-Lawvere axiom) and an infinitesimal version of simplicial objects (used for defining the dg-algebra of differential forms) appear to be of so fundamental and vast a scope as a priori conditions for mathematical physics and other branches of natural science. And how are we to understand Kant's concept of space in the light of synthetic differential geometry, it being  point-free ? Seeing a vector field on $X$ as an infinitesimal deformation of the identity map on $X$ seems very intuitive.

Kant's theory of schematism could be interpreted in particular as implying that any line-shape $R \rightarrow X$ must be seen as a solution of a differential equation; we give the Weil algebra a generative dynamic interpretation.

Let us look at the first two antinomies of pure reason in the transcendental dialectic, involving time and composition. The point-free nature of space is presupposes for the argument to work. Can causality involving a previous moment of time be captured in terms of the infinitesimal path $D$ via a prolongation principle ?  The infinity which cannot be object of a synthesis appears to be best captured by a set dense set for $<$.   Otherwise why cannot for instance the points $\frac{1}{n}$ be objects of a synthesis ?

Kant appears to be saying that every state of the universe must have a temporally previous state but at the same time there cannot be infinitely many previous states to a given state.  

We must compare the antinomies to various intuitionistic principles and classical principles not intuitionistically valid.

It could be tempting to see a sheaf as a derived concept relative to an abstract bundle (which is a very Kantian notion). Sheaves are sections of some bundle. But we must check if this carries over for Grothendieck topologies too, that there is an equivalence of categories between sheaves over a site and étale bundles defined not in terms of ordinary topology but in terms of Grothendieck topologies (as in étale cohomology).  However there are other entities in geometry which are more naturally seen as sheaves than as a bundles: for instance the sheaf of continuous or holomorphic functions. A sheaf is essentially a phase space, a space of phenomenal possibilities which expresses how these possibilities flow locally and cohere (se our post https://chryssipus.blogspot.com/2024/06/cognition-and-states-of consciousness.html). But in actual situations the number of different sheaves (over a given space) is very definite and determinate. For instance, sheaves of smooth functions on a manifold and smooth $k$-forms and other functions relevant to analysis. Many of these actual sheaves form natural complexes, so much so, that the complex itself can be seen as adequate kind of generalized sheaf (cf. the derived category).  Since it is the cohomology of the sequence that mostly of interest, complexes are identified if they have the same cohomology : this is the basis of the derived category constructions which turns morphisms of complexes module homotopy into 'fractions' where  in the denominator quasi-isomorphisms are inverted.

But how are actual sheaves, sheaves of concrete interest, 'generated' , beyond the basic ones discussed above ? By functors generated by continuous maps $f: X \rightarrow Y$.  Of great interest is the study of neighbourhoods of fibers $f^{-1}(x)$ as $x$ varies.  $Y$ is often seen as parametrization space or base space. Hence the presheaf  on $Y$ given bt $V \rightarrow \Gamma(f^{-1}(V), F)$ for $F$ a sheaf on $X$. This carries over to a derived functor $Rf_{\star}$ taking complexes of sheaves on $X$ to complexes of sheaves on $Y$ by which is studied the cohomological variation of the fibers along $Y$. This is a main source of the generation of interesting sheaves, the sheaves used in practice.  Another source of is the functor $f^{-1}$ by which is studied the cohomology in an infinitesimal neighbourhood of the images of open sets via $f$. These two functors are abstract versions of integration and differentiation. Another important operation is the "restriction" composition $R_\star j j^{-1}$ for an inclusion $j : V \rightarrow X$. This takes a sheaf $F$ and yields a new sheaf which, roughly speaking, takes into account only the nature of $F$ on $V$ or in infinitesimal neighbourhoods of $V$.

Thus a category of sheaves becomes interesting and intelligible by its relation to other categories of sheaves.  So considering at once the 2-category of sheaf-categories, or more generally, toposes, is very natural and imposes itself in the nature of things. The adjunction between toposes and locally presentable categories is discussed in Marta Bunge, Aurelio Carboni, The symmetric topos, Journal of Pure and Applied Algebra 105:233-249, (1995).

Having a whole category of sheaves leaves a vast amount of elbow-room. A category of sheaves represents a spectrum of different spaces of possible manifestation (cf. how locally constant sheaves can be identified with covering spaces). The category of sheaves over a given topological space represents every mode of phenomenal possibility space of that space - thus in a way the category can be identified with the space itself.

A remarkable property of sheaves is their homogeneity for scaling. Given an open set $U \subset X$ we get automatically from a category of sheaves on $X$ a new category of sheaves on $O$, $\Gamma_U : Shv(X) \rightarrow Shv(U)$.

Categories do not apparently have the vertical hierarchical structure of the classical genus-species classification. For instance: group is a species of monoid and abelian group is a species of group. There are corresponding categories Mon, Grp and Ab which form a chain of subcategories. Our construction from our paper can be interpreted in terms of successively taking equivalence classes of equivalence classes of equivalence classes.  In category theoretic terms this translates as a sequence of categories $C_0, C_1,C_2,...,C_n$ and a sequence  $F_1, F_2,...,F_n$ of sets of morphisms in $C_{i-1}$ such that $C_i$ is the quotient category of $C_{i-1}$ via $F_i$.  Consider how a given infima species might be described in terms of a protoype $P$, membership  of an object $X$ being ensured by the existence of a deformation $f : P \rightarrow X$. We can also think of an interpretation in terms of $\infty$-groupoids but it is more subtle; it is a top-down approach using connected components and $n$-contractibility.

But sections of a sheaf are like individuals of a given infima species. The category of sheaves is like the genus of the infima species. Then the 2-category of (small) sheaf categories is like the yet higher genus of this genus. Functor categories are like the category of relations.

So: section $->$ sheaf $->$ sheaf category $->$ 2-category of sheaf categories. Aquinas (in De Ente et Essentia) views genus as a space of possibilities rather than as a minimal matter to which difference is added as a form. Difference determines or 'picks out' a latent possibility of the genus. This agrees with the concept of sheaf and category: we choose a section of a sheaf or a definite sheaf in the category of sheaves.

Formal axiomatic metaphysics in the sense of Zalta all depends on a multi-valued logic having truth-values $\Omega^W$ were $W$ represents possible worlds.  Or rather, consider that $W$ must be endowed with a topology or Grothendieck topology so we can consider $\Omega$ for $Shv(W)$.

Thus we have a fundamental axiom of metaphysics: the set of possible worlds must be endowed with a topology. The idea of alternative situation always depends on a more or less strong continuous deformation of the actual world. If a sentence $P$ holds at a world $w$ then it must hold in some neighbourhood $U(w)$ of $w$. Thus it is natural to capture modal logic by the topos of sheaves over $W$, $Shv(W)$.

To do: in the section on understanding in the phenomenology of spirit there are many interesting considerations about 'force' and exteriorization and the unity of motion determined by a law.  The germ of a sheaf is the unity of between interiority and exterior manifestation (for there is not definition for the point itself independently of its neighbourhood).   Let $W$ be a category $\mathbb{W}$ with a Grothendieck topology.  Then take modal logic to be a functor $ m: \mathbb{W} \rightarrow \mathbf{Cat}$.

Topos theory is (very roughly) the study of how logic coheres and varies with space and time and  possibility. All predicates $X\rightarrow \Omega$ have a domain which is a 'type' $X$.  But in general $X$ is a sheaf. Types as spaces can be expressed as 'types as sheaves'.  For instance a type parametrized by possible worlds. Thus predicates are in this case coherently parametrized according to possible worlds. But notice how in general the objects $X(U)$ can vary for $U$ in the base category. If $X$ were an atomic type of sets of individuals then the actual individuals could very according to the possible world. An equivalent way to see this is as the subobject fibration in which we view propositions on $X$ in correspondence with Sub(X).

Zalta's encoding could be descriped as a morphism $enc_X : PX \rightarrow PX$.

Jean Hyppolite lays emphasis on Hegel's positing of  'the identity of identity and difference'. But it is difficult not to think here of the univalence axiom: 'the equivalence of equivalence and identity'. Hegel's logic, despite this going against the surface intention of Hegel himself, may well be capable of a formal axiomatic treatment. This will depend on a proper account of identity and equality.

In Jacobs' Categorical Logic and Type Theory there is the idea of giving a categorical semantics for untyped lambda calculus ($\lambda$-categories on p.155-156) related to Scott's reflexive objects. We take an object $\Omega$ in a Cartesian closed category for which $\Omega = \Omega \rightarrow \Omega$. This expresses that $\Omega$ has a mediation within itself, is self-mediating. Is in and for itself.

Cognition and States of Consciousness

Husserl wanted us to develop a state of consciousness which also, of course, has a cognitive aspect - indeed the cognitive aspect might be seen as its raison d'être. But it is more than this. A state of consciousness implies a permanent habit, a transformation of character. Both Husserl and the oldest Buddhist texts dwell on (analytic) insight, disidentification, suspension and distancing (abgeschiedenheit).

If conscious experience is normally present in unreflected 'globs' , the goal of analytic insight is to unmask and be continuously aware of the ternary structure present in consciousness $\bullet \rightarrow \bullet$ and its subsequent higher-order unfoldings.

We mentioned before the archetypal structures and processes of consciousness. Here is an incomplete tentative list (with an implicitly Kantian basis):

Synthesis - gluing, covers, the sheaf-condition = extensibility on $j$-dense objects for a topos with a Joyal-Tierney topology.

- different orders of wholes (higher groupoids)

Self-reflection - a system which can represent (partially at least) higher order aspects of itself within itself.  This is the original synthetic unity of apperception = I know that I am knowing. This is found in recursive definitions, inductive types, the successive powers of the $\lambda$-cube wherein external aspects of the system become internalized and internally represented, also the subobject classifier, truth-value object $\Omega$ in a topos. See also our post on the meaning of the logical connectives.

Return-to-self, that is, Kant's trinary structure in the CPR.  This is related to the negation of the negation, double negation as the third (synthesis).  In topos theory this relates to the dense topology and in particular to forcing.  The idea is simple. In rough terms it is as follows: consider $U\Vdash\phi$ as signifying that the sentence $\phi$ holds in region $U$. We define $U \Vdash \neg \phi$ if $\phi$ does not hold on any subregion $V\subset U$.  Then $U\Vdash \neg\neg \phi$ means that for any subregion $V\subset U$ we choose we must have that there exists a $W\subset V$ such that $W\Vdash \phi$.

Double-negation can also be connected to temporality: something must pass to reveal itself, ti to einai, quod quid erat.

But this is assuming a static consciousness, a fixed state of consciousness with its corresponding archetypal structures and processes. But what about the transformation into other states (such as found in Schopenhauer and Hegel) ? Do the archetypes change ? Or must we find further higher-level archetypes that govern and characterize this transformation ? To self-reflection we should add self-negation and self-transcendence whereby the correlative self of consciousness is abrogated and transposed to more universal and wide-encompassing modalities and states.

Kant also had a Leibnizean dream, a complete axiomatic-deductive system of the pure a priori concepts and principles of the understanding. What is not clear is how he envisioned deduction and the interplay of the analytic and the synthetic.  Could the synthetic be exhausted in a finite set of axioms and all the rest be entirely analytic, Frege-style ? How could Kant explain that in mathematics there is often a convergence between intuition and formal deduction ?

The history of transcendental idealism is yet to be written, specially as regards to the time between Kant and Husserl. Schopenhauer, Von Hartmann and Spir are far more important than Fichte or even Schelling. Tolstoy wrote of Spir: "reading Thought and Reality has been a great joy for me. I do not know a philosopher so profound and at the same time so precise, I mean scientific, accepting only what is strictly necessary and clear for everybody. I am sure that his doctrine will be understood and appreciated as it deserves and that the destiny of his work will be similar to that of Schopenhauer, who became known and admired only after his death".

We can view Husserl' transcendental subjective idealism and Fregean-Leibnizean objective platonism as not mutually opposing by complementary or at least compatible. Also these two need not be considered exhaustive of reality,  as an important place should be given to ethics, to philosophy of art and to naturphilosophie and above all the practice and psychological basis of meditation (higher ethics).

Theory of theories

Given a theory, a systematic theory, we can analyze i) its intrinsic logical-conceptual structure, ii) the process by which a person comes to learn and understand the theory, and iii) the historical or personal biographical genesis of the theory (which of course can involve i) and ii) at previous times).

Regarding approach i) we can ask to what extent is the organization of the theory drawn by necessity and each logical step or 'development' (in an asynchronic sense) guided by implicitness or inner necessity ? (These considerations seem to have played an important role for Fichte and Schelling).

Speculation: can ii) and iii) somehow shed light on this question regarding i) ?  What is the relationship of this to Aristotle's distinction between things clear to us and things clear in themselves and his methodology of starting with the former ?

Speculation: can the study of biological organization or general systems theoretic concepts help with i) ?  What are the most important metatheoretic concepts we need to consider (for instance the idea of something external and ad hoc becoming internalized, the discussions in our post about reflection-into-self, etc.) . Category theory and categorical logic offer a very important paradigm and key. The bare concept of category (and higher category) functions like a supreme genus. As more properties are added these are mirrored in the nature of the internal logic. They way successive relevant properties emerge is certainly not arbitrary but seems to conform to basic meta-theoretic archetypes, if we are careful to unfold them in a gradual and ordered way.

But let us look at the processes and archetypes of consciousness (such as unification, return-to-self, negation, intentionality, etc.). How are they reflected in or determine theories ? Does the logical-conceptual structure of theories reveal the structure and processes of the mind and vice-versa ? The formula for Aufhebung $A \rightarrow \neg\neg A$. However this process stops after the first iteration. Subpresheaves (subfunctors) of a presheaf over a category $C$ form a Heyting algebra. It is interesting to look at $\neg\neg A$. This is related to density (the dense or double negation Grothendieck topology).  Given a subset $A$ of a set $X$ we can look for the smallest set $B$ for which $A$ is dense in $B$, that is $B$ is the closure or completion of $A$.

Sheaf theory recalls Kantian schematism: it is the synthetic realization (in particular topological) of an abstract category. The sheaf axioms express Kantian synthesis.

Of particular importance are theories of wholes, of different kinds of wholes, in particular non-distributive (mass-noun-like) and constructive/computational wholes.  All quantifiers (in dependent type theory) are intensions related to wholes and it is important to know what kind of whole is under consideration.

On a Formalization of Kant

The paper by Van Lambalgen and Pinosio 'The logic and topology of Kant's temporal continuum' (which is just one of a series of papers by Van Lambalgen on Kant)  opens with a nice discussion and careful justification of the general idea of the formalization of philosophical systems. The coined expression 'virtuous circle'  is particularly fortunate. In this post, which will be continuously updated, we will critically explore the above paper and make some connections with our own work on Aristotle's theory of the continuum.

The primitives are called 'events', self-affectations of the mind, which must be brought into order by fixed rules.  The authors work over finite sets of events which is justified by textual evidence from the CPR (we will return to this later).  Their task is to formalize relations between events - and to thus develop a point-free theory of the linear temporal continuum.

We find that that their notation could be improved and the axioms better justified. Instead of the confusingly asymmetric (all for the sake of the substitution principle, I suppose, or for the transitivity axiom) $aR_- b$ and $cR_+ d$  let us write $a{}_\bullet \leq b$ and $d\leq_\bullet c$. Instead of $a\oplus b$ we write $a\leftarrow b$ and insead of $a\ominus b$ we write $a\rightarrow b$.

The basic idea is that : $x{}_\bullet\leq y$ does not need to imply that $x\leq_\bullet y$ or vice-versa.

Kant's concept of causality implies that in order for a part $x$ of $a$ to influence $b$ we must have $x{}_\bullet\leq b$.  Thus the following axiom is expected

\[  a\ominus b{}_\bullet\leq b\]

But let us look at axiom 4 for event structures (in our notation):

\[ cOb\,\&\, a\leq_\bullet c \,\&\, b{}_\bullet \leq a \Rightarrow aOb \]

Our task is to make sense of this by offering a more satisfactory account of the primitive relations. Let us consider the set of connected (hence simply connected) subsets of the real line $\mathbb{R}$ and the interpretations:

\[ a{}_\bullet\leq b \equiv \forall x \in a. \exists y\in b. x\leq y  \]

\[ a \leq_\bullet b \equiv \forall x \in b. \exists x\in a. x\leq y  \]

But this does not work for  $a{}_\bullet\leq b \Rightarrow a\leq_\bullet b$. But let us take our events to be bounded open intervals $(a,b)$ and consider

\[ (a,b){}_\bullet\leq (c,d) \equiv  b < d  \]

\[ (a,b) \leq_\bullet (c,d) \equiv a < c \]

\[(a_1,a_2)O(b_1,b_2) \equiv a_2 > b_1\,\&\, a_1 < b_2\]

Then if we consider $(0,1)$ and $(0,2)$ we have that $(0,1){}_\bullet\leq (0,2)$ but not $(0,1)\leq_\bullet (0,2)$. The inequalities must be strict for allowing  $(a,b){}_\bullet\leq (a,b)$ is absurd, for then we could not associate any clear or definite Kantian philosophical concept with the relation.

Now let us look at axiom 4:

\[ (c_1,c_2)O(b_1,b_2)\,\&\, (a_1,a_2)\leq_\bullet (c_1,c_2) \,\&\, (b_1,b_2){}_\bullet \leq (a_1,a_2) \Rightarrow (a_1,a_2)O(b_1,b_2) \] which becomes

\[ c_2 > b_1\,\&\,  c_1 < b_2   \,\&\,a_1< c_1\,\&\, b_2 < a_2 \Rightarrow a_2 > b_1\,\&\, a_1 < b_2\]

But this follows immediately, using in addition the fact that $b_2 > b_1$. The condition $c_2 > b_1$ appears not to be needed.

We could try defining $(a_1,a_2)\rightarrow (b_1,b_2) := (a_1,b_2)$ when $a_1 < b_2$ and $(a_1,a_2)\leftarrow (b_1,b_2) :=  (b_1,a_2)$ when $b_1 < a_2$.

This models should be introduced right at the start of the paper to motivate the the definition of event structure. Notice that the set of events is here identified with the (infinite) subset $E \subset \mathbb{R}\times\mathbb{R} = \{(x,y): x < y\}$ but we could take only a finite subset.

We must check the axioms for event-structures for our model and also give a geometrical interpretation of the relations and operations above in terms of the identification of $E$ as a subset of the plane above.