9)V1_Ch_5_Sheaf Theory.pdf
8)Significance of sheaf theory in CS.pdf
7)V1_Ch_4_Topoi.pdf
6)Significance of Topoi in CS.pdf
5)V1 Ch 3 Natural Transformations.pdf
4)Ch 2 Functors.pdf
3)Application of Abstract mathematics to handle Chaos.pdf
2)Amazon through Rings,Spec(A), Structured Sheaf, Affine Scheme Lens.pdf
1)Main ch1 categories (1).pdf
Knowledge can be structured as Spec(A), where A is the Ring of Embeddings.
We already treat the set of Embeddings, as a d - dimensional
Real vector space. We make it into a Commutative Ring by defining addition as the usual addition of two d-dimentional vectors and multiplication as the Hadamard multiplication, that is, component wise multiplication. Then the vector (1, 1,.....,1) is the multiplicative Identity.
Now Spec(A) is the Topological Space, endowed with the Zarisky Topology, on the space of all Prime Ideals of A. Here the Prime ideals are preciously those who have a zero in some component. One such prime ideal, is for example, v(k) whose kth component is zero. Here, all prime ideals are closed, under this topology.
The Quotient Ring A/v(k) is isomorphic to R, the ring of reals and extracts the feature whose kth component is 0.
Spec(A) is a Topological space whose points are the prime ideals of the Ring A. Each point in Spec(A) thus corresponds to a " Feature Axis" or a Dimension of the Embedding space. This Topology would reveal the most fundamental axis of meaning that the Embedding space captures. For instance, the prime ideal corresponding to " Color " Dimension will be distinct from that associated with " Shape" Dimension, reflecting a clear separation of these fundamental features.
The points of Spec(A) could thus represent these fundamental irreducible feature axis or highly abstract invarient properties that the AI has learned to identify within it's data.
In ML we can use the K- means clustering approximations to chose open sets. In essence, prime ideals are the Fundamental Knowledge of an AI.
We still have only a Static set of prime ideals. If we involve a Sheaf ( Algebraic Geometry), it provides the Dynamic, Local-to- Global machinery
needed to fully leverage the Ring Structure of Embeddings. It allows an AI to not only identify
Fundamental truths but also to understand how "knowledge functions " behave locally ( on an open set of Spec(A), that is compliments of Prime ideals of A, which are closed under the Zariski Topology).
how they can be consistently composed , and where global inconsistencies and gaps may lie. This is a massive leap in formality and capability for representing and reasoning about complex knowledge structures.
Endowing Spec(A) with a Sheaf, makes it into an Afine Scheme, the fundamental building block of Algebraic Geometry. This Structured Sheaf is what makes an Algebraic Spec(A) into a Geometric object.
This allows us to translate Algebraic properties of "Embeddings" to the Geometric properties of
" Knowledge Spaces".