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.
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). 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.
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